Relativistic Kinetic Theory: With Applications in Astrophysics and Cosmology 9781107261365

Relativistic kinetic theory has widespread application in astrophysics and cosmology. The interest has grown in recent y

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Relativistic Kinetic Theory: With Applications in Astrophysics and Cosmology
 9781107261365

Table of contents :
Contents......Page 5
Acknowledgments......Page 10
Acronyms and Definitions......Page 11
Introduction......Page 14
Part I Theoretical Foundations......Page 20
1.1 Nonrelativistic Kinetic Theory......Page 22
1.2 Special Relativistic Kinetic Theory......Page 23
1.3 General Relativistic Kinetic Theory......Page 24
1.4 One-Particle Distribution Function......Page 25
1.5 Invariance of One-Particle Distribution Function......Page 26
1.6 Macroscopic Quantities......Page 27
2.1 Formulation of Kinetic Equation......Page 29
2.2 Collision Integral for Particle Scattering......Page 31
2.3 Boltzmann Equation in General Relativity......Page 32
2.5 Radiative Transfer......Page 34
2.6 Cross Section......Page 37
2.7 Relaxation Time......Page 38
3.1 Covariant Statistical Averaging......Page 39
3.3 The Role of Averaging in Kinetic Theory......Page 41
4.1 Conservation Laws and Relativistic Hydrodynamics......Page 43
4.2 H-theorem......Page 45
4.3 Equilibrium......Page 46
4.4 Relativistic Maxwellian Distribution......Page 49
4.5 Generalized Continuity Equation......Page 50
5.1 The Hierarchy of Kinetic Equations......Page 53
5.2 The First and Second Approximations in Relativistic Transport Equations......Page 57
5.3 The Vlasov-Maxwell System......Page 58
5.4 The Vlasov-Einstein System......Page 61
6.1 Plasma Frequency......Page 63
6.2 Correlations in Plasma......Page 64
6.3 Coulomb Collisions......Page 65
6.4 Characteristic Distances......Page 66
6.5 Microscopic Scales in Kinetic Theory and Hydrodynamics......Page 68
6.6 Relativistic Degeneracy......Page 69
Part II Numerical Methods......Page 72
7.1 Finite Differences and Computational Grids......Page 74
7.2 Stability and Accuracy of Numerical Schemes......Page 76
7.3 Numerical Methods for Partial Differential Equations......Page 79
7.5 ODE Systems and Methods of Their Solution......Page 96
7.6 Stiff Systems and Gear’s Method......Page 98
7.7 Numerical Methods for Linear Algebra......Page 102
8.1 Finite Differences and the Method of Lines......Page 108
8.2 Monte Carlo Method......Page 112
9 Multidimensional Hydrodynamics......Page 119
9.1 High-Order Godunov Methods......Page 120
9.2 Multidimensional Multitemperature High-Order Godunov Code......Page 122
9.3 Riemann Problem Solver in Special Relativity......Page 136
9.4 Particle-Based Methods......Page 139
Part III Applications......Page 146
10 Wave Dispersion in Relativistic Plasma......Page 148
10.1 Collisionless Plasma......Page 150
10.2 Response in an Isotropic Case......Page 151
10.3 Dispersion in Relativistic Thermal Plasma......Page 152
10.4 Landau Damping......Page 154
10.5 Plasma Instabilities......Page 156
10.6 Weibel Instability......Page 157
10.7 Two-Stream Instability......Page 160
10.8 Collisionless Shock Waves......Page 161
11.1 Pair Plasma in Astrophysics and Cosmology......Page 164
11.2 Qualitative Description of the Pair Plasma......Page 166
11.3 Collision Integrals......Page 167
11.4 Relativistic Boltzmann Equation on the Grid......Page 179
11.5 Thermalization Process......Page 180
11.6 Thermalization Timescales......Page 186
11.7 Dynamics and Emission of Mildly Relativistic Plasma......Page 190
11.8 Kinetic Equilibrium and Chemical Potential of Photons......Page 194
12 Kinetics of Particles in Strong Fields......Page 195
12.1 Avalanches in Strong Crossing Laser Fields......Page 196
12.2 Creation and Thermalization of Pairs in Strong Electric Fields......Page 199
12.3 Emission from Hot Bare Quark Stars......Page 211
13.1 The Boltzmann Equation for Compton Scattering......Page 216
13.2 Mean Number of Scatterings......Page 217
13.3 Kompaneets Equation......Page 219
13.4 Sunyaev-Zeldovich Effect......Page 223
13.5 Comptonization in Static Media......Page 227
13.6 Comptonization in Relativistic Outflows......Page 229
13.7 Monte Carlo Simulations of the Photospheric Emission from Relativistic Outflows......Page 233
14 Self-Gravitating Systems......Page 239
14.1 Kinetic Theory of Self-Gravitating Systems......Page 241
14.2 Gravitational Instability......Page 249
14.3 Collisionless (Violent) Relaxation......Page 263
14.4 Quasi-stationary States......Page 266
14.5 Self-Gravitating Systems in Equilibrium......Page 268
14.6 Cosmic Structure Formation......Page 273
15.1 Supernova Models and Neutrinos......Page 275
15.2 Spherically Symmetric Collapse of a Stellar Iron Core with Neutrino Transport......Page 278
15.3 Supernova Explosion Mechanism with Large-Scale Convection and Neutrino Transport......Page 287
Appendix A Hydrodynamic Equations in Orthogonal Curvilinear Coordinates......Page 291
B.1 Collision Integrals for Binary Interactions......Page 294
B.2 Collision Integrals for Binary Reactions with Protons......Page 300
B.3 Collision Integrals for Triple Interactions......Page 302
B.4 Mass Scaling for the Proton-Electron/Positron Reaction......Page 304
C.1 Scattering of Neutrinos on Electrons......Page 306
C.2 Absorption of Neutrinos by Neutrons......Page 308
C.3 Creation of Neutrinos......Page 310
Bibliography......Page 312
Index......Page 339

Citation preview

RELATIVISTIC KINETIC THEORY With Applications in Astrophysics and Cosmology

Relativistic kinetic theory has widespread application in astrophysics and cosmology. The interest has grown in recent years, as experimentalists are now able to make reliable measurements on physical systems where relativistic effects are no longer negligible. This ambitious monograph is divided into three parts. Part I presents the basic ideas and concepts of this theory; equations and methods, including derivation of kinetic equations from the relativistic BBGKY hierarchy; and discussion of the relation between kinetic and hydrodynamic levels of description. Part II introduces elements of computational physics, with special emphasis on numerical integration of Boltzmann equations and related approaches as well as multicomponent hydrodynamics. Part III presents an overview of applications ranging from covariant theory of plasma response, thermalization of relativistic plasma, and comptonization in static and moving media to kinetics of self-gravitating systems, cosmological structure formation, and neutrino emission during the gravitational collapse. gregory v. vereshchagin is Professor at the International Center for Relativis-

tic Astrophysics Network (ICRANet), Pescara, Italy. He graduated from Belarusian State University and received a Candidate of Science degree (PhD) in theoretical physics from the National Academy of Sciences of Belarus. He also holds a PhD in relativistic astrophysics from Sapienza University of Rome and was awarded the NATO-CNR fellowship. Author of more than 30 refereed papers, his research interests include cosmological singularity and inflation, loop quantum cosmology, the role of neutrinos in cosmology, thermalization of relativistic plasma, and photospheric emission from relativistic outflows. alexey g. aksenov is Senior Researcher at the Institute for Computer-Aided

Design, Russian Academy of Sciences (ICAD RAS) in Moscow. He graduated from the Moscow State Engineering Physics Institute (Technical University) and holds a PhD in astrophysics from the Space Research Institute of the Russian Academy of Sciences. He is author of more than 30 refereed publications in different topics in astrophysics and plasma physics, gravitational collapse, neutrino transport, inertial confinement fusion, numerical solution of kinetic Boltzmann equations, and hydrodynamic simulations.

R E L AT I V I S T I C K I N E T I C T H E O RY With Applications in Astrophysics and Cosmology G R E G O RY V. V E R E S H C H AG I N International Center for Relativistic Astrophysics Network, Pescara, Italy

A L E X E Y G . A K S E N OV Institute for Computer-Aided Design, Russian Academy of Sciences, Moscow, Russia

University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 4843/24, 2nd Floor, Ansari Road, Daryaganj, Delhi - 110002, India 79 Anson Road, #06-04/06, Singapore 079906 Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781107048225

© Gregory V. Vereshchagin and Alexey G. Aksenov 2017 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2017 A catalogue record for this publication is available from the British Library ISBN 978-1-107-04822-5 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

Contents

Preface Acknowledgments Acronyms and Definitions Introduction

page ix x xi 1

Part I Theoretical Foundations 1

Basic Concepts 1.1 Nonrelativistic Kinetic Theory 1.2 Special Relativistic Kinetic Theory 1.3 General Relativistic Kinetic Theory 1.4 One-Particle Distribution Function 1.5 Invariance of One-Particle Distribution Function 1.6 Macroscopic Quantities

9 9 10 11 12 13 14

2

Kinetic Equation 2.1 Formulation of Kinetic Equation 2.2 Collision Integral for Particle Scattering 2.3 Boltzmann Equation in General Relativity 2.4 Quantum Corrections to the Collision Integral 2.5 Radiative Transfer 2.6 Cross Section 2.7 Relaxation Time

16 16 18 19 21 21 24 25

3

Averaging 3.1 Covariant Statistical Averaging 3.2 Spacetime Averaging 3.3 The Role of Averaging in Kinetic Theory

26 26 28 28

v

vi

Contents

4

Conservation Laws and Equilibrium 4.1 Conservation Laws and Relativistic Hydrodynamics 4.2 H-theorem 4.3 Equilibrium 4.4 Relativistic Maxwellian Distribution 4.5 Generalized Continuity Equation

30 30 32 33 36 37

5

Relativistic BBGKY Hierarchy 5.1 The Hierarchy of Kinetic Equations 5.2 The First and Second Approximations in Relativistic Transport Equations 5.3 The Vlasov-Maxwell System 5.4 The Vlasov-Einstein System

40 40 44 45 48

Basic Parameters in Gases and Plasmas 6.1 Plasma Frequency 6.2 Correlations in Plasma 6.3 Coulomb Collisions 6.4 Characteristic Distances 6.5 Microscopic Scales in Kinetic Theory and Hydrodynamics 6.6 Relativistic Degeneracy

50 50 51 52 53 55 56

6

Part II Numerical Methods 7

The Basics of Computational Physics 7.1 Finite Differences and Computational Grids 7.2 Stability and Accuracy of Numerical Schemes 7.3 Numerical Methods for Partial Differential Equations 7.4 The Method of Lines 7.5 ODE Systems and Methods of Their Solution 7.6 Stiff Systems and Gear’s Method 7.7 Numerical Methods for Linear Algebra

61 61 63 66 83 83 85 89

8

Direct Integration of Boltzmann Equations 8.1 Finite Differences and the Method of Lines 8.2 Monte Carlo Method

95 95 99

9

Multidimensional Hydrodynamics 9.1 High-Order Godunov Methods 9.2 Multidimensional Multitemperature High-Order Godunov Code 9.3 Riemann Problem Solver in Special Relativity 9.4 Particle-Based Methods

106 107 109 123 126

Contents

vii

Part III Applications 10

Wave Dispersion in Relativistic Plasma 10.1 Collisionless Plasma 10.2 Response in an Isotropic Case 10.3 Dispersion in Relativistic Thermal Plasma 10.4 Landau Damping 10.5 Plasma Instabilities 10.6 Weibel Instability 10.7 Two-Stream Instability 10.8 Collisionless Shock Waves

135 137 138 139 141 143 144 147 148

11

Thermalization in Relativistic Plasma 11.1 Pair Plasma in Astrophysics and Cosmology 11.2 Qualitative Description of the Pair Plasma 11.3 Collision Integrals 11.4 Relativistic Boltzmann Equation on the Grid 11.5 Thermalization Process 11.6 Thermalization Timescales 11.7 Dynamics and Emission of Mildly Relativistic Plasma 11.8 Kinetic Equilibrium and Chemical Potential of Photons

151 151 153 154 166 167 173 177 181

12

Kinetics of Particles in Strong Fields 12.1 Avalanches in Strong Crossing Laser Fields 12.2 Creation and Thermalization of Pairs in Strong Electric Fields 12.3 Emission from Hot Bare Quark Stars

182 183 186 198

13

Compton Scattering in Astrophysics and Cosmology 13.1 The Boltzmann Equation for Compton Scattering 13.2 Mean Number of Scatterings 13.3 Kompaneets Equation 13.4 Sunyaev-Zeldovich Effect 13.5 Comptonization in Static Media 13.6 Comptonization in Relativistic Outflows 13.7 Monte Carlo Simulations of the Photospheric Emission from Relativistic Outflows

203 203 204 206 210 214 216

Self-Gravitating Systems 14.1 Kinetic Theory of Self-Gravitating Systems 14.2 Gravitational Instability 14.3 Collisionless (Violent) Relaxation 14.4 Quasi-stationary States

226 228 236 250 253

14

220

viii

15

Contents

14.5 Self-Gravitating Systems in Equilibrium 14.6 Cosmic Structure Formation

255 260

Neutrinos, Gravitational Collapse, and Supernovae 15.1 Supernova Models and Neutrinos 15.2 Spherically Symmetric Collapse of a Stellar Iron Core with Neutrino Transport 15.3 Supernova Explosion Mechanism with Large-Scale Convection and Neutrino Transport

262 262 265 274

Appendix A Hydrodynamic Equations in Orthogonal Curvilinear Coordinates

278

Appendix B Collision Integrals in Electron-Positron Plasma B.1 Collision Integrals for Binary Interactions B.2 Collision Integrals for Binary Reactions with Protons B.3 Collision Integrals for Triple Interactions B.4 Mass Scaling for the Proton-Electron/Positron Reaction

281 281 287 289 291

Appendix C Collision Integrals for Weak Interactions C.1 Scattering of Neutrinos on Electrons C.2 Absorption of Neutrinos by Neutrons C.3 Creation of Neutrinos

293 293 295 297

Bibliography Index

299 326

Preface

The endeavor of writing this book started from a series of lectures given by the first author for students of the International Relativistic Astrophysics PhD program (IRAP PhD) supported by the Erasmus Mundus program of the European Commission. For this book the material has been expanded and more topics incorporated. It soon became clear that an updated and systematic presentation of relativistic kinetic theory and its numerous applications in astrophysics and cosmology is lacking in the literature. Some existing monographs, presenting fundamental aspects of kinetic theory, are focused on selected applications. Others, which contain applications of kinetic theory in relativistic astrophysics and cosmology, lack the presentation of fundamental concepts of relativistic kinetic theory. Moreover, none of the existing monographs discussed in depth various numerical methods developed and successfully applied in kinetic theory in the recent decades. This last observation urged us to bridge this gap in the literature. This effort eventually resulted in the current monograph, divided in three parts. Parts I and III, with the sole exception of the last chapter, were written by the first author. Part II and the last chapter of Part III were written by the second author. Gregory V. Vereshchagin Pescara, Italy Alexey G. Aksenov Moscow, Russia June 2016

ix

Acknowledgments

This book is a joint effort of the two authors. Their collaboration was initiated and supported throughout by Professor Remo Ruffini, to whom they are deeply indebted. This multiyear collaboration was made possible by constant support from ICRANet. The authors are grateful to ICRANet faculty, V. A. Belinski, C. L. Bianco, J. A. Rueda, and S.-S. Xue, for numerous discussions on different topics related to the book. ICRANet provides an extraordinary environment where constant interaction with experts in theoretical physics, astrophysics, and cosmology from all over the world is made possible. Both authors acknowledge the discussions at different stages of writing this book with F. Aharonian, G. S. Bisnovatyi-Kogan, S. I. Blinnikov, S. K. Chakrabarti, J. Chluba, J. Ehlers, I. D. Feranchuk, V. G. Gurzadyan, I. M. Khalatnikov, H. Kleinert, M. V. Medvedev, J. Reinhardt, L. G. Titarchuk, E. Waxman, and R. M. Zalaletdinov. We are also thankful to our coauthors: M. M. Basko, D. Bégué, A. Benedetti, M. D. Churazov, V. M. Chechetkin, M. Lattanzi, M. Milgrom, D. K. Nadyozhin, I. A. Siutsou, and V. V. Usov. Some results obtained in joint publications are included in the book. The first author dedicates this work to his father, Victor Vereshchagin, who attracted his attention to the brilliant stars in the marvelous night sky of Petrunino when he was five years old. The encouragement, patience, and devotion of his wife, Alina, are invaluable. Without her support the book would not have been written. The second author acknowledges V. Ya. Arsenin, K. V. Brushlinskii, B. L. Rozhdestvenskii, and I. M. Sobol. Their lectures in MEPhI with original approaches were used in Part II of the book. He also acknowledges the support of the Russian Science Foundation Grant No. 16-11-10339, used for preparation of Chapter 15. Last, but not least, we thank Cambridge University Press and, in particular, Content Manager Lucy Edwards for guidance and advice during the writing of this book. Our special gratitude goes to Editorial Director Dr. Simon Capelin for his great patience and constant support. x

Acronyms and Definitions

AGN BBGKY CM CMB DF GRB KT LHS LP MC ODE PDE PIC QED qSS RHS SN SPH SW SZ

active galactic nuclei Bogoliubov-Born-Green-Kirkwood-Yvon (hierarchy) center of momentum cosmic microwave background distribution function gamma-ray burst kinetic theory left-hand side large particle Monte Carlo (method) ordinary differential equation partial differential equation particle-in-cell quantum electrodynamics quasi-stationary state right-hand side supernova smoothed-particle hydrodynamics shock wave Sunyaev-Zeldovich (effect)

xμ = (ct, x) jμ = (cn, j) pμ =  (p0 , p) p0 = p2 + m2 c2  = cp0 = γ mc2 = εmc2 p2 ≡ pμ pμ = m2 c2 gμν = diag (1, −1, −1, −1)

coordinate four-vector four-current momentum four-vector relativistic energy-momentum relation particle energy on shell condition Minkowski metric tensor xi

xii

Acronyms and Definitions

  μ ν 1/2 ds = gμν dx = cdτ  0 dx  dτ = mc/ p (t ) dt = dt/γ  −1/2  −1/2 γ ≡ 1 − (v/c)2 = p0 pμ pμ uμ ≡ dxμ /dτ = diag(γ c, γ v) = pμ /m U μUμ = c2 0 v = cp/p  −1 = p/ (γ m) ∂μ = c ∂/∂t, ∇ ρ = c−2 T μν UμUν P = − 13 T μν μν μν = gμν − c−2U μU ν kμ = (ω/c, k) dη = dt/a H ≡ d ln a/dt = a−2 da/dη

interval proper time Lorentz factor particle four-velocity velocity normalization condition three-velocity vector four-gradient energy density pressure projection operator four–wave vector conformal time Hubble parameter

Introduction

This book presents the subject of relativistic kinetic theory (KT). It starts from fundamental concepts and ideas and arrives at a vast spectrum of applications through the bridge of various numerical methods. It is not by chance that we adopted such an approach. KT of gases is perhaps the most fundamental theory in the classical (nonquantum) domain. It has been developed with the goal to derive the properties of matter at the macroscopic level, which is accessible to direct experiments and observations, based on the study of properties of microscopic particles and their mutual interactions, to which one has no direct access. The atomic picture of the world has emerged in this way. Indeed, KT was born in the nineteenth century, the golden age of classical physics. Based on the atomic picture [1], such properties as heat and electrical conductivity as well as viscosity and diffusion found natural explanations. The term originates from the Greek, where κινησις means “motion.” In fact, all the properties of the medium may be understood from the analysis of its microscopical structure and motions. Nowadays KT has to be considered in a wider context of statistical mechanics, which appeared at the end of the nineteenth century, essentially in the works of Maxwell, Boltzmann, and Gibbs. It should be emphasized that the ideas and principles of KT influenced the development of many other branches of science, including mathematics (probability theory, ergodic theory), biology (evolutionary biology, population genetics), and economics (financial markets, econophysics). Within physics, KT is closely related to statistical physics, thermodynamics, and hydro- and gasdynamics. Today one can say that the main task of kinetic theory is the explanation of various macroscopic properties of a medium based on known microscopic properties and interactions. In a general context, KT is a theory of nonequilibrium systems. Indeed, all the above-mentioned fields of physics assume that the medium is in its most probable microphysical state, called equilibrium. Clearly, any macroscopic manifestation of deviations from this microscopic equilibrium should be considered within KT. Because basic phenomena in 1

2

Introduction

the microworld are described in a quantum language, KT uses extensively quantum theory. In fact, the basic principles and equations of KT may be derived from quantum field theory [2]. The first classical applications of KT concerned gases. A successful description of ideal and nonideal gases has been reached within the framework of Newtonian mechanics. At the same time, the progress in stellar dynamics [3] led to the formulation of the collisionless Boltzmann equation with the mean gravitational potential, satisfying the Poisson equation. The latter was later rediscovered in the context of plasma physics. With the appearance of special relativity, KT had to be reconciled with the existence of the limiting speed, the speed of light. In particular, equilibrium distributions, i.e., Maxwell-Boltzmann distribution of velocities, had to be modified. These developments resulted in the work of Jüttner on relativistic equilibrium distribution function already in 1911 [4]. It soon became clear that there is another natural arena for the application of KT, which is plasma physics [5, 6]. The major difference between plasma and gas is the long-range nature of electromagnetic interaction, which has been accommodated by introduction of the mean field description [7]. While classical KT theory of plasmas developed rapidly in the 1930s, it was essentially nonrelativistic. Even the Landau damping phenomenon was discussed within the nonrelativistic framework, despite the fact that its analysis requires the use of the Vlasov equation, which is Lorentz invariant. The formulation of KT within special relativity was completed in the 1960s, and it is presented in several monographs (see, e.g., [2, 8, 9]). Relativistic astrophysics emerged in the same period. The main triggers were the discoveries of the cosmic background radiation (CMB), pulsars, and quasars. Observation of the CMB confirmed the hot model of the universe. It urged the development of models for matter at extreme densities and temperatures, characteristic of the early universe. In this way the kinetics of thermonuclear reactions was analyzed, leading to the big bang nucleosynthesis (BBN) theory. Similarly, the discovery of pulsars and their interpretation as rapidly rotating magnetized neutron stars urged the formulation of models of matter under extreme densities and in strong gravitational and electromagnetic fields. Hence, it is natural that most applications of relativistic KT are in the fields of astrophysics and cosmology. Both fields are somewhat special in physics. They lack the very essential feature of traditional physics: the possibility to set up and control physical experiments. In both fields the available experimental data originate essentially from observations, and the observer has no power whatsoever to influence or change the conditions under which the observed phenomenon takes place. For this reason, numerical simulations appear to be the unique tool for development

Introduction

3

of theoretical models in astrophysics and cosmology, which are eventually tested against observations. Nevertheless, relativistic KT is becoming more accessible to direct tests due to the recent progress in two fields: inertial fusion and ultra-intense lasers. Operation of ultra-intense lasers is approaching such intensities that the creation of electronpositron plasma in the laboratory is becoming technologically feasible [10]. Hence, the application of relativistic KT to electron-positron plasmas, discussed in the book, becomes of great importance. The focus of the book is mainly on KT within special relativity. A general relativistic kinetic equation is formulated in Part I, while general relativistic effects are discussed only in Part III in relation to the gravitational instability phenomenon as well as gravitational collapse. The formulation of KT is presented in Part I in Lorentz-invariant fashion. In Chapter 1 the evolution of basic concepts of KT, such as phase space and distribution functions, from nonrelativistic to special and general relativistic frameworks is outlined. The relation between mechanical and kinetic pictures is presented. The physical meaning of the one-particle distribution function is given and its Lorentz invariance is demonstrated. Then the most useful macroscopic quantities, such as four-current, entropy four-flux, energy-momentum tensor, and hydrodynamic velocity, are obtained. These concepts are essential to proceed with the formulation of kinetic equations and to understand the relation between KT and hydrodynamics, discussed in the following chapters. In Chapter 2, an axiomatic approach to derive kinetic equations for the one-particle distribution function is adopted, and special attention is given to the advection part. First, the kinetic equation in special relativity is presented by considering particle world lines. Then the Boltzmann equation in general relativity is derived using the Klimontovich random function. The particularly important case of scattering of two particles is considered, for which a collision integral is derived. Quantum corrections to the collision integral are also considered. The relation between KT and the radiative transfer theory is outlined. The connection between collision integrals and cross section is presented. Finally, the notion of relaxation time is introduced. The role of averaging as one of the fundamental instruments of KT is discussed in Chapter 3. While in nonrelativistic physics, averaging appears to be straightforward, it does not prove to be so in relativistic generalization, where time and space averaging, considered separately, are not Lorentz invariant, because space and time are no longer absolute. Within general relativity, averaging is an even more complicated issue, with a fully covariant formulation of KT still missing. In Chapter 4, equations of relativistic hydrodynamics are derived from the Boltzmann equation. It is shown that microscopic conservation of energy and momentum at each interaction between particles implies the existence of conservation laws for

4

Introduction

macroscopic quantities such as four-current and energy-momentum tensor. Then H-theorem is proved and conditions for local thermodynamic equilibrium are formulated. The one-particle distribution function as well as some useful macroscopic quantities in equilibrium, such as density, pressure, and entropy, are obtained. The generalized continuity equation for nonequilibrium systems is also derived. In Chapter 5 the derivation of the Bogolyubov-Born-Green-Kirkwood-Yvon hierarchy for relativistic plasma is presented. The basic idea in this approach is that any many-body system can be characterized by the set of equations of motion under the given interaction. Applying averaging to Klimontovich distribution functions, one can derive the chain of equations for many particle distribution functions. In order to obtain tractable kinetic equations, this hierarchy can then be truncated at a certain level, using expansion in small parameters or other physical considerations. In this way Maxwell-Vlasov and Belyaev-Budker equations are derived. In the last chapter of Part I, kinetic properties of dilute gas and plasma are considered. In the relativistic domain, many qualitatively new phenomena, such as particle-antiparticle production, occur in plasma. To understand these phenomena, as well as to provide the physical foundations for the derivation of the Boltzmann and Vlasov equations, it is very useful to discuss the characteristic quantities in both gases and plasmas. In particular, the plasma parameter, Coulomb logarithm, Debye length, degeneracy parameter, and Knudsen number are introduced. Physics is an empirical science, and all its concepts are verified in experiments. By analogy, computer simulation of a physical process can be considered as numerical experimentation with all the necessary methodology, setup, and data analysis. Owing to physical limitations for both the computer memory and CPU or GPU speed, such simulations have limited space and time resolution for the simulated problems, very much like traditional physical instruments in experiments. In comparison with the physical setup the computer and the numerical method can be considered as a universal tools. During the last several decades the power of computers increased exponentially with the doubling of computing power every 18 months, following Moore’s law. This provides conditions for the unprecedented development of numerical techniques and their application to various physical problems. Numerical methods applied in relativistic KT and in hydrodynamics are discussed in Part II. In Chapter 7 an informal introduction to computational physics is presented. Although an analytic solution completely describes the problem, it is not available for most nonlinear problems. New results in modern physics are often obtained in numerical simulations. The chapter describes standard types of equations of classical mathematical physics and existing methods of their solution, focusing mainly on finite difference techniques. Systems of ordinary differential equations and problems of linear algebra are considered as well. Stability and accuracy of numerical schemes are addressed, providing the convergence of the numerical solution to the exact solution of the underlying differential equation.

Introduction

5

In Chapter 8, numerical integration of Boltzmann equations is discussed. The approach is illustrated by the finite difference scheme on a fixed grid in the 4D phase space, and it is based on the method of lines. This method reduces the integration of partial differential equations to the solution of the system of ordinary differential equations. The latter are solved by the implicit Gear’s method. The method is suitable for both optically thick and optically thin regions and is especially useful for describing neutrino transport in gravitational collapse. The Monte Carlo approach for solution of the Boltzmann equation is discussed as well. This approach is universally applied when the optical depth is small, especially in multidimensional problems. Finally, Chapter 9 describes classical shock-capturing hydrodynamic transport in multidimensional space. The modern high-order Godunov-type methods are described. For multicomponent systems, kinetic Boltzmann equations in 7D phase space are replaced by hydrodynamics with diffusion and flux limiters in 5D phase space. The interpolation of fluxes of spectral energy density in the intermediate case between the transparent (free flow) and the nontransparent (diffusion or heat conduction) cases is introduced. A special relativistic Riemann solver is also discussed. The last section of the chapter briefly describes smooth particle hydrodynamics (SPH) and particle-in-cell (PIC) methods. Such particle-based simulations are especially useful in describing advection of a smooth flow. The common idea in this chapter is the multidimensional hydrodynamics and explicit methods for advection. In Part III, applications of relativistic KT in astrophysics and cosmology are considered. In Chapter 10, one of the most important domains of application of relativistic KT, the theory of waves in relativistic plasma, formulated in a gaugeinvariant fashion, is presented. After a brief introduction to this theory, several important applications, such as Landau damping and relativistic plasma instabilities, are considered. Collisionless shocks and their relevance to astrophysics are discussed. In Chapter 11, relaxation of nonequilibrium optically thick relativistic plasma is discussed. Collision integrals are represented as integrals over matrix elements, provided by quantum electrodynamics, describing various two-particle interactions between photons, electrons, positrons, and protons. Collision integrals for three-particle interactions are also introduced. Then a theory of thermalization, including the concepts of kinetic and thermal equilibria, is presented. Time scales for relaxation toward thermal equilibrium as functions of the total energy density and baryonic loading are reported. At the end of the chapter, a dynamical kinetic description for a mildly relativistic plasma ball is presented, including its radiation properties. Chapter 12 is dedicated to discussion of kinetic effects related to pair creation out of a vacuum in strong electromagnetic fields. Nonlinear effects relevant for ultra-intense lasers are briefly discussed. Then the entire dynamics of energy

6

Introduction

conversion from an initial strong electric field, ending up with thermalized optically thick electron-positron-photon plasma, is studied. It is crucial that pair creation involves back reaction of pairs onto an external field. Accounting for such back reaction is imperative in this problem. As an application, emission of an electronpositron pair wind from a hot bare quark star is considered. In Chapter 13, some essential aspects of Compton scattering are discussed and various processes in which this scattering plays an important role are illustrated. In particular, one of the most important astrophysical implications, the SunyaevZeldovich effect, is addressed. The Kompaneets equation is derived. The theories of comptonization in static and relativistically moving media are reviewed. Photospheric emission from relativistic outflows is also considered. In Chapter 14, kinetic properties of self-gravitating systems are discussed and contrasted with kinetic properties of gases and plasmas discussed in previous chapters. Here the Lorentz-invariant formulation is abandoned in favor of clarity and simplicity of presentation. First, Boltzmann equations are derived out of the Bogolyubov-Born-Green-Kirkwood-Yvon hierarchy under different approximations. Then relativistic theory of gravitational instability on the kinetic level is briefly reviewed. Collisionless relaxation and quasi-stationary states are also discussed. Finally, self-gravitating systems in equilibrium and their instability are addressed. In the last chapter, Chapter 15, an example of accurate neutrino treatment in a spherically symmetric collapse is given. The role of multidimensional effects is discussed. These results are of interest for the multidimensional models with large-scale convection as well as for the ongoing experimental search for neutrinos from supernovae. In the Landau and Lifshitz course of theoretical physics, volume 10, “Physical Kinetics,” is the last. Students are indeed expected to master all branches of physics before proceeding to this subject. Similarly, it is expected that graduate students in physics and astrophysics who wish to get acquainted with relativistic KT have already learned both special and general relativity and cosmology as well as quantum electrodynamics. Only with this broad and solid background will it be possible for students to make their way, employing numerous techniques and methods, to the applications of relativistic KT and find novel specific problems to be addressed and, eventually, solved. We offer in this book our vision of the foundations, numerical methods, and vast series of applications of the modern relativistic kinetic theory.

Part I Theoretical Foundations

1 Basic Concepts

In this chapter the evolution of basic concepts of KT, such as phase space and distribution functions, from nonrelativistic to special and general relativistic frameworks is outlined. The relation between mechanical and kinetic pictures is presented. The physical meaning of the one-particle distribution function is given, and its Lorentz invariance is demonstrated. Then the most useful macroscopic quantities, such as four-current, entropy four-flux, energy-momentum tensor, and hydrodynamic velocity, are obtained. These concepts are essential to proceed with the formulation of kinetic equations as well as to understand the relation between kinetic theory and hydrodynamics discussed in following chapters.

1.1 Nonrelativistic Kinetic Theory In classical (nonrelativistic) mechanics a complete description of a system composed of N interacting particles is given by their N equations of motion. In nonrelativistic KT one deals with a space of positions and velocities of these particles, called configuration space or the space of canonical variables: positions and momenta of particles, called the phase space M. Often this mechanical description can be formulated in the language of Hamilton equations, and then an equivalent description of the system is given by a function F ( , t ) of time and 6N independent variables, defined on M. An equation can be formulated for this function, called the Liouville equation, that can be written in an apparently very simple form: dF ( ) = 0, dt

(1.1)

where the derivative is over time. Its complexity, however, is equivalent to the complexity of the original N-body problem, and in the majority of cases, it cannot be addressed directly.

9

10

Basic Concepts

A tremendous simplification occurs for such systems, where N is very large. One may define the s-particle distribution function (DF) of states depending on 6s variables with 1 ≤ s ≤ N and time by integrating out the remaining 6(N − s) degrees of freedom in F ( , t ). The hierarchy of integro-differential equations, which connects the s-particle DFs with the s + 1-particle ones, called the Bogolyubov-BornGreen-Kirkwood-Yvon (BBGKY) hierarchy, is obtained in this way. Among these s-particle DFs, the one-particle DF plays a central role in KT, as it describes the probability of finding the particle in a state with momentum in the range (p, p + d 3 p) and position in the range (x, x + d 3 x) at the moment t. The s-particle DFs describe joint probabilities, i.e., particle correlations. Formally, this hierarchy can be truncated at a given level (usually at s = 1, 2) by specifying the functional form of the s + 1-particle DF. This is the way kinetic equations for such systems as gases or plasmas were derived out of this hierarchy, and it is called the Bogolyubov method, after his monograph [11]. The power of Bogolyubov’s method is in its observation that the truncation of the hierarchy may be justified considering the expansion of the DF either in powers of density (for short-range interactions) or in powers of interaction energy (particularly for Coulomb interactions). Remarkably, these kinetic equations coincided with the ones derived previously on a phenomenological basis by Boltzmann and Landau, respectively. Hence the BBGKY hierarchy allows establishing kinetic equations out of the first principles. 1.2 Special Relativistic Kinetic Theory At first glance, special relativity brings few modifications to kinetic theory. Indeed, the usual distribution function appears to be Lorentz invariant, as does the Boltzmann equation. Deep analysis shows, however, that conceptual changes are required. First, the theory must be consistent with the existence of the limiting speed, the speed of light c. The first attempt to adopt special relativistic treatment within KT is due to Jüttner back in 1911 [4], who established the equilibrium DF in the form consistent with special relativity. Second, the whole theory must be proven Lorentz invariant. It took some time to formulate the problem and to prove the Lorentz invariance of the one-particle DF. For the final settlement of the question, see the monograph [2] and more recent paper [12]. Third, the meaning of initial data and dynamics has to be reconsidered, following the revision of the concepts of space and time in special relativity. As one of the consequences of the modifications mentioned earlier, the Liouville equation (1.1) must be reformulated [13].

1.3 General Relativistic Kinetic Theory

11

Nevertheless, the generalization of KT for relativistic gas appeared to be rather straightforward [2, 8]. Some new phenomena have been discovered; in particular, it was shown that the heat flow is determined not by the temperature gradient but by the gradient of thermal potential, the relativistic gas possesses a bulk viscosity [14], and the specific heat anomaly of an ideal relativistic Bose gas was found [15]. The case of plasma, i.e., charged particles with long-range electromagnetic interaction, presented serious challenges due to the retarded action of the electromagnetic signal. Nevertheless, kinetic equations for collisionless [7] as well as collisional [16] relativistic plasma have been obtained. Only later was the relativistic BBGKY hierarchy formulated and kinetic equations derived from it [17, 18]. A very useful concept introduced for this purpose is the so-called random function of Klimontovich. This derivation is reviewed in Chapter 5.

1.3 General Relativistic Kinetic Theory Within general relativity the formulation of KT is evidently more complicated. Although general relativistic Boltzmann equations have been obtained in the literature [19, 20], their validity is limited to fixed background geometry described by a given metric. Several applications, mainly in cosmology, have been considered (see [9, 13, 21, 22]). However, within all these treatments, the back reaction of particle kinetics on the background metric is not taken self-consistently into account. It is important to emphasize that KT, similar to hydrodynamics, is a macroscopic theory that represents the system of particles as a continuous medium. The KT microscopic bases are the equations of motion of individual particles, on one hand, and the field equations, on the other. Both (nonquantum) electrodynamics and gravitation theory are field theories, and both of them are microscopic. The connection between microscopic and macroscopic electrodynamics has been established by Hendrik Lorentz [23] by virtue of the corresponding averaging over space and time. Due to linearity of microscopic equations, the averaging appears to be straightforward (see, e.g., [24]). This issue is further discussed in Chapter 3. The same principle of averaging can be extended to fulfill general covariance. However, the nonlinear character of equations of general relativity presents serious challenges in this approach. An attempt to construct macroscopic gravity in these lines has been undertaken by Zalaletdinov [25, 26, 27, 28]. Notwithstanding the lack of justification from first principles, Einstein-Vlasov and Einstein-Boltzmann equations are used in the literature [9, 13, 21, 29], with applications in cosmology and galactic dynamics, although their physical meaning is unclear. In this book the focus is mainly on special relativistic KT. Discussion of general relativistic KT is limited to the case of fixed background geometry.

12

Basic Concepts

1.4 One-Particle Distribution Function The basic function used in KT for which master equations are formulated is the oneparticle DF. In fact, all macroscopic information about the evolution of the system can be obtained from this function by suitable integration in the phase space. Two approaches exist in the literature on relativistic KT regarding the definition of the phase space for one-particle DF. In the first approach, particle coordinate four-vector xμ and particle momentum four-vector pμ are considered as basic independent variables, and the one-particle DF F (x, p) is defined on the 8D phase space M8 with the help of particle four-current jμ (x) as  μ (1.2) j (x) ≡ c F (x, p)pμ d 4 p, where c is the speed of light. For brevity in what follows, denote1 the coordinates in the phase space as x ≡ xμ = (ct, x) and p ≡ pμ = (p0 , p), where t is time, x is a coordinate, and p is momentum three-vectors, respectively. The DF defined in eq. (1.2) is manifestly Lorentz invariant. However, this function, unlike the nonrelativistic DF, does not have the meaning of the density of particles in some space. In the second approach, the one-particle DF f (xμ , pμ ) is defined on the 6D phase space M6 . This function depends on only seven variables: three space coordinates, three momentum components, and time. This choice is evident because particle energy  = cp0depends on particle momentum, and p0 satisfies the relativistic equation p0 = p2 + m2 c2 , where m is particle mass. The one-particle DF is then defined such that the integral  N≡ f (p, x, t )d 3 pd 3 x (1.3) M6

gives the total number of particles. Notice that the integral is clearly Lorentz invariant. The invariance of the DF itself is not obvious from such a definition and has to be demonstrated explicitly (see Section 1.5). From the definition (1.3) one observes that f (x, p)d 3 pd 3 x is an average number of particles having momenta in the range (p, p+d 3 p) and coordinates in the range (x, x+d 3 x) at the moment t, and the integral (1.3) is taken in the whole phase space M6 . While the function f (p, x, t ) normalized to unity can be interpreted as a density of probability, this is not so for the function F (x, p), defined in eq. (1.2) (for discussion, see [30]). 1

In this book Greek indices run from 0 to 3, whereas Latin ones run from 1 to 3. The Einstein summation rule is adopted. Dummy indices are also used with the summation symbol , but they are always indicated below it, and this should not give rise to confusion.

1.5 Invariance of One-Particle Distribution Function

13

Notice that, despite the symmetrical form of f (x, p), there is a conceptual difference between x and p. In particular, the integral  +∞ n(x, t ) = f d3 p (1.4) −∞

is assumed to be finite, leading to certain restrictions on f (p). When the DF is isotropic in momentum space, p2 f (|p|) should decrease with increasing momentum for |p|  1 fast enough, at least faster than 1/ |p|; it also should not increase with decreasing momentum for |p|  1 faster than 1/ |p|.

1.5 Invariance of One-Particle Distribution Function The one-particle DF defined by eq. (1.3) is not written in a Lorentz-invariant way. However, it is an invariant, as demonstrated following [2, 31] (see also [12]). Consider the system of particles of equal mass m with coordinates xi (t ) and momenta pi (t ), where index “i” enumerates particles. By definition (see eqs. (1.3) and (1.4)), from the statistical point of view, the one-particle DF is the averaged particle density in momentum space (see, e.g., [32]), that is, 

  f (p, x, t ) = δ (3) p − pi (t ) δ (3) [x − xi (t )] , (1.5) i

ens

where δ (3) (x) is the 3D Dirac δ-function and the subscript “ens” denotes the averaging over the statistical ensemble. As discussed in Section 1.4, in a relativistic context, it is natural to introduce an 8D one-particle phase space M8 . In such phase space the variable p0 is not necessarily related to p; likewise t is not related to x. At the end of any calculations involving M8 the physical results can be recovered by restricting every equation to the submanifold of the mass shell, defined by pμ pμ = m2 c2 , with the condition p0 > 0. Introducing in this way a function F (x, p) = 2(p0 )δ(pμ pμ − m2 c2 ) f (p, x, t ),

(1.6)

where (x) is the Heaviside step function and the term (p0 )δ(pμ pμ − m2 c2 ) is Lorentz scalar, one has to show that this function is a Lorentz scalar. Recalling the identity dZ −1 δ (x − xi ) , (1.7) δ [Z(x)] = dx i

14

Basic Concepts

where xi are the roots of the equation Z(x) = 0, rewrite eq. (1.6) using eq. (1.5) as 

1     3 3 0 δ p − pi (t ) δ p − pi (t ) δ [x − xi (t )] F (x, p) = . (1.8) p0i (t ) i ens

Introducing additional integration over a delta function as   1   δ [t − ti ] δ 3 p − pi (ti ) F (x, p) = dt 0 pi (ti ) i

  3 0 × δ p − pi (ti ) δ [x − xi (ti )]

(1.9)

ens

and using the relation between the time dti and the proper time dτi = mcdti /p0i (ti ), one obtains that F (x, p) is indeed a scalar, because 

 1 4 4 F (x, p) = dτ δ [x − xi (τ )] δ [p − pi (τ )] , (1.10) mc i ens

where coordinates xi (τ ) and momenta pi (τ ) correspond to trajectories in M8 . The last expression can be understood2 as the ensemble-averaged and time-integrated Klimontovich random function F K (x, p) (see, e.g., [21] and Section 2.3). Note that being averaged, the quantity F (x, p) is regular and smooth, while F K (x, p) is the random quantity described in terms of the generalized functions. 1.6 Macroscopic Quantities It is important to keep in mind that the DF defined by eq. (1.3) is not accessible to measurements directly. In any experiment or observation one has to deal with averaged quantities. Particularly useful is averaging over momentum variables. For a function a(x, p) depending on a microscopic state defined on the phase space M6 , it is useful to introduce a macroscopic quantity A(x) as  +∞ A(x) ≡ a(x, p)d 3 p. (1.11) −∞

Particle density (1.4) is the simplest example. By definition the macroscopic quantity does not depend on momentum but only on coordinates and time. In this way one may construct moments of DF. It comes out that the first two moments of DF prove to be useful macroscopic quantities and help to establish a relation between KT and hydrodynamics (see Chapter 4). 2

The averaging operation, on one hand, and integration or differentiation, on the other, do commute (see Chapter 3).

1.6 Macroscopic Quantities

15

Recalling eq. (1.2), one can define an invariant quantity, instead of eq. (1.4), as  d3 p μ j (x, t ) ≡ c pμ f 0 , (1.12) p where both f and d 3 p/p0 are scalars. This first moment of the DF is the particle four-flux. Its spatial part represents usual three-vector flux  j(x, t ) = c v f d 3 p, (1.13) where v = cp/p0 is the velocity vector of a relativistic particle with momentum p, while its velocity four-vector is uμ = dxμ /dτ . Analogously, the second moment can be constructed as  d3 p μν T (x, t ) ≡ c pμ pν f 0 , (1.14) p and so on. The quantity T μν is a symmetric tensor by construction. It represents an energy-momentum tensor of the system of particles. It should be noted that in eq. (1.14), only rest mass energy and kinetic energy of particles are taken into account, excluding their potential energy due to interactions. One more important quantity helping to establish the connection of KT to thermodynamics is the entropy flux, defined as   d3 p    μ S (x, t ) ≡ −kB c pμ f 0 log h3 f − 1 , (1.15) p where two constants appear: kB is Boltzmann’s constant and h is a dimensional parameter needed to make the argument of the logarithm dimensionless. Unlike nonrelativistic KT, in its relativistic counterpart, macroscopic velocity can be defined in different ways. Two widespread definitions are due to Eckart [33] and Landau and Lifshitz [34]: c jμ UEμ ≡  , j μ jμ

cT μν Uν μ ULL ≡ , Uρ T ρσ Tσ τ U τ

(1.16)

respectively. Whereas in Eckart’s definition, UEμ can be interpreted as the average μ velocity of particles, in Landau and Lifshitz’s definition, ULL can be understood as the average velocity of energy-momentum transfer. With the help of the hydrodynamic velocity, a useful reference frame, the Lorentz reference frame, can be defined as such a frame where U μ = ULμ = (c, 0, 0, 0). This reference frame is used frequently in the following chapters.

2 Kinetic Equation

In this chapter, the relativistic kinetic equation for a one-particle DF is formulated, and special attention is given to its left-hand side. First, the kinetic equation in special relativity is presented by considering particle world lines. Then, the Boltzmann equation in general relativity is derived using the Klimontovich random function. The particularly important case of scattering of two particles is considered, for which a collision integral is derived. Quantum corrections to the collision integral are also considered. The relation between KT and the radiative transfer theory is given. The connection between collision integrals and a cross-section is presented. Finally, the notion of relaxation time is introduced.

2.1 Formulation of Kinetic Equation First, assume for simplicity that particles do not interact. One can introduce a scalar quantity 1 J = c



μ

3 σ





d σμ j = 3

d σμ 3

3 σ

d3 p μ p f, p0

(2.1)

where the timelike four-vector d 3 σμ is an oriented three-surface element of a plane spacelike surface σ , the quantity 3 σ is a small three-surface element, and the last equality follows from eq. (1.12). In the Lorentz reference frame, where d 3 σμ is purely timelike, it has components (d 3 x, 0, 0, 0). In this frame,  J =

 f (x, p)d 3 pd 3 x,

(2.2)

3 σ

which is just an average number of world lines crossing the segment 3 σ . Considering those world lines, which have momenta in the range 3 p around p, one 16

2.1 Formulation of Kinetic Equation

17

can get 

 J =

f (x, p)d 3 pd 3 x. 3 σ

(2.3)

3 p

Accepting this interpretation, consider world lines given by eq. (2.1) that later cross another three-surface element 3 σˆ . Since there are no collisions, it is possible to write     d3 p μ d3 p μ 3 3 d σμ p f − d σ p f = 0, (2.4) μ 0 0 3 σˆ 3 p p 3 σ 3 p p or, in differential form, 

 d σμ 3

3 x

3 p

d3 p μ p f = 0, p0

(2.5)

where 3 x is the surface of Minkowski space element 4 x. Applying the Gauss theorem, one gets   d3 p μ 4 d x p ∂μ f = 0, (2.6) 0 4 x 3 p p where ∂μ = (c−1 ∂/∂t, ∇ ) and 3 x and 3 p are some arbitrary hypersurfaces in the phase space. The master equation represents the time evolution of the DF due to microscopic interactions in the system. In the absence of any interactions between particles, it represents continuity of the four-vector pμ f , and it follows from eq. (2.6) as pμ ∂μ f = 0.

(2.7)

Written in vector notation (and dividing by p0 /c), df ∂f ≡ + v · ∇ f = 0. dt ∂t

(2.8)

In the general case, both collisions and external forces alter eq. (2.7). Since such alteration is essentially local, the kinetic equation becomes pμ ∂μ f + mF μ

∂f = St [ f ], ∂ pμ

(2.9)

where F μ represents an external four-force and St [ f ] is the collision integral. This is the relativistic transport equation. One of the main goals of KT is to establish the form of the collision integral.

18

Kinetic Equation

2.2 Collision Integral for Particle Scattering The simplest interaction between particles is scattering. Consider an elastic collision 1 + 2 −→ 1 + 2 ,

(2.10)

where particles 1 and 2 have masses m1 and m2 and momenta pμ1 and pμ2 , which μ changed after the collision to p μ 1 and p2 , respectively. Energy-momentum conservation gives μ pμ1 + pμ2 = p μ 1 + p2 .

(2.11)

The average number of such collisions is proportional to (1) the number of particles per unit volume with momenta pμ1 in the range d 3 p1 , (2) the number of particles per unit volume with momenta pμ2 in the range d 3 p2 , and (3) the intervals d 3 p μ 1 , 3 μ 4 d p2 , and d x. The proportionality coefficients, depending only on four-momenta 0 before and after the collision, are represented as W (p1 , p2 | p 1 , p 2 )/(p01 p02 p 0 1 p2 ). The quantity W (p1 , p2 | p 1 , p 2 ) is called the transition rate, and it is a scalar. By this process, particles leave the phase volume d 3 p1 around pμ1 . Collisions also bring particles back into this volume by the inverse process with the corresponding rate W (p 1 , p 2 | p1 , p2 ). Then, the Boltzmann equation can be written as      3 d p1 d 3 p 1 d 3 p2 d 3 p 2 d 3 p1 4 1 μ p ∂μ f 0 d x = (2.12) 2 V P p1 p01 p 0 p02 p 0 V P 1 2 × [ f (x, p 1 ) f (x, p 2 )W (p 1 , p 2 | p1 , p2 ) − f (x, p1 ) f (x, p2 )W (p1 , p2 | p 1 , p 2 )]d 4 x, where V and P are volumes in coordnate and momentum spaces, respectively, or, in differential form,  3 3 d p1 d p2 d 3 p 2 1 μ p ∂μ f = (2.13) 2 p 0 p02 p 0 1 2 × [ f (x, p 1 ) f (x, p 2 )W (p 1 , p 2 | p1 , p2 ) − f (x, p1 ) f (x, p2 )W (p1 , p2 | p 1 , p 2 )]d 4 x. The same equation in vector notation becomes  ∂f 1 +v·∇f = d 3 p 1 d 3 p2 d 3 p 2 [ f (x, p 1 ) f (x, p 2 )w p 1 p 2 ;p1 p2 ∂t 2 − f (x, p1 ) f (x, p2 )w p1 p2 ;p 1 p 2 ],

(2.14)

2.3 Boltzmann Equation in General Relativity

19

0 where w p1 p2 ;p 1 p 2 = cW (p1 , p2 | p 1 , p 2 )/(p01 p02 p 0 1 p2 ). If, in this expression, particle momenta are substituted by their velocities, this equation will coincide with the one derived first by Boltzmann [1]. Hence the equation of the form (2.13) is called the Boltzmann equation. In what follows, also the more general equation of the form (2.9) is called the Botzmann equation. Notice that the factor 1/2 in front of the collision integral is due to the fact that particles are indistinguishable.

2.3 Boltzmann Equation in General Relativity The derivation of the Boltzmann equation in general relativity is presented here following [21]; for more details, see [19, 20, 32, 35] and also [9]. Let us start by introducing the 8D phase space M8 . The random function K F (x, p) is defined on this space by Klimontovich1 [17]:  1 dτ δ 4 [x − xi (τ )]δ 4 [p − pi (τ )], (2.15) F K (x, p) = mc i Notice that eq. (1.10) is nothing but the ensemble-averaged function (2.15), namely, F (x, p) = F K (x, p) ens .

(2.16)

The equations of motion for each particle in the gravitational field are (see, e.g., [36]) mc

dxμ = pμ ds

mc

d pμ μ ν λ = − νλ p p , ds

(2.17)

where ds = (gμν dxμ dxν )1/2 = cdτ

(2.18)

μ is the element of spacetime interval, gμν is the metric tensor, and νλ are the Christoffel symbols. Using the property of δ-functions

d d dg δ[x − g(s)] = − δ[x − g(s)] ds dx ds and the identity

 ds

1

d 4 {δ [x − xi (s)]δ 4 [p − pi (s)]} = 0, ds

(2.19)

(2.20)

Actually, Klimontovich used a slightly different definition with his N(xμ , pμ ) = mcF K (x, p). Also, for his oneparticle DF defined on M8 , one has f (xμ , pμ ) = mcF (x, p). Similarly, Hakim [13] used R(x, p) = mcF K (x, p) and D(x, p) = mcF (x, p).

20

Kinetic Equation

one obtains ∂ (pμ F K ) ∂  μ ν λ K − μ νλ p p F = 0. μ ∂x ∂p

(2.21)

∂ pμ ∂  μ ν λ − p p =0 ∂xμ ∂ pμ νλ

(2.22)

Using the identity

(see [21]), and applying to eq. (2.21), the averaging procedure defined by eq. (2.16), one finally gets pμ

∂F μ ν λ ∂F − p p = 0. νλ ∂xμ ∂ pμ

(2.23)

This is the collisionless kinetic equation for the distribution function F defined in M8 by eq. (1.10). As for the DF f (p, x, t ) defined in M6 , the corresponding equation can be obtained using eq. (1.6) and integrating eq. (2.23) over p0 . As the result, one has ∂f ∂f i pμ μ − νλ pν pλ i = 0. (2.24) ∂x ∂p Finally, assuming that it is possible to introduce a local Lorentz reference frame and define the expressions for St [ f ] in that frame, one can write by analogy with eq. (2.9) the general expression for the Boltzmann equation as pμ

∂f ∂f i − νλ pν pλ i = St [ f ]. ∂xμ ∂p

(2.25)

This equation has to be compared with eq. (2.9): in general relativity the curved nature of spacetime results in a term similar to the external force in eq. (2.9). Another form of Boltzmann equation can be written in a different way, similar to eq. (2.13), by introducing the Cartan covariant derivative (see [21]) ∂ λ ν ∂ ∇˜ μ (x, p) ≡ μ − μν p . ∂x ∂ pλ

(2.26)

Then, for the ensemble-averaged DF, one has pμ ∇˜ μ f (x, p) = St [ f ].

(2.27)

Notice that definitions (1.12) and (1.14) have to be modified (see, e.g., [9]), because the volume element in momentum space is changed as  d3 p d3 p −→ det g μν 0 . p0 p

(2.28)

It is appropriate to stress once more that eq. (2.17), and consequently eq. (2.25), assume a fixed background geometry.

2.5 Radiative Transfer

21

2.4 Quantum Corrections to the Collision Integral When particles follow quantum statistics, it is still possible to use the collision integral [37, 38], which is phenomenologically modified as follows:  3 3 d p1 d p2 d 3 p 2 1 St [ f ] = (2.29) 2 p 0 p02 p 0 1 2 ¯ p1 )][1 + θ n(x, ¯ p2 )]W (p 1 , p 2 | p1 , p2 ) × { f (x, p 1 ) f (x, p 2 )[1 + θ n(x, − f (x, p1 ) f (x, p2 )[1 + θ n(x, ¯ p 1 )][1 + θ n(x, ¯ p 2 )]W (p1 , p2 | p 1 , p 2 )}, where f (x, p) = g¯n(x, p)/(2π h¯ )3 ; g is the degeneracy factor, which counts the number of states available with the same momentum and position; θ = ±1, 0 for, respectively, Bose-Einstein, Fermi-Dirac, and Boltzmann statistics; and h¯ = h/(2π ) is the reduced Planck’s constant. Comparing this expression to eq. (2.13), one finds additional multipliers 1 ± (2π h¯ )3 f (x, p)/g, which guarantee that equilibrium distribution functions are indeed Bose-Einstein and Fermi-Dirac ones, respectively (see, e.g., [9, 39, 29]). The quantity n(x, ¯ p) is dimensionless and is called the occupation number. It is clear that classical kinetic equations are limited to the condition that quantum correlations between particles can be neglected. When the concept of particle cannot be used, quantum KT, which is not considered in this book, should be adopted. 2.5 Radiative Transfer It is instructive to present the connection between kinetic and radiative transfer formalisms. The problem of radiation transfer in a medium is solved using the formalism similar to KT. The main difference is that KT describes macroscopic properties of media and radiation through microscopic interactions of constituent particles, while radiation transport is a formal approach that attributes given macroscopic properties to a medium without considering elementary interactions. In addition, while in KT, the distribution function is a local variable, the radiation field is, generally speaking, nonlocal in the sense that this field is not determined only by local conditions. On one hand, when the medium is opaque, radiation transport is a diffusion process and the properties of radiation are determined by the local conditions in the medium. In this case the two formalisms are equivalent. On the other hand, when the medium is transparent, photons stream out without interaction with the medium. Radiative transfer equations properly account for both these asymptotic cases and describe the intermediate case of a finite optical depth. Radiative transfer theory has been developed in the second half of the twentieth century and is presented in several monographs [40, 41, 42, 43, 44]. It has a specific

22

Kinetic Equation

terminology, but its master equation can be derived from the Boltzmann equation for photons (see, e.g., [45]). The central quantity in radiative transfer theory is specific intensity2 of photons Iν . Following [46], consider the energy of photons dEν in the frequency interval (ν, ν + dν ) passing through an area dA having the normal vector n at the point x from the direction s, with the element of solid angle d, during time interval dt. This energy is related to the specific intensity of radiation as follows: dEν = Iν (x, s,ν, t )μdAddνdt,

(2.30)

where μ = (n · s) = cos ϑ and ϑ is the angle between the direction of photon propagation s and the normal n to the area dA. The specific intensity Iν characterizes the radiation field from the macroscopic point of view. From the microscopic point of view, the radiation field is composed of photons, which can be characterized using the one-particle DF f . Recalling the definition (1.3) and writing d 3 p = p2 d pd, d 3 x = μdAcdt, p = (hν/c)s, one obtains the relation between these two quantities h4 ν 3 f (x, p, t ) = C1 f (x, p, t ). (2.31) c2 Given the Lorentz invariance of the DF, it is clear from this equation that the quantity Iν /ν 3 is Lorentz invariant too. Using this relation, the Boltzmann equation (2.9) can be rewritten as

  1 ∂Iν 1 ∂Iν + c(s · ∇ )Iν = , (2.32) C1 ∂t C1 ∂t St Iν (x, s,ν, t ) =

where the RHS of this equation describes local interactions of the photon field with the medium. As photons can be absorbed, emitted, or scattered, the RHS in the general case contains three terms responsible for each of the processes. Quantum electrodynamics describes elementary processes of interaction between photons and electrons, positrons, protons, photons, and other particles. Some of these processes are discussed in detail in Chapter 11 in Part III. The radiative transfer theory defines these processes formally through the corresponding coefficients. One may define the mass absorption coefficient κν so that the energy dE in the frequency interval (ν, ν + dν ) and time interval dt of a beam with intensity Iν incident normal to the area dA contained within an element of solid angle d is absorbed by a volume element dV = dAdl with mass density ρ of cross section dA and length dl along the ray as dE = κν ρIν ddAdνdtdl. 2

(2.33)

Here and subsequently in this section, the subscript "ν" refers to frequency, not to be confused with the tensor index.

2.5 Radiative Transfer

23

Similarly, the scattering coefficient σν is defined as the energy scattered from the same volume from the same beam: dE = σν ρIν ddAdνdtdl.

(2.34)

The total energy loss from the beam is called the extinction coefficient and is the sum of absorption and scattering coefficients: ν = κν + σν .

(2.35)

The mass emission coefficient jν is defined by the energy added in the same volume element and emitted in the direction of the beam: dE = jν ρddAdνdtdl,

(2.36)

which is the sum of emitted and scattered energy. The energy balance between these processes gives the RHS of eq. (2.32), which can be multiplied by the coefficient C1 to bring it to the standard form of the radiative transfer equation: ∂Iν + c(s · ∇ )Iν = −ν ρIν + jν ρ. ∂t

(2.37)

By introducing the differential optical depth3 dτ = ν ρdl

(2.38)

and the source function Sν = jν /ν , the radiative transfer equation can be rewritten in a more compact form: dIν = −Iν + Sν . dτ This gives a well-known formal solution in the absence of sources: Iν ∝ Iν (0) exp(−τ ).

(2.39)

(2.40)

The optical depth is an invariant quantity (see eq. (13.7)), so it is clear from the derivation that the form (2.39) of the radiative transfer equation is Lorentz invariant. In Part III the relativistic Boltzmann equation (2.9) is represented in the form à la radiative transfer as 1∂f 1 + + v · ∇ f = η(x, k) − χ (x, k) f (x, k), (2.41) c ∂t c where η and χ are also called emission and absorption coefficients. This form is convenient, because it represents creation (emission) and destruction (absorption) of states, described by the one-particle DF. 3

Not to be confused with the proper time defined after eq. (1.9).

24

Kinetic Equation

2.6 Cross Section An important concept describing the strength of particle interactions is the cross section. It plays an important role in the case of two-particle collisions, which is the most simple and hence the most studied case. This concept will be illustrated for the process of scattering (2.10). It is possible to introduce [47] the following invariant variables, called Mandelstam variables:  2 2 2   s = pμ1 + pμ2 , u = pμ1 − p 2 μ , t = pμ1 − p μ . (2.42) 1 They prove technically useful, but two of them also possess a physical interpretation: sc2 is the square of the energy in the center of momemtum (CM) reference system, where pμ = {p0 , 0, 0, 0}, t is related to the scattering angle in this system: cos ϑ − 1 = 2t/(s − 4m1 m2 c2 ). Then one may rewrite  μ  W (p1 , p2 | p 1 , p 2 ) = sσ (s, ϑ )δ 4 pμ1 + pμ2 − p μ (2.43) 1 − p2 , where σ (s, ϑ ) is the differential cross section for a given process. Recall that the differential cross section dσ is defined [48] through the probability dw of the process per unit time and unit volume as dw = jdσ,

(2.44)

where j=

cI p01 p02V

(2.45)

is the quantity related to the relative velocity of particles, V is normalization volume, and  2  2 1/2 I = pμ1 pμ2 − m1 m2 c2 (2.46) is invariant flux of particles in the initial state. Here summation over μ is assumed. It is possible to show [2] that  3 3   d p1 d p2 1 W (p1 , p2 | p1 , p2 ) = σ d = dσ, (2.47) p 0 p 0 1 2 j where d is the element of the solid angle in the CM reference frame. Then, using the detailed balance condition W (p1 , p2 | p 1 , p 2 ) = W (p 1 , p 2 | p1 , p2 ),

(2.48)

2.7 Relaxation Time

one may write the Boltzmann equation as  3 d p2 1 μ p ∂μ f = σ [ f (x, p 1 ) f (x, p 2 ) − f (x, p1 ) f (x, p2 )]d, 0 2 p2

25

(2.49)

or in vector notation as  ∂f 1 +v·∇f = d 3 p2 σ v[ f (x, p 1 ) f (x, p 2 ) − f (x, p1 ) f (x, p2 )]d, (2.50) ∂t 2 where vrel = cI/(p01 p02 ) is the relative velocity of particles, also called Møller velocity. 2.7 Relaxation Time One of the useful concepts of KT is the relaxation time. It gives an estimate of the timescale on which the system is driven toward equilibrium. The RHS of the Boltzmann equation can be modeled [49, 50] as 1 St [ f ] = −pμUμ [ f (x, p, t ) − feq (x, p)], t∗

(2.51)

where feq (x, p) stands for equilibrium DF, which is a solution of the kinetic equation (2.9), and t∗ is the characteristic timescale of the relaxation process, called relaxation time. Given eq. (2.49), an order of magnitude estimation for the relaxation time is 1 t∗ ∼ , (2.52) n σ v  where brackets σ v = n−1 σ v f d 3 p denote the averaging with the DF. Particularly simple is the timescale for particles moving with relativistic relative velocities and interacting with constant cross section, in which case one has t∗ ∼ (nσ c)−1 . The relaxation time is of practical use, as in many applications, average particle velocities are known.

3 Averaging

The main object of KT, the one-particle DF, is not a directly observable quantity. To relate this quantity to observable ones, some kind of averaging is required. Hence averaging is one of the fundamental instruments of KT. The averaging has already been discussed in Section 1.6, when macroscopic quantities as moments of oneparticle DF were introduced. While in nonrelativistic physics, averaging appears to be straightforward, it does not prove so in relativistic generalization, where simple time or space averaging are not Lorentz invariant, since space and time are not absolute anymore. As invariance of the DF, which has been proven in Section 1.5, the invariance of averaging is required as well. Within general relativity, averaging is an even more complicated issue, with fully covariant self-consistent formulation of KT still missing. One of the most interesting applications of this issue is the study of back-reaction of small-scale inhomogeneities on the global dynamics in cosmology. In this chapter, covariant averaging is discussed. The role of averaging in KT is also addressed. 3.1 Covariant Statistical Averaging In this section the approach developed in [24] to construct the covariant statistical averaging is adopted. The covariant one-particle DF defined in eq. (1.10) (see also eq. (2.15)) is related to the Klimontovich random function as F (x, p) =

F K (x, p) ens . The Klimontovich random function is singular, as it contains δfunctions. It can be smoothed out, or coarse grained, by employing a weighted  average, with weights satisfying the condition k wk = 1, as follows:  1 w F (x, p) = wk dτ δ 4 [x − xik (τ )]δ 4 [p − pik (τ )]. (3.1) mc k i Such a weighted DF replaces the one-particle DF introduced in eq. (1.10). Similarly, s-particle DFs can be smoothed out. 26

3.1 Covariant Statistical Averaging

27

The useful microscopic quantities appearing in KT, which require averaging, are usually sums of one-particle or two-particle quantities. The sums of one-particle quantities have the form  A(x) = dτ a{xi (τ ), pi (τ ); x}, (3.2) i

where a(x, p) is some function, while those of two-particle quantities have the form  dτ b{xi (τ ), pi (τ ), x j (τ ), p j (τ ); x}, (3.3) B(x) = i= j

where b(x, p) is some function. The statistical (or ensemble) averaging is performed by taking weighted averages. Hence the average of the quantity (3.2) can be defined using eq. (3.1) as  1

A(x) ens ≡ a(x, p)F (x, p)d 4 pd 4 x. (3.4) N M8 Similarly, the average of the quantity (3.3) is  1

B(x) ens ≡ b(x, p)F (x, p, x , p )d 4 pd 4 xd 4 p d 4 x , N(N − 1) M8 where F2w (x,

1 p, x , p ) = (mc)2





 dτ





δ 4 [x − xi (τ )]δ 4 [p − pi (τ )]

(3.5)

(3.6)

i, j

× δ 4 [x − x j (τ )]δ 4 [p − p j (τ )] is the two-particle DF. It is crucial that these definitions imply that averaging and derivatives commute, namely,

∂μ a(x) ens = ∂μ a(x) ens .

(3.7)

This property is important and is used later, in Chapter 5. Covariant formulation of classical electrodynamics in media requires covariant averaging of Maxwell equations with discrete sources, which has been performed in [24] using the notions of synchronous, retarded, and advanced DFs. In this way, macroscopic Maxwell equations were derived together with the polarization tensor, representing the material relations between macroscopic average fields and induction.

28

Averaging

3.2 Spacetime Averaging As spacetime averaging is not unique, there are several proposals for its definition in the literature [25, 51] – for a review, see [52]. There are necessary conditions to be satisfied by such definitions, i.e., covariance and independence on the volume and shape of the averaging region of spacetime. A particular definition of the covariant spacetime averaging is given in [25]:    ν  1 Tμ (x) ≡ Aμμ (x, x )Aνν (x, x )Tνμ (x ) −g(x )d 4 x , (3.8) V where g(x ) = det gμν (x ) and where   −g(x )d 4 x V =

(3.9)

is the finite volume of spacetime region over which the averaging is performed,

Tμν (x) is the averaged tensor field defined in the supporting point x and Aμμ (x, x ) are some bilocal transport operators satisfying two conditions, limx →x Aμν (x, x ) = δνμ and Aμν Aνλ = δλμ . It is clear that in Minkowski space the averaging reduces to    ν  1 Tμ (x) = Tνμ (x ) −g(x )d 4 x . (3.10) V In the special case of a scalar quantity on the three-dimensional hypersurface, one can simplify the averaging to   1

A(xi , t ) = A(xi , t ) − det gi j (x )d 3 x , (3.11) VS S where VS is the 3-volume of the region S and gi j is the 3-metric defined on this hypersurface. Recently a new proposal for covariant averaging has been put forward [53]. It involves definition of averaged quantities by means of surface integrals along the backward-lightcones related to a given event of the observation, as the most straightforward generalization of the averaging procedure in special and general relativity. Such backward-lightcones are Lorentz invariant and accessible to measurements for any inertial observer. In the nonrelativistic limit c → ∞, the lightcone flattens so that measurements on it become isochronous in any reference frame. 3.3 The Role of Averaging in Kinetic Theory In nonrelativistic KT and in statistical physics, ensemble averaging is closely related to space and time averaging. While experiments deal with space- and

3.3 The Role of Averaging in Kinetic Theory

29

time-averaged quantities, theory usually works with ensemble-averaged ones. Hence the connection between macroscopic and microscopic quantities, on one hand, and also spacetime-averaged quantities and ensemble averaged ones, on the other hand, is required. An important concept called statistical equilibrium requires that any macroscopically large part of the system have macroscopic physical quantities equal to their statistical average values. For a microscopic quantity this statement can be represented as follows:   +∞   +∞ A(x, p) f (x, p)d 3 pd 3 x 1 3 3 , (3.12) A(x, p)d pd x = V −∞ +∞ 3 3 V V −∞ V −∞ f (x, p)d pd x where V is an arbitrary macroscopic 3-volume. One of the most important theorems in statistical mechanics states that for ergodic systems the time-averaged quantity should be equal to its ensemble average. In a specific case when the system is described by one-particle DF, this statement is reduced to   1 T 1 lim A(x, p)dt = A(x, p) f (x, p)d 3 pd 3 x. (3.13) T →∞ T 0 N M6 It is difficult to prove ergodicity of a given physical system. Nevertheless, ergodicity is often assumed in practice.1 However, it is clear that the notions of separate space and time averaging are not Lorentz invariant. For this reason the notion of erodicity has to be reconsidered in relativistic KT. 1

In systems with long-range interactions, such as self-gravitating systems, the ergodicity is broken (see Chapter 14).

4 Conservation Laws and Equilibrium

Microscopic conservation of energy and momentum at each interaction between particles implies the existence of conservation laws for macroscopic quantities such as four-current and energy-momentum tensors, which represent basic hydrodynamic equations. In this chapter, equations of relativistic hydrodynamics are derived from the Boltzmann equation. Then the H-theorem is proven and conditions for local thermodynamic equilibrium are formulated. Useful macroscopic quantities as well as one-particle DF in equilibrium are obtained. The generalized continuity equation for nonequilibrium systems is also derived.

4.1 Conservation Laws and Relativistic Hydrodynamics In this chapter, following [2], the conservation laws fulfilled by the macroscopic quantities are derived, namely, the particle number conservation, the entropy conservation, and the energy-momentum conservation (see also [54, 55]). Consider a mixture of D components (sorts) whose particles may interact by elastic or inelastic collisions, conserving the total number of particles. In the absence of external fields, Boltzmann equations (2.9) for each DF fk (x, pk ) read1 pμk ∂μ fk =

D

Ckl (x, pk ),

(4.1)

l=1

where D   d 3 pl d 3 pi d 3 p j  1 Ckl (x, pk ) = f W − f f W f . i j i j|kl k l kl|i j 2 i, j=1 p0l p0i p0j

1

(4.2)

In this chapter, Latin indices denote the sorts of particles and should not to be confused with tensor indices.

30

4.1 Conservation Laws and Relativistic Hydrodynamics

31

An important property of collision integrals follows from the microscopic conservation laws fulfilled at each interaction, namely, D  d 3 pk ψk (x)Ckl (x, pk ) = 0, (4.3) F= p0k k,l=1 where ψk (x) are so-called summational invariants ψk (x) = ak (x) + pμk bμ (x);

(4.4)

they are arbitrary functions, except for the constraint that ak (x) is additively conserved in all reactions, i.e., ai (x) + a j (x) = ak (x) + al (x),

(4.5)

and bμ (x) is an arbitrary vector. The proof of eq. (4.3) is based on eq. (4.5) and on energy-momentum conservation in a binary reaction pμi + pμj = pμk + pμl . In particular, for elastic scattering, one has  3 d pk Ckl (x, pk ) = 0. (4.6) p0k Now it is possible to show how the basic equations of relativistic hydrodynamics, namely, the particle number conservation (continuity) equation and the energymomentum conservation equations, arise from the Boltzmann equation. Consider the case when, in eq. (4.4), bμ (x) = 0 and ak (x) = qk a(x), where a(x) is an arbitrary function. Then, from eqs. (4.1) and (4.2), one has  3 D d pk μ qk p ∂μ fk = 0. (4.7) p0k k k=1 Recalling the definition (1.12) for each component  3 d pk μ μ jk = c p fk , p0k k

(4.8)

one gets μ

∂μ J = 0,

μ

J =

D

qk jkμ ,

(4.9)

k=1

where qk is a charge (e.g., electric, leptonic, baryonic). In particular, with q = 1, this is just particle number conservation. Similarly, the conservation law for the total particle number can be obtained. In particular, for elastic scattering, using eq. (4.6), one finds ∂μ jkμ = 0.

(4.10)

32

Conservation Laws and Equilibrium

Consider now the case ak (x) = 0. Then, from eq. (4.3), one finds D  d 3 pk μ pk Ckl = 0. 0 p k k,l=1

(4.11)

Substituting this into eqs. (4.1) and (4.2) and recalling the definition (1.14), one gets ∂ν T μν = 0, where T

μν

D  d 3 pk μ ν =c p p fk p0k k k k=1

(4.12)

(4.13)

is the energy-momentum tensor of the mixture. This is the energy-momentum conservation equation. Equations (4.9) and (4.11) represent basic equations of relativistic hydrodynamics (see, e.g., [44, 56]). These equations are widely employed in astrophysics and cosmology as well as in relativistic heavy-ion collisions [57, 58, 59]. Notice that the system of eqs. (4.9) and (4.11) is not closed. In particular, for a single component, these are 5 equations for 14 unknown quantities: 4 components of the four-current J μ and 10 components of the energy-momentum tensor T μν . Other equations needed for the closure are determined by matching the hydrodynamics to the underlying KT. Traditionally, two schemes are used for this purpose: the Chapman-Enskog or Grad method. Both have drawbacks, and so far the consistent relativistic hydrodynamics derivation out of KT is lacking. For recent results, see, e.g., [60, 61]. Regarding multicomponent systems or mixtures, where inelastic collisions (interactions) between components are possible, eqs. (4.10) are no longer valid and a nonvanishing RHS appears, which comes from interactions (see Section 4.5 and Chapter 9). 4.2 H-theorem First, one has to show that the quantity defined as divergence of four-vector (1.15) as σ (x) ≡ ∂μ Sμ

(4.14)

can never decrease. This section follows [2]; for an alternative derivation, see [20]. From eqs. (1.15) and (4.14) it follows that  3 d p   3  μ σ = −kB c (4.15) log h f p ∂μ f . p0

4.3 Equilibrium

33

Substituting Boltzmann equation (2.9) into this expression, one gets  3  3 d p   3  d p   3  μ ∂ f σ = −kB c log h log h f F f St f + k c . (4.16) [ ] B p0 p0 ∂ pμ μ

Assume that the force satisfies the following properties: pμ Fμ = 0 and ∂F = 0. ∂ pμ The former condition means that the force is mechanical and does not alter particle rest mass. Then the second contribution to the integral in eq. (4.16) can be written as       ∂  2kB c d 4 p μ (p0 )δ(pμ pμ − m2 c2 ) f log h3 f − 1 F μ , (4.17) ∂p and it vanishes, provided that the integrand is decreasing fast enough for large momenta in the sense defined in Section 1.4. The first contribution can be rewritten as  d 3 pi d 3 p j d 3 pk d 3 pl  fk fl  1 σ = − kB c log fi f jWi j|kl . (4.18) 0 0 0 0 4 f f p p p p i j i j k l i, j,k,l Now using the property  d 3 pi d 3 p j d 3 pk d 3 pl   fk fl − fi f j Wi j|kl = 0, 0 0 0 0 pi p j pk pl i, j,k,l which follows from the bilateral normalization condition [2],  d 3 pi d 3 p j  d 3 pi d 3 p j W = Wi j|kl , kl|i j 0 0 0 0 p p p p i j i j i, j i, j

(4.19)

(4.20)

one finally gets

 d 3 pi d 3 p j d 3 pk d 3 pl 1 A (y) fi f jWi j|kl , σ = kB c 4 p0i p0j p0k p0l i, j,k,l

(4.21)

where A (y) = y − log y − 1 ≥ 0,

y=

fk fl . fi f j

(4.22)

Since A(y) is a nonnegative function, eq. (4.21) implies that σ ≥ 0. This completes the proof of the Boltzmann H-theorem. 4.3 Equilibrium Notice that σ = 0 holds if and only if   fi (x, pi ) f j x, p j = fk (x, pk ) fl (x, pl ) .

(4.23)

34

Conservation Laws and Equilibrium

This condition is satisfied, as can be seen from eq. (2.49), when the collision integral in the RHS of the Boltzmann equation vanishes. This case is identified as local equilibrium. In fact, the equilibrium DF is characterized by the following macroscopic quantities as parameters: density, temperature, 4-velocity. It is possible to show this by turning to a simple system with binary collisions and rewriting the condition (4.23) as         log h3 f1 + log h3 f2 = log h3 f1 + log h3 f2 . (4.24) It is clear that the quantity log(h3 f ) is a summational invariant (c.f. eq. (4.5)). The most general summational invariant, as discussed earlier, is a linear combination of a constant and pμ (see eq. (4.4)). Then one-particle DF in equilibrium is f eq =

  1 exp a(x) + bμ (x)pμ 3 h

(4.25)

with arbitrary space- and time-dependent parameters a(x) and bμ (x). However, DF f eq will be a solution of the Boltzmann equation if and only if it turns to zero also its LHS. Then the parameters of eq. (4.25) should satisfy pμ ∂μ a(x) + pμ pν ∂μ bν (x) + mbμ (x)F μ (x, p) = 0,

(4.26)

which should be an identity for arbitrary pμ . When the DF satisfies eq. (4.26), it is called global equilibrium DF f EQ . In the absence of an external field, f EQ reduces to the Jüttner [4] momentum distribution

1 φ − pμUμ EQ , (4.27) f (p) = 3 exp h kB T where φ, T , and Uμ are parameters, U μUμ = c2 . There have been long-standing debates in the literature whether other possible forms of equilibrium distribution of particles are possible (see, e.g., [62, 63]). It has been shown recently, out of numerical simulations based on fully relativistic molecular dynamics, that the correct equilibrium distribution is indeed given by Jüttner, in eq. (4.27) (see [63, 64, 53]), for one-, two-, and three-dimensional cases, respectively. It is now possible to compute such important macroscopic quantities as the number density, the energy density, and the pressure of a system in local equilibrium. Using the definition (1.12) and jμ = nU μ , one has  3   ν  d p μ 1 φ −p Uν jμUμ . (4.28) p Uμ exp n = 2 = 3 exp c ch kB T p0 kB T The integral, being a scalar, can be evaluated in the Lorentz reference frame, where U μ = (c, 0, 0, 0), by introducing polar coordinates and dimensionless variables

4.3 Equilibrium

35

 θ = kB T/(mc2 ), ν = φ/(mc2 ) and y = c p2 + m2 c2 /(kB T ). The result is ν    1 (4.29) exp neq = θ K2 θ −1 , 3 2 θ 2π λC where λC =

h mc

and

  2n−1 (n − 1)! −n z Kn θ −1 = (2n − 2)!





n− 3  dy y2 − θ −2 2 y exp (−y)

(4.30)

z

is the modified Bessel function of the second kind. In full analogy, using the definition of the energy-momentum tensor and T μν = c−2 ρU μU ν − p μν , where μν = gμν − c−2U μU ν is the projection operator, one can compute the energy density ρ and pressure P as follows:  3 T μν UμUν d p  μ 2 EQ 1 ρ= p Uμ f = (4.31) 2 c c p0  3 d p μ ν c 1 μν p p μν f EQ . (4.32) P = − T μν = − 3 3 p0 Performing the integrals, one finally gets ν       mc2 exp ρ= 3θ 2 K2 θ −1 + θ K1 θ −1 3 2 θ 2π λC  2 ν  2  −1  mc . exp P= θ K2 θ θ 2π 2 λC3

(4.33) (4.34)

Introducing the enthalpy as he = (ρ + p)/n, one obtains he = mc2

K3 (θ −1 ) . K2 (θ −1 )

(4.35)

The entropy density is given by   3   ν   SμUμ φ d p μ −p Uν ϕ − pν Uν kB s= − 1 exp . = − exp p Uμ 2 0 c c kB T p kB T kB T (4.36) Taking into account eqs. (4.29) and (4.33), this gives 1 (ρ − φn) + kB n. (4.37) T Finally, for the adiabatic index ≡ cP /cV , which is the ratio of specific heat capacities     ∂he ∂ (ρ/n) cP = cV = , (4.38) ∂T p ∂T v s=

36

Conservation Laws and Equilibrium

1.8 1.7 1.6 1.5 1.4 1.3 1.2 0.001

0.01

0.1

1

10

100

q

Figure 4.1 The adiabatic index of relativistic gas as function of dimensionless temperature.

one has

    he 2 he = θ −2 + 5 − , −1 mc2 θ mc2 θ

and the limiting cases are

5 →

3

,

4 , 3

θ →0 θ →∞

.

(4.39)

(4.40)

Combining expressions (4.29), (4.33), (4.34), (4.35), and (4.37), one finds the perfect gas laws P = nkB T, P = ( − 1) ρ,

(4.41)

φ = he − T s. In Figure 4.1 the dependence (θ ) computed using eqs. (4.39) and (4.35) is shown. Nonrelativistic and ultrarelativistic asymptotes are shown with dashed lines. Interestingly, at temperatures kB T ∼ mc2 , usually considered mildly relativistic, this function is already close to its ultrarelativistic value. Note that the traditional scheme of thermodynamics is recovered if one identifies T as temperature and φ as the chemical (Gibbs) potential. 4.4 Relativistic Maxwellian Distribution It is instructive to consider the relativistic Maxwellian distribution of particles in more detail. Considering eq. (4.27) in the local rest frame and assuming ν = 0, one

4.5 Generalized Continuity Equation

37

5

dP[b]

4 3 2 1 0 0.0

0.2

0.4

0.6

0.8

1.0

b

Figure 4.2 Relativistic Maxwellian distribution function for selected values of dimensionless temperature. From left to right, the temperature increases, taking the values θ = {0.02, 0.06, 0.18, 0.56}.

has f LEQ =

 γ 1 exp − , h3 θ

(4.42)

where γ = p0 /(mc) is particle Lorentz factor. Using eq. (1.4) and comparing it with eq. (4.29), one gets the probability distribution for particle energy (or Lorentz factor)  γ  1   γ γ 2 − 1 exp − dP (γ ) = dγ . (4.43) θ θ K2 θ −1 The probability distribution for the velocity β = |v|/c is   1 1 5 2   γ β exp −  dβ. dP(β ) = θ K2 θ −1 1 − β 2θ

(4.44)

The function dP(β ) is shown in Figure 4.2 for selected values of the dimensionless temperature θ = {0.02, 0.06, 0.18, 0.56}. While the DF with the lowest temperature is reminiscent of a classical Maxwellian, the one with the highest temperature is already far from it: the effect of limiting velocity is clearly visible. For more details, see [65], where the transition from nonrelativistic to ultrarelativistic form in eq. (4.27), and consequently in eq. (4.43), is considered as a critical phenomenon. 4.5 Generalized Continuity Equation So far only such interactions where particle conservation is satisfied have been discussed. An obvious example is scattering. However, there are processes where

38

Conservation Laws and Equilibrium

particle conservation does not hold. This is possible either for self-interacting particles or for mixtures with reactions. The simplest examples are the annihilation of particle-antiparticle pairs in two photons and the inverse process of pair creation from two photons. This process is discussed in detail in Chapter 11. Even if the total number of particles (both pairs and photons) is conserved, the individual number of particles in each component can change. Consider in more detail the process e+ + e− ←→ γ1 + γ2 ,

(4.45)

with the corresponding energy-momentum conservation p− + p+ = k1 + k2 , where subscripts “−” and “+” refer to electrons and positrons, respectively. For a positron (electron) from eq. (2.13), one has2  3 d p∓ d 3 k 1 d 3 k 2 μ p ∂μ f ± = [ f1 f2W (k1 , k2 | p± , p∓ ) − f± f∓W (p± , p∓ | k1 , k2 )], p0∓ k10 k20 (4.46) where f± = f (x, p± ), etc. From eqs. (2.44) and (2.46), one gets

vrel

d 3 k1 d 3 k2 W (p± , p∓ | k1 , k2 ) = jdσ k10 k20  c = (v− − v+ )2 − (v− × v+ )2 = I 0 0 , p± p∓

(4.47) (4.48)

where vrel is the relative velocity between electron and positron. Then, integrating 3 eq. (4.46) over d p0p± and using eq. (4.47), one obtains  ∂μ

±

d 3 p± μ p f± = p0±



d 3 p± d 3 p∓ d 3 k 1 d 3 k 2 f1 f2W (k1 , k2 | p± , p∓ ) p0± p0∓ k10 k20  3 d p± d 3 p∓ f± f∓ jdσ. − p0± p0∓

In view of eqs. (1.4) and (4.48), the annihilation rate is defined as  3 d p± d 3 p∓ n± n∓ σ v ann ≡ c f± f∓ jdσ. p0± p0∓

(4.49)

(4.50)

This is an invariant quantity, as can be seen from the analysis of the RHS. Notice that the LHS in eq. (4.49) is nothing but the derivative of the particle four-flux (1.12). In equilibrium this quantity is conserved (see eq. (4.10)). So, in equilibrium,  3 d p± d 3 p∓ d 3 k 1 d 3 k 2 eq c f1 f2W (k1 , k2 | p± , p∓ ) = neq (4.51) ± n∓ σ v ann . p0± p0∓ k10 k20 2

Electrons and positrons are distinguishable particles, hence there is no factor 1/2 in front of the collision integral.

Thus one can write

4.5 Generalized Continuity Equation

39

  eq μ ∂μ j ± = σ v ann neq + n− − n+ n− ,

(4.52)

which reduces to eq. (4.10) in thermal equilibrium, since in equilibrium, particle nonconserving interactions balance each other. This equation, first obtained by Zeldovich [66, 67], finds numerous applications, especially in cosmology, since it is much easier to solve than the integro-differential eq. (2.13). In particular, it describes freeze-out of particles decoupling from the thermal plasma during expansion of the universe, e.g., electron-positron pairs and neutrinos.

5 Relativistic BBGKY Hierarchy

In this chapter the derivation of the Bogolyubov-Born-Green-Kirkwood-Yvon hierarchy for relativistic plasma is given following [18]. The basic idea in this approach is that any N-body system can be characterized by the set of equations of motion under the given interaction. Applying averaging to Klimontovich random functions, one can derive the chain of equations (hierarchy) for many particle distribution functions. To obtain tractable kinetic equations, this hierarchy can then be truncated at a certain level, using expansion in small parameters or other physical considerations. 5.1 The Hierarchy of Kinetic Equations For definiteness, consider a system of N charged particles of equal mass m and charge q, interacting via the electromagnetic field. In this case it is convenient to transform the four-momentum as pμ −→ pμ − qc Aμ , where Aμ is the vector potential of the electromagnetic field. The equations of motion are d pμ q = F μν uν , dτ c

pμ = muμ ,

uμ =

dxμ , dτ

(5.1)

where F μν is the electromagnetic field tensor and uμ is the particle four-velocity. For a point particle the corresponding four-current is   jμ = q d 4 puμ dτ δ 4 [xν − xν (τ )]δ 4 [pν − pν (τ )]. (5.2) Recalling the definition (2.15) of the Klimontovich random function, one may proceed in analogy with Section 2.3. Using eq. (5.1), one arrives at the Klimontovich equation: pμ ∂μ F K + 40

∂F K ∂ q q pν F μν μ = pμ ∂μ F K + pν μ (F μν F K ) = 0. c ∂p c ∂p

(5.3)

5.1 The Hierarchy of Kinetic Equations

41

This equation has to be supplemented by the Maxwell field equations ∂μ F μν = 4π J ν

εμνσ ρ ∂ ν F σ ρ = 0,

(5.4)

where εμνσ ρ is the completely antisymmetric Levi-Civita symbol and J μ = N μ i=1 ji is the four-current of electric charge (see eq. (1.2)), defined for the Klimontovich random function (2.15). Equations (5.3) and (5.4) are the basis for derivation of the relativistic BBGKY hierarchy. Notice that these equations are used in numerical simulations, in particular in particle-in-cell algorithms, discussed in Section 9.4.2. The solutions of eqs. (5.3) and (5.4) are approximate solutions to the Vlasov-Maxwell system of equations [68] with accuracy O(μ), where g p = (nλ3D )−1 is the plasma parameter (see eq. (6.21)), λD is the Debye length (see eq. (6.9)), and n is particle density. One may start from basic equations for the Klimontovich random function and the electromagnetic field. Maxwell equations can be formulated for the vector potential, defined by Fμν = ∂μ Aν − ∂ν Aμ .

(5.5)

Equations (5.3) and (5.4) then may be written, using eq. (5.1), as follows: ∂F K (x, p) q =0 pμ ∂μ F K (x, p) + pν Fμν c ∂ pμ  4π q ν pμ F K (x, p)d 4 p, ∂μ Aμ = 0. ∂ ∂ ν Aμ = mc

(5.6) (5.7)

These equations are valid for a one-component system, e.g., electrons described by their DF (2.15), embedded in a positive charge that is distributed uniformly. Generalization to a system composed of ions and electrons is straightforward. Equation (5.7) can be solved neglecting radiation effects by virtue of the Green’s function formalism [36] (see also [69], p. 183). The solution to eq. (5.7) in infinite space with no boundaries is   q 4 Aμ (x) = d x G(x, x ) d 4 p p μ F K (x , p ), (5.8) mc where G(x, x ) is Green’s function, corresponding to the d’Alembert operator ∂ ν ∂ν G(x, x ) = 4π δ 4 (x − x ).

(5.9)

Recall that here and in what follows, x = xμ , x = xμ . Fourier transform of both sides of this equation gives     c 1 4 μ G(x, x ) = − d x , (5.10) k exp −ik − x μ μ (2π )4 k ν kν

42

Relativistic BBGKY Hierarchy

where the four-vector kμ = (ω/c, k) is introduced. Then the solution of eq. (5.7) is   4     d k  K 1 q λ d 4 x d 4 p p x . F (x , p ) exp −ik − x Aμ (x) = − 3 λ μ λ 4π mc k ν kν (5.11) This integral has a singularity at kν kν = 0, and its evaluation leads to the concept of retarded and advanced potentials (see, e.g., [36, 69]). Now eq. (5.11) can be used to eliminate the vector potential from eq. (5.6). Define the operator  q2  pν ∂μ − p μ ∂ν G(x, x ) (5.12) 2 mc       d4k   1 q2 λ λ ∂ p − ∂ p . exp −ik x exp −ik x =− 3 2 μ λ ν λ ν μ 4π mc k ρ kρ

Lμν =

Inserting eq. (5.11) into eq. (5.6) and using eq. (5.5) one arrives at the single equation for the Klimontovich random function:   ∂ μ K 4 p ∂μ F (x, p) + d x d 4 p Lμν (x − x , p )pν [F K (x , p )F K (x, p)] = 0. ∂ pμ (5.13) It is the basis for the chain of relativistic BBGKY equations for particle DF. Alternatively, one can start from the joint system of equations for DFs and transverse components of electromagnetic fields (see, e.g., [17, 70, 71]). This alternative approach, however, requires additional assumptions about electromagnetic fields. At this point, one has to introduce many-particle DFs (see, e.g., [13]). The sparticle Klimontovich random function is defined as F K (x1 , p1 ; x2 , p2 ; . . . xs , ps )   s  1 dτ . . . dτ δ 4 [x − xi (τ j )]δ 4 [p − pi (τ j )], = 1 s (mc)s i ...i j=1 1

(5.14)

s

where in the sum all ia with a = 1, . . . , s are different. The corresponding s-particle DF is defined as F (x1 , p1 ; x2 , p2 ; . . . xs , ps ) = F K (x1 , p1 ; x2 , p2 ; . . . xs , ps ) ens .

(5.15)

The first equation of the hierarchy is obtained by applying ensemble averaging to eq. (5.13). Using the definition (2.16), one gets   ∂ μ 4 p ∂μ F (x, p) + N d x d 4 p Lμν pν

F K (x, p)F K (x , p ) ens , (5.16) ∂ pμ

5.1 The Hierarchy of Kinetic Equations

where



F (x, p)F (x , p ) ens K

K

1 = (mc)2



 dτ





43

δ 4 [x − xi (τ )]δ 4 [p − pi (τ )]

i, j

    × δ 4 x − x j (τ ) δ 4 p − p j (τ )

.

(5.17)

ens

This expression contains two parts, which can be schematically represented [70, 13] as   

= + . (5.18) i= j

i, j

i= j

While the first term is just the two-particle DF, the second term represents the backreaction of a particle on itself (self-action), namely,  K  F (x, p)F K (x , p ) ens = N(N − 1)F2 (x, p, x , p ) (5.19)  + N δ 4 [x − x (τ )]δ 4 [p − p (τ )]dτ F (x, p). The second term, being infinite in classical electrodynamics, is discarded. This term is accounted for when the BBGKY hierarchy is derived for the case including radiation (see [13]). Then, applying ensemble averaging, one obtains   ∂ μ 4 p ∂μ F (x, p) + N d x d 4 p Lμν pν F2 (x, p, x , p ). (5.20) ∂ pμ Similarly, for the two-particle DF, the dynamical equation can be obtained by multiplying eq. (5.13) by F K (x , p ) and averaging it:   ∂ μ 4

F K (x, p)F K (x , p )F (x , p ) ens , p ∂μ F2 (x, p, x , p ) + d x d 4 p Lμν pν ∂ pμ (5.21) where

F K (x, p)F K (x , p )F (x , p ) ens = N(N − 1)(N − 2)







× F3 (x, p, x , p , x , p ) + N(N − 1)  × δ 4 [x − x (τ )]δ 4 [p − p (τ )]dτ F2 (x, p, x , p )  + δ 4 [x − x (τ )]δ 4 [p − p (τ )]dτ F2 (x, p, x , p )  + δ 4 [x − x (τ )]δ 4 [p − p (τ )]dτ F2 (x, p, x , p )   +N δ 4 [x − x (τ )]δ 4 [p − p (τ )]δ 4 [x − x (τ )] × δ 4 [p − p (τ )]dτ dτ F (x, p).

(5.22)

44

Relativistic BBGKY Hierarchy

For t = t one obtains μ





p ∂μ F2 (x, p, x , p ) +



4

d x



4

d p



dτ Lμν pν

∂F2 (x, p, x , p ) ∂ pμ

(5.23)

× δ 4 [x − x (τ )]δ 4 [p − p (τ )]   ∂F3 (x, p, x , p , x , p ) 4 d 4 p Lμν pν . +N d x ∂ pμ This is the second equation in the BBGKY hierarchy. Similarly, the dynamical equation for the two-particle DF can be obtained, and so on. Equations (5.20) and (5.23) represent the first two equations in the chain of BBGKY equations. Once more, notice that Klimontovich, in his original derivation [18] of the relativistic kinetic equation neglecting radiation, used the solution of Maxwell equations for the four-potential Aμ . Hence, in his chain of equations, only particleparticle correlation functions such as F2 (x, p, x , p ) appear (see also [72]).

5.2 The First and Second Approximations in Relativistic Transport Equations The BBGKY hierarchy is a system of coupled integro-differential equations. It is equivalent in complexity to the original system of equations (5.3) and (5.4). To find approximate solutions to this system, additional assumptions are required. When a gas is considered, the molecular chaos hypothesis (Stosszahlansatz) [1] is adopted: the absence of correlation between particles before the collision is assumed. In plasma physics with electromagnetic interactions between particles, the principle of rapid attenuation of correlations [11, 73] is applied. Notice that such a principle may be considered as a consequence of ergodicity of the system [74]. Systems with long-range interactions, such as self-gravitating systems lacking ergodicity, are more complicated. For the derivation of the kinetic equations out of the BBGKY hierarchy in that case, see Chapter 14. In this way, the assumption of absence of correlation between particles is F2 (x, p, x , p ) = F (x, p)F (x , p ) F3 (x, p, x , p , x , p ) = F (x, p)F (x , p )F (x , p ),

(5.24)

... It implies that eq. (5.20) becomes   ∂F1 (x, p) μ 4 d 4 p pν Lμν , F1 (x , p ) . p ∂μ F (x, p) + N d x ∂ pμ

(5.25)

5.3 The Vlasov-Maxwell System

45

Recalling the definitions (5.12) and (5.8), the Vlasov-Maxwell equations for the DF F (x, p) and for the corresponding mean field F μν ens are recovered. The system of self-consistent Vlasov-Maxwell equations can be obtained also in a formalism where DF and electromagnetic field are treated independently (see, e.g., [13, 17]). Both DF and electromagnetic field can be considered as random functions. Their average values are F (x, p) = F K (x, p) ens and F μν ens , while deviations from average values can be denoted as δF K (x, p) and δF μν , respectively. For instance, the average F K (x, p)F μν ens can be written as F K (x, p)F μν ens = F (x, p) F μν ens + δF K (x, p)δF μν . Equations for fluctuations δF μν and δF K can be formulated. The theory of fluctuations in plasma is developed in [75].

5.3 The Vlasov-Maxwell System Direct averaging of eqs. (5.3) and (5.4) gives ! " ∂F K q pμ ∂μ F + pν F μν μ =0 c ∂ p ens  μν ∂μ F ens = 4π qc F (x, p)pν d 4 p.

(5.26)

When the second central moments are neglected, i.e., the absence of correlations between fields and particles is assumed,  μν K    F F = F μν F K , (5.27) one arrives at the Vlasov-Maxwell equations

∂μ F μν

∂F q pμ ∂μ F + pν F μν μ = 0 c ∂p  = 4π qc pν Fd 4 p, εμνσ ρ ∂ ν F σ ρ = 0,

(5.28) (5.29)

where the average operator of the electromagnetic field is dropped and eq. (2.16) is used. The system of equations (5.28) and (5.29) describes relativistic plasma under conditions that particle collisions can be neglected. This is the case when the mean free path exceeds the size of the system. In such collisionless plasma, particles interact not through collisions, as in gases, but through an electromagnetic field, which in turn depends on the distribution and motion of particles themselves. This is the system of nonlinear integro-differential equations for coupled distribution function and electromagnetic tensor. Notice that these equations are completely time reversible. This implies the absence of any relaxation; in particular, the Htheorem discussed in Section 4.2 for the Boltzmann equation does not hold for the

46

Relativistic BBGKY Hierarchy

system described by eqs. (5.28) and (5.29). Equations (5.28) and (5.29) are studied in the monograph [76] from the mathematical point of view, where the Cauchy problem formulation is discussed. In the limiting case c → ∞, the Vlasov-Poisson equations follow [77]. The Maxwell equations imply that the electric field is a gradient of a potential E = −∇. Using the vector notation (2.8), the system (5.28) and (5.29) then reduces1 to ∂f ∂f ∂f +v· − q∇φ · =0 ∂t ∂r ∂p  ∇ 2 ϕ = −4π q f d 3 p,

(5.30) (5.31)

where φ is elecrostatic potential and the electric charge is −q for electrons, while for protons and positrons, it is +q. Collisionless plasma is considered further in Chapter 10, where a theory of linear response to perturbations is presented and various plasma instabilities as well as the Landau damping phenomenon are discussed.

5.3.1 Belyaev-Budker and Landau Equations To obtain the kinetic equation that takes into account particle collisions, one has to consider that at least two-particle correlation functions are not vanishing. This is possible, assuming that F2 (x, p, x , p ) = F (x, p)F (x , p ) + g pG2 (x, p, x , p ),













(5.32)



F3 (x, p, x , p , x , p ) = F (x, p)F (x , p )F (x , p ), ...,

(5.33)

where G2 (x, p, x , p ) is the relativistic correlation function and the parameter g p  1 characterizes the strength of correlations. With these conditions, eqs. (5.20) and (5.23) after integration over x and p become

  ∂G2 (x, p, x , p ) μ 4 4 ν ∂F (x, p) d p p Lμν F (x , p ) + p ∂μ F (x, p) + N d x ∂ pμ ∂ pμ pμ ∂μ G2 (x, p, x , p ) +

1



(5.34)   ∂F1 (x, p) dτ Lμν pν F1 x (τ ), p (τ ) . ∂ pμ

Vlasov-Poisson equations are also valid in Newtonian gravity, with the substitutions ∇ ←→ q2 ←→ −Gm2 , where  is the gravitational potential (see eqs. (14.14) and (14.15)).

(5.35)

q m ∇φ

and

5.3 The Vlasov-Maxwell System

47

It turns out that the correlation function is determined by the DF of unprimed particle and of primed particle at different time moments. Since particles with unprimed and primed variables are identical, another equation for G2 (x, p, x , p ) is obtained by interchanging these variables in eq. (5.35): μ







p ∂μ G2 (x, p, x , p ) +

dτ Lμν (x − x, p)p ν F1 [x(τ ), p(τ )]

∂F1 (x , p ) . (5.36) ∂ p μ

A symmetric combination of solutions to eqs. (5.35) and (5.36) should be used in eq. (5.34). Writing the equation for G2 (x, p, x , p ) explicitly and approximating the motion of particles by straight world lines, one has μ







p ∂μ G2 (x, p, x , p ) +

dτ Lμν pν

∂F1 (x, p) F1 [x (τ ), p (τ )] ∂ pμ

(5.37)



p (τ − τ ) 4 δ [p − p ] = 0 × δ 4 x − x − m

and a similar equation for the second particle. This equation is solved using either Fourier transform or the causal Green’s function: #  $       p τ − τ G x − x , p − p , τ − τ = θ τ − τ δ 4 x − x − δ 4 p − p . m (5.38) One should also neglect initial correlations by letting the initial proper time go to minus infinity and integrating over τ and τ . The solution for the relativistic correlation function G2 is then inserted back into eq. (5.34). Tedious calculations [18] led to obtainment of the Belyaev-Budker equation [16]: uμ ∂ μ F = −

 2   uλ uλ ∂F ∂F F , (5.39) d p  B − F  32 μν  2 ∂ p ν ∂ pν c u λ uλ − 1 %  2  & = uλ u λ − 1 δμν − uμ uν − u μ u ν − uλ u λ uμ u ν + u μ uν ,

2π q2 q 2  Kμ = c2 Bμν

∂Kμ , ∂ pμ



4

where  is Coulomb logarithm (see eq. (6.16)), primed and unprimed values correspond to two incoming particles, and the mean field is neglected. In the

48

Relativistic BBGKY Hierarchy

nonrelativistic case, this equation reduces to the famous Landau equation [5, 6]   ∂f ∂f 1 ∂f ∂sa +v +q E+ v×B =− (5.40) ∂t ∂r c ∂p ∂ pa 2       v − v δab − va − va vb − vb 3 ∂ f 2 2 ∂f sa = 2π q q  f −f d p. ∂ pb ∂ pb (|v − v |)3 Landau derived his kinetic equation in the general case, valid both for neutral and charged plasmas. He pointed out that for neutral plasma, the divergence at large distances (small deflection angles) can be cured by introducing the cutoff at the Debye length (see eq. (6.9)). Instead, for charged plasma, the cutoff corresponds to the linear size of the system R [5]. For more discussion on this point, see Chapter 14. For alternative derivations of the Belyaev-Budker equation from the BBGKY hierarchy, see also [13, 70, 78, 79]. Recall that in dilute plasma, collisions with small momentum transfer dominate. For this reason the Coulomb collision integral in nonrelativistic plasma is usually approximated by the Fokker-Planck diffusive term. Such approximation actually becomes invalid for relativistic plasma with kB T  me c2 , where me is electron mass, since at these temperatures, pairs of electrons and positrons form (see Chapter 11). The description of such relativistic plasma requires the full Boltzmann collision integral.

5.4 The Vlasov-Einstein System The system of particles interacting via gravitational mean field is described [9, 21] by the Einstein-Vlasov system of equations  3 1 μν 8π G d p μ ν K μν μν R − g R+g = 4 c p pF (5.41) 2 c p0 pμ

K ∂F K i ν λ ∂F − p p = 0, νλ ∂xμ ∂ pi

(5.42)

where F K is the Klimontovich random function defined by eq. (2.15) and G is Newton’s constant. For mathematical aspects, see a recent review [80]. These equations form the basis for microscopic gravity. These equations are well suited for numerical experiments. Statistical averaging may be implemented numerically after performing many experiments with different microscopic initial conditions. This possibility, however, is yet unexplored, given the much more complicated and nonlinear structure of these equations. It has to be noted again (see discussion after eq. (1.14)), that the RHS of eq. (5.41) contains only rest mass energy and kinetic energy of particles and does not account for interaction energy.

5.4 The Vlasov-Einstein System

49

Among the most interesting applications of eqs. (5.41) are attempts to shed light on the occurrence of singularities in general relativity. It has been shown that perfect fluid models of matter do develop naked singularities [81], while the VlasovEinstein system does not [82], at least in the spherically symmetric case with small data. Macroscopic gravitation theory should be derived from these equations by applying the averaging procedure. Following the discussion in this chapter, it is expected that correlations should appear in both equations after the averaging and represent matter-field, matter-matter, and field-field correlations. An attempt to construct such equations up to the second-order terms in interaction is made in [21].

6 Basic Parameters in Gases and Plasmas

Having derived basic kinetic equations in Chapters 2 and 5, it is possible to turn to their applications. Dilute gas and plasma are traditionally considered as primary applications for KT. The generalization to the relativistic case of KT for gas required mainly terminological changes [8]. However, KT of plasma had to be built on a relativistic basis from the beginning since the Maxwell equations, being Lorentz invariant, are a natural part of it. Besides, in the relativistic domain (at relativistic temperatures), many qualitatively new phenomena, such as particleantiparticle production, occur in plasma. To understand these phenomena, as well as to provide the physical foundations for the derivation of the Boltzmann and Vlasov equations discussed in the previous chapter, it is useful to introduce the characteristic quantities of both gases and plasmas.

6.1 Plasma Frequency Consider the Maxwell equations (5.4) and assume that particles move collectively with velocity v given by the equation of motion m

∂U μ q = − Fνμ . ν ∂x c

(6.1)

Taking 0-1 components in eq. (5.4), one has m

d (γ β ) = qE, dt

dE = −4π qnβ, dt

(6.2)

where n is particle density and E is electric field. Differentiating the first equation with respect to time, one gets (see, e.g., [83]) d 2 u 4π q2 n u + = 0, √ dt m 1 + u2 50

u = γ β.

(6.3)

6.2 Correlations in Plasma

51

This equation describes nonlinear Langmuir oscillations (reducing to harmonic ones for v  c) with the frequency given by ω2p0 =

4π q2 n . mγ

(6.4)

This is one of the fundamental plasma parameters, and it is called the proper plasma frequency. Here the factor γ refers to collective motion of particles. It corresponds to the quantity n/γ , which is called the proper number density and is an invariant. An alternative definition of plasma frequency is 4π q2 n . (6.5) m It refers to the number density, which is not invariant, being a time component of the four-vector jμ (x) (see eq. (1.12)). Despite this fact, the definition (6.5) is widely used. ω2p =

6.2 Correlations in Plasma To determine other characteristic quantities of plasma, one needs to consider the notion of correlation in plasma. It is well known that the correlation function in a medium composed of particles interacting via the Coulomb (long-range) potential is divergent. However, since a neutral plasma contains both positive and negative charges in equal amounts, the field of a charged particle in plasma is different from the Coulomb field. To illustrate this point, consider a charged particle at rest in the origin (see, e.g., [84, 85]). The system of equations (5.30) and (5.31) simplifies for this case: ∂ fi ∂φ ∂ fi vi − qi = 0 φ = −4π qi ni − 4π qδ (r) , (6.6) ∂r ∂r ∂pi i where φ is electrostatic potential. To solve these equations, one has to set up the boundary conditions. Assume that the electric field vanishes at infinity, i.e., φ(∞) = 0. Assume also that the DF far from the origin is the Maxwell-Boltzmann one (4.42), that is,   γi mi c2 + qi φ(r) fi (γi ) ∝ ni exp − . (6.7) kB T  Then, taking into account charge conservation i qi ni = 0, one can get for the potential   qi φ φ = −4π qδ (r) + 4π . (6.8) qi ni 1 − exp − kB T i

52

Basic Parameters in Gases and Plasmas

At large radii |qi φ|  kB T and instead of eq. (6.8), a linear equation is obtained: φ −

1 φ = −4π qδ (r) λ2D

kB T . 2 i 4π qi ni

λ2D = 

(6.9)

The quantity λD is referred to as Debye length. It gives the solution for the electric potential in equilibrium plasma:   q r φ = exp − . (6.10) r λD This result implies that at large distances, the Coulomb field of the point charge is screened. Define now a two-particle spatial correlation function in equilibrium as P2 (1, 2) = P1 (γ1 ) P1 (γ2 ) g(r),

(6.11)

where the probabilities Pi (γi ) are given by eq. (4.43), r = |x2 − x1 |, and g(r) is called the radial correlation function. Using the normalization fi (γi )dγi = 1, one can introduce the total correlation function ξ (r) = g(r) − 1,

(6.12)

where ξ (r) is zero for uncorrelated particles. For dilute plasma in a state close to equilibrium [86], this function is   1 q2 r ξ (r) = − exp − , (6.13) kB T r λD which means that the correlation radius for this plasma is rcor ∼ λD .

6.3 Coulomb Collisions Consider a Coulomb collision with large impact parameter and, consequently, small deflection angle ϑ, measured in the CM system. The transport cross section in the nonrelativistic case [87] is   1 σt = (1 − cos ϑ ) dσ  ϑ 2 dσ. (6.14) 2 Here the differential cross section with small angles is given by the Rutherford cross section  2 8π qq dϑ dσ = , (6.15) 4 ϑ3 2 μ (v − v )

6.4 Characteristic Distances

53

where prime denotes the second particle and μ = cos ϑ. Then the total cross section is  2  4π qq dϑ σt = , (6.16)   = 4 2 ϑ μ (v − v ) where  is referred to as Coulomb logarithm. This result shows that in nonrelativistic plasma, due to the long-range nature of the electromagnetic interactions, “collisions” at large impact parameters are more important than close encounters. Consider now Coulomb logarithm in relativistic plasma. Electron-electron or electron-positron collisions are then described by Møller and Bhabha cross sections, instead of by (6.15). In this case the Born approximation has to be used since the relative velocity vrel between particles is larger than αc, where α = q2 /(¯hc) is the fine structure constant, and then m vγ −1 λD  ϑmin . h In thermal equilibrium, using eqs. (6.9) and (4.29), one finds   K1 θ −1   −→ 3θ

γ = 3θ + K2 θ −1 θ→∞ ' v ( 2 exp θ −1  1 + 3θ + 3θ 2    γ = −→ 3θ , θ→∞ c θ K2 θ −1 =

(6.17)

(6.18) (6.19)

so in the relativistic case, one has =

− 21 3 3  2 αλ3 n θ −→ O(1). C θ→∞ 4π 3/2

(6.20)

This result implies that the mean free path due to Compton scattering lC ∼ σT1 n , where σT is Thomson cross section, and the one due to Coulomb scattering lC ∼ 1 become comparable in the ultrarelativistic case. This formula shows also that σT n for relativistic plasmas, when   O(1), the momentum transfer in Coulomb collisions is no longer small, so the Fokker-Planck approximation (5.39) does not hold. These results are used in Chapters 10 and 11.

6.4 Characteristic Distances Following [75, 88], compare the characteristic distances in gas and plasma: the correlation radius rcor , the average distance between particles rav , and the mean free path l. For dilute gas, interactions between particles occur when they approach each other, so the correlation radius is rcor ∼ r0 , where r0 is the particle (atom or

54

Basic Parameters in Gases and Plasmas g 10

2

10

3

10

4

0.1

0.2

0.5

1.0

2.0

5.0

10.0

q

Figure 6.1 The plasma parameter of relativistic plasma in thermal equilibrium as a function of dimensionless temperature θ = kB T/(mc2 ).

molecule) size. The average distance between particles is determined from particle density n as rav ∼ n−1/3 . The mean free path, i.e., the average distance that particles travel without interactions, is l ∼ (nσ )−1 ∼ (nr02 )−1 , where, in the last relation, the fact that the cross section in gas is typically σ ∼ r02 is used. For dilute plasma, as discussed earlier, rcor ∼ λD . The mean free path is instead l ∼ (nλ2D )−1 . From these quantities it is possible to construct dimensionless parameters characterizing a given medium, respectively: gp =

1 nλ3D

gg = nr03 ,

(6.21)

where g p is referred to as plasma parameter. For dilute plasma and ideal gas, respectively, g p  1 and gg  1. Relativistic plasma in thermal equilibrium is always dilute (see Figure 6.1). The following inequalities usually hold for gas and plasma: rcor  rav  l

(gas)

rav  rcor  l

(plasma).

(6.22)

It is clear that in dilute gas, interaction occurs only when two particles encounter or “collide” with each other. Correlations between particles may be neglected before and after the collision. In dilute plasma the situation is quite opposite. A given particle interacts simultaneously with many particles located in the Debye sphere around this particle with radius λD . This means that particles move in the mean electromagnetic field, created by many other particles. This field has to be averaged over some volume smaller than the Debye volume λ3D but larger than the interpar3 ticle volumes rav , as well as associated time. The Vlasov approximation (5.28) and (5.29) is valid when the rate of particle collision is smaller than the rate of change

6.5 Microscopic Scales in Kinetic Theory and Hydrodynamics

55

of this averaged electromagnetic field. In other words, the relaxation length is much larger than the size of the system L. Having defined these basic parameters, it is possible to justify the derivation of the Boltzmann equation in Chapter 2, where only binary interactions have been considered and interactions between three particles, four particles, and so on, were neglected. Indeed, triple collisions in gas are much less probable than binary collisions, since the correlation function C(1, 2) is small (see, e.g., [89]): C (1, 2) = f2 (1, 2) − f1 (1) f1 (2) ∼ gg f2 (1, 2) ,

(6.23)

where f2 (1, 2) is two-particle DF and where f1 (1) and f1 (2) stand for one-particle DF of particle 1 and 2, respectively. Analogously, C(1, 2, 3) ∼ gg f3 (1, 2, 3), and so on. In the same way, for dilute plasma, C(1, 2) ∼ g p f2 (1, 2). Since in dilute plasma g p  1, one can neglect the three-particle correlations. This justifies the choice made in Chapter 5 (see eq. (5.32)). 6.5 Microscopic Scales in Kinetic Theory and Hydrodynamics Now discuss the notion of infinitesimally small scales in kinetic and hydrodynamic approximations following [88]. From the kinetic point of view, physically infinitesimally small scales should satisfy the inequalities r ph  L

nr3ph  1,

(6.24)

where L is the characteristic size in the problem (the size over which DF changes significantly). Then one has for such kinetic infinitesimally small scales rK  l rK  λD

(gas)

(6.25)

(plasma).

From the hydrodynamic point of view, the relaxation timescale is a function of the characteristic size L and of one of the three dissipation coefficients: diffusion D, viscosity ν, ˜ and heat conductivity χ˜ . The corresponding physically infinitesimally small scale satisfies the following inequality: vL2 D∗ = max (D, ν, ˜ χ˜ ) . (6.26) D∗ The transition from the kinetic level of description to the hydrodynamic one is realized by the introduction of the physical Knudsen number: r ph Kn =  1. (6.27) L The approximate methods of solution of the Boltzmann equation (such as Hilbert and Chapman-Enskog; see, e.g., [9] and [89]) use Kn as a small parameter for rHD 

56

Basic Parameters in Gases and Plasmas Log q 2 0 2 4 6

10

5

0

5

Log nlc 3

Figure 6.2 The temperature-density diagram for relativistic plasma. Solid line corresponds to the condition D = 1. To the right of this curve, D < 1 and plasma is degenerate. The dashed curve corresponds to the condition g p = 1. Above this curve, g p < 1 and plasma is ideal. The dotted curve corresponds to thermal electron-positron plasma.

expansion of kinetic equations. The ratio of the two infinitesimally small scales (kinetic and hydrodynamic ones) is rK ∼ g3/10 Kn6/5 ≤ 1, g rHD

(6.28)

where equality corresponds to the maximal Knudsen number (minimal scale L) when a common hydrodynamic and kinetic description of the system is still possible [88] (see Chapter 9).

6.6 Relativistic Degeneracy If the temperature of plasma decreases for a given density of particles, it may become degenerate [90]. The same phenomenon occurs when particle density increases but the temperature is fixed. It is useful to construct a temperature-density diagram (see Figure 6.2). The characteristic temperature that separates non-degenerate from degenerate systems is defined by θF ≡

F , mc2

(6.29)

where F is Fermi energy, corresponding to the Fermi momentum  1 pF = 3π 2 n 3 h¯ .

(6.30)

For a relativistic gas the total energy and momentum are related by  2 = p2 c2 + m2 c4 . Equating the kinetic energy  − mc2 to the Fermi energy, the degeneracy

6.6 Relativistic Degeneracy

57

temperature is obtained: θF =





2

 23 

λC n

1 3

2

1/2 +1

− 1.

(6.31)

Define the degeneracy parameter D=

θ . θF

Note that it is related to the degeneracy parameter introduced in [2] as D =

(6.32) θ3 nλC3



D . Definition (6.32) takes into account both nonrelativistic and ultra-relativistic asymptotics in eq. (6.31). As can be seen from Figure 6.2, even in thermal equilibrium, relativistic plasma becomes degenerate, though this degeneracy is weak. 3

Part II Numerical Methods

7 The Basics of Computational Physics

This chapter provides a brief informal introduction to computational physics. While analytic results contain all information, often it is impossible to obtain analytic solutions of complex differential equations, especially nonlinear ones. Then, new results can be revealed from numerical simulations. The numerical method is an important ingredient of computational simulations. No universal set of numerical receipts is applicable to the whole range of physical problems. Nevertheless, some numerical algorithms have now been made part of such computational programs as Mathematica1 and MATLAB2 [91]. It has to be emphasized that basic ideas in computational physics come from theoretical and mathematical physics. This chapter describes the standard types of equations of classical mathematical physics along with methods for their solutions. The focus is on finite difference methods. Systems of ordinary differential equations and problems of linear algebra are also considered. In addition, order of accuracy and stability of the scheme, providing the convergence of the numerical solution to the actual solution of the partial differential equation, are discussed. 7.1 Finite Differences and Computational Grids Computational technology cannot represent smooth mathematical functions or infinite values. Computers operate with a finite set of discrete objects. Moreover, the integer numbers and even the real numbers on a computer are finite sets with finite cardinality N, whereas in mathematics, real numbers are an infinite set with cardinality of the continuum ℵ, and the integer numbers and rational numbers are infinite sets with the cardinal number of the ordered set ℵ0 [92]. Among different computational methods, the finite difference methods are among the most widely used. Such methods operate with the finite set of small 1 2

https://www.wolfram.com/mathematica/ https://www.mathworks.com/products/matlab/

61

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The Basics of Computational Physics

elements of the continuous physical value. The method solves the differential equations by approximating the derivative by the finite difference. Let independent variable x be defined in the interval (X1 , X2 ). Replace the continuum interval by  the finite grid constructed from points xl = X1 + lj=1 x j , where 0 ≤ l ≤ J and x = (X2 − X1 )/J. An arbitrary function f (x) can be replaced in the grid by a finite set of values fl = f (xl ). Such a description is not complete, but one can use the interpolation for f (x), for example, the linear interpolation (x j − x) f j−1 + (x − x j−1 ) f j f (x) = , (7.1) x j − x j−1 inside the interval x ∈ (x j−1 , x j ). One can expect this discrete representation to be appropriate for slowly variable function f (x) at some selected intervals. In other words, is it not possible to describe the variation of the function f (x) inside the discrete intervals [93]. The approximation of the derivative on a finite grid is illustrated with discrete Fourier transforms. Assume the function f (x) is periodic in the interval (X1 = 0, X2 = 2π ) [94]. If the intervals are x j = const = 2π /J, then the discrete Fourier transform is defined as J fˆk e2πik j/J , fj = (7.2) k=1

where the hat denotes Fourier amplitudes 1 −2πi jk/J fˆk = f je . J j=1 J

(7.3)

The scalar product is defined as 1 (f · g) ≡ f j gˆ j . J j=1 J

(7.4)

To prove expressions (7.2)–(7.3), one needs to verify the orthogonality of functions φk,i = e2πik/J : (φk · φl ) = =

J J 1 2πi jk/J −2πi jl/J 1  2πi(k−l)/J  j e e e = J j=1 J j=1  1 2πi(k−l)/J 1−e2πi(k−l) e , k=  l J 1−e2πi(k−l)/J

 =

1, k = l sin πi(k−l) = 1J e(J+1)πi(k−l)/J sin(πi(k−l)/J) ,

k = l

1, k = l

because the sum of geometric progression is

J j=1

xj =

x(1−xN ) 1−x

= δkl , for x = 1.

(7.5)

7.2 Stability and Accuracy of Numerical Schemes

63

This means Fourier functions φk form the basis. The representation by discrete Fourier transform is exact. This implies the functional completeness [92] of functions φk . The discrete Fourier transform has a limited wavelength resolution on the finite set of numerical data f j , 1 ≤ j ≤ J, given by λmin = 2π /J. 7.1.1 Representation of Derivatives on the Grid One can use the expansion in Taylor series [95] of the function to obtain necessary derivatives of function f (x) in the point xi : n f (k) (x j ) (x − x j ). f (x) = k! k=0

(7.6)

To obtain n-order derivative f j(k) on the grid, one needs to take into account n + 1 points x j from the system of linear algebraic equations in this equation. For example, for the equidistant grid, one has f (x j ) =

f (x j+1 ) − f (x j−1 ) + O(( x)2 ). 2 x

(7.7)

The grid step in power 2 in the error of the approximation O(( x)2 ) means the second order of accuracy of the method. In this case it is said that the formula (7.7) has second order. Using the discrete Fourier transform (7.2), one can rewrite the representation (7.7) as f (x j+1 ) − f (x j−1 ) ˆ eikx j+1 − eikx j−1 = fk 2 x 2 x k=1 J

=

J k=1

2i sin k x ≈ fˆk eikx j 2 x

J

fˆk eikx j k = f (x j ),

(7.8)

k=1

if k x  1. The numerical approximation of a derivative on the grid (7.7) is acceptable if sin k x is close to k x, or k x  1.

(7.9)

Notice that it is not possible to obtain the derivative numerically of the fast oscillating function with the small wave number on the scarse grid r > k−1 . 7.2 Stability and Accuracy of Numerical Schemes This section introduces simplified concepts of stability and accuracy. The rigorous definitions for finite differences are presented after classification of the types of equations in mathematical physics.

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The Basics of Computational Physics

7.2.1 Formulation of the Cauchy Problem At this point one can formulate the problem with the initial data for the differential (ordinary or partial) equations, the so-called Cauchy problem. Consider vectors defined on the computational grid. For the initial state u(r, t0 ) the problem to be solved is ˙ t ) = Lu, u(r,

(7.10)

where L is an operator acting on the vector u and a dot denotes differentiation with respect to time. The operator L can be nonlinear. Example 1. Newton’s law for the material point is.  0 T u = (x, v) , L = −F/m

 1 , 0

(7.11)

where F is force and m is particle mass. Example 2. The heat conduction equation in one dimension is ut − kuxx = 0, where subscripts denote partial derivatives with respect to coordinate and time. For the vector defined on the computational grid, one has u j−1 − 2u j + u j+1 ∂ u = Lu, where u j (t ) = k = kuxx (x j , t ) + O(( x)2 ). ∂t ( x)2 (7.12) The finite difference approximates the derivative uxx in the point of grid x j at time moment t. The main problem in the computational physics is expressed by the question, how should the finite difference equations be formulated? For example, one can introduce the grid for the time coordinate t n = t0 + n t and rewrite the preceding equations for both cases using an explicit scheme: un+1 − un = ut |t n + O(( t )1 ) = L(un ) + O(( t )1 ). (7.13) t In Example 1, such a scheme is accurate on the fine grid with small enough parameters x, t. In Example 2 the explicit scheme cannot give appropriate results for any grid parameters x, t, as the numerical solution diverges for a finite number of time steps. This is the problem with stability of the scheme [94, 93]. 7.2.2 Accuracy, Stability, and Convergence The scheme is said to be stable if errors do not grow during integration of the evolutionary problem. Given the error  n on the nth time step, it should remain

7.2 Stability and Accuracy of Numerical Schemes

65

small at the next time step  n+1 for nongrowing solutions [94, 93] | n+1 | ≤ | n |,

(7.14)

where the error  n is the difference of the numerical solution from the exact solution of the differential equation at the time step t n . It is simple to check the errors for the linear transfer factor to a new time step:  n+1 = g n ,

(7.15)

where g is called the transfer factor and the stability condition requires |g| ≤ 1.

(7.16)

If the transfer operator to the new time step is nonlinear, it is possible to analyze the linealized transfer operator. Consider an ordinary differential equation (ODE) du + f (u, t ) = 0, dt

(7.17)

with the initial data u(t 0 ) = u0 . The simplest numerical scheme is the first-order explicit Euler method, which is

First order here means

un+1 = un − f (un , t n ) t.

(7.18)

un+1 − un du 1 n = dt n + O(( t ) ). t t t

(7.19)

What are the stability criteria of such a method? For the error  n+1 at the new time step, one has un+1 +  n+1 = un +  n − f (un +  n , t n ) t   ∂ f n n n n n n n  + o( ) t. (7.20) = u +  − f (u +  , t ) + ∂u un Then

 ∂ f n = 1− t + o( )  n , ∂u un 



n+1

and the transfer factor is

∂ f t. g=1− ∂u un

(7.21)

(7.22)

66

The Basics of Computational Physics ImΔtλ 1

−2

−1

1

ReΔtλ

−1

Figure 7.1 Euler’s method is stable in the filled region for the equation y˙ = λy.

For the equations with the decay3 ∂ f /∂u < 0 one has the following stability condition (see Figure 7.1): the time step should be limited by the condition 1 − ∂ f t ≤ 1. (7.23) ∂u n u

The approximation of derivatives by finite differences is another requirement of the numerical scheme. In general, the first-order scheme can give a finite error for any small steps. But usually errors at every time step have a random “phase,” so the first-order scheme can converge to the actual solution. But obviously the second-order scheme is preferred. The high-order scheme guarantees the convergence of the solution and usually has the calculation error ∝ t a and ∝ xa , where a ≥ 2 [93] when the stability condition is satisfied. 7.3 Numerical Methods for Partial Differential Equations In this section numerical methods for solution of hyperbolic, elliptic, and parabolic partial differential equations are discussed. 7.3.1 The Classification of Linear Partial Differential Equations of the Second Order Recall the classification of the second-order linear partial differential equations (PDEs). Consider two independent variables x, y and linear second-order 3

The growing solutions in the case ∂ f /∂u > 0 are useless for the analysis of the scheme.

7.3 Numerical Methods for Partial Differential Equations

67

differential equations [96, 97, 98] a11 uxx + 2a12 uxy + a22 uyy + F (x, y, u, ux , uy ) = 0,

(7.24)

where subscripts denote the derivatives with respect to variables and the coefficients ai j depend on x and y. Replacing the variables as ξ = ξ (x, y),

η = η(x, y),

(7.25)

the equation in new variables becomes α11 uξ ξ + 2α12 uξ η + α22 uηη + F (x, y, u, ux , uy ) = 0,

(7.26)

where the coefficients are α11 = a11 ξx2 + 2a12 ξx φy + a22 ξy2 ,

(7.27)

α12 = a11 ξx ηx + a12 (ξx ηy + ξy ηx ) + a22 ξy ηy ,

(7.28)

α22 = a11 ηx2 + 2a12 ηx ηy + a22 ηy2 .

(7.29)

One can obtain the relation 2 α12

− α11 α22 =



a212

 ∂ (φ, ψ ) 2 − a11 a22 . ∂ (x, y)

(7.30)

This means the sign of the discriminant ≡ a212 − a11 a22 is invariant. One can introduce the classification that, when in some region D the discriminant of the second-order PDE is positive, > 0, the equation (7.24) is the hyperbolic equation in the region D. For < 0, the equation is elliptic, while for = 0, it is parabolic. One can transform different types of equations to the canonical forms of the second-order PDE. For the hyperbolic equation, it is [96] uξ η + F1 (uξ , uη , u, ξ , η) = 0.

(7.31)

For the elliptic equation, it is uξ ξ + uηη + F1 (uξ , uη , u, ξ , η) = 0.

(7.32)

For the parabolic equation, it is uηη + F1 (uξ , uη , u, ξ , η) = 0.

(7.33)

For the linear equations, the canonical forms are uξ η + β1 uξ + β2 uη + γ u = f (ξ , η),

(7.34)

uξ ξ + uηη + β1 uξ + β2 uη + γ u = f (ξ , η),

(7.35)

uηη + uηη + β1 uξ + β2 uη + γ u = f (ξ , η).

(7.36)

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The Basics of Computational Physics

Consider in detail the second-order equation without mixed derivative a11 uxx + a22 uyy + b1 ux + b2 uy + cu = f (x, y).

(7.37)

The type of equation is defined according to the signs of the coefficients a11 (x, y) and a22 (x, y). When a11 = 0, a22 = 0, and the sign of a11 (x, y) differs from the sign of a22 (x, y), the equation is of hyperbolic type. When a11 = 0, a22 = 0, and the sign of a11 (x, y) is the same as the sign of a22 (x, y) and it is different from zero, the equation is of elliptic type. When a11 (x, y) = 0 or a22 (x, y) = 0, the equation is of parabolic type. For the linear equations with many independent variables (x1 , . . . , xn ) aii uxi xi + bk uxk + cu = f (x1 , . . . , xn ), (7.38) i

k

the following classification holds. When all coefficients are different from zero aii = 0, and all have the same sign, the equation is elliptic. When aii = 0, and all except for one coefficient have the same sign, the equation is hyperbolic. When all aii = 0, except one, and coefficients have the same sign, the equation is parabolic. The famous examples of different types of equations in physics are the wave equation (hyperbolic) in the 3D case uxx + uyy + uzz − a2 utt = f (x, y, z, t ),

(7.39)

the heat conduction or diffusion equation (parabolic) uxx + uyy + uzz − a2 ut = f (x, y, z, t ),

(7.40)

and the Poisson equation (elliptic) uxx + uyy + uzz = f (x, y, z, t ).

(7.41)

In Part III, all these types of equations are encountered.

7.3.2 Finite Difference Methods for the Wave Equation The type of finite difference method depends on the type of equation. First, consider the linear hyperbolic equation utt − a2 uxx = 0,

(7.42)

which is equivalent to the system of first-order PDEs ut + a2 ux = 0 vt + ux = 0.

(7.43)

7.3 Numerical Methods for Partial Differential Equations

69

Changing variables to α = u + av and β = u − av, one has αt + aαx = 0 βt − aβx = 0.

(7.44)

This allows reduction of the linear second-order hyperbolic equation (7.42) to the first-order wave equations, like ut + aux = 0.

(7.45)

One can write the general solution of eq. (7.42) as u = F (x − at ),

(7.46)

as the wave propagating with the velocity a in the plane (x, t ). Along the characteristics of eq. (7.45), dt dx = ; 1 a with x − at = const, the function u is consatnt:

(7.47)

du = const. (7.48) dt The general solution of eq. (7.42) is the sum of two waves propagating in different directions, u = F (x − at ) + G(x + at ),

(7.49)

where functions F and G correspond to different characteristics dx/dt = ±a. The Cauchy problem for the wave equation in the region −∞ < x < ∞ is the problem to find u(x, t > 0) with the initial data ut + aux = 0 u(x, 0) = φ(x).

(7.50)

In the finite region 0 ≤ x ≤ l, the Cauchy problem also includes boundary conditions, for example, specified function u: u(0, t ) = g0 (t ), u(l, t ) = gl (t ). To solve the wave equation numerically, one can introduce the grid for the independent variables (x, t ): xm = xm, 0 ≤ m ≤ M, tn = tn, 0 ≤ n and define the grid function unm (see Figure 7.2). One has to write finite difference equations that approximate the PDE (see Figure 7.3), for example, n unm − unm−1 mn+1 m − um +a = fmn , 0 < m ≤ M, t x un0 = gn0 , 0 ≤ n,

u0m = φm , 0 ≤ m ≤ M.

(7.51)

70

The Basics of Computational Physics t

unm

tn = Δtn

dx /d t=

a

P

Q

xm = Δxm

x

Figure 7.2 Example of the computational grid. The bold curve dx/dt = a is the characteristic for the hyperbolic equation.

un+1 m

unm−1

unm

Figure 7.3 The upwind asymmetric explicit template for the hyperbolic equation.

The grid, the finite difference equations, and the additional equations are referred to as the finite difference scheme. The scheme described by eq. (7.51) is called the explicit scheme, because one has the explicit relations for the definition of the set un+1 m , 0 ≤ m ≤ M at the new time step. The Accuracy of the Finite Difference Scheme In the finite difference scheme ∂u un+1 − unm (xm , t n ) = m + O( t ) ∂t t unm − unm−1 ∂u (xm , t n ) = + O( x), ∂x x

(7.52)

the finite difference equations approximate the first-order PDE for the grid intervals t, x.

7.3 Numerical Methods for Partial Differential Equations

71

un+1 m

unm−1

unm

Figure 7.4 The forward difference explicit template for the hyperbolic equation. un+1 m

unm−1

unm

unm+1

Figure 7.5 The symmetric explicit template. un+1 m−1

un+1 m

un+1 m−1

unm−1

unm

unm+1

Figure 7.6 The implicit second-order template

Other types of schemes are possible: r The explicit asymmetric forward difference scheme (see Figure 7.4) n un − unm un+1 m − um + a m+1 = fmn t x

(7.53)

has the error of approximation O( t ) + O( x).

r The explicit symmetric scheme (see Figure 7.5)

n un − unm−1 un+1 m − um + a m+1 = fmn t 2 x

(7.54)

has the error of approximation O( t ) + O(( x)2 ). r The implicit central difference scheme (see Figure 7.6) n+1 n a unm+1 − unm−1 a un+1 un+1 m+1 − um−1 m − um + + = fmn t 2 2 x 2 2 x

(7.55)

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The Basics of Computational Physics

has the error of approximation O(( t )2 ) + O(( x)2 ). However, it requires solving the set of linear algebraic equations to find the set un+1 m , 0 ≤ m ≤ M at the new time step t n+1 . Such scheme is called the implicit scheme.4 Characteristics and Stability For the wave equation in more general form, ut + aux = f (x, t ),

(7.56)

dx du dt = = , 1 a f

(7.57)

one has the relation

or along the characteristic

dx dt

= a [98], du = f. dt

(7.58)

The unknown function u at the point P = (t, x) depends on the region Q on the characteristic x − at = const. The region Q is called the domain of dependence of the solution in the point P: u(P) = φ(Q) (see Figure 7.2). One can predict the stability for the explicit schemes for hyperbolic equations. When the domain of dependence of the finite difference equation contains the domain of dependence of the PDE, the scheme can be stable. On the contrary, when the domain of dependence of the finite difference equation does not contain the domain of dependence of the PDE, the scheme is unstable. For instance, the explicit “upwind” (the integration against the flow) scheme (7.51) illustrated in Figure 7.3 is stable for small enough time step t ≤ a x (un+1 depending on m n n the set of points um−1 , um ), while the forward difference scheme given by eq. (7.53) and illustrated in Figure 7.4 is unstable. The numerical solution of the explicit symmetric scheme given by eq. (7.54) and Figure 7.5 does not include the domain of dependence, while un+1 should depend on unm . The characteristic properties and m the stability of the implicit scheme (7.55), shown in Figure 7.6, are not obvious: n+1 n n n un+1 depends on the set of points un+1 m m−1 , um+1 um−1 , um , um+1 . Also the stability of the explicit scheme (7.54) shown in Figure 7.5 is not obvious. Both schemes in Figure 7.5 and 7.6 include the domain of dependence. Such schemes are stable, but the explicit scheme (7.54) is stable only for small time steps t.

4

Implicit schemes were invented in the struggle to avoid problems with numerical solutions, which emerged with the Soviet Atomic project. This fact was communicated to one of us (G.V.) by Isaak Khalatnikov (see also [99], p. 49).

7.3 Numerical Methods for Partial Differential Equations

73

Von Neumann Spectral Method and Stability In numerical analysis, von Neumann stability analysis, also known as Fourier stability analysis, is a procedure used to verify the stability of finite difference schemes applied to linear PDEs [100, 101, 102]. One can consider the evolutionary scheme as the linear operator A: U n+1 = AU n ,

(7.59)

transforming the solution U n at the moment t n to the solution U n+1 at the moment t n+1 . For a stable scheme the norm of the operator A ≡ supU

AU U

(7.60)

should be limited: An  ≤ C, n t ≤ T

(7.61)

A ≤ 1 + C t,

(7.62)

or

where C is a constant. One can estimate the norm of the operator A as the maximal eigenvalue λ: AU = λU,

(7.63)

max |λ| ≤ A.

(7.64)

because

The expected necessary condition for the stability is |λ| ≤ C, n t ≤ T, or |λ| ≤ 1 + C t.

(7.65)

If the eigenvectors constitute the basis, the conditions for the eigenvalues (7.65) are sufficient for the stability. The numerical solution on the infinite plane ∞ < m < ∞ can be presented as  2π 1 1 W (φ)eimφ dφ, where W (φ) = √ um e−imφ (7.66) um = √ 2π 0 2π m and vm = eimφ are eigenfunctions of the operator A. One can use Parseval’s identity [92]  2π |um |2 = |W (φ)|2 dφ. m

0

(7.67)

(7.68)

74

The Basics of Computational Physics Imλ 1

−1

1

Reλ

−1

Figure 7.7 The spectra λ(φ) for the upwind asymmetric explicit scheme (7.51): 0 < a t/ x < 1 (solid curve), 1 < a t/ x (dashed curve), and |λ| = 1 (dotted curve).

Then, one can write

  1 1 Aum = √ W (φ)Avm dφ = √ W (φ)λeimφ dφ. (7.69) 2π 2π Parseval’s identity gives  2π  2π 2 2 2 |Aum | = |W (φ)| |λ| dφ ≤ (1 + O( t )) |W (φ)|2 dφ 0 0 2 |um | ; (7.70) = (1 + O( t )) m

relations |λ| ≤ 1 + O( t ) and A ≤ 1 + O( t ) are equivalent. Then, the necessary and sufficient conditions for the stability are the limited eigenvalue |λ(φ)| ≤ 1 + O( t )

(7.71)

for the eigenfunction (7.67) for any φ. To define the eigenvalue λ(φ) of the scheme, one can search for the solution in the form unm = λn eimφ .

(7.72)

Now one can analyze the stability of the explicit schemes (7.51), (7.53), (7.54) and the implicit scheme (7.55) for the hyperbolic equation. For the scheme (7.51) of Figure 7.3, one has the equation for the eigenvalue a t −iφ λ(φ) = 1 + (e − 1). (7.73) x On the plane with coordinates (Reλ, Imλ) the eigenvalue λ(φ) is the circle of the radius a t/ x with the center (1 − a t/ x, 0) in Figure 7.7. Obviously

7.3 Numerical Methods for Partial Differential Equations

75

Imλ 1

−1

Reλ

1

−1

Figure 7.8 The spectra λ(φ) for the forward unstable explicit scheme (7.53): for any time steps t > 0, the solid curve is outside the region |λ| = 1 (dotted curve).

|λ(φ)| ≤ 1 if a t/ x ≤ 1 for the positive value a ≥ 0. This condition is expected for the scheme (7.51) from the characteristic properties of the hyperbolic equation. The explicit difference scheme with the forward difference (7.53) gives λ(φ) = 1 −

a t iφ (e − 1). x

(7.74)

For φ = π λ = 1 + 2a t/ x and |λ| > 1 + O( t ), see Figure 7.8. This implies that the scheme is unstable. For the explicit scheme with the central difference (7.54) one has λ(φ) = 1 − i

a t sin φ, x

(7.75)

and |λ(φ)| = 1 + (a t/ x)2 sin2 φ (Figure 7.9). In practice the scheme is useless, because it is stable, |λ(φ)| ≤ 1 + O( t ), only for very small time steps t = O(( x)2 ). For the implicit scheme (7.55) one has λ(φ) =

a t 1 − i 2 x sin φ a t 1 + i 2 x sin φ

,

(7.76)

and |λ|2 = 1. The implicit scheme is stable for any time steps and also has secondorder accuracy. However, the second-order accuracy is not always useful for the hyperbolic equation because of its monotonic property. The spectral method for the investigation of the stability of finite difference schemes is more useful. In the case of nonlinear equations one can try to investigate the stability of linealized equations with “frozen” coefficients. The next

76

The Basics of Computational Physics Imλ 1

−1

1

Reλ

−1

Figure 7.9 The spectra λ(φ) for the symmetric explicit scheme (7.54) are the solid curve near Reλ = 1 and |λ| = 1 (dotted curve).

step is investigation of the stability and the convergence of the scheme in the numerical experiment. Other methods for investigations of the stability can be found in the literature, for example, the energetic method [103]. Monotonic Property for the Hyperbolic Equation and High-Order Schemes The scheme is said to be monotonic if it produces the monotonic solution from the monotonic state [104, 105]. This property of the scheme is important for the hyperbolic equations. Such hyperbolic equations describe, for example, hydrodynamics of ideal gas. In the first approximation, hydrodynamic equations can be considered as a set of the wave equations for density, velocities, and pressure. In the case of strong jumps (shock wave (SW), contact discontinuity) it is important to avoid artificial oscillation near such a jump. For the nonmonotonic scheme the value of density and pressure on the grid can even become negative and, consequently, unphysical. Consider the explicit scheme for the wave equation un+1 = ck unm+k (7.77) m k

with constant coefficients ck . The criterion of the monotonic scheme is the condition for all coefficients ck ≥ 0. The proof is the following. Write the difference: n+1 un+1 = ck (unm+1+k − unm+k ). m+1 − um k

(7.78)

(7.79)

7.3 Numerical Methods for Partial Differential Equations

77

Obviously, if the initial state at t n is a monotonic, for example, all unm+1+k − unm+k ≤ 0, then the solution at new time step t n+1 is not monotonic either; in the selected n+1 example, un+1 m+1+k − um+k ≤ 0 holds as well. Proof by contradiction is as follows. Let the coefficient be negative, ck0 < 0. Then, if one takes  1, k ≤ k0 n (7.80) uk = 0, k > k0 ,  n+1 the result is un+1 = k ck (unm+1+k − unm+k ) = ck0 −(m+1) , and un+1 m+1 − um −1 − un+1 = c < 0. It implies that the scheme is not monotonic. k0 −2 It is interesting to stress two important properties of the scheme (7.77). The first, obvious property is that the implicit scheme ak un+1 = bk unm+k (7.81) m k

k

can be reduced to the explicit scheme (7.77) by the explicit solution of the system of linear equations. Another interesting property of the scheme (7.77) is its stability. When to the monotonic property ck ≥ 0 one adds the natural requirement  k ck = 1, max |un+1 ck max |unm | ≤ max |unm |, (7.82) m |≤ m

k

m

m

= const, if initially unm = const. one new constant solution appears, un+1 m The absence of the monotonic second-order scheme for the wave equation is another important property [105]. Consider again the general scheme (7.77). One can select the parabola   x 1 2 1 u(x, 0) = − (7.83) − , x 2 4 where h is the grid size, as an initial state. The solution of the wave equation is a shifted parabola   1 2 1 x+t − (7.84) u(x, t ) = − . x 2 4 The second-order (or higher-order) scheme has the error O(( x)2 + ( t )2 ); it will reproduce this parabola in the numerical solution. The initial finite difference function is positive,   1 2 1 0 um = m − (7.85) − ≥ 0, 2 4

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but the second-order scheme gives   1 2 1 t n um = m + − − . x 2 4

(7.86)

For m = −1 and t n = x the numerical solution becomes negative: 1 un−1 = − . 4

(7.87)

But the monotonic scheme (7.77) should have coefficients ck ≥ 0 and give all unm ≥ 0. Then one arrives to an important conclusion. In the general case, it is useful to construct high-order-resolution schemes for hyperbolic equations only if the flow is smooth. It is necessary to reduce the order of the scheme to the first in the region with high gradients or discontinuities.

7.3.3 Finite Difference Methods for the Parabolic Equation Consider now the parabolic equation ut = a2 uxx

(7.88)

and adopt the explicit scheme represented in Figure 7.5, n un − unm + unm+1 un+1 m − um 2 m−1 =a , t ( x)2

(7.89)

having the error of the approximation O( t + ( x)2 ). The test solution vmn = λn eimφ

(7.90)

gives λ(φ) − 1 = 2

a2 t φ a2 t (cos φ − 1) = −4 sin2 . 2 2 ( x) ( x) 2

(7.91)

The explicit scheme is stable (see Figure 7.10) only for a small enough time step, given by a2 t 1 ≤ . 2 ( x) 2

(7.92)

More adequate for parabolic equations is the implicit second-order O(( t )2 + ( x)2 ) scheme shown in Figure 7.6 [100]: n+1 n+1 n un − unm + unm+1 + un+1 un+1 m−1 − um + um+1 m − um = a2 m−1 , t 2( x)2

(7.93)

7.3 Numerical Methods for Partial Differential Equations

79

Imλ 1

−1

Reλ

1

−1

Figure 7.10 The spectra λ(φ) for the symmetric explicit scheme (7.89) (solid a2 t 1 curve) and |λ| = 1 (dotted curve) for ( x) 2 = 2.

which gives the coefficient λ as φ 2 2 φ a2 t 2 ( x)2 sin 2

2 a t 1 − 2 ( x) 2 sin 2

λ(φ) =

1+

.

(7.94)

Such a scheme is stable |λ(φ)| ≤ 1 for any time steps. 7.3.4 Finite Difference Methods for the Elliptic Equation Consider the system u + f = 0,

(7.95)

where is the Laplace operator, with the boundary condition u|bound = φ.

(7.96)

One can solve the evolutionary problem for the parabolic equation vt = v + f

(7.97)

v|bound = φ,

(7.98)

with some arbitrary initial value v(t = 0) = v0 , until the stationary state is obtained. For the difference w = v − u one has wt = w

(7.99)

w|bound = 0,

(7.100)

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w(t = 0) = v0 − u, and w ∝ e−λt ,

(7.101)

where λ is the minimal eigenvalue of the operator . Consider the following example: ut + u = ut + uxx + uyy = 0

(7.102)

in the region 0 ≤ x ≤ π , 0 ≤ y ≤ π with the boundary condition u|bound = 0.

(7.103)

U n+1 − U n = 11U n + 22U n , t

(7.104)

Write down the explicit scheme,

where the operators 11 and 22 are ul+1,m − 2ulm + ul−1,m ( x)2 ul,m+1 − 2ulm + ul,m+1 22U = , ( x)2 11U =

(7.105)

and U n+1 = AU n , where A = I + t(11 + 22 ).

(7.106)

The solution can be searched in the form U ∼ eilφ eimψ

(7.107)

with ulm = 0 on the boundary for l, m = 0, L. The eigenvalues are a11 = 2 φ t ±ilφ − ( x) (the corresponding real eigenfunction 2 sin 2 for the eigenfunctions e −ilφ

2 ψ t ) of the operator t11 , and a22 = − ( x) for the eigensin(lφ) = e −e 2 sin 2i 2 function sin(mψ ) of the operator t22 . For the operator A = I + t(11 + 22 ) one has the eigenfunctions ilφ

uklm1 k2 = sin(k1 lh) · sin(k2 mh),

(7.108)

with 1 ≤ k1 ≤ L − 1 and 1 ≤ k2 ≤ L − 1 and eigenvalues ρ( t, k1 , k2 ) = 1 − a11 (k1 ) − a22 (k2 ) t τ 2 k1 x 2 k2 x − . sin sin =1− ( x)2 2 ( x)2 2

(7.109)

7.3 Numerical Methods for Partial Differential Equations

81

One can approximate the norm of the operator for a small h: A( t ) = max |ρ( t, k1 , k2 )| = |1 − 2a(1)| k1 ,k2 t 2 x sin = 1 − 2 · 4 ≈ |1 − 2 t|. ( x)2 2

(7.110)

t 1 2 The stability of the explicit scheme requires ( x) 2 ≤ 2 , and A = 1 − ( x) /2. One can find the number of time steps to obtain the small enough numerical solution U N  ≤ U 0  for any small : (1 − ( x)2 /2)N ≤ . It implies that the required number of the steps in the explicit relaxation scheme is proportional to the large value N ≥ −L2 π2 ln  ∝ L2 . For this explicit scheme, there exists a so-called Chebyshev set of variable time steps t to achieve sufficient accuracy with a reduced number of time steps [93]. For the implicit scheme with alternate directions (the time step consists of two time steps with the integration along only one direction) 1 the number of time steps is small, N ≥ −L 2π ln . An alternative approach to the elliptic equation is the solution the system of linear algebraic equations arising after the replacement of the derivatives by the finite differences for the grid functions or implementation of Fourier transforms in the case of the periodic boundary conditions.

7.3.5 Finite Difference Methods for Multidimensional Problems and Nonlinear Equations So far, one-dimensional finite difference schemes for different types of equations of the mathematical physics have been considered. Often numerical solution of multidimensional problems is required. The multidimensional problem can be split into the set of one-dimensional problems for the application of a finite difference method: the so-called dimensional splitting method. For the three-dimensional case, for example, at every time step, one can consider three separate problems, taking into account the derivatives along one direction [93]. This approach is effective and stable and has satisfactory accuracy. This approach can be especially useful for implicit schemes, for example, the Crankicolson method for heat conduction [100]. Another important problem is the stability of the scheme for nonlinear differential equations. To study the stability, one can consider linearized problems with the assumption of constant coefficients in front of derivatives. The stability of the problem with such “frozen” coefficients can be verified by means of the spectral method [93]. The stability of the scheme for the nonlinear problem can be verified by computational experiment.

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7.3.6 Fast Fourier Transform The discrete Fourier transform for the discrete function fi and the Fourier amplitudes fˆk were introduced in eq. (7.2). Writing J = 2J1 , it is possible to reduce the calculations of Fourier transforms to the Fourier transforms on the two subgrids uj =

J1

uˆk e2πik j/J1 =

k=1

vj =

J1

J1

fˆ2k e2πi2k j/J1 , uˆk = fˆ2k ,

k=1

vˆ k e2πik j/J1 =

k=1

J1

fˆ2k+1 e2πi(2k+1) j/N1 e−2πi j/J1 ,

k=1

vˆ k = fˆ2k+1 , j = 1, . . . , J1 .

(7.111)

One obtains a set of relations f j = u j + e−2πi j/J1 v j , j = 1, . . . , J1 .

(7.112)

Analogously, one can obtain the following set of transformations: fJ1 + j = u j − e−2πi j/J1 v j , j = 1, . . . , J1 − 1.

(7.113)

The calculations on the subgrid require the number of operations ∼ J1 = J/2, so the number of operations on the grid J is T (J) ≤ 2T (J/2) + 2J. The optimal grid from the viewpoint of the minimization of the calculations is the power of 2: J = 2s , where s is a natural number. Then, the fast Fourier transform allows reduction of  the number of calculations from J 2 to Jk=1 2k T (J/2k ) ∝ 2J log2 J. 7.3.7 Variational Methods Consider the linear self-adjoint positively defined operator L with the following properties: the scalar product (Lu · v) = (u · Lv) and (Lv · v) ≥ 0. In Euclidian space the matrix corresponding to the self-adjoint operator is symmetric, A = AT . The problem Lu∗ = f

(7.114)

I[u∗ ] = min I[U], where I[u] = (Lu, u) − 2(u, f ),

(7.115)

and the variational problem u

are equivalent. To prove this statement, one can evaluate the following expresion d ∗ with the new parameter t and any function v: dt I[u + tv] t=0 = 0. One has I[u∗ + tv] = (Lu∗ + tLv, u∗ + tv) − 2(u∗ , f ) − 2t(v, f ),

(7.116)

7.5 ODE Systems and Methods of Their Solution

83

d I[u∗ + tv]

and = (Lv, u∗ ) + (Lu∗ , v) − 2(v, f ) = 2((Lu∗ , v) − (v, f )) = dt t=0 2(Lu∗ − f , v) = 0. Since the function v is arbitrary, the proof that the assumption (7.115) gives the relation (7.114) is completed. To prove the relations (7.114) and (7.115), one can evaluate the expression I[u∗ + v] = (Lu∗ , u∗ ) − 2(u∗ , f ) + (Lu∗ , v) + (Lv, u∗ ) − 2(v, f ) + (Lv, v) = (Lu∗ , u∗ ) + (Lv, v). Because (Lv, v) ≤ 0, one obtains I[u∗ + v] ≤ I[u∗ ]. The Dirichlet problem for the Laplace equation ∇u = f in the region with the given value on the boundary u|B = u0 is the appropriate example for the variational problem. The variational method in the numerical simulation can be useful for the suitable selection of the base functions and the representation of the solution as the linear combination of the base functions with coefficients to be defined. The Ritz-Galerkin variational method converts a continuous operator problem, such as a differential equation, to a discrete problem (see examples in [93, 106]). 7.4 The Method of Lines The method of lines is a method for solving PDEs in which all but one dimension is discretized [107, 108]. Obviously, if one has the evolutionary problem (Cauchy problem), it is possible to introduce the grid in the space and replace all partial derivatives (except time derivatives) by finite differences of the grid functions. The necessary integrals in the phase space are replaced by sums. After this procedure, one has the system of ODEs describing the evolution of grid functions in time. Then one can apply the standard methods for the solution of ODE systems. The realization of the method of lines for Boltzmann equations is discussed in Chapter 8. 7.5 ODE Systems and Methods of Their Solution Consider the initial value problem y(0) = y0 for a system of ODEs y˙ = f(y, t ).

(7.117)

In the beginning of the chapter the first-order explicit Euler method (7.18) was considered as the simplest illustration for the solution of the problem (7.117). The conditions for the convergence of the numerical solution to the mathematical solution were formulated: the accuracy and the stability of the method. The expansion in Taylor series gives the accuracy of the method. The stability can be verified by the test equation y˙ = λy. The scheme is stable if the errors of the approximation do not grow. The function f(y, t ) should satisfy the Lipschitz condition [109]: there exists a constant L such that f(y, t ) − f(y∗ , t ) ≤ Ly − y∗ .

(7.118)

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The Basics of Computational Physics

If this condition is satisfied, the initial value problem has a unique solution. For the explicit one-step Euler method yn+1 = yn + tf(yn , t n )

(7.119)

with the time step t = t n /n, the definition of the convergence is yn −→ y(t )

(7.120)

for all 0 ≤ t ≤ b as n −→ ∞ and y0 −→ y(0). Consider the stability of the numerical method for the special model problem with the complex value λ: y˙ = λy.

(7.121)

The reason for consideration of such a simple problem is the following. For the system of ODEs (7.117), one can consider the local linealized problem y˙ =

∂f y ∂y

(7.122)

∂f with a constant Jacobi matrix J = ∂y . By the definition of right eigenvectors Jr j = λ j r j and R = diag(λ1 , . . . , λn )R, R−1 JR = , where matrix R consists of right eigenvectors R = (r1 , . . . , rn )T and  = diag(λ1 , . . . , λn ). If the Jacobi matrix J is nondegenerate (all eigenvalues are different from zero), the system can be transfered to the diagonal form:

d −1 (R y) = (R−1 y), dt

(7.123)

so one can investigate the stability of the method for one ordinary differential equation. As an example, consider the widely used classical Runge-Kutta method, developed around 1900 by the German mathematicians C. Runge and M. W. Kutta [110]. It is a four-stage explicit method of fourth order: k0 = t f (yn , t n ),  1 n n k1 = t f y + k0 , t + t/2 , 2   1 k2 = t f yn + k1 , t n + t/2 , 2 k3 = t f (yn + k2 , t n+1 ), 1 yn+1 = yn + (k0 + 2k1 + 2k2 + k3 ). 6 

(7.124)

7.6 Stiff Systems and Gear’s Method

85

This method can be viewed [109] as an attempt to extend the Simpson quadrature     t n+1 t t n n n+1 f (t ) + 4 f t + + f (t ) f (t )dt ≈ . (7.125) 6 2 tn If f (y, t ) is a function of t only, the Runge-Kutta formula and the Simpson’s integral are equivalent [109]. To verify the order of the method, one can use the expansion in Taylor series around t n with the accuracy O(( t )4 ). For the investigation of the stability one can examine the formula for the equation y˙ = λy. One has   1 k1 = tλ 1 + tλ yn , 2    1 1 1 yn , k2 = tλ 1 + tλ + tλ 1 + tλ 2 2 2     1 1 1 n yn , k3 = tλ 1 + y + tλ 1 + tλ + tλ 1 + tλ 2 2 2   1 1 1 n+1 2 3 4 y = 1 + tλ + ( tλ) + ( tλ) + ( tλ) yn . (7.126) 2 6 24  4 . Thus the In accordance with the fourth-order approximation yn+1 = yn 4n=0 ( tλ) n! region of the stability is that area in which the growth factor is limited: 4 ( tλ)4 (7.127) < 1. n! n=0 The region of the stability in the complex plane for coordinates  4 (Re( tλ), Im( tλ)) limited by the curve 4n=0 ( tλ) = eiφ (|eiφ | = 1) is shown in n! Figure 7.11. For the explicit method, time steps are limited to −2.8  Re( tλ)  0 for Im( tλ) = 0. For the implicit method it is possible to have the absolutely stable (A-stable) method for any Re( tλ) ≤ 0. 7.6 Stiff Systems and Gear’s Method One can approximate the Cauchy problem (7.117) near the exact solution y(t ), y˙ − Jy − f (y, t ) = 0,

(7.128)

with Jacobi matrix J(t ). If the variation of J(t ) is small in the vicinity of fixed t, the solution can be represented in this vicinity as y ≈ y(t ) + ci eλit ri , (7.129) i

The Basics of Computational Physics

Δλ

86

Δλ

 4 Figure 7.11 The filled region 4n=0 ( tλ) ≤ eiφ is the region of stability for the n! explicit fourth-order Runge-Kutta method.

where ci = const, ri is the eigenvector of J, and λi is the corresponding eigenvalue. Consider the ODE system with decay Re(λi ) < 0, with the corresponding timescales 1/Re(−λi ). The system with decay is called stiff [109] if all timescales are considerably different: maxi Re(−λi )  1. mini Re(−λi )

(7.130)

Such a stiff system describes an interesting class of physical problems with decays and different time scales. For example, the reaction rates of elementary particles in the electron-positron plasma have very different time scales (see Chapter 11).

7.6 Stiff Systems and Gear’s Method

87

ImΔtλ 1

−2

−1

1

ReΔtλ

−1

Figure 7.12 The implicit Euler method is stable in the filled region.

To ensure the stability, one has to use such time a step that for all eigenvalues, tλi is in the region of stability. For the methods with the finite region of stability the time step is limited by the smallest timescale of the system, while the total time of the integration is similar to the maximal timescale. This implies that explicit methods are not adequate here. To avoid limits on the time step, the A-stable methods are useful. The method is called A-stable if its region of stability includes all the half-plane Re( tλ) < 0 [109]. The linear multistep implicit methods can be stable only for the order less or equal to 2 [109]. An example of the second-order implicit method is the trapezoidal rule yn+1 − yn f (yn , t n ) + f (yn+1 , t n+1 ) = . t 2 For the equation y˙ = λy the trapezoidal rule gives yn+1 = the stability is tλ tλ 1 + ≤ 1 − , 2 2

1+λ t/2 n y. 1−λ t/2

(7.131) The region of

(7.132)

or when tλ/2 is closer to −1 than to 1, which is the entire left complex half-plane Im tλ ≤ 0 (Figure 7.12). Another A-stable method is the implicit Euler one, yn+1 − yn = f (yn+1 , t n+1 ), t

(7.133)

The Basics of Computational Physics

Δλ

88

Δλ

Figure 7.13 The implicit trapezoidal rule is stable in the filled region.

with the stability condition −1 1 − tλ ≤ 1. 2

(7.134)

Its stability region is the exterior of the circle of radius 1 centered at (1,0) in the complex plane tλ (see Figure 7.13). A-stability is a very strong condition, and in practice less strong conditions are adopted. The method is called A(a) stable with a ∈ (0, π /2) if the region of the stability is |arg(− tλ)| < a (see Figure 7.14). Gear [111] introduced the definition of stiff stability: r the method is absolutely stable in the region R1 (Re( tλ) ≤ D) and r is exact in the region R2 (D < Re( tλ) < a, |Im( tλ)| < θ ).

Gear’s method is the implicit predictor-corrector method with backward differences: y

n+1

=

k

α j yn− j+1 + tβ0 f (t n+1 , yn+1 ).

(7.135)

j=1

Coefficients of the method α j , β can be found in [111] for the different orders of the method k, 1 ≤ k ≤ 6. The difference (7.135) gives the system of nonlinear equations for yn+1 to be solved with Newton’s iterations. It is possible to variate the order k from 1 to 6 to achieve the maximal time step for the given accuracy of

7.7 Numerical Methods for Linear Algebra

89

ImΔtλ 1

α −2

−1

1

ReΔtλ

−1

Figure 7.14 A(α) stable method in the filled region.

the integration. The method also utilizes the Jacobi matrix from the previous time steps when possible to reduce the number of operations for the matrix inversions. The regions of stability of Gear’s method [111] are illustrated in Figure 7.15. The curve  −eiqθ + qj=1 α j ei(q− j)θ tλ(θ ) = − , 0 ≤ θ ≤ 2π (7.136) β0 eiqθ is the boundary of the region of the stability in Figure 7.15. 7.7 Numerical Methods for Linear Algebra Once the numerical solution of the system of PDEs is reduced to the system of ODEs, the methods of linear algebra can be utilized. In this section a brief discussion of such methods is given. 7.7.1 Exact Solution Consider a system of linear algebraic equations Au = F,

(7.137)

where A is a n × n square matrix. It can be solved by the Gauss elimination method, which is the exact method to find the inverse matrix A−1 and the solution for vector u. First, the n × n identity matrix E = diag{1, . . . , 1} is augmented to the right of A, forming a n × 2n block matrix (A|E ). By means of elementary transformations, the composite matrix (A|E ) is reduced to the form (E|A−1 ), where A−1 is the inverse

The Basics of Computational Physics

Δλ

Δλ

90

Δλ

Δλ

Δλ

Δλ

Δλ

Δλ

Δλ

Δλ

Δλ

Δλ

Figure 7.15 Regions of the stability for Gear’s methods for different orders k = 1, 2, 3, 4, 5, 6.

7.7 Numerical Methods for Linear Algebra

91

matrix. The elementary transformations are the linear combinations of the matrix rows. The unknown vector u can then be found from the relation u = A−1 F. The number of operations is ∝ N 2 . Often at the three-point approximation of space derivatives in implicit schemes, one deals with the system of linear equations with tridiagonal-type matrix A, ai ui−1 + bi ui + ci ui+1 = fi , 1 ≤ i ≤ N,

(7.138)

for the unknown vector ui . For such a system the cyclic reduction method is useful [94, 93]. In this method one searches the relations between unknown components of the solution in the linear form ui+1 = xi+1 ui + yi+1 .

(7.139)

It is possible to derive coefficients xi , yi from coefficients xi+1 and yi=1 by inserting the solution (7.139) into the initial equation (7.138): ai ui−1 + bi ui + ci (xi+1 ui + yi+1 ) = fi , 1 ≤ i ≤ N.

(7.140)

One obtains ui = −

ai fi − ci yi+1 ui−1 + = xi ui−1 + yi bi + c i x i bi + ci xi+1

(7.141)

ai fi − ci yi+1 , yi = . bi + ci xi+1 bi + ci xi+1

(7.142)

and xi = −

The right boundary condition, aN uN−1 + bN uN = fN ,

(7.143)

allows for calculation of coefficients xN , yN . The number of operations is proportional to the rank of matrix ∝ N. The generalization to block tridiagonal matrix A is straightforward. The cyclic reduction method is used in the calculations in Part III. It is possible to realize a parallel algorithm for the tridiagonal matrix cyclic solver [112] (see also [113]). 7.7.2 Iterative Methods In most cases one needs to solve algebraic equations with a large number of unknowns, as large as the number of volumes on the computational grid. The iterative methods can be useful in some cases. Jacobi Method The Jacobi method is a simple algorithm for determination of the solutions of a diagonally dominant system of linear equations [93]. The initial matrix A is

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represented by the sum of two matrices– the diagonal matrix D and the matrix R with zero elements on the diagonal: ⎛ ⎞ a11 0 ... 0 ⎜ 0 a22 . . . 0 ⎟ ⎜ ⎟ A = D + R, where D = ⎜ . .. .. ⎟ .. ⎝ .. . . . ⎠ 0 0 . . . ann ⎛ ⎞ 0 a12 . . . a1n ⎜a21 0 . . . a2n ⎟ ⎟ ⎜ (7.144) R=⎜ . .. .. ⎟ . . . . ⎝ . . . . ⎠ an1 an2 . . . 0 The iteration procedure for the new k + 1th iteration is u(k+1) = D−1 (x − Rx(k) ), or for the elements ui(k+1) =

 1  j = iai j ukj . bi − aii

The diagonal dominance means that for each row, the absolute value of the diagonal term is greater than the sum of absolute values of other terms: |ai j |. |aii > j=i

Gauss-Seidel Method The idea of the iterative Gauss-Seidel method is connected with the simplicity of the inverse matrix for the triangular matrix. For the equation Au = f

(7.145)

L∗ u(k+1) = f − Uu(k) .

(7.146)

one can write the iteration process

The decomposition of A into the matrices is performed: ⎛ a11 ⎜a21 ⎜ A = L∗ + U, where L∗ = ⎜ . ⎝ .. an1

lower triangle L∗ and strictly upper triangle U 0 a22 .. .

... ... .. .

an2

...





0 ⎜0 ⎟ ⎜ ⎟ ⎟ , U = ⎜ .. ⎝. ⎠ 0 ann 0 0 .. .

a12 0 .. .

... ... .. .

0

...

⎞ a1n a2n ⎟ ⎟ .. ⎟ . . ⎠ 0 (7.147)

7.7 Numerical Methods for Linear Algebra

93

The new (k + 1) iteration in components is ⎛ ⎞ 1 ⎝ fi − ⎠ ui(k+1) = ai j x(k+1) − ai j u(k) j j aii ji

(7.148)

The procedure is known to converge [93] if either A is symmetric and positivedefinite, zT Az > 0,

(7.149)

for nonzero vector z or A is strictly or irreducibly diagonally dominan: |ai j |. |aii | ≥

(7.150)

j=i

Successive Overrelaxation Method In the successive overrelaxation method one adopts the decomposition of A into a diagonal D and strictly lower and upper triangle components L, U: A = D + L + U, where



a11 ⎜0 ⎜ D=⎜ . ⎝ .. 0

0 a22 .. .

... ... .. .

0 0 .. .

0

...

ann





0 ⎟ ⎜a21 ⎟ ⎜ ⎟ , L = ⎜ .. ⎠ ⎝ .

(7.151)

0 0 .. .

an1 an2 ⎛ 0 a12 ⎜0 0 ⎜ U = ⎜. .. ⎝ .. . 0 0

... ... .. .

⎞ 0 0⎟ ⎟ .. ⎟ , .⎠

...

0

... ... .. . ...

⎞ a1n a2n ⎟ ⎟ .. ⎟ . . ⎠

(7.152)

0

The system of linear equations may be rewritten as (D + ωL)u = ωf − [ωU + (ω − 1)D]u

(7.153)

with the relaxation factor ω > 1. The new iteration is u(k+1) = (D + ωL)−1 [ωf − (ωU + (ω − 1)D)u(k) ] = Lω x(u) + c, or for elements ui(k+1) = (1 − ω)ui(k) +

⎛ ω ⎝ bi − aii

ji

(7.155)

94

The Basics of Computational Physics

If A is symmetric and positively definite, then ρ(Lω ) < 1 for 0 < ω < 2 and the convergence of the iteration follows [93, 114]. In general, for ω = 1, one should choose the Seidel method, for ω > 1, the successive overrelaxation one, and for ω < 1, the successive down-relaxation one. Iterative Methods for the Sparse Matrix For the computation of a number of problems one needs the inverse of a sparse matrix. A sparse matrix is a matrix in which most of the elements are zero. In contrast, when most of the elements are nonzero, the matrix is said to be dense. The fraction of nonzero elements over the total number of elements (i.e., that can fit into the matrix, say, a matrix of dimension m × n can accommodate m × n total number of elements) in a matrix is called the sparsity (density). In some cases the structure of the matrix is well known. For example, in the direct integration of the kinetic Boltzmann equation by the implicit code, one has block matrices like the three-diagonal matrices. In that case the cyclic reduction method is effective. Numerous problems operate with the sparse matrix with the common structure. For example, in [115], a Newton’s iteration method for obtaining equilibria of rapidly rotating axially symmetric stars was developed in the base sparse matrix solver. The Bernoulli integral for the barotropic equation of state and the Poisson equation for the gravitational potential in the computational grid were reduced to the set of finite difference equation for the dimensionless gravitational potential. The system of nonlinear equations for the gravitational potential was solved by Newton’s iteration method. The system of linear algebraic equations at every step of the iteration was solved by the elimination technique developed by Zlatev [116]. This approach allows for operation with huge sparse matrices and uses the LU decomposition iteration process.5 Such a common solver for the sparse matrix is less effective in comparison with matrix elimination of the sparse matrix with known structure. 5

See also the realization of the NAG Library at www.nag.com/numeric/numerical_libraries.asp

8 Direct Integration of Boltzmann Equations

In this chapter, our experience of the integration of Boltzmann equations by the finite difference method is presented. In spherically symmetric geometry the approach operates with finite differences on the fixed grid in 4D phase space for particles (r, μ, , t ), and it is based on the method of lines. The ODE system is solved by implicit Gear’s method suitable for integration of stiff equations. The method takes into account all reactions and is applicable to both optically thick and optically thin regions. This chapter also includes description of Monte Carlo (MC) methods for integration of the Boltzmann equation. For small and moderate optical depth, MC is a universal approach able to describe different reactions and is especially suitable for multidimensional geometry. It is shown how the linear MC method is extended for large optical depths with simple reaction rates such as in Compton scattering. 8.1 Finite Differences and the Method of Lines This section is limited to 1D spatial geometry (spherically symmetric case) with full physical processes and implicit methods. A large number of physical problems requires solution of Boltzmann equations. Analytical solutions are available in exceptional cases, and in the general case one should rely on effective numerical integration. As discussed in the previous chapter, the finite difference technique [117] represents one such effective method. Finite differences are widely employed in astrophysical problems. Specific examples, considered in Part III, include thermalization of nonequilibrium optically thick electron-positron plasma and neutrino transport in the core collapse supernovae. Finite differences for the kinetic equation in astrophysical applications were introduced in [118]. In particular, in the gravitational collapse of the iron core of a massive star to a neutron star, neutrinos play a crucial role. Neutrino transport 95

96

Direct Integration of Boltzmann Equations

should be described by Boltzmann equations. The core on the verge of collapse has the radius 108 cm, and it is transparent for neutrinos. The collapsed hot protoneutron star has a radius of 106 –107 cm, and neutrinos are trapped with the optical depth  103 . It is impossible to treat such problems by explicit methods due to the presence of different timescales characterizing different processes. The relativistic Boltzmann equations (2.41) for the distribution function in the spherically symmetric flow fι (|p|, μ, r, t ) for the particle of sort ι (see, e.g., [2, 44, 118]) is    q  1 ∂ fι ∂ fι 1 − μ2 ∂ fι + βι μ = η˜ι − χιq fι , (8.1) c ∂r ∂r r ∂μ q where μ = cos ϑ, ϑ is the angle between the radius vector r and the particle momentum p. In addition, η˜ιq is the emission coefficient for the production of a particle of sort ι via the physical process labeled by q, and χιq is the corresponding absorption coefficient. The summation runs over all considered physical processes that involve a particle of sort ι. Gravity and external forces are neglected in (8.1). Particle densities are determined from eq. (1.4). In numerical simulations it is convenient to represent hyperbolic PDEs in socalled conservative form. The “conservative” numerical method can provide exact conservation of energy on a finite computational grid. The Boltzmann equations are said to be in conservative form when the advection terms on the LHS are represented as a derivative of a flux divided by a volume [119]. To take this form, the quantities Eι (ι , μ, r, t ) =

2π ι3 βι fι c3

(8.2)

are used instead of fι . Since 2π ι3 βι fι drdι dμ = Eι drdι dμ, (8.3) c3 it is clear that Eι is the spectral energy density in the {r, μ, ι } phase space, in which the volume element is drdι dμ. Using eqs. (8.1) and (8.3), the Boltzmann equations can be written in terms of the spectral energy density Eι as  q 1 ∂  1 ∂Eι μ ∂ 2 + (r βι Eι ) + (1 − μ2 )βι Eι = (ηι − χιq Eι ), (8.4) c ∂r r ∂r r ∂μ q ι fι drdp =

where ηιq = 2π ι3 βι η˜ιq /c3 . This form of Boltzmann equation is the basis for the conservative finite difference code used for numerical simulations reported in Part III. It allows for carrying out calculations with large time steps even for large optical depths.

8.1 Finite Differences and the Method of Lines

97

A computational grid in the {r, μ, } phase space is defined as follows. The r domain (R < r < rext ) is divided into jmax spherical shells whose boundaries are designated with half integer indices. The jth shell (1 ≤ j ≤ jmax ) is between r j−1/2 and r j+1/2 , with r j = r j+1/2 − r j−1/2 . The μ-grid is made of kmax intervals μk = μk+1/2 − μk−1/2 : 1 ≤ k ≤ kmax . The energy grids for different kinds of particles are different. The quantities to be computed are the energy densities averaged over phase-space cells  1 Eι,ω,k, j (t ) = Eι ddμr2 dr, (8.5) X ω , μk , r j where X ≡ ω μk (r3j )/3 and (r3j ) ≡ r3j+1/2 − r3j−1/2 . Replacing the space and angle derivatives in the Boltzmann equations (8.4) by finite differences, one arrives at the following set of ODEs for the quantities Ei,ω,k, j specified on the computational grid:   ! " (1 − μ2 )Eι,ω, j k (r2 μk Eι,ω,k ) j dEι,ω,k, j 1 + + βι,ω βι,ω dt r j μk (r3j )/3 q = [ηι,ω,k, j − (χ E )qι,ω,k, j ], (8.6) q

where βγ ,ω = 1 and βe,ω = [1 − (me c2 /ι,ω )2 ]1/2 . To achieve the second-order approximation, the following quantities are replaced by their mean values over the volume or over the corresponding coordinate on the grid or the central differences: ι,ω−1/2 + ι,ω+1/2 ι,ω = , (8.7) 2 μk−1/2 + μk+1/2 μk = , (8.8) 2   2 ! " r j+1/2 − r2j−1/2 /2 1  , = 3 (8.9) r j r j+1/2 − r3j−1/2 /3 r1−1/2 + r j+1/2 , 2  1 Eι,ω,k (r) = Eι (, μ, r)ddμ, ω μk ω μk  3 Eι (, μ, r)drd, Eι,ω, j (μ) = ω r3j ω r j rj =

Eι,k, j () =

3 μk r3j

(8.10) (8.11)

(8.12)

 μk r3j

Eι (, μ, r)drdμ,

(8.13)

98

Direct Integration of Boltzmann Equations

(r2 μk Eι,ω,k ) j = r2j+1/2 (μk Eι,ω,k )r=r j+1/2 − r2j−1/2 (μk Eι,ω,k )r=r j−1/2 ,

(8.14)

  βι Eι,k, j ω = ι,ω+1/2 βι,ω+1/2 (Eι,k, j )ω+1/2 − ι,ω−1/2 βι,ω−1/2 (Eι,k, j )ω−1/2 , 

(1 − μ )Eι,ω, j 2

(8.15)

 k

= (1 −

μ2k+1/2 )(Eι,ω, j )μ=μk+1/2

− (1 −

μ2k−1/2 )(Eι,ω, j )μ=μk−1/2 . (8.16)

The LHS of the Boltzmann equation in the quasi-linear form is the PDE of the hyperbolic type. For numerical solution one can use the “upwind” derivative approximation1 for derivatives along μ and  without introducing the artificial viscosity: ⎧ ⎪ Eι,ω+1,k, j ⎪ ⎪ ⎪ (E −E ⎪ j )(ω+1/2 −ω+1 ) ⎪ , ⎪+ ι,ω+2,k, j ι,ω+1,k, ⎪ ω+2 −ω+1 ⎨ (8.17) (Eι,k, j )ω+1/2 = μ ≥ 0, ⎪ ⎪ ⎪ Eι,ω,k, j ⎪ ⎪ ⎪ (Eι,ω,k, j −Ei,ω−1,k, j )(ω+1/2 −ω ) ⎪ ⎪ , ⎩+ ω −ω−1 μ < 0,   μk (Eι,ω,k, j − Eι,ω,k−1, j ) Eι,ω, j μ=μk+1/2 = Eι,ω,k, j + , (8.18) μk−1 + μk which should be restricted to nonnegative values. For the approximation of the derivative over r, the second-order “upwind” differences cannot be used. This is because the method employs the cycling reduction for the solution of the ODE system. Instead, the combination of the first-order approximation “upwind” difference and the second-order central difference is adopted: (μk Eι,ω,k )r=r j+1/2 = (1 − χ˜ι,ω,k, j+1/2 )   μk − |μk | μk + |μk | Eι,ω,k, j + Eι,ω,k, j+1 × 2 2 Eι,ω,k, j + Eι,ω,k, j+1 + χ˜ι,ω,k, j+1/2 μk 2 with the coefficient 1 1 −1 χ˜ι,ω,k, + . j+1/2 = 1 + χι,ω,k, j r j χι,ω,k, j r j+1

(8.19)

(8.20)

The dimensionless coefficient χ˜ is introduced to describe correctly both the optically thin and the optically thick computational cells by means of a compromise 1

This is integration along characteristics in the one-dimensional case.

8.2 Monte Carlo Method

99

between the high-order method and the monotonic transport scheme without the artificial viscosity (see [103, 118, 120]). In the optically thin region the scheme is first-order “upwind.” In the optically thick region the scheme is different. Numerical oscillations characteristic for the nonmonotonic scheme are suppressed by high reaction rates, which on the grid become q ηι,ω,k, j

(χ E )qι,ω,k, j

1 = X 1 = X

 ω , μk , r j

ηιq ddμr2 dr

(8.21)

χι Eιq ddμr2 dr.

(8.22)



ω , μk , r j

q The physical processes included in the simulations the expressions ηι,ω,k, j and (χ E )qι,ω,k, j , discussed in Part III and the appendices. Clearly in the numerical q q scheme ηι,ω,k, j and (χ E )ι,ω,k, j are sums instead of integrals of the Boltzmann equations. The numerical method of the conversion of the evolutionary PDEs into the ODEs is the so-called method of lines [108]. The method of the numerical solution of the ODEs can be either explicit or implicit. When there are different timescales in the physical problem, one should use implicit schemes, such as Gear’s method, suitable for the numerical integration of stiff ODE systems. For the inverse of the sparse matrix with the structure I + tJ, where J is a Jacobi matrix, the cyclic reduction method is useful for the three-diagonal matrix with rank jmax composed from the matrices kmax × ωmax × ιmax . Note that expansion of the presented method to 2D geometry in coordinate space is an open question due to limitations related to the use of the cyclic reduction method with three-diagonal matrices for solution of the system of linear algebraic equations.

8.2 Monte Carlo Method The MC method was proposed in 1949 [121] and since then has found applications in many physical problems, due to its universality. The MC method can be applied for any problem characterized by the presence of a stochastic (random) process. MC methods have the following main features: r Usually it operates with calculations having simple structure. r The error of calculations is ∝ N −1/2 , where N is the number of sampled points,

regardless of the number of dimensions. This fact makes the MC method particularly suitable for multidimensional problems.

100

Direct Integration of Boltzmann Equations

There are the following main types of MC methods [106]: r direct MC, in which random numbers are used to model the effect of complicated

stochastic processes. An example is the modeling of traffic, where the behavior of cars is determined in part by random numbers. r MC integration, which is a method for numerical evaluation of integrals using random numbers. This method is particularly efficient for multidimensional integrals. r The Markov chain MC method [122] based on the Metropolis-Hastings algorithm [123, 124], in which a sequence of distributions of a system is generated in a socalled Markov chain. This method allows study of the static properties of classical and quantum many-particle systems. In a cosmology Markov chain MC methods are employed in the determination of cosmological parameters from observations of large-scale structure and cosmic microwave background (CMB) radiation. 8.2.1 The Probability and the Mathematical Expectation The basic terminology of the probability theory is introduced following [125]. The random value ξ is discrete if it is in the discrete set ξ ∈ (x1 , .., xn ), with the corresponding probabilities (p1 , .., pn ), i.e., P(ξ = xi ) = pi , with pi > 0 and n i=1 pi = 1. The mathematical expectation of ξ is the average value of ξ : Mξ =

n

ξ i pi .

(8.23)

i=1

Mathematical expectation has the following properties: constancy, M(c) = c,

(8.24)

where c is a constant; proportionality, M(cξ ) = cM(ξ );

(8.25)

M(ξ + η) = M(ξ ) + M(η),

(8.26)

and additivity, where ξ and η are two random values. The dispersion of the random value ξ is D(ξ ) = M[(ξ − M(ξ ))2 ] = M(ξ 2 ) − (M(ξ ))2 > 0.

(8.27)

For two independent random values ξ and η, one has M(ξ η) = M(ξ ) × M(η)

(8.28)

D(ξ + η) = D(ξ ) + D(η).

(8.29)

and

8.2 Monte Carlo Method

101

Suppose for the continuous value ξ in some interval (a, b), one has the probability density of ξ or the density of the distribution – the function p(x); i.e., the probability to find ξ in the interval (a , b ) ∈ (a, b) is the integral  b P[a < ξ < b ] = p(x)dx. (8.30) a

The density p(x) has the following properties: 1. p(x) is positive p(x) > 0; b 2. the integral over the total region gives unity, a p(x)dx = 1. The mathematical expectation of ξ is the average value of ξ :  b Mξ = xp(x)dx.

(8.31)

a

The mathematical expectation of any function f (x) is the average value of ξ :  b M f (ξ ) = f (x)p(x)dx. (8.32) a

A classic example is the normal (Gauss) distribution in the interval (−∞, ∞),   √ (x − a)2 , (8.33) pG (x) = σ 2π exp − 2σ 2 where a, σ > 0. Its mathematical expectation is Mξ = a,

(8.34)

and the dispersion is Dξ = σ 2 . (8.35)  a+3σ The integral over the normal distribution a−3σ pG (x)dx = 0.997 demonstrates the 3σ rule: P(a − 3σ < ξ < a + 3σ ) = 0.997.

(8.36)

The probability 0.997 is close to unity, and in a single test, it is practically impossible to obtain ξ different from Mξ by more than 3σ , or three standard deviations. 8.2.2 Central Limit Theorem The central limit theorem states that, given certain conditions, the arithmetic mean of a sufficiently large number of iterates of independent random variables will be approximately normally distributed, regardless of the underlying distribution [125].

102

Direct Integration of Boltzmann Equations

Let ξ1 , .., ξN be N independent random values with the same probability distributions, so Mξ1 = .. = MξN = m

(8.37)

Dξ1 = .. = DξN = b2 .

(8.38)

Let ρN be the sum ρN =

N

ξi ;

(8.39)

i=1

then, from additivity properties (8.26) and (8.29), one has MρN = M(ξ1 + .. + ξN ) = Nm

(8.40)

DρN = D(ξ1 + .. + ξN ) = Nb2 .

(8.41)

Let ζN be a normal random value with the same expectation and dispersion, a = Nm, σ 2 = Nb2 . The central limit theorem states that lim |PρN (x) − PζN (x)| = 0;

(8.42)

N→∞

the probability density of the sum ρN corresponds to a normal distribution. As soon as one is dealing with the effect of large numbers of the random factors, the normal distribution is obtained as a result. A six-line proof of the central limit theorem based on the inverse Fourier transform of the probability is given in [126]. It is possible to say that data influenced by many small and unrelated random effects are approximately normally distributed.

8.2.3 The General Monte Carlo Scheme Suppose one needs to compute the mathematical expectation m. Introduce the random value ξ with the properties Mξ = m and Dξ = b2 . Consider N random values ξ1 , .., ξN with the distribution of the value ξ . If the number N is large enough, in accordance with the central limit theorem, the distribution of the sum ρN = ξ1 + .. + ξN is approximately normal with the parameters a = Nm, σ 2 = Nb2 , and one has % √ √ & P Nm − 3b N < ρN < Nm + 3b N ≈ 0.997 (8.43) or

 3b √ 3b ρN P m− √ 1 and is useless. Recently, the generalization of the method within special relativity was presented in [162, 163].

9.4.2 Particle-in-Cell Methods A mixture between grid-based and mesh-free methods is illustrated by particle-incell (PIC) algorithms (see, e.g., [164, 165, 166, 167] for reviews). In this method, individual particles, or fluid elements, in a Lagrangian frame are tracked in continuous phase space, whereas moments of the DF, such as densities and currents, are computed simultaneously on Eulerian (stationary) mesh points. PIC methods were used as early as 1955 in Los Alamos National Laboratory [168]. The main advantage of the method is the simple possibility of the consideration of different types of particles in the same computational grid. Examples of physical problems treated with this method include nuclear explosions in the atmosphere and meteorite impacts on the Earth. This method is particularly useful for the solutions of the Vlasov-Maxwell equations, hence its primary application is plasma physics [169]. In particular, it is widely employed for simulations of collisionless plasma (see Chapter 10). A number of fully relativistic 3D PIC codes are available [170, 171, 172]. The set of assumptions in the physical model of PIC simulation is reduced to the minimum. Indeed, the code tracks individual “particles” by solving their equations of motion. Given that, in practice, the number of particles is very large, each “particle,” called a quasi-particle, is associated with a collection of actual particles. All macroscopic quantities, such as density and currents, are computed from the

130

Multidimensional Hydrodynamics

position and momenta of particles. The way of assignment of macroscopic quantities to the particles used in simulation represents the core of the approach. The method is split into several parts: r a “particle solver,” which solves equations of motion of particles; r sources and boundary effects, which specify boundary conditions and inject new

particles in the volume or at the boundaries of the computational domain; r “particle weighting,” which calculates the macroscopic quantities associated with

particles; r a “field solver,” which provides solution of Maxwell equations whose sources are

computed in the previous items. The numerical solution of Maxwell equations, in turn, can be performed by a variety of methods, among them the finite difference method and solution in the Fourier space using the fast Fourier transform technique. Consider the realization of the electromagnetic PIC method [172]. It solves the Maxwell-Vlasov equations (5.28) and (5.29). The basic idea of PIC methods is that DF is approximated (particle weighting) as fι (x, p, t ) =

Ns

  Nsi S [x − xi (t )] δ 3 p − pi (t ) ,

(9.88)

i=1

where NlS is the number of actual particles that each quasi-particle i of species s represents and S is called the shape function. Comparing this expression to the Klimontovich random function given by eq. (2.15), one observes that PIC methods follow closely the Klimontovich approach. Equations of motion are derived by taking moments of the Vlasov equation (5.28), dNsi = 0, dt

dxsi = vsi , dt

  dpsi = qs Ei + vsi × Bi , dt

(9.89)

where the first equation represents the particle number conservation of each species. Here psi = ms γis vsi . The electric and magnetic fields are given by    3    s Ei = ES x − xi d x Bi = BS x − xsi d 3 x, (9.90) so the fields are also averaged (field weighting) over the quasi-particle. From the computational point of view the computation of field equations on the grid brings an important advantage for PIC methods: the CPU time scales with the number of particles as ∝ N, unlike an exact N-body, which scales as ∝ N log N. The typical number of quasi-particles in current PIC simulations is 108 –109 .

9.4 Particle-Based Methods

131

The Eulerian grid in the coordinate space equations of motion (5.1) can be solved by a standard leapfrog method: xn+1/2 − xn−1/2 pni i i = vni = t ms γin   − pni pn+1 n+1/2 n+1/2 i = qs En+1/2 + v × B . i i i t The Boris scheme [173, 174] is usually adopted for velocity calculations, vn+1/2 = i

pni + pn+1 i 2ms γin+1/2

,

(9.91) (9.92)

(9.93)

and the momentum update is split into a half-step acceleration by electric field E, a rotation by magnetic field B, and another half-step acceleration by E. Similarly, for the computation of the electromagnetic fields, the leapfrog method can be implemented. A particular choice is the finite difference time domain method, which employs the staggered Yee grid to represent magnetic fields on faces, electric fields and current densities on edges, and charge densities on corners of the grid. The discrete curl operators are defined on the Cartesian grid with the x-components given by  +  Ez,i, j+1,k+1/2 − Ez,i, j,k+1/2 ∇ × E x,i, j+1/2,k+1/2 = (9.94) y Ey,i, j+1/2,k+1 − Ey,i, j+1/2,k − , z  −  Bz,i+1/2, j+1/2,k − Bz,i+1/2, j−1/2,k (9.95) ∇ × B x,i+1/2, j,k = y By,i+1/2, j,k+1/2 − By,i+1/2, j,k−1/2 − , z and other components obtained by cyclic permutation. The discrete divergence operators are  +  Bx,i+1, j+1/2,k+1/2 − Bx,i, j+1/2,k+1/2 ∇ · B i+1/2, j+1/2,k+1/2 = (9.96) x By,i+1/2, j+1,k+1/2 − By,i+1/2, j,k+1/2 + y Bz,i+1/2, j+1/2,k+1 − Bz,i+1/2, j+1/2,k + z  −  Ex,i+1/2, j,k − Ex,i−1/2, j,k (9.97) ∇ · E i, j,k = x Ey,i, j+1/2,k − Ey,i, j−1/2,k Ez,i, j,k+1/2 − Ez,i, j,k−1/2 + + . y z

132

Multidimensional Hydrodynamics

Maxwell’s equations are then n−1/2 En+1/2 i, j,k − Ei, j,k

t n+1 Bi, j,k − Bni, j,k t where

= ∇ − × Bni, j,k − 4π jni, j,k

(9.98)

= −∇ + × En+1/2 i, j,k ,

(9.99)

  Ei, j,k = Ex,i+1/2, j,k , Ey,i, j+1/2,k , Ez,i, j,k+1/2   Bi, j,k = Bx,i, j+1/2,k+1/2 , By,i+1/2, j,k+1/2 , Bz,i+1/2, j+1/2,k .

(9.100) (9.101)

The discretized divergence equations (∇ − · E)i, j,k = 4π ρi, j,k

(∇ + · B)i+1/2, j+1/2,k+1/2 = 0

(9.102)

are satisfied provided that the discretized charge conservation n−1/2 ρi,n+1/2 j,k − ρi, j,k

  + ∇ − · jni, j,k i, j,k = 0

(9.103) t is also satisfied. The stability of the scheme is ensured by the Courant condition. Computation of the fields closes the loop of the PIC scheme. The choice of shape functions S[x − xi (t )] determines the properties of the numerical method. These functions are usually modeled with B-splines [165]. In current PIC simulations the weighting scheme assigns quantities to the two nearest grid points, which is referred to as the cloud-in-cell (CIC) scheme. It is important to bear in mind that the charge-sharing scheme automatically expands the particle to a grid size. At the same time, for ultra-relativistic particles, the Lorentz contraction can become so large that both particle and electric field are compressed in the direction of motion [164]. These effects should be correctly accounted for in the fully relativistic PIC method. Notice that in addition to solving Vlasov-Maxwell equations, modern PIC codes such as iPiC3D2 are able to describe particle collisions, which are accounted for with statistical MC models. 2

https://github.com/CmPA/iPic3D

Part III Applications

10 Wave Dispersion in Relativistic Plasma

One of the most important domains of application of relativistic KT is the theory of waves in relativistic plasma. The study of waves in thermal plasma has a long history, starting with the fundamental works of Vlasov [7] and Landau [175], being extended to the relativistic domain [176, 177, 178, 179, 180] and many other works later on. Recently, a covariant and gauge-independent theory of response has been formulated [181]. In this chapter, after a brief introduction to this theory, several important applications, such as Landau damping and plasma instabilities, are considered, following [181]. The general approach is described subsequently. One assumes that the plasma is perturbed. The perturbation is represented as a plane wave. In particular, for the electromagnetic potential, one assumes Aμ (x) ∝ exp(−ikμ xμ ) ≡ exp(−ikx) = exp[−i(ωt − kx)],

(10.1)

μ

where k denotes the four-vector k ≡ k = (ω/c, k), ω is frequency, and k is a wave vector. Solution of the Vlasov-Maxwell equations (5.28) and (5.29) is searched, where the perturbation in the DF and current is also considered to be of the form (10.1). In this approach it is natural to Fourier transform all quantities, as follows:  d4k ˆ G(x) = G(k) exp(−ikx), (10.2) (2π )4 ˆ where G(k) are amplitudes. The response of the plasma to an electromagnetic perturbation described by the four-potential Aμ (k) can be associated to the induced four-current J μ (k). The final goal is to obtain the dispersion relation, namely, the relation between the frequency and wave number ω(k). Fourier transform (10.2) is applied1 to the relation (5.5) between the tensor of electromagnetic field and the four-potential, which gives F μν (k) = −i[kμ Aν (k) − kν Aμ (k)]. 1

(10.3)

In what follows the hat symbol above the Fourier-transformed quantities is omitted.

135

136

Wave Dispersion in Relativistic Plasma

One may verify that the Maxwell equation εμνσ ρ ∂ ν F σ ρ = 0 is satisfied identically in the Fourier space. The transform of ∂μ F μν = 4π J ν gives 

 k2 gμν − kμ kν Aν (k) = −4π J μ (k) .

(10.4)

The key idea of the response theory is the separation of the current into an induced part, which corresponds to the response of the plasma, and external part, which describes the source: μ μ (k) + Jext (k) . J μ (k) = Jind

(10.5)

If the induced current is sufficiently weak, the response is described by μ (k) = "μν (k) Aν (k) , 4π Jind

(10.6)

where "μν (k) is referred to as the linear response tensor. Various physical requirements on this tensor are reflected in mathematical constraints on it. In particular, the charge continuity equation (4.9), being Fourier transformed, becomes μ kμ Jind (k) = 0, and it results in the condition kμ "μν (k) = 0. The condition of gauge independence of the response results in another condition kν "μν (k) = 0. Particularly important is the separation of the linear response tensor into Hermitian (H) and anti-Hermitian (A) parts: "Hμν (k) =

 1  μν " (k) + "∗νμ (k) 2

"Aμν (k) =

 1  μν " (k) − "∗νμ (k) , 2 (10.7)

where the asterisk denotes a complex conjugate. While the Hermitian part describes a time-reversible response, the anti-Hermitian part corresponds to dissipation. The Kramers-Kronig relations on "Hμν (k) and "Aμν (k) follow from the causality condition. The dispersion relation is obtained by substituting eqs. (10.5) and (10.6) into the Maxwell equations (10.4), producing   μ (k) . (10.8) μν (k) Aν (k) ≡ k2 gμν − kμ kν + "μν (k) Aν (k) = −4π Jext The dispersion relation is the condition of existence of the homogeneous equation, μ obtained from eq. (10.8) by neglecting the source term Jext (k), and the dissipation Aμν is described by " (k). Owing to the properties μν kν = kμ "μν (k) = 0, the dispersion relation cannot be obtained directly from the equation det[μν (k)] = 0, as it vanishes identically. Instead, one may notice [182] that using an identity kλ gλν = 0, one can rewrite the

10.1 Collisionless Plasma

homogeneous equation, corresponding to eq. (10.8), as   kμ kλ ¯ μν μ μν  (k) Aν (k) = gλ − 2  (k) Aν (k) = 0 k  2 λν  μν ¯  (k) = k g + "Aλν (k) .

137

(10.9) (10.10)

The first term in the round brackets leads to the vanishing determinant of μν (k). Therefore the dispersion relation is obtained from the equation  μν  ¯ (k) = 0. det  (10.11) The problem of determining the wave properties can be addressed by solving eq. (10.11).

10.1 Collisionless Plasma The kinetic approach is remarkably useful in studying collisionless systems. The statistical theory of collisionless plasmas is based on the Vlasov-Maxwell system of equations (5.28) and (5.29). The plasma consists of particles and electromagnetic fields. In the response theory it is useful to reformulate the plasma description in terms of background plasma and wave subsystems. Several different approaches exist, including the forward-scattering, Vlasov, and cold plasma approaches [181]. While they are based on different physical arguments, there is a relation between these approaches, and they give the same results. Here the Vlasov approach is discussed in some detail. Consider a particle that moves in a fluctuating electromagnetic field (5.5) described by the vector potential. The equation of motion (5.1) for particle trajectory x(τ ) can be written by Fourier transforming this fluctuating field as  d pμ q d 4 k iq = F μν uν = exp[−ikx(τ )](k u)Gμν (k , u)Aν (k ), (10.12) dτ c m (2π )4 where Gμν (k, u) = gμν −

k μ uν . ku

(10.13)

The Fourier transform is performed, which is equivalent to assuming perturbations to be harmonic, i.e., of the form (10.1). The perturbed DF F (k, p) = F (p)(2π )4 δ 4 (k) + F1 (k, p) satisfies the Fourier-transformed Vlasov equation (5.28), which becomes

d pμ ∂ μ −ip kμ + m F (k, p) = 0, (10.14) dτ ∂ pμ

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Wave Dispersion in Relativistic Plasma

where the term d pμ /dτ is given by eq. (10.12). The solution for the perturbed DF is found: ∂F (p) F1 (k, p) = qGμν (k, u) Aν (k) . (10.15) ∂ pμ Recalling definitions (1.10) and (4.9), the associated four-current is  μ J1 (k) = c q F1 (k, p)pμ d 4 p. This current defines the linear response tensor (10.6) as q2  ∂F (p) μν d 4 pGλν (k, u) pμ . " (k) = c m ∂ pλ

(10.16)

(10.17)

This general expression is now considered for a special case: an isotropic plasma. 10.2 Response in an Isotropic Case Consider a plasma that is isotropic in its Lorentz reference frame. Recall that this frame, also called the rest frame, is defined such that the hydrodynamic velocity is U μ = (c, 0, 0, 0). The derivative of the distribution function with respect to four-momentum is reduced to the derivative over particle energy in the rest frame, represented by the invariant pμUμ = (pU ), so eq. (10.17) can be rewritten as

q2  (kU ) uμ uν ∂F (p) μν 4 μ ν d p (uU ) U U − " (k) = c . (10.18) (ku) m ∂ (pU ) The linear response tensor in an isotropic medium can be decomposed into longitudinal, transverse, and rotatory parts. The latter in the classical plasma vanishes, so one has "μν (k) = "L (k) Lμν (k, U ) + "T (k) T μν (k, U ) ,

(10.19)

where second-rank tensors Lμν (k, U ) and T μν (k, U ) are constructed2 from the available four-vectors kμ and U μ , as follows: Lμν = −Lμ Lν , (kU )2 μ ν kμ kν μν L L + g − , k2 k2 kλ Gλμ Lμ =  1/2 , k2 − (kU )2

T μν =

2

Not to be confused with the energy-momentum tensor.

(10.20)

10.3 Dispersion in Relativistic Thermal Plasma

139

and invariant functions "L (k) and "T (k) completely describe the linear response of the isotropic plasma. Tensor μν (k) defined in eq. (10.8) in isotropic medium can be decomposed exactly in the same way as "μν (k) in eq. (10.19), namely, μν (k) = L (k) Lμν (k, U ) + T (k) T μν (k, U ) ,

(10.21)

where L (k) =

(kU )4 Lμν μν k4

1 T (k) = Tμν μν . 2

(10.22)

The dispersion relation is obtained from the conditions3  2 L (k) = T (k) = 0 (see eq. (10.8)), which finally gives (kU )2 + "L (k) = 0

k2 + "T (k) = 0.

(10.23)

These two equations describe longitudinal and transverse waves, respectively.

10.3 Dispersion in Relativistic Thermal Plasma The linear response tensor determines the properties of wave modes, including the dispersion relation, the polarization vector, the energetics, and damping of waves. In the general case, plasma can contain both thermal and nonthermal populations of particles. The nonthermal component usually excites waves. Instead, the properties of waves are determined mainly by the thermal particles. Consider the equilibrium DF (4.27). One can normalize this DF to the equilibrium number density neq using eq. (4.29), such that the integral in the Lorentz reference frame gives unity

n φ − pμUμ N   exp P (p) = . (10.24) kB T 4π (mc)3 θ K2 θ −1

3

The solutions of L (k) correspond to longitudinal waves, while the solutions of T (k) correspond to two degenerate transverse modes.

140

Wave Dispersion in Relativistic Plasma

Using this normalized DF, one can perform angular integration in eq. (10.17). The result is "L (k) = −ω2pL (z, θ ) ,

"T (k) = −ω2pT (z, θ ) ,  ∞ z2 z3 L  (z, θ ) = − + 2  −1  dχ sinh χ cosh2 χ A (θ , χ ) , θ 2θ K2 θ 0  ∞   z2 z T dχ sinh χ z2 cosh2 χ − sinh2 χ A (θ , χ ) ,  (z, θ ) = − + 2  −1  2θ 4θ K2 θ 0 (10.25)     z cosh χ + sinh χ cosh χ ln , A (θ , χ ) = exp − θ z cosh χ − sinh χ where the new variables z and χ are introduced as follows: (kU ) z = − 1/2 k2 − (kU )2

cosh χ = 

p0 pμ pμ

1/2 .

(10.26)

Note that the quantity z characterizes the phase velocity, and in the Lorentz reference frame, it reduces to z = ω/|k|. Equation (10.23) together with eq. (10.25) describe wave properties in relativistic thermal plasmas. They may be rewritten as ω = 2

ω2pL

(z, θ )

ω = 2

z2 ω2p 1 − z2

T (z, θ ) .

(10.27)

In the absence of a medium the dispersion relation is k2 = 0, and only transverse waves are possible. For pure (electrically neutral) electron-positron plasma the only longitudinal waves are Langmuir waves, and the plasma frequency includes equal contributions from electrons and positrons. Other modes are possible when ions are present. First, consider the limiting cases of eq. (10.27), which determine the Debye length (with z → 0) and the cutoff frequency ωc (with z → ∞). The Debye length is obtained by analyzing the static longitudinal response of the plasma, and it is defined by ω2p L (z, θ ) . = z→0 z2 c2 θ

λ−2 D = lim

(10.28)

This expression is valid for all temperatures, both nonrelativistic and ultrarelativistic. Recall that waves with the frequency ω < ωc cannot propagate in plasma. The cutoff frequency is obtained from the dispersion relation in the limit z → ∞. In

10.4 Landau Damping w wp

141

2

1.0

0.5

0.5

1.0

1.5

2.0

z

0.5

Figure 10.1 The functions L (z, θ ) and z2 T (z, θ )/(1 − z2 ) for θ = 1 are shown. The solid curve shows the real part of L (z, θ ), and the dashed curve shows the imaginary part of L (z, θ ). The real part of the function z2 T (z, θ )/(1 − z2 ) is shown by the dash-dotted curve, while its imaginary part is shown by the dotted curve.

this limit, from eq. (10.25), it follows that ωc2

 "  2 ! ω2p  v 2 ω p 1 − 52 θ , θ  1   → = ω2p 3θ1 − 6θ1 3 , θ  1. 3θ c

(10.29)

The dispersion curves ω2 /ω2p for Langmuir and transverse waves are shown in Figure 10.1 for θ = 1. Qualitatively, this figure is the same for any θ . First, let us discuss transverse waves. The real part of T (z, θ ) is negative for subluminal waves, which implies that they cannot exist. In contrast, superluminal waves can exist. The imaginary part of T (z, θ ) is zero, so transverse superluminal waves are undamped. This result is independent of temperature θ .

10.4 Landau Damping Now turn to longitudinal waves. The dominance of the imaginary part over the real part for z  1 implies that all waves are strongly damped. Only near the maximum of Re[L (z, θ )] is there a narrow region of weakly damped subluminal waves. This region becomes narrower with increasing temperature. Since for z > 1, the imaginary part of L (z, θ ) vanishes, all superluminal waves are undamped. In the nonrelativistic domain the peak of the function Re[L (z, θ )] shifts to lower values of the variable z. This gives two effects: first, the weakly damped subluminal waves become possible, and second, the damping becomes exponentially weak. Recall that the dispersion relation for Langmuir waves with kλD  1 in nonrelativistic

142

Wave Dispersion in Relativistic Plasma

plasma is given by [183] ω2 k 2 c2 = 1 + 3θ = 1 + 3 (kλD )2 , ω2p ω2p

(10.30)

where eq. (10.28) is used in the second expression. The decay of such waves occurs with the law exp(−δt ), where the rate δ is given by [175] 8

π ωp 1 3 . (10.31) δ= exp − − 8 2 (kλD )3 2 (kλD )2 2 Particularly interesting are ultra-relativistic expressions for dispersion relations in isotropic plasma [180, 184]. For θ  1 the superluminal waves with kλD  1 fulfill the relation

ω2 9 1 2 ) (kλ 1 + , (10.32) = D ω2p 3θ 5 and those near the light limit with 1  (kλD )2  log θ satisfy the relation    (10.33) ω = kc 1 + 2 exp −2 (kλD )2 − 2 . For the waves with the phase velocity approaching the speed of light one has   k − k0 ω=c k− log θ , (10.34) 6θ 2 √ where k0 = log θ /λD . For k < k0 the dispersion relation matches eq. (10.33) and waves are superluminal without damping. In the opposite case, k0 < k  k0 + λ−1 D , the oscillations become damped with the decay rate [184]

π c 3 δ= . (10.35) exp − (k − k0 ) λD 16 λD θ 2 (k/k0 − 1) The second condition k  k0 + λ−1 D is clearly required for weakness of damping. For waves with larger k the damping is strong, and one can no longer use the notion of a wave. Damping of electromagnetic waves in collisionless plasma was discovered by Landau [175]. The physical meaning of this damping is the presence of electrons with velocities equal to the phase velocity of the wave, ve = ω/k. For such electrons the electromagnetic field is stationary and its average over time does not vanish. Thus the energy of the field may be converted into the energy of electrons, resulting in the damping of the wave. Nonrelativistic linear Landau damping is well described in textbooks (see, e.g., [87]). More details, including nonlinear damping, are given in the mathematical

10.5 Plasma Instabilities

143

treatise [185]. Actually, the damping of plasma oscillations in the nonrelativistic case is exponentially weak everywhere apart from the region with kλD ∼ 1. In contrast, in the relativistic case, there are two possibilities: when phase velocity of the wave is smaller than the speed of light, the damping is strong; in the opposite case, the damping is totally absent [180]. The damping is weak only in the region with k  k0 + λ−1 D . These results are generally confirmed for pure electronpositron plasma [186]. Landau damping occurs also for relativistic plasma with intense radiation or turbulence [187]. Note that the Landau damping phenomenon is not restricted to plasmas. Considering eqs. (5.30) and (5.31) and the corresponding footnote, it is clear that a similar phenomenon is present in self-gravitating systems (see Chapter 14).

10.5 Plasma Instabilities In Section 10.4, damping of waves in collisionless plasma is discussed. This process suppresses the amplitude of initial perturbations, thus bringing the system to equilibrium. The opposite can also happen, namely, initially small perturbations can grow with time: this process is generally referred to as instability. Instability in a self-gravitating system is believed to be the cause of the structure formed at the largest scales of the universe. This gravitational instability is discussed in Chapter 14. In the linear case the instability causes exponential growth of electromagnetic fields and currents on different scales of space and time. These currents and fields tend to restore the equilibrium state of the plasma. Saturation of instability occurs in the nonlinear phase, where perturbations are no longer small, which follows the linear phase. Many kinds of plasma instabilities occur when different plasma flows, particles with different masses, and electromagnetic fields interact (see, e.g., [188, 189, 190]). Most plasma instabilities were discovered in laboratory experiments. Coping with plasma instabilities is essential for performance of fusion reactors based on the tokamak technology being planned and built, such as ITER.4 In general, they are classified as either kinetic or hydrodynamic instabilities. Kinetic or microscopic instabilities are associated with a momentum space nonequilibrium, unlike hydrodynamic or macroscopic instabilities, which are associated with coordinate space nonequilibrium. To illustrate these concepts, the focus will be on instabilities occurring in the presence of anisotropies and streams in plasma. In the nonrelativistic case, such a situation corresponds to Weibel and two-stream instabilities, respectively.

4

https://www.iter.org/

144

Wave Dispersion in Relativistic Plasma

The Weibel instability [191] has been discovered in nonrelativistic plasma which possesses an anisotropy in momentum space. In a broader context, the Weibel instability in an unmagnetized plasma occurs either due to anisotropy in momentum distribution or due to the presence of counterstreaming beams. The two-stream instability (see, e.g., [87]), can be thought of as the inverse of Landau damping, where the existence of a greater number of particles that move more slowly than the wave phase velocity, as compared with those that move faster, leads to an energy transfer from the wave to the particles. When streams are present, instabilities are classified with respect to the orientation of the wave vector [192]. For the wave vector orthogonal to the beam direction, it is referred to as filamentation instability; for the wave vector parallel to the beam, it is referred to as the two-stream instability. Such instabilities are widely considered to be important astrophysical contexts [193]. They might operate in astrophysical environments, such as pulsar magnetospheres, relativistic jets, collisionless shocks in gamma-ray bursts, and supernova remnants. Such instabilities may be responsible, in particular, for the generation of electromagnetic fields and particle acceleration in relativistic collisionless shocks. It has been recently proposed [194] to model occurrence of collisionless shocks in the laboratory, which will allow laboratory experiments with such shocks, providing insights into astrophysical conditions. Some qualitative differences occur when temperatures and streaming velocities become relativistic. In the next section the Weibel instability in relativistically streaming plasma [182] is discussed in detail. For relativistic two-stream instability, see [195, 196], while for streaming instabilities in plasma with relativistic temperatures, see [197, 198]. A recent review of the beam-plasma instabilities in the relativistic regime is given in [199]. 10.6 Weibel Instability Following [182], consider the background plasma in the Lorentz reference frame with the four-velocity U μ = (c, 0, 0, 0), having ι components (e.g., ions and electrons) and two counterstreaming beams of charged particles moving along the zaxis with the four-velocity Ubμ = γb (c, 0, 0, ±vb ), γb = [1 − (vb/c)2 ]−1/2 , where the subscript b denotes the beams and the sign ± corresponds to the choice of the direction. The counterstreaming beams are assumed to be equal, with the same density and equal and opposite velocity. As discussed earlier, the modes orthogonal to the beam direction are considered, so the wave four-vector in the rest frame is chosen: kμ = (ω/c, K, 0, 0). Generally speaking, there is a coupling between the transverse and longitudinal response of the plasma. However, when the beams are completely symmetric, such coupling disappears and Weibel instability is purely transverse. Application of the covariant

10.6 Weibel Instability

145

theory of plasma response described earlier to this case results in the dispersion relation K 2 c2 − ω2 + "33 = 0.

(10.36)

For nonrelativistic temperatures of the background plasma the thermal motion or other random motions of the particles can be neglected. It is well known that, in this case, the kinetic and hydrodynamic (fluid approximation) approaches to plasma give essentially the same result [181]. The dispersion relation for nonrelativistic beams is  2  1 K2 K 2 vι2 2 2 2 2 2 K c −ω + − 2 2 ωι + 2 ωι vι = 0, (10.37) 2 γ ω ω ω ι ι ι ι ι where the summation is performed over all components of background and beams with plasma frequencies given by ωι2 = 4π q2ι nι /mι , where qι are charges, nι are proper number densities, and mι are masses of the components. The same relation holds for weak relativistic beams when the plasma frequency associated with beams is much smaller than the one associated with the background. The growth rate defined by ω = Im δ for unstable modes with |ω2 |  K 2 c2 in the nonrelativistic limit with vb  c is 2 δW ≈

ωb2 K 2 vb2 , K 2 c2 + ωb2 + ωB2

(10.38)

 where ωB2 = j 4π q2j n j /m j and the index j runs for background particles only. Then ωB and ωb stand for the background and beam plasma frequency, respectively. This result reduces to the one obtained by Weibel [191] with ωB = 0. It turns out that for relativistic temperatures the results of kinetic and hydrodynamic approaches are again similar. This is shown first for the waterbag DF [200]. There is a caveat, however, that although the waterbag DF is frequently used in the analysis of instabilities, it is suffering from the neglect of Landau damping [196]. Nevertheless, similar results follow from both kinetic and hydrodynamics approaches, and this conclusion appears to be generic [201, 202]. For this reason the consideration here is limited to the fluid approximation. The dispersion relation is [182] #  $  2 2 ω /c − K 2 v˜ 2 2 2 2 2 2 K c − ω + ωB + ωb 1 − = 0, (10.39) ω2 − K 2C˜ 2 C˜ =



Cb

γb 1 − Cb2 vb2 /c4

,

v˜ =



vb

γ˜b 1 − Cb2 vb2 /c4

,

−1/2  γ˜b = 1 − (Cb/c)2 , (10.40)

146

Wave Dispersion in Relativistic Plasma

where Cb2 = c2 ∂Pb/∂ρb is the sound speed of the beam, Pb is its pressure, and ρb is internal energy density. Equation (10.39) can be rewritten as a biquadratic equation: ω4 − B (K) ω2 + C (K) = 0,



2

  v˜ C˜ 2 2 2 2 2 B (K) = ωB + ωb 1 − 2 + K c 1 + 2 , c c  2  2 2  2  2 2 2 2 C (K) = ωB + K c K C˜ − ωb K v˜ − C˜ 2 .

(10.41) (10.42) (10.43)

The instability occurs for 1 2 1 2 = B− B − 4C < 0, ω− 2 2 provided that C < 0. It defines the maximum unstable wave number   Kmax = ωb2 M2 − 1 − ωB2 M=

v˜ γbvb , = γ˜bCb C˜

(10.44)

(10.45) (10.46)

where M is referred to as effective Mach number. This is one of the key parameters describing Weibel instability. It is instructive to consider the ultra-relativistic case with the beam velocity vb ∼ c and γb  1. In this limit, hydrodynamic and kinetic approximations again coincide. It implies also M  1. Another key parameter is the measure of the strength of the beams: H=

ωb2 . ωB2

(10.47)

The dispersion relation in the ultra-relativistic case is    κ2  2 δ˜4 + 1 + κ 2 δ˜2 − κmax − κ 2 = 0, 2 M

(10.48)

2 where dimensionless quantities δ˜2 = −ω2 /ωB2 and κ = Kc/ωB are used and κmax  2 2 HM − 1. The instability occurs if HM > 1. In Figure 10.2 the dimensionless growth rate δ˜ is shown for M = 100 and H = 1 (solid curve) and for M = 100 and H = 0.01 (dotted curve). It is clear that for small κ  1, the growth rate √ increases ˜ linearly with κ. If κmax  1, the growth rate saturates at κ > 1 at δ  H. Finally, at κ  κmax , the growth rate rapidly decreases to zero. It is interesting to compare the ultra-relativistic and nonrelativistic cases described by eq. (10.48) and (10.38), respectively. One can see that there is no limiting wave number κmax in the nonrelativistic case. The maximum growth rate in the nonrelativistic case is δmax,NR  ωbvb/c, while in the ultra-relativistic case, it is δmax,UR  ωb2 /ωB .

10.7 Two-Stream Instability

147

1

Im d wB

0.1

0.01

0.001 0.01

0.1

1 Kc wB

10

100

Figure 10.2 Dimensionless growth rate versus dimensionless wave number for Weibel instability with ultra-relativistic symmetric beams. The solid curve corresponds to M = 100 and H = 1, while the dotted curve corresponds to M = 100 and H = 0.01.

Analysis of Weibel instability in magnetized plasma [182] leads to the constraint on the magnetic field KB =

|qb| B    Kmax . b mbCb 1 + ρmb +P b nb

(10.49)

This result implies that the magnetic field presence is unimportant for the development of Weibel instability unless the field is so strong that the beam plasma frequency and the Larmor frequency of the beam particles become comparable. 10.7 Two-Stream Instability Electrostatic instability occurring for waves propagating along the beam direction is called the two-stream instability. Hence, in this section, the modes parallel to the beam direction are considered, so the wave four-vector in the Lorentz refer¯ This instability was predicted in the ence frame is chosen: kμ = (ω/c, K, 0, K). nonrelativistic case in [203, 183] and discovered experimentally soon after. Relativistic linear theory of two-stream instability was developed later [204, 205]. For a monochromatic beam, initial instability occurs in the hydrodynamic phase, where thermal spread V is relatively small: K V  δ. The growth of perturbations leads to an increase of the velocity spread and transition to the kinetic phase, where the velocity spread cannot be neglected. The dispersion relation has the form [206] # $  ωb2 ωB2 K 2 vb2 ωb2 ωb2 ωB2 2 2 2 2 1− 2 − − c + ω + − ω = 0. k     B ω γb ¯ b 2 ¯ b 2 ω2 γb3 ω − KV γb ω − KV (10.50)

148

Wave Dispersion in Relativistic Plasma

If the density of the beam is small compared to the density of the background, nb  nB , the largest growth rate occurs at ω ∼ ωB , with √  

3 nb 1/3 K¯ 2 K2 . (10.51) δT S = ωB 4/3 + 2 γbnB γ 2 k2 k2 One can compare these results to the nonrelativistic case (see, e.g., [87]), where the dispersion relation is 2  ω 2  ω B b + =1 (10.52) ¯ b ω ω − KV and the growth rate is ωb  2 , ωB 1 − KV ¯ b

δT S = ± 8

(10.53)

¯ b  ωB . It is clear that the growth of perturbations with K ∼ k leads where KV to departure from the “monochromaticity” condition K V  δ and changes the growth rate. 10.8 Collisionless Shock Waves While most results on plasma instabilities, including the relativistic domain, were obtained long ago, only recently the interest in such instabilities has been revived due to development of inertial confinement fusion at the National Ignition Facility5 or HiPER,6 on one hand, and astrophysical applications such as collisionless shocks in gamma-ray bursts (GRB), on the other [207, 199, 208]. In the GRB context, interaction between two streams moving relativistically with respect to each other is expected [209]. Similar plasma instabilities are important also in experiments with ultra-intense lasers [194]. As one can see from eqs. (10.38) and (10.51), the growth rates of Weibel and two-stream instabilities in the limit of weak beams are  1/2  1/3 nb nb δW ∝ δT S ∝ , (10.54) nB nB respectively. The typical wavelengths for both instabilities in the relativistic case are similar: c vb λW  λT S  . (10.55) ωB ωB 5 6

https://lasers.llnl.gov/ www.hiper-laser.org/

10.8 Collisionless Shock Waves

149

Figure 10.3 Image of a region containing collisionless shock from the 2D PIC simulation. In this simulation, relativistic isotropic plasma with the bulk Lorentz factor γ = 15 is sent against the wall (on the left), is reflected, and creates a SW, which generates a turbulent magnetic field. The field decays downstream from the shock. Density of plasma n and averaged magnetization εB (the fraction of magnetic energy density in terms of particle energy density) are shown. ©AAS 2008. Reproduced from [213] with kind permission of the authors.

There are different types of instabilities relevant in the relativistic regime. It is remarkable that, currently, numerical experiments based on particle-in-cell methods (see Section 9.4) in two and three dimensions (see, e.g., [210, 211, 212]) allow not only study of the development of instabilities at their linear stage but also the ability to follow them on much longer timescales, where saturation occurs and complex electromagnetic field patterns emerge. Relativistic shocks occur on the length scale comparable to the plasma skin depth c/ω p. To study the long-term and large-scale evolution of instabilities and their role in magnetic field generation, the dimensionality of the problem should be reduced. With this goal, 2D simulations are performed [213]. An example of such a simulation is in Figure 10.3. Although instabilities generate turbulent near-equipartition magnetic fields near the shock with dimensionless magnetic energy density

εB ∼ 0.1, the field decays downstream from the shock. The linear response theory

150

Wave Dispersion in Relativistic Plasma

applied to this case gives the dispersion relation for normal modes in the plasma with electromagnetic field as follows [213]: k 2 c2 − (1 + 4π χ ) = 0, ω2 where the plasma susceptibility is   ω2p,ι  ∂ ∂ f0s 3 kvx vx + d p. 4π χ = 2 ω ∂ px ω − kvy ∂ py ns ι

(10.56)

(10.57)

Here index ι denotes electrons or positrons, f0s stands for isotropic DFs, and nι are densities; the electric field perturbation is aligned with the x-axis, the wave vector lies along the y-axis, and magnetic field perturbation is along the z-axis. As discussed, the subluminal waves are damped. Evaluation of the integral (10.57) for the neutral electron-positron plasma with relativistic Maxwellian DFs (4.43), which fit well the DFs in numerical simulations, gives [213] 4π χ  i

ω2p |k| cω

,

(10.58)

while the decay rate is δ  −|kc|3 /ω2p. The simulations confirm these results, with the magnetic field energy decreasing inversely proportional to the distance from the shock, εB ∝ x−1 , in the downstream. This result puts strong constraints on the models of GRBs involving synchrotron emission from relativistic shocks, as the magnetic fields do not remain amplified in the macroscopic volume in the downstream.

11 Thermalization in Relativistic Plasma

In this chapter, relaxation of nonequilibrium optically thick relativistic plasma is discussed. First, basic parameters describing relativistic plasma, such as the plasma parameter, the classicality parameter, the Coulomb logarithm, and the optical depth, are introduced. Then collision integrals as integrals over matrix elements describing various two-particle interactions between photons, electrons, positrons, and protons are given. Collision integrals for three-particle interactions are also discussed. The theory of thermalization, including the concepts of kinetic and thermal equilibria, is presented. Then thermalization timescales as functions of the total energy density and baryonic loading are reported. At the end of the chapter, a kinetic description for a mildly relativistic plasma ball is given, including its dynamics and emission properties. 11.1 Pair Plasma in Astrophysics and Cosmology Relativistic plasma is understood as plasma in which the mean particle kinetic energy is near to or higher than the characteristic value given by the electron rest mass energy:1

  mc2 = 0.511 MeV,

(11.1)

where m is electron mass. This kinetic energy can be of two kinds: collective motion with respect to some given reference frame, e.g., a beam of relativistic electrons in the laboratory, or random motion of particles. In the former case one may consider the reference frame co-moving with the beam and find no relativistic motion in this frame. For this case to be truly relativistic, interactions of the streaming plasma either with a target at rest or between streams having relativistic relative motion are required. In the latter case, particle relative velocities are relativistic in any 1

In the following two chapters, m stands for electron mass, while M denotes proton mass.

151

152

Thermalization in Relativistic Plasma

reference frame. Particle interactions result in a number of physical phenomena that are absent in nonrelativistic plasmas. In particular, at such energies, positrons are present in plasma, and they may even outnumber ions. An electron-positron plasma is of interest in many fields of physics and astrophysics [10]. In the early universe [214, 215, 216, 217] during the lepton era, ultrarelativistic electron-positron pairs contributed to the matter contents of the universe. In GRBs, electron-positron pairs play an essential role in the dynamics of expansion [218, 219, 220]. Indications exist that the electron-positron plasma is present also in active galactic nuclei (AGN) [221], in the center of our galaxy [222], around hypothetical quark stars [223]. In the laboratory, pair plasma is expected to appear in the fields of ultra-intense lasers [224] (see [225] for review), where particle production may serve as a diagnostic tool for high-energy plasma [226]. From the theoretical point of view, particularly interesting are pure electronpositron (pair) plasmas, when ions are absent. The mass symmetry between the plasma components results in the absence of both acoustic modes and Faraday rotation; waves and instabilities in such plasma differ significantly from asymmetric electron-ion plasma (see, e.g., [227] and the previous chapter). Besides, theoretical progress in understanding quark-gluon plasma in the high-temperature limit is linked to understanding QED plasma since the results in these two cases differ only by trivial factors containing the QCD degrees of freedom (color and flavor) [228]. Impressive progress made with ultra-intense lasers [229] has led to the creation of positrons at an unprecedented density of 1016 cm−3 using ultra-intense short laser pulses in a region of space with dimensions on the order of the Debye length. However, such densities have not yet reached those necessary for the creation of an optically thick pair plasma [230, 231]. Particle pairs are created at the focal point of ultra-intense lasers via the Bethe-Heitler conversion of hard X-ray bremsstrahlung photons [232] in the collisionless regime [233]. The approach to an optically thick phase may well be envisaged in the near future. In many stationary astrophysical sources, the pair plasma is assumed to be in thermodynamic equilibrium. A detailed study of the relevant processes [234, 235, 236, 237, 238, 239], radiation mechanisms [240], possible equilibrium configurations [236, 241, 242], and spectra [243] in an optically thin pair plasma has been carried out. Particular attention has been given to collisional relaxation processes [244, 245], pair production and annihilation [246], relativistic bremsstrahlung [247, 248], and double Compton scattering [249, 250]. As discussed in chapter 4, an equilibrium occurs when the sum of all reaction rates vanishes. For instance, electron-positron pairs are in equilibrium when the net pair production (annihilation) rate is zero. This can be achieved in a variety of ways, and the corresponding condition can be represented as a system of algebraic equations [251]. However, the main assumption made in all the mentioned works is that electrons, positrons, protons, and photons obey, respectively, Fermi-Dirac or

11.2 Qualitative Description of the Pair Plasma

153

Bose-Einstein distributions. The latter is shown to be possible, in principle, in the range of temperatures up to 10 MeV [234, 245]. The main goal in this chapter is to show how, independently of a wide set of initial conditions, thermal equilibrium forms for the phase space DFs are recovered during the process of thermalization by two-particle and three-particle direct and inverse interactions in optically thick plasmas. In many cases mentioned earlier, the electron-positron plasma can be optically thick. Although moderately thick plasmas have been considered in the literature [242], until recently, only qualitative description [234, 241] was available for large optical depths. In particular, optically thick plasma is expected in the source of GRBs. In fact, detailed study of the pair plasma equilibrium configurations, as performed in [241], is of little use here, because, essentially, nonequilibrium processes have to be considered. In addition, particles may not reach an equilibrium in other rapidly evolving systems, such as the early universe or transient events, when the energy is released on a very short timescale compared to a dynamical timescale in the plasma. Notice that there are additional substantial differences between the ion-electron plasma, on one hand, and electron-positron plasma, on the other. First, the former is collisionless in the wide range of parameters [252], while collisions are often essential in the latter. Second, when collisions are important, a relevant interaction in the former case are Coulomb scattering of particles, which is usually described by the classical Rutherford cross section. In contrast, interactions in the pair plasma are described by quantum cross sections even if the plasma itself can still be treated as classical. Motivated by the study of initial states of the pair plasma in GRB sources, analysis of thermalization in relativistic plasma has been performed in [253, 254] in the case of pure pair plasma. Then, in [255], details about the computational scheme adopted in [253] were presented, and a more general case, the pair plasma loaded with baryons, was considered. Timescales of thermalization in optically thick electron-positron plasma with proton loading were systematically explored in [256]. Also, dynamical evolution and emission of pair plasma initially confined to a spherical region were studied in [257]. This chapter reviews all these studies. 11.2 Qualitative Description of the Pair Plasma First of all, it is important to specify the domain of parameters characterizing the pair plasma considered in this chapter. It is convenient to use dimensionless parameters usually adopted for this purpose. Consider mildly relativistic pair plasma, in which the average energy per particle  brackets the electron rest mass energy:  0.1   10. (11.2) mc2

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Thermalization in Relativistic Plasma

The lower boundary is required for significant concentrations of pairs, while the upper boundary is set to avoid substantial production of other particles, such as muons and neutrinos. The plasma parameter g p = (n− λ3D )−1 is given by eq. (6.21), where n− is electron number density and λD is the Debye length. To ensure applicability of the kinetic approach, it is necessary that the plasma parameter be small, g p  1. This condition implies that kinetic energy of particles dominates their potential energy due to mutual interaction. In Figure 6.1 the plasma parameter as a function of electron temperature θ− for pair plasma in thermal equilibrium is shown. It is clear that the condition g p  1 is always satisfied in the temperature range under consideration, hence it is sufficient to consider only one-particle DFs. Furthermore, the classicality parameter, defined as $ = q2 /(¯hvrel ) = α/βrel , where vrel = βrel c, is the relative velocity of colliding particles. The condition $  1 means that particle collisions can be considered classically, while for $  1, quantum description is required. In the case under consideration, both for pairs and protons, quantum cross sections are used, since $ < 1. The strength of screening of the Coulomb interactions is characterized by the ˜ rel vrel d/¯h  kT d/(¯hc), where M ˜ is the reduced mass. Coulomb logarithm  = Mγ For electron-electron or electron-positron scattering the reduced mass is approximately equal to m/2, while for electron-proton or positron-proton scattering the ˜  M; for proton-proton scattering, M ˜  reduced mass is just the proton mass M M/2. Coulomb logarithm varies with mean particle velocity and Debye length, and it cannot be set a constant, as is usually done in most studies of the pair plasma (see discussion in Chapter 6). Finally, it is assumed that the pair plasma has linear dimension R, which greatly exceeds the mean free path of photons l = (n− σ )−1 , where σ is the corresponding total cross section. Thus the optical depth τ  1 is large, and interactions between photons and other particles have to be taken in due account. These interactions are discussed in the next section. Note that natural parameters for perturbative expansion in the problem under consideration are the fine structure constant α and the electron-proton mass ratio m/M. 11.3 Collision Integrals For simplicity, first consider pure pair plasma composed of electrons e− , positrons e+ , and photons γ . A more general case, including protons p, is considered subsequently. Assume that pairs of photons appear by some physical process in the region with a size R. Further assume that DFs of particles depend neither on spatial

11.3 Collision Integrals

155

coordinates nor on the direction of momenta. Then, DFs are functions of energy and time only, fi = fi (, t ); they describe isotropic in the momentum space and spatially uniform plasma. One may verify whether the classical kinetic description is adequate for description of relativistic plasma. Recall the definition of the degeneracy parameter defined in eq. (6.32) in Section 6.6. As can be seen from Figure 6.2, thermal electronpositron plasma becomes degenerate at kT  3mc2 . It should be noted, however, that the average occupation numbers for the kinetic equilibrium state are not high even in the ultra-relativistic limit with chemical potential μ = 0: 8.7 percent for electrons and positrons and 36.8 percent for photons. Therefore, as a first approximation, one can still apply the classical kinetic approach and neglect quantum corrections. The conditions discussed here justify the computational approach based on classical relativistic Boltzmann equation (2.9). At the same time the RHS of the Boltzmann equation contains collision integrals as functions of quantum matrix elements, as discussed later. For homogeneous and isotropic DFs of electrons, positrons, and photons, eq. (2.9) written in the form of (2.41) reduces to  1 ∂ fι  q = ηι − χιq fι , c ∂t q

(11.3)

which is a coupled system of integro-differential equations. In eq. (11.3) also the Vlasov term is explicitly neglected, since energy densities of fluctuations of the electromagnetic field are many orders of magnitude smaller than the energy densities of particles [258]. Therefore, the LHS of the Boltzmann equation is reduced to a partial derivative of the distribution function with respect to time. The RHS of the Boltzmann equation contains collision integrals with matrix elements describing various interactions in the pair plasma. For all binary interactions, exact QED matrix elements are used. These matrix elements can be found in the standard textbooks on QED (e.g., in [48, 259, 260]), and are given in Appendix B in Sections B.1 and B.2. Before describing the calculation of these integrals, recall the definition of the matrix element and its relation to the differential cross section.

11.3.1 The Definition of Matrix Elements Following [48], define the scattering matrix, being composed of real and imaginary parts   S f i = δ f i + i (2π h¯ )4 δ (4) p f − pi T f i , (11.4)

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Thermalization in Relativistic Plasma

where δ f i is the unity matrix, δ (4) stands for the four-momentum conservation, the elements of T f i are scattering amplitudes, and p≡pμ . The transition probability of a given process per unit time is then w f i = c (2π h¯ )4 δ (4) (p f − pi )|T f i |2V,

(11.5)

where V is the normalization volume. For a process involving a outgoing particles and b incoming particles the differential probability per unit time is defined as # $ $#  h¯ c  dp h¯ c a dw = c(2π h¯ )4 δ (4) (p f − pi )|M f i |2V , (11.6) 2bV (2π h¯ )3 2a a b where p a and a are, respectively, momenta and energies of outgoing particles; b are energies of particles before interaction; M f i are the corresponding matrix elements; and dp ≡ d 3 p. The matrix elements are related to the scattering amplitudes by  $−1/2 #  h¯ c  h¯ c Mfi = Tf i . (11.7) 2bV 2a V a b For a binary process, e.g., scattering (eq. (2.10)) with two incoming and two outgoing particles, it is convenient to introduce the differential cross section. In fact, the differential probability for incoming particles with four momenta p1 and p2 , energies 1 and 2 , and masses m1 and m2 , respectively, is just the product of the differential cross section and the flux density (see eq. (2.44)). In the CM reference frame the relation between the cross section and |M f i |2 acquires its simplest form if the cross section does not depend on the azimuth of p 1 relative to p1 ; then dt h¯ 2 c4 |M f i |2 , 64π I 2 t = (p1 − p2 ) ,

(11.9)

2|p1 ||p 1 |d

(11.10)

dσ =

dt =

cos ϑ,

(11.8)

where ϑ is the angle between p1 and p 1 . 11.3.2 Example of Collision Integral for Binary Interactions The form of collision integrals is illustrated for an absorption coefficient in Compton scattering of photons:  χ cs fγ = dk dpdp Wk ,p ;k,p fγ (k, t ) f± (p, t ), (11.11)

11.3 Collision Integrals

157

where p and k are momenta of electron (positron) and photon, respectively, dp = d± do±2 β± /c3 , dk = dγ γ 2 do γ /c3 , and the transition function Wk ,p ;k,p is related to the transition probability differential dwk ,p ;k,p per unit time as Wk ,p ;k,p dk dp ≡ V dwk ,p ;k,p .

(11.12)

The differential probability dwk ,p ;k,p = wk ,p ;k,p dk dp is given by eq. (B.3). Given the momentum conservation, one can perform one integration over dp in (11.11) as  dp δ(k + p − k − p ) → 1, (11.13) but it is necessary to take into account the momentum conservation in the next integration over dk , so one has    (11.14) dγ δ γ + ± − γ − ±      1 6 δ γ + ± − γ − ± = d γ + ±  ∂  + ± ∂ γ

γ

1 6 ≡ Jcs , →  ∂  + ± ∂ γ γ where the Jacobi matrix of the transformation is Jcs =

1

1−

, β± b γ ·b ±

and bi = pi /p, b i = p i /p , b ± = (β± ± b± + γ bγ − γ b γ )/(β± ± ). Finally, for the absorption coefficient, one has  γ |M f i |2h¯ 2 c2 cs χ fγ = − doγ dp Jcs fγ (k, t ) f± (p, t ), 16± γ ±

(11.15)

(11.16)

where the matrix element here is dimensionless. This integral is evaluated numerically as described in Section B.1.

11.3.3 Cutoff in the Coulomb Scattering It is customary in the literature (see, e.g., [261]) to use the Fokker-Planck approximation for Coulomb collisions in the pair plasma. Following [262], consider the Coulomb logarithm in relativistic plasma. As discussed in Section 6.3, the mean free path due to Compton scattering lC = nσ1T and the one due to Coulomb scattering lC = 1/(log()σT n) become equal in the ultra-relativistic case. It implies that

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Thermalization in Relativistic Plasma

the Fokker-Planck approximation does not hold for Coulomb collisions in relativistic plasma. To take into account Debye screening, the cutoff of the cross section for small angles in Coulomb collisions is adopted. This allows one to apply the same scheme for the computation of emission and absorption coefficients for Coulomb scattering. Haug [263] gives the minimal scattering angle in the CM system as θmin =

γr 2¯h , √ ˜ Mcb (γr + 1) 2(γr − 1)

(11.17)

˜ as before, is the reduced mass, γrel is the Lorentz factor of relative where M, motion, and b is the maximum impact parameter (neglecting the effect of protons), given by b=

c2 p0 , ω 10

(11.18)

where the quantities in the CM reference frame are denoted with index 0. The invariant Lorentz factor of relative motion (e.g., [263]) is γrel = 

1 1 2 − p1 p2 c2 = .  2 m1 m2 c4 1 − vcrel

(11.19)

Then, one finally obtains in the CM frame

  2   10 2 tmin = 2 (mc)2 − 1 − β10 cos θmin . c Since this quantity isinvariant, one can replace t in the denominator of |M f i |2 in 2 /t 2 to implement the cutoff scheme; for details, see (B.32) by the value t 1 + tmin [255].

11.3.4 The Role of Triple Interactions For such an optically thick plasma, collision integrals in (11.3) should include not only binary interactions, having order α 2 in Feynman diagrams, but also triple ones, having order α 3 [48]. As an example for triple interactions, consider relativistic bremsstrahlung e1 + e2 ↔ e 1 + e 2 + γ .

(11.20)

11.3 Collision Integrals

159

For the time derivative, for instance, of the distribution function f2 in the direct and in the inverse reactions (11.20), one has    f˙2 = dp1 dp 1 dp 2 dk Wp 1 ,p 2 ,k ;p1 ,p2 f1 f2 fk − Wp1 ,p2 ;p 1 ,p 2 ,k f1 f2 (11.21)

 6 3 1 ¯ δ (4) (Pf − Pi )|M f i |2 c h f1 f2 fk − = dp1 dp1 dp2 dk f1 f2 , (2π )2 25 1 2 1 2 γ (2π h¯ )3 where dp1 dp2Wp 1 ,p 2 ,k ;p1 ,p2 ≡ V 2 dw1

(11.22)

dp 1 dp 2 dk Wp1 ,p2 ;p 1 ,p 2 ,k ≡ V dw2 ,

(11.23)

and where dw1 and dw2 are given by eq. (11.6) for the inverse and direct process (11.20), respectively. The matrix element here has dimensions of the length squared (see Section 11.3.1). The analysis performed in [253, 255] shows that thermalization in relativistic plasma occurs in two stages. First, binary interactions such as Compton scattering reach a detailed balance. At this stage the distribution function of particles already takes the form   2 ε − νι fι (ε) = , (11.24) exp − (2π h¯ )3 θι ϕι kB Tι  with chemical potential νι ≡ mc 2 and temperature θι ≡ mc2 , where ε ≡ mc2 is the energy of the particle. It is remarkable that in this quasi-stationary state referred to as kinetic equilibrium [215, 261, 29], all particles (except for protons; see later) have the same temperature and nonzero chemical potentials. Given the fact that triple interactions have rates smaller than the binary ones by a factor α, it is expected that triple interactions will not be important for reaching kinetic equilibrium. Once the distribution functions acquire the form (11.24), each collision integral contains a multiplier proportional to

Fι = exp

νι , θι

(11.25)

called fugacity. The calculation of emission and absorption coefficients can then be performed using the well-established thermal equilibrium rates [251], multiplied by the corresponding fugacities. Although there is no conceptual difficulty, in principle, in computations using exact matrix elements for triple reactions as well, this simplified scheme allows for much faster numerical computation. The corresponding reaction rates for triple interactions are given in Section B.3.

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Table 11.1 Microphysical processes in the pair plasma Binary interactions

Radiative and pair-producing variants

Møller and Bhabha ± ± ± e± 1 e2 −→ e1 e2 e± e∓ −→ e± e∓

Bremsstrahlung ± ± ± e± 1 e2 ↔ e1 e2 γ ± ∓ e e ↔ e± e∓ γ

Single Compton e± γ −→ e± γ

Double Compton e± γ ↔ e± γ γ

Pair production and annihilation γ γ ↔ e± e∓

Radiative pair production and three-photon annihilation γ γ ↔ e± e∓ γ e± e∓ ↔ γ γ γ e± γ ↔ e± e∓ e±

Indeed it is found that triple interactions become essential after kinetic equilibrium is already established [253, 255]. Thermal equilibrium is then reached by three-particle interactions, since chemical potential cannot be changed in binary collisions. Strictly speaking, the sufficient condition for reaching thermal equilibrium is when all direct reactions are exactly balanced with their inverse ones. Therefore, in principle, not only triple but also four-particle, five-particle, and so on, reactions have to be accounted for in eq. (11.3). The timescale for reaching thermal equilibrium will then be determined by the slowest reaction that is not balanced with its inverse. Notwithstanding, the necessary condition is the detailed balance at least in triple interactions, since binary reactions do not change chemical potentials. It is worth mentioning the physical meaning of the chemical potential νk in kinetic equilibrium entering the formula (11.24). In the case of pure pair plasma a nonzero chemical potential represents deviation from the thermal equilibrium through the relation νk = θ ln(nk /nth ),

(11.26)

where nth are concentrations of particles in thermal equilibrium. Consider all possible binary and triple interactions between electrons, positrons, and photons, as summarized in Table 11.1. Each of these reactions is characterized by the corresponding timescale and optical depth. For Compton scattering of an electron, for instance, one has tcs =

1 σT n± c

τcs = σT n± R,

(11.27)

11.3 Collision Integrals

161

h¯ 2 where σT = 8π α 2 ( mc ) is the Thomson cross section. There are two timescales in 3 the problem that characterize the condition of detailed balance between direct and inverse reactions: tcs for binary and α −1tcs for triple interactions, respectively. To explore the thermalization process, arbitrary initial distribution functions can be chosen and the system can be evolved numerically until thermal equilibrium is established. Notice that a method similar to the one described earlier was applied in [261] to compute spectra of particles in kinetic equilibrium. Although the approach was similar, the computation was never carried out in order to actually observe the reaching of thermal equilibrium. This is because MC simulations cannot be extended for long timescales, compared to the shortest timescale of the problem, unlike finite difference implicit schemes (see Chapter 8).

11.3.5 Energy Loss for Neutrino-Antineutrino Production An electron-positron pair can also annihilate into a neutrino channel with the main contribution from the reaction e± e∓ −→ν ν¯ (see, e.g., [10]). By this process, the energy could leak out from the plasma if it is transparent for neutrinos. The optical depth and energy loss for this process can be estimated following [264] by using Fermi theory; see also [265, 266] for calculations within electro-weak theory. The optical depth is given by eq. (11.27) with the cross section   g2 h¯ 2 σν ν¯ ∼ , (11.28) π mc where g  10−12 is the weak interaction coupling constant and it is assumed that typical energies of electron and positron are ∼mc2 and their relative velocities are v ∼ c. Numerically, one has σν ν¯ /σT = 8π3 2 (g/α)2  7 × 10−22 . For astrophysical sources the plasma may be both transparent and opaque to neutrino production. The energy loss when pairs are relativistic and nondegenerate is  3  mc2  dρ 128g2 9 2 mc = . (11.29) ζ (5)ζ (4)θ mc dt π5 h¯ h¯ The ratio between the energy loss due to neutrinos and the energy of photons in thermal equilibrium is then  2 1 dρ 128g2 t 5 mc t = t  3.6 × 10−3 θ 5 . (11.30) ζ (5)ζ (4)θ 3 ργ dt π h¯ 1 sec For astrophysical sources with the dynamical time t ∼ 10−3 s, such as GRBs, the energy loss due to neutrinos becomes relevant [267] for high temperatures θ > 10. However, on the timescale of relaxation to thermal equilibrium t ∼ 10−12 s [253, 255], the energy loss is negligible.

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Thermalization in Relativistic Plasma

11.3.6 Proton Loading So far only plasma consisting of leptons, having the same mass but opposite charges, has been addressed. In that case the condition of electric neutrality is identically fulfilled. Electrons and positrons may be described by the same DF, and this fact allows for significant gain in the computational time. The situation becomes more complicated when an admixture of protons is allowed. Since charge neutrality n− = n+ + n p

(11.31)

is required, the number of electrons is not equal to the number of protons. In such a case, a new dimensionless parameter, the baryonic loading B, can be introduced as B=

n pMc2 NMc2 = , E ρr

(11.32)

where N and n p are the number and the concentration of protons, E, and ρr = ργ + ρ+ + ρ− are radiative energy and energy density, respectively. Since, in relativistic plasma, electrons and positrons move with almost the speed of light, both photons and pairs in thermal equilibrium behave as relativistic fluid with equation of state Pr  ρr /3 (see, e.g., [268]). At the same time, protons are relatively cold particles in the energy range (11.2), with negligible pressure and dustlike equation of state P  0. In this way, by introducing parameter B, one distinguishes a radiation-dominated (B < 1) from a matter-dominated (B > 1) plasma. For electrically neutral plasmas, there exists an upper limit on the parameter B defined by (11.32), which is B ≤ M/m. In the range of energies (11.2) the radiative energy density can be approximated m as ρr ∼ n− mc2 , and then one has for concentrations n p ∼ n− B M . If protons and electrons are at the same temperature, then from the equality of the kinetic energy m Mv 2 v of a proton k,p = 2 p and the one of an electron k,− ∼ mc2 , one has cp ∼ M ; therefore protons are indeed nonrelativistic. In the presence of protons, additional binary reactions consist of Coulomb collisions between electrons (positrons) and protons, scattering of protons on protons, and Compton scattering of protons. Additional triple reactions are radiative variants of these reactions (see Table 11.2 and Section B.2). Protons can be thermalized first by proton-proton collisions and then by electron/positron-proton collisions, or alternatively just by the latter mechanism, depending on corresponding timescales. The rate of proton-proton collisions  the m np m 3/2 is a factor M ∼ B( M ) smaller than the rate of electron-electron collin− sions (see eq. (B.58)). The rate of proton-electron/positron collisions is a factor  m ∼M smaller than the one of electron-electron collisions (see eq. (B.54)). Mc2

11.3 Collision Integrals

163

Table 11.2 Microphysical processes in the pair plasma involving protons Binary interactions

Radiative and pair-producing variants

Coulomb scattering p1 p2 −→ p 1 p 2 pe± −→ p e±

Bremsstrahlung p1 p2 ↔ p 1 p 2 γ pe± ↔ p e± γ

± ± ∓ pe± 1 ↔ p e1 e e

Single Compton pγ −→ p γ

Double Compton pγ ↔ p γ γ pγ ↔ p e± e∓

m m Therefore, for B > M , proton-proton collisions are faster, while for B < M , proton-electron/positron ones predominate. 11.3.7 Conservation Laws A number of conservation laws are present in the problem. On one hand, these conservation laws must be ensured in the numerical calculation. On the other hand, they can be used for determination of such important quantities as temperature and chemical potentials of particles in kinetic equilibrium. Conservation laws consist of baryon number, charge, and energy conservations. In addition, in binary reactions, particle number is conserved. Energy and particle number conservation laws imply d ρι = 0 (11.33) dt ι d nι = 0. (11.34) dt ι Since baryonic number is conserved, the number density of protons is a constant. 11.3.8 Determination of Temperature and Chemical Potentials in Kinetic Equilibrium Consider DFs for photons and pairs in the most general form (11.24). If one supposes that the reaction rate for the Bhabha scattering vanishes, i.e., there is equilibrium with respect to reaction e+ + e− ↔ +e+ + e− ,

(11.35)

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Thermalization in Relativistic Plasma

then the corresponding condition can be written in the following way: f+ (1 − f+ ) f− (1 − f− ) = f+ (1 − f+ ) f− (1 + f− ),

(11.36)

where Bose enhancement along with Pauli blocking factors are included for generality. It can be shown that electrons and positrons have the same temperature, θ+ = θ− ≡ θ± ,

(11.37)

and that they have arbitrary chemical potentials. With (11.37), analogous consideration for the Compton scattering e± + γ ↔ e± + γ

(11.38)

f± (1 − f± ) fγ (1 + fγ ) = f± (1 − f± ) fγ (1 + fγ )

(11.39)

gives

and leads to equality of temperatures of pairs and photons, θ± = θ γ ≡ θ k ,

(11.40)

with arbitrary chemical potentials. If, in addition, reaction rate in the pair-creation and annihilation process e± + e ∓ ↔ γ + γ

(11.41)

vanishes too, i.e., there is equilibrium with respect to pair production and annihilation, with the corresponding condition, f+ f− (1 + fγ )(1 + fγ ) = fγ fγ (1 − f+ )(1 − f− ),

(11.42)

it turns out that also chemical potentials of pairs and photons satisfy the following condition: ν+ + ν− = 2νγ .

(11.43)

However, since, generally speaking, νγ = 0, the condition (11.43) does not imply ν+ = ν− . These considerations are known as detailed balance conditions (see, e.g., [20, 29]). Analogous consideration of the detailed balance conditions in different reactions leads to the relations between temperatures and chemical potentials summarized in Table 11.3. The timescales of pair production and annihilation processes as well as Compton scattering are nearly equal in the range of energies of interest and are given by eq. (11.27). Therefore kinetic equilibrium is first established simultaneously for electrons, positrons, and photons. They reach the same temperature, but with chemical potentials different from zero. Later on, the temperatures of this electronpositron plasma and the one of protons reach a common value.

11.3 Collision Integrals

165

To find temperatures and chemical potentials, one has to implement the following constraints: energy conservation (11.33), particle number conservation (11.34), charge conservation (11.31), and condition for the chemical potentials (11.43). Given eq. (11.24), one finds for photons   ργ νγ 1 2θγ3 ; = 3θγ nγ = exp (11.44) 2 nγ mc V0 θγ for pairs ρ± = j2 (θ± ), n± mc2

  1 ν± j1 (θ± ); n± = exp V0 θ±

(11.45)

and for protons ρp 3m θp =1+ Mn pc2 2M  8   νp − 1 π M 3/2 exp np = V0 2 m θp

(11.46) M m

 3

θ p2 ,

(11.47)

where it is assumed that protons are nonrelativistic, the Compton volume is defined as  3 h¯ 2 V0 = π , (11.48) mc and functions j1 and j2 are defined as   π − 1 3/2 e θ θ , θ → 0, −1 2 j1 (θ ) = θ K2 (θ ) → 2θ 3 , θ → ∞,  −1 −1 3K3 (θ ) + K1 (θ ) 1 + 3θ2 , θ → 0, j2 (θ ) = → 3θ , θ → ∞, 4K2 (θ −1 )

(11.49) (11.50)

For pure electron-positron plasma in kinetic equilibrium, summing up energy densities in eqs. (11.44) and (11.45) and using eqs. (11.37), (11.40), and (11.43), one obtains    2mc2 νk  4 3θ + j1 (θk ) j2 (θk ) , ρι = exp (11.51) V0 θk ι and analogously for number densities, one gets    2 νk  3 θk + j1 (θk ) . nι = exp V0 θk ι From eqs. (11.51) and (11.52), two unknowns, νk and θk , can be found.

(11.52)

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Thermalization in Relativistic Plasma

When protons are present, in most cases the electron-positron plasma reaches kinetic equilibrium first, while protons thermalize with the plasma later. In that case, the temperature of protons θ p is different from the rest of particles, so while θ+ = θ− = θγ = θk , θ p = θk . Then, summing up energy densities in eqs. (11.44) and (11.45), one obtains   1    n pV0 ν+ 2 4 mc2 ν+ (11.53) exp − ρι = 6θk exp 1− V0 j1 (θk ) θk θk ι

  ν+ + 2 j1 (θk ) exp − n pV0 j2 (θk ) , θk and analogously for number densities, one gets   1     < n pV0 ν+ 2 4 1 ν+ ν+ + 2 j1 (θk ) exp . exp − 1− nι = 6θk exp V0 j1 (θk ) θk θk θk ι (11.54) From eqs. (11.53) and (11.54), two unknowns, ν+ and θk , can be found. Then, the rest of the chemical potentials are obtained from     n pV0 ν− ν+ exp (11.55) = exp + θk θk j1 (θk )       1 n pV0 νγ ν+ 2 ν+ exp exp − = exp 1+ . (11.56) θk θk j1 (θk ) θk The temperature and chemical potentials of protons can be found separately from eqs. (11.46) and (11.47). In thermal equilibrium, νγ vanishes, and one has

n pV0 ν− = θk arcsinh ν+ = −ν− , (11.57) 2 j1 (θk ) which both reduce to ν− = ν+ = 0 for n p = 0. At the same time, for n p > 0, one always has ν− > 0 and ν+ < 0 in thermal equilibrium. The chemical potential of protons in thermal equilibrium is determined from eq. (11.47) for θk = θth , where θth is the temperature in thermal equilibrium. 11.4 Relativistic Boltzmann Equation on the Grid To solve equations (11.3), a finite difference method described in Chapter 8 is used. The goal is to construct the scheme implementing energy, baryon number, and electric charge conservation laws (see Section 11.3.7). For this reason the spectral energy densities defined by eqs. (8.2) and (8.3) are employed instead of DFs.

11.5 Thermalization Process

One can rewrite Boltzmann equations (11.3) in the form  1 ∂Eι  q = η˜ι − χιq Eι , c ∂t q

167

(11.58)

where η˜ιq = (4π ι3 βι /c3 )ηιq . As discussed in Chapter 8, the method used is conservative and the exact energy conservation law is satisfied, so it implies that eqs. (11.33) and (11.34) can be rewritten as d d Yι,ω Yι,ω = 0 = 0, (11.59) dt ι,ω dt ι,ω ι,ω where

 Yι,ω =

ι,ω + ι,ω /2 ι,ω − ι,ω /2

Eι d.

(11.60)

For binary interactions on the grid the particle number conservation law is satisfied as interpolation of grid functions Eι,ω inside the energy intervals is adopted. In Appendix B the discretization of collision integrals and the corresponding formulas for matrix elements are given. 11.5 Thermalization Process In this section, several examples of thermalization processes with different initial conditions are presented. It is shown that, independently of initial conditions, interactions between particles lead eventually to complete relaxation, and the state of thermal equilibrium is obtained. 11.5.1 Pure Electron-Positron Plasma The results of numerical simulations for pure electron-positron plasma reported in [253] are described below. Two initial conditions with flat spectra Ei (i ) = const are chosen: (1) electron-positron pairs with a 10−5 energy fraction of photons and (2) the reverse case, i.e., photons with a 10−5 energy fraction of pairs. The numerical grid in the phase space consists of 60 energy intervals and 16 × 32 intervals for two angles characterizing the direction of the particle momenta. In both cases the total energy density is ρ = 1024 erg/cm3 . In the first case, initial concentration of pairs is 3.1 × 1029 cm−3 ; in the second case the concentration of photons is 7.2 × 1029 cm−3 . In Figure 11.1, concentrations of photons and pairs as well as their sums for both initial conditions are shown. After calculations begin, concentrations and energy

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Thermalization in Relativistic Plasma

Figure 11.1 Dependence on time of concentrations of pairs (dashed), photons (solid), and both (dotted). Upper (lower) figure corresponds to the case when initially there are mainly pairs (photons). Reproduced from [253].

density of photons (pairs) increase rapidly with time, due to annihilation (creation) of pairs by the reaction γ γ ↔ e± e∓ . Then, in the kinetic equilibrium phase, concentrations of each component stay almost constant, and the sum of concentrations of photons and pairs remains unchanged. Finally, both individual components and their sum reach stationary values. If one compares and contrasts both cases as reproduced in Figure 11.1, one can see that, although the initial conditions are drastically different, in both cases the same asymptotic values of the concentration are reached. First let us discuss in detail the case when initially pairs dominate. One can see in Figure 11.2 that the spectral density of photons and pairs [269] dρι 4π = 3 f (, t ) 3 βι , dε c

(11.61)

 where β± = 1 − (mc2 /)2 for pairs and βγ = 1 for photons, can be fitted already at tk ≈ 20tcs  7 × 10−15 s by DFs (11.24) with definite values of temperature θk (tcs ) ≈ 1.2 and chemical potential ϕk (tk ) ≈ −4.5, common for pairs and photons. As expected, after tk , the DFs preserve their form (11.24) with the values of temperature and chemical potential changing in time, as shown in Figure 11.3. As one can see from Figure 11.3, the chemical potential evolves with time and reaches zero at the moment tth ≈ α −1tk  7 × 10−13 s, corresponding to the final stationary solution.

11.5 Thermalization Process

169

Figure 11.2 Spectra of pairs (upper figure) and photons (lower figure) when initially only pairs are present. The black curve represents the results of numerical calculations obtained successively at t = 0, t = tk , and t = tth (see the text). Both spectra of photons and pairs are initially taken to be flat. The gray thick curves indicate the spectra obtained from eq. (11.24) at t = tk , and the final spectra at t = tth , as thermal equilibrium is reached. The perfect fit of the two curves is most evident in the entire energy range. Reproduced from [253].

Consider the DFs (11.24) with different temperatures θi and chemical potentials ϕi for pairs and photons. The requirement of vanishing reaction rate for the Compton scattering f± fγ = f± fγ leads to the equal temperature of pairs and photons θ± = θγ ≡ θk (see also [261, 29]). In this way the detailed balance between any direct and corresponding inverse reactions shown in Table 11.1 leads to the relations between θ and ϕ collected in Table 11.3. These relations are not imposed but are verified through the numerical calculations. From Table 11.3 one can see that the necessary condition for thermal equilibrium in the pair plasma is a detailed balance between direct and inverse triple interactions. This point is usually neglected in the literature, where there are claims that the thermal equilibrium may be established with only binary interactions [245, 270]. The existence of a nonnull chemical potential for photons indicates the departure of the distribution function from the one corresponding to thermal equilibrium. A negative (positive) value of the chemical potential generates an increase (decrease) in the number of particles to approach the one corresponding to the thermal equilibrium state. Then, since the total number of particles increases (decreases), the

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Table 11.3 Relations between parameters of equilibrium DFs fulfilling detailed balance conditions for each of the reactions shown in Table 11.1 Interaction

Parameters of distribution functions

Compton scattering Pair production Tripe interactions

θγ = θ± , ∀ϕγ ,ϕ± ϕγ = ϕ± , if θγ = θ± ϕγ , ϕ± = 0, if θγ = θ±

Figure 11.3 Time dependence of temperatures, measured on the left axis (black), and chemical potentials, measured on the right axis (gray), of electrons (dotted) and photons (solid). The dotted vertical lines correspond to the time when the kinetic (∼10−14 s) and the thermal (∼10−12 s) equilibria are established. Upper (lower) figure corresponds to the case when initially there are mainly pairs (photons). Reproduced from [253].

energy is shared between more (less) particles and the temperature decreases (increases) (see Figure 11.3). Clearly, as thermal equilibrium is approached, the chemical potential of photons is zero. In the example considered, with the energy density 1024 erg/cm3 , the thermal equilibrium is reached at ∼7 × 10−13 s with the final temperature Tth = 0.26 MeV. For a larger energy density the duration of the kinetic equilibrium phase, as well as of the thermalization timescale, is smaller.

11.5 Thermalization Process

171

Figure 11.4 Dependence on time of energy densities of electrons and positrons (dashed), photons (solid), and protons (dash-dotted). Total energy density is shown by the dotted black line. Interaction between pairs and photons operates on very short timescales up to 10−23 s. Quasi-equilibrium state is established at tk  10−14 s, which corresponds to kinetic equilibrium for pairs and photons. Protons start to interact with them as late as tth  10−13 s. Reproduced from [255].

11.5.2 Electron-Positron Plasma with Proton Loading Now describe the case of electron-positron plasma with proton admixture reported in [255]. The computational grid consists of 60 energy intervals and 16 × 32 intervals for two angles ϑ and φ characterizing the direction of the particle momentum. The following initial conditions are chosen: flat initial spectral densities Ei (i ) = const and total energy density ρ = 1024 erg/cm3 . Plasma is composed of photons with a small amount of electron-positron pairs, with the ratio between energy densities in photons and electron-positron pairs ζ = ρ± /ργ = 10−5 . Baryonic loading parameter B = 10−3 , corresponding to ρ p = 2.7 × 1018 erg/cm3 . The energy density in each component of plasma changes, as can be seen from Figure 11.4, keeping constant the total energy density shown by the dotted line in Figure 11.4, as the energy conservation requires. As early as 10−23 s, the energy starts to be redistributed between electrons and positrons, on one hand, and photons, on the other, essentially by the pair-creation process. This leads to equipartition of energies between these particles at 3 × 10−15 s. Concentrations of pairs and photons equalize at 10−14 s, as can be seen from Figure 11.5. From this moment, temperatures and chemical potentials of electrons, positrons, and photons tend to be equal, and it corresponds to the approach to kinetic equilibrium. This is a quasi-equilibrium state since the total number of particles is still approximately conserved, as can be seen from Figure 11.5, and triple interactions are not yet efficient. At the moment t1 = 4 × 10−14 s the temperature of photons and

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Thermalization in Relativistic Plasma

Figure 11.5 Dependence on time of concentrations of electrons and positrons (dashed), photons (solid), and protons (dash-dotted). Total number density is shown by a dotted black line. In this case, kinetic equilibrium between electrons, positrons, and photons is reached at tk  10−14 s. Protons join thermal equilibrium with other particles at tth  4 × 10−12 s. Reproduced from [255].

Figure 11.6 Spectral density as a function of particle energy shown before, at initial, and at final moments of the computations. The final photon spectrum is the black body one. Reproduced from [255].

pairs is θk  1.5, while the chemical potentials of these particles are νk  −7. Concentration of protons is so small that their energy density is not affected by the presence of other components; also proton-proton collisions are inefficient. In other words, protons do not interact yet and their spectra are not yet of equilibrium form. The temperature of protons starts to change only at 10−13 s, when proton-electron Coulomb scattering becomes efficient. The chemical potentials of electrons, positrons, and photons evolved by that time due to triple interactions.

11.6 Thermalization Timescales

173

Since chemical potentials of electrons, positrons, and photons were negative, the particles were in deficiency with respect to the thermal state. This caused the total number of these particles to increase and consequently the temperature to decrease. The chemical potential of photons reaches zero at t2 = 10−12 s, which means that electrons, positrons, and photons are now in thermal equilibrium. However, protons are not yet in equilibrium with other particles since their spectra are not thermal. Finally, the proton component thermalizes with other particles at 4 × 10−12 s, and from that moment plasma is characterized by unique temperature, θth  0.48. Protons have final chemical potential ν p  −12.8. This state is characterized by thermal distribution of all particles, as can be seen from Figure 11.6. There initial flat as well as final spectral densities are shown together with fits of particles spectra with the values of the common temperature and the corresponding chemical potentials in thermal equilibrium.

11.6 Thermalization Timescales Thermalization timescales for optically thick plasmas are estimated in the literature by order of magnitude arguments using essentially just the reaction rates of the dominant particle interaction processes (see, e.g., [244, 245]). They have been computed using various approximations. In particular, electrons have been considered ultra-relativistic, and the Coulomb logarithm has been replaced by a constant. The accurate determination of such timescales as presented here is instead accomplished by solving the relativistic Boltzmann equations including the collisional integrals representing all possible particle interactions. In this case the Boltzmann equations become highly nonlinear coupled partial integro-differential equations, which can only be solved numerically. In this section the systematic results obtained in [256] by exploring the large parameter space characterizing pair plasmas with baryonic loading are reported. The two basic parameters are the total energy density ρ and the baryonic loading parameter (see eq. (11.32)). The following range of parameters is chosen: 1023 ≤ ρ ≤ 1033 erg/cm3 10

−3

≤ B ≤ 10 , 3

(11.62) (11.63)

where B is defined in eq. (11.32), allowing to treat also the limiting cases of almost pure electron-positron plasma with B  1, and almost pure electron-ion plasma with B  M/m, respectively. The temperatures in thermal equilibrium corresponding to eq. (11.62) also bracket the electron rest-mass energy 0.1  kB T  10 MeV. The approachto complete thermal equilibrium depends on the baryon loading. For B  m p/m, protons are rare and thermalize via proton-electron

174

Thermalization in Relativistic Plasma

 (positron) elastic scattering, while in the opposite case, B  m p/m, protonproton Coulomb scattering dominates over the proton-electron scattering and brings protons into thermal equilibrium first with themselves. Then protons thermalize with the pair plasma through triple interactions; for details, see [255]. The two-particle timescales involving protons should be compared with the threeparticle timescales, bringing the electron-positron plasma into thermal equilibrium. In fact it is found that for B  1, the electron-positron plasma reaches thermal equilibrium at a given temperature, while protons reach thermal equilibrium with themselves at a different temperature; only later the plasma evolves to complete thermal equilibrium with the single temperature on a timescale τth  Max[τ3p, Min(τep, τ pp )],

(11.64)

where m pc , e σT ne 8 mp (σT n pc)−1 ,  m  (ασT ne c)−1

τep 

(11.65)

τ pp

(11.66)

τ3p

(11.67)

are the proton-electron (positron) elastic scattering timescale, the proton-proton elastic scattering timescale, and the three-particle interaction timescale, respectively. In eqs. (11.65)–(11.67) the energy dependence of the corresponding timescales is neglected. The relaxation (or thermalization) timescale is usually computed as   <  dFι −1 τι = lim [Fι (t ) − Fι (∞)] , (11.68) t→∞ dt where Fι = exp (ϕι /θι ) is the fugacity of a particle of type ι. Instead of Fι , one of the quantities θι , ϕι , nι , or ρι is used in this computation. Boltzmann equations (11.3) are solved with parameters (ρ, B) in the range given by eqs. (11.62) and (11.63). In total, 78 models were computed, starting from a nonequilibrium configuration until reaching a steady state solution on the computational grid with 20 intervals for the particle energy and 16 intervals for the angles; for details, see [255]. For each model the corresponding timescales for all particles of the ιth kind are computed. For practical purposes, instead of (11.68), the following approximation is used:  −1  t f in 1 dθ τth = dt, (11.69) [θ (t ) − θ (tmax )] t f in − tin tin dt with tin < t f in < tmax , where tmax is the moment of time where the steady solution is reached and tin and t f in are the boundaries of the time interval over which the averaging is performed; for details, see [271]. The thermalization timescale of

11.6 Thermalization Timescales

3

2

175

Log B 3 1 2 1 0

10 12 Log tg 14 16 18 20 23

25

27 29 Log r

31

Figure 11.7 The thermalization timescale of the electron-positron-photon component of plasma as a function of the total energy density and the baryonic loading parameter. The energy density is measured in erg/cm3 , time in seconds. Reprinted with permission from [257].

the electron-positron-photon component is shown in Figure 11.7 as a function of the total energy density of the plasma and the baryonic loading parameter. The timescales of electrons, positrons, and photons coincide. The final thermalization timescale of pair plasma with baryonic loading is shown in Figure 11.8. Its dependence on either variable cannot be fit by a simple power law, although it decreases monotonically with increasing total energy density, while it is not even a monotonic function of the baryonic loading parameter. In Figure 11.9 the final thermalization timescale is shown for all the models computed, along with the “error bars,” which mark one standard deviation of the timescale computed with eq. (11.69) away from the average value τth in the averaging interval tin ≤ t ≤ t f in . The largest source of error comes from the small values of the time derivative in (11.69), although errors are typically below a few percent. It is clear that the relaxation to thermal equilibrium for the total energy density in the range (11.62) always occurs on a timescale less than 10−9 s. It is interesting that the electron-positron-photon component and/or proton component can thermalize earlier than the time at which complete thermal equilibrium is reached. These results may be of relevance for the ongoing and future laboratory experiments aimed at creating electron-positron plasmas. Current optical lasers producing pulses during ∼10−15 s carrying energy ∼102 J= 109 erg are capable of producing positrons with the number density 1016 cm−3 [229]. These densities today are yet far from 1028 cm−3 , required for the plasma with the size r0  μm to be optically

176

Thermalization in Relativistic Plasma

3

2

Log B 3 1 2 1 0

10 12 Log tb 14 16 18 20 23

25

27 29 Log r

31

Figure 11.8 The final thermalization timescale of a pair plasma with baryonic loading as a function of the total energy density and the baryonic loading parameter. The energy density is measured in erg/cm3 , time in seconds. Reprinted with permission from [257]. Log t 10 12 14 16 18 20

23 24 25 26 27 28 29 30 31 32 33 34 35

Log r

Figure 11.9 The final thermalization timescale of pair plasma with baryonic loading as a function of the total energy density for selected values of the baryonic loading parameter B = (10−3 , 10−1.5 , 1, 10, 102 , 103 ). The energy density is measured in erg/cm3 , time in seconds. Error bars correspond to one standard deviation of the timescale (11.69) away from the average value τth over the interval tin ≤ t ≤ t f in . Reprinted with permission from [257].

thick [230]. Notice that the expansion timescale of such plasma is r0 /c ∼ 10−14 s, while the timescale to establish kinetic equilibrium for the number density considered is of the same order of magnitude. These arguments show that theoretical results obtained assuming thermal or kinetic equilibrium, such as in [228], cannot

11.7 Dynamics and Emission of Mildly Relativistic Plasma

177

be applied directly to pair plasma generated by ultra-intense lasers. Nevertheless, conceptual understanding of evolution of optically thick pair plasma is required. For this reason, in the next section, an example application of kinetic theory to optically thick relativistic plasma with macroscopic dimensions is given. Results presented here are important for understanding also astrophysical systems in which optically thick electron-positron plasmas play an essential role. As a specific example, recall that electron-positron pairs play a crucial role in the dynamics of GRB sources. Considering typical energies and initial radii for GRB progenitors [219] 1048 erg < E0 < 1054 erg 107 cm < R0 < 108 cm,

(11.70)

the range for the energy density in GRB sources is 1023

erg 32 erg < ρ < 10 , cm3 cm3

(11.71)

which coincides with eq. (11.62). As for the baryonic loading of GRBs, it is typically in the lower range of (11.62), namely, B < 10−2 [272]. Such high-energy density leads to a large number density of electron-positron pairs in the source of GRB, of the order of 1030 cm−3 < n < 1037 cm−3 ,

(11.72)

making it opaque to photons with huge optical depth of the order of 1013 < τ < 1018 .

(11.73)

In fact, the radiative pressure of optically thick electron-positron plasma in these systems is responsible for the effect of accelerated expansion [220, 273, 274], leading to unprecedented Lorentz factors of bulk motion attained  B−1 , up to 103 (see, e.g., [275]). The role of the baryon admixture in electron-positron plasma in GRBs is to transfer internal energy of pairs and photons into kinetic energy of the bulk motion, thus giving origin to afterglows of GRBs [219, 272]. Notice that in GRBs the timescales of thermalization are much shorter than the dynamical timescales R0 /c ∼ 10−3 s, which implies that expanding electron-positron plasma even in the presence of baryons is in thermal equilibrium during the accelerating optically thick phase [276].

11.7 Dynamics and Emission of Mildly Relativistic Plasma So far only homogeneous isotropic plasma has been considered. It is clear that relativistic spherically symmetric plasma is dynamically unstable. If such optically thick plasma initially has a radius R, it is expected to expand on a timescale ∼R/c, cool down, and eventually become transparent for radiation, producing the

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Thermalization in Relativistic Plasma

characteristic flash of quasi-thermal radiation, the so-called photospheric emission. Such emission is considered to be a characteristic phenomenon of GRBs. Many recent works study this phenomenon; for a recent review, see [277]. Several methods of the computation of spectra and light curves of photospheric emission have been proposed, in particular, integration over the photospheric equitemporal surfaces [278, 279]; integration over volume with attenuation factors [280, 281]; approximations to the radiative transfer [282, 279]; Monte Carlo simulations of photon scattering [283, 284, 285, 286]; Fokker-Planck approximation to the collision integral with anisotropic photon field (generalized Kompaneets equation) [287]; and relativistic Boltzmann equations [288, 289]. In particular, Monte Carlo techniques are based on well-known reaction rates [234, 236, 241, 242] (see also Section 13.7). In the optically thick case, usually a hydrodynamic approach is postulated [219]. Such an approximation is justified for large optical depth or in the beginning of expansion. Owing to complexity of the calculations, only very few works adopt the kinetic approach for the description of the plasma and try to calculate the spectra when the optical depth is not very far from unity [283]. In this section the study, reported in [257], of mildly relativistic plasma that is initially optically thick is presented. The description of plasma is based on relativistic Boltzmann equations. By means of this instrument, the applicability of hydrodynamic description to the plasma is verified. The focus is on the difference between the hydrodynamic description and more detailed kinetic one. DFs fι (|p|, μ, r, t ) where ι = e± , γ satisfy the relativistic Boltzmann equations (2.41), which in the spherical symmetric case are reduced to eq. (8.1). All twoparticle interactions and corresponding triple interactions listed in Tables 11.1 and 11.2 are taken into account. As in previous cases, the method of lines is used to solve it. The grid in the phase space is introduced {ω+1/2 , μk+1/2 , r j+1/2 }. After replacing all derivatives except the derivative with respect to time in Eq. (8.1) by finite differences and collisional integrals by sums, one has the set of ordinary differential equations for grid values. The reaction cannot be resolved with a coarse angle grid for very large optical depths. For this reason the following method for computing reaction rates in a such a region with τ  1 is adopted. First, hydrodynamic velocity β is computed from the known DFs. Then, all quantities are transformed to the co-moving reference frame by using the Lorentz transformations (see, e.g., [44], p. 414) ⎡ ⎤ V    pμ + c c v (, μ, φ, r, t ) = ⎣  + cp μ ,  , φ , r , t ⎦ , (11.74)  c  V 2 m2 c2 + c p μ − 2 c where v is hydrodynamic velocity (1.16). The average values n ι , which do not depend on angles, are computed in the co-moving reference frame. Then the

ng, n e

11.7 Dynamics and Emission of Mildly Relativistic Plasma 10

30

10

29

10

28

10

27

10

26

10

25

10

24

10

23

10

22

10

21

10

20

10

179

,e

2

10

3

4

10

r, cm

Figure 11.10 Photon number density (solid) and pair number density (dashed) as a function of the radius at different time moments from left to right: 2 × 10−15 s, 7 × 10−9 s, 3 × 10−8 s, 1.0 × 10−7 s, and 1.7 × 10−7 s. Reproduced from [257] with permission. © 2012 World Scientific.

average absorption coefficients χ = const are evaluated in this reference frame. The emission coefficients η in the co-moving reference frame are taken to be proportional to equilibrium intensities. Finally, the emission and absorption coefficients are transformed back into the laboratory frame by the transformations 2 p I ( , μ ),  2 p p η(, μ) = η ( , μ ), p I(, μ) =

E(, μ, r, t ) ≡ χ (, μ) =

2π  3 β f ∝ I, c3

 χ ( , μ ), 

(11.75) (11.76)

which fulfill exactly energy and momentum conservation on the finite grid. Consider kinetic evolution of nonequilibrium optically thick plasma consisting at the moment t = 0 of electron-positron pairs with number density n = 1030 cm−3 in the small region with radius R0 = 200 cm and average particle energy 600 keV. Although such parameters are far from both laboratory conditions for ultra-intense lasers or the GRB sources, this choice of parameters is important since it provides some new insights with respect to the traditional hydrodynamic description adopted previously in [218, 290, 291, 231]. The plasma evolution with time is shown in Figures 11.10 and 11.11, and spectra of photons near the maximum of emission at t = 7 × 10−7 s, crossing the sphere with radius 2.2 × 104 cm, are shown in Figure 11.12. Initially, nonequilibrium plasma relaxes to the thermal state on the timescale 5 × 10−11 s, and it starts to expand on the dynamical timescale R0 /c  6 × 10−9 s (see Figure 11.11). The concentrations (see Figure 11.10) and the optical depths of both electrons/positrons and photons decrease with time. The bulk velocity of plasma

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Thermalization in Relativistic Plasma 4

,e 3

2

1

10

2

10

3

4

10

r, cm

Figure 11.11 Photon (solid) and pair (dashed) radial velocity as a function of the radius at different time moments, as in Figure 11.10. Reproduced from [257] with permission. © 2012 World Scientific.

Figure 11.12 Energy spectra of photons (solid thick) and electrons (dashed thick) together with their thermal fits (solid and dashed, respectively) in the laboratory reference frame, at the radius 2.2 × 104 cm at time moment 7 × 10−7 s, near the maximum luminosity. Reproduced from [257] with permission. © 2012 World Scientific.

becomes relativistic and the density in the laboratory frame becomes distributed in a shell. As the temperature becomes smaller than mc2 , the energy density in pairs starts to decrease exponentially, since they remain in thermal equilibrium. The departure of pairs from thermal equilibrium, or freeze-out (see Section 4.5), does not occur in the optically thick phase of expansion. When the optical depth of photons is large, the hydrodynamic description is accurate enough to calculate the photon spectra. At t = 1.5 × 10−7 s the optical depth for photons decreases below unity. After this moment the radial distribution of photons in the expanding shell becomes fixed. Photon spectra, reproduced in Figure 11.12 around the maximal luminosity, are nearly thermal. The spectrum emerging from the photosphere is

11.8 Kinetic Equilibrium and Chemical Potential of Photons

181

expected to deviate from Planck shape due to a number of effects [277], but with the adopted grid in energy space, the deviations are not resolved. Recently [268] photospheric emission from an ultra-relativistic plasma ball has been considered. This problem was formulated long ago [218], but appropriate methods have been developed only recently. Deviations from Planck spectrum were found to be larger with respect to previous studies. 11.8 Kinetic Equilibrium and Chemical Potential of Photons In this chapter, recent results obtained in the study of relativistic plasma out of equilibrium are presented. These results clearly show the existence of two types of equilibrium: the kinetic and the thermal. Kinetic equilibrium in pair-photon plasma occurs when detailed balance conditions (11.36), (11.39), and (11.42) are satisfied so that pair-creation, Compton and Bhabha/Møller scattering processes, all come in detailed balance. The electron-positron plasma then is described by common temperature and nonzero chemical potentials which are given by eqs. (11.53), (11.54), and (11.55), (11.56). Protons at this stage may or may not have yet established equilibrium. When B is small, proton-proton collisions are inefficient, and the proton spectrum is shaped up by the proton-electron/positron collisions, reaching equilibrium, when other particles are already thermalized. When the baryonic loading B is large, protons establish their equilibrium temperature prior to the moment when kinetic equilibrium in the pair-photon plasma is established. The meaning of nonzero chemical potentials in kinetic equilibrium can be understood as follows. The existence of a nonnull chemical potential for photons indicates the departure of the DF from the one corresponding to the thermal equilibrium. A negative value of the chemical potential generates an increase in the number of particles to approach the one corresponding to the thermal equilibrium state. A positive value of the chemical potential leads to the opposite effect, decreasing the number of particles. Then, since the total number of particles increases (or decreases), the energy is shared between a larger (or smaller) number of particles and the temperature decreases (or increases). Clearly, as thermal equilibrium is approached, the chemical potential of photons tends to zero, while the chemical potentials of electrons and positrons are given by (11.57), to guarantee an overall charge neutrality. Effective chemical potential of photons can also be induced in Compton scattering of photons off electrons at different temperatures (see discussion of the Sunyaev-Zeldovich effect in Chapter 13).

12 Kinetics of Particles in Strong Fields

In the previous chapter, thermalization of nonequilibrium pair plasma was discussed. Such plasma is expected to be present in the early universe as well as in astrophysical sources. Recent development of ultra-intense laser technology opens up the way for pair plasma creation under laboratory conditions. Hence this chapter is dedicated to discussion of pair creation out of a vacuum in strong electromagnetic fields. Quantum electrodynamics predicts that vacuum breakdown in a strong electric field E comparable to the critical value1 Ec = m2 c3 /q¯h results in nonperturbative electron-positron pair production [292, 293, 294]. Nonlinear effects in highintensity fields can be observed already in undercritical electric fields (see, e.g., [295, 296]). One way of creating such a strong electric field able to produce electron-positron pairs is dynamical amplification, and it involves focusing of counterpropagating ultra-intense laser beams. Invention of the chirped pulse amplification (CPA) technology [297] and its further development allowed for the design of petawatt lasers generating pulses with intensities up to 1022 W/cm2 [298]; for a review, see [299]. Considerable effort has been made over the last two decades to increase the intensity of high-power lasers in order to explore these high field regimes. Strong electric fields up to several percent of the critical value are expected to be reached by advanced laser technologies in laboratory experiments [300, 301, 302]; X-ray free electron laser facilities;2 or optical high-intensity laser facilities, such as Vulcan, the Extreme Light Infrastructure,3 or XCELS.4 Electron beam-laser interaction also seems to be promising for reaching high–Lorentz transformed electromagnetic

1 2 3 4

In this chapter, as well as in the previous one, m stands for electron mass. www.xfel.eu/ https://eli-laser.eu/ www.xcels.iapras.ru/

182

12.1 Avalanches in Strong Crossing Laser Fields

183

fields capable of multiple pair production [303]. Besides, strong laser fields can efficiently accelerate particles [304]. Yet, the Schwinger critical field Ec is far from being reached; for recent reviews, see [10, 225]. There are indications that such technology is limited to undercritical fields due to the occurrence of avalanches and cascades [305, 306, 307], which deplete the external field faster than it can potentially grow. It has been suggested [308, p. 4] that “the critical QED field strength can be never attained for a pair creating electromagnetic field.”The physical conditions leading to avalanches are discussed in Section 12.1 following a recent review [309]. Another mechanism for producing electron-positron pairs from an electric field is based the on possible existence of an overcritical electric field in which pair production is blocked [223]. Such conditions are widely discussed in astrophysical contexts or compact stars, e.g., hypothetical quark stars [310, 223] (see Section 12.3) or neutron stars [311]. Pair production in such overcritical fields may occur for several reasons, e.g., heating [312] or gravitational collapse of the compact object [313]. Assuming the existence of such an overcritical electric field in Section 12.2, the issue of conversion of energy from the initial electric field into electron-positron plasma once the blocking is released is discussed. Finally, in Section 12.3, the fate of electron-positron plasma emitted by a hypothetical quark star is discussed. It is crucial that both mechanisms of pair creation involve the back-reaction of pairs onto an external field. Accounting for such a back-reaction, as shown subsequently, is imperative in this problem.

12.1 Avalanches in Strong Crossing Laser Fields Interaction of particles with an electromagnetic field is characterized by two dimensionless parameters (see, e.g., [314, 225])   2 q2  μ  q¯h 2  2 2 ξ = − 2 4 A Aμ χe = (12.1) Fμν pν , 3 4 mc mc where Aμ is four-potential of the electromagnetic field and brackets denote averaging over suitable time and space domains. For laser field and particle moving with the Lorentz factor γ these parameters can be represented as ξ=

λ E λC Ec

χe = γ

E⊥ , Ec

(12.2)

where λ is the laser wavelength, λC is the Compton wavelength, and E⊥ is the component of the field in the direction orthogonal to the direction of particle velocity, and similarly for a photon χγ = (εγ /mc2 )(E⊥ /Ec ).

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Kinetics of Particles in Strong Fields

Two types of cascades occur in external electromagnetic fields. The first type, called an A-cascade (A stands for “avalanche,” in analogy with dielectric breakdown) is generated even when there is one seed charged particle, e.g., an electron, at rest at the center of the laser pulse focal area. The electric field in the focal area works both as target and as accelerator. It is predicted to occur at laser field intensities I > 1024 W/cm2 [305], and it operates until the energy of the field is depleted. The second type, called an S-cascade (S stands for “showers,” in analogy with atmospheric cosmic-ray showers) occurs when a high-energy charged particle radiates a hard photon in an external field due to nonlinear Compton scattering. This photon in turn creates a pair in the external field, due to the nonlinear Breit-Wheeler process. This process, observed already in the experiment [295, 296], operates until the particle loses its energy. Both these cascade processes require the conditions ξ > 1 and χe,γ > 1. The process can be described by the relativistic Boltzmann equations (2.9) for pairs and photons in an external electromagnetic field: ∂ f± ∂ f± ∓ q [E + v × H] (12.3) ∂t ∂p±      = wrad p± + pγ → pγ f± r, t, p± + pγ d 3 pγ      − Wrad (p± ) f± (r, t, p± ) + wcr pγ → p± fγ r, t, pγ d 3 pγ    ∂ fγ = wrad pe → pγ [ f+ (r, t, p± ) + f− (r, t, p± )] d 3 p± (12.4) ∂t     − Wcr pγ fγ r, t, pγ , where dWrad (pe → pγ ) = wrad (pe → pγ )d 3 pγ is the differential probability of photon creation and dWcr (pγ → p± ) = wcr (pγ → p± )d 3 p± is the differential probability of pair creation in an external electromagnetic field; Wrad and Wcr are total probabilities of these processes. These equations are similar to those in atmospheric cosmic-ray showers theory [315], with addition of the Lorentz force in eq. (12.3). It is important to note that the radiation reaction force is included in the RHS of eq. (12.3) (see [307, 225]). The probabilities of radiation and pair creation are calculated in [316, 317, 318] and can be estimated in the limit of large χe,γ as 

We,γ

mc2 ∼α h¯

2

2/3 χe,γ . εe,γ

(12.5)

The necessary condition for the A-cascade is the hierarchy of timescales tacc  −1 We,γ  tesc , where tacc is the acceleration time of the particle, and tesc ∼ λ/(2c) is

12.1 Avalanches in Strong Crossing Laser Fields

185

the time for which the particle remains in the laser focal area. Acceleration time, computed for a rotating electric field, gives  −1 8 2 h¯ mc E tacc ∼ , (12.6) 2 mc Ec h¯ ω where ω is the rotation frequency. Then, from the inequality for timescales, it follows that A-cascades may occur already in electric fields E ∼ αEc , corresponding to the laser intensity I ∼ 1025 W/cm2 [308]. The total number of pairs produced in one laser shot can be estimated as N± ∼ exp(We,γ tesc ). For large enough intensities the energy stored in pairs can become comparable to the energy of the laser field, which gives the upper limit to the number of pairs:  5/4  2 5/2 E mc −1/4 Nmax ∼ α . (12.7) Ec h¯ ω Consider now the S-cascade occurring when a high-energy particle moving with large Lorentz factor γ collides with the field of a standing electromagnetic wave. The electric field reached in the rest frame of the particle is Erest ∼ 2γ Elaser [10]. Initially the parameter given by eq. (12.2) of the particle injected into the focal region is χi . The multiplicity of the cascade can be estimated in the same way as in [319]. The duration of the cascade tS should be smaller than the duration of the laser pulse tL and is estimated [320] as   χi h¯ γ −2/3 tS ∼ χ , (12.8) log 2 i mc2 α χf where χ f is the final value of the parameter χe , which should be set to the value below unity. Owing to logarithmic dependence, the exact value of χ f is not essential. It has been shown [320] that if the laser field is strong enough, the S-cascade is followed by the A-cascade produced due to the strong field E⊥ , which curves the trajectory and leads to radiation of hard photons, which in turn produce pairs, as described earlier. Numerical studies of avalanches in counterpropagating laser beams [321, 307] of a wide range of laser pulse shapes and polarizations confirm that they occur already at intensities I > 1024 W/cm2 . It is important to stress that both types of cascades do not occur in plane waves or constant uniform electric fields, as they require that the charged particle radiates. The condition for radiation is not fulfilled when the parameter χe is an integral of motion. Most calculations have been performed so far [305, 308, 320] for a homogeneous rotating electric field. Such a field is formed at the antinodes of a

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Kinetics of Particles in Strong Fields

circularly polarized standing monochromatic wave, which can in turn be produced by counterpropagating laser beams.

12.2 Creation and Thermalization of Pairs in Strong Electric Fields The most general framework for considering the problem of back-reaction of created matter fields on an initial strong electric field is QED. Up to now the problem has been treated in QED in the 1+1 dimension case for both scalar [322] and fermion [323] fields. It was shown there that pair creation is followed by plasma oscillations due to back-reaction of pairs in the initial electric field. The results were compared with the solutions of the relativistic Vlasov-Boltzmann equations and shown to agree very well. The Vlasov-type kinetic equation for description of e+ e− plasma creation under the action of a strong electric field was used previously, e.g., in [324, 325, 326]. The back-reaction problem in this framework was considered in [327, 328]. The KT has been applied to describing the vacuum quark creation under action of a supercritical chromo-electromagnetic field [329, 330]. Kinetic equations for electron-positron plasma in a strong electric field were obtained in [331] from the BBGKY hierarchy. A much simpler model was developed later, starting from Vlasov-Boltzmann equations [332] and assuming that all particles are in the same momentum state at a given time, which allowed consideration of pair-photon interactions. In this way the system of partial integro-differential equations was reduced to the system of ordinary differential equations, which was integrated numerically. This model was studied in detail in [333, 83], where existence of plasma oscillations was confirmed also for undercritical electric fields. It was also shown that photons are generated and reach equipartition with pairs on a timescale much longer than the oscillation period. In the work [289], for the first time, the entire dynamics of energy conversion from an initial strong electric field, ending up with thermalized optically thick electron-positron plasma, is studied. With this goal, previous treatments [322, 332, 333, 83] are generalized. In particular, the delta-function approximation of particle momenta adopted in [332] is relaxed. A kinetic approach in which collisions can be naturally described, assuming invariance under rotations around the direction of the electric field, is adopted. In this perspective the relativistic Vlasov-Boltzmann equations are solved numerically. Collision integrals are computed from the exact QED matrix elements for the two-particle interactions [255], namely, electron-positron annihilation into two photons and its inverse process, Bhabha, Möller, and Compton scatterings (see Chapter 11 for details).

12.2 Creation and Thermalization of Pairs in Strong Electric Fields

187

12.2.1 Relativistic Boltzmann Equation with Axial Symmetry in the Phase Space So far only spherically symmetric phase space has been considered, and it has been assumed that DFs depend only on particle energy and time. In the problem under discussion, there exists a preferred direction, namely, the direction of the external electric field. Based on this symmetry of the problem the axially symmetric momentum space is considered. Hence the momentum of the particle is described by two components: parallel p and orthogonal p⊥ to the direction of the initial electric field. Then the azimuthal angle φ describes the rotation of p⊥ around p . These momentum space coordinates are defined in the following intervals: p ∈ (−∞, +∞), p⊥ ∈ [0, +∞), φ ∈ [0, 2π ]. Within the chosen phase space configuration,  3  +∞for the integral over the entire momentum space is  2π theprescription +∞ d p → 0 dφ −∞ d p 0 d p⊥ p⊥ and the relativistic energy is given by the equation   = c p2 + p2⊥ + m2 c2 , (12.9) where for photons, obviously, m = 0. The number density is given by its integral over the entire momentum space  +∞   +∞ 3 nA = d p fA = 2π d p d p⊥ p⊥ f A , (12.10) −∞

0

where index A denotes the kind of particle. Owing to the assumed axial symmetry, the DF does not depend on φ and consequently is a function of the two components of the momentum only, fA = fA (p , p⊥ ) . The Boltzmann equation is written in conservative form for the function FA = 2π  fA . The energy density is  +∞  +∞ d p d p⊥ FA . (12.11) ρA = −∞

0

In isotropic momentum space this DF is reduced to the spectral energy density EA defined in eq. (8.2). The relativistic Boltzmann-Vlasov equation (2.41) takes the following form: ∂F± (p , p⊥ ) ∂F± (p , p⊥ ) ± eE ∂t ∂ p   ∗q = η± (p , p⊥ ) − χ±q (p , p⊥ ) F± (p , p⊥ ) + S(p , p⊥ , E ) ,

(12.12)

q

where the source term S is the rate of pair production. In particular, the electronpositron DFs in eq. (12.12) vary due to the acceleration by the electric field, the creation of pairs due to vacuum breakdown, and the interactions. Indeed, here

188

Kinetics of Particles in Strong Fields

the Vlasov term describes the mean field produced by all particles as well as the external field. Particle motion between interactions is assumed to be subject to the external field only,5 and the mean field is neglected. The rate of pair production defines particles distribution in the momentum space according to [10] S(p , p⊥ , E ) = −

2π |q E|

 p⊥   c π (m2 c2 + p2⊥ ) δ(p ) . × log 1 − exp − h¯ |q E| m2 c2 (2π h¯ )3

(12.13)

For E < Ec this rate is exponentially suppressed. Besides, eq. (12.13)already indicates that pairs are produced with orthogonal momentum, up to about m (E/Ec ), but are at rest along the direction of the electric field. The Boltzmann equation for photons is  ∂Fγ (p , p⊥ )  ∗q = ηγ (p , p⊥ ) − χγq (p , p⊥ ) Fγ (p , p⊥ ) , (12.14) ∂t q and their DF changes due to the collisions only. In more detail, photons must be produced first by annihilating pairs, then they affect the electron-positron DF through Compton scattering. Besides, also photon, annihilation into electron-positron pairs becomes significant at later times. Equations (12.14) and (12.12) are coupled by means of the collision integrals; therefore they are a system of partial integrodifferential equations that must be solved numerically. The method for solving such equations in the optically thick case is presented in Chapter 11. It has to be noted that the avalanches discussed in the previous section do not occur in uniform electric fields. It is well known [332, 333] that both acceleration and pair creation terms in eq. (12.12) operate on much shorter timescales than interactions with photons described by collision terms in eqs. (12.12) and (12.14). For this reason, two different classes of simulations are discussed: one neglecting collision integrals, which is referred to as collisionless, and another one including them, called interacting. The discretization of the phase space is done defining a finite number of elementary volumes which are uniquely identified by triplets of integer numbers (i, k, l). Their values run over the ranges {1, 2, ..., I − 1, I}, {1, 2, ..., K − 1, K}, and {1, 2, ..., L − 1, L}, respectively. Since one deals with an axially symmetric phase space with respect to the direction of the electric field, the parallel momentum is aligned with it, while the orthogonal component lies on the plane orthogonal to this preferential axis. Each elementary volume encloses only one momentum vector, which can be written explicitly in cylindrical coordinates as (pi , p⊥k , φl ). The 5

The backreaction due to plasma polarization is, however, taken into account.

12.2 Creation and Thermalization of Pairs in Strong Electric Fields

189

corresponding boundaries are marked by semi-integer indices [pi−1/2 , pi+1/2 ], [p⊥k−1/2 , p⊥k+1/2 ], [φl−1/2 , φl+1/2 ]. Owing to axial symmetry, the DFs do not depend on the azimuthal angle φ, and the index l is used only to identify angles explicitly. From these definitions, the energy of a particle with mass mA corresponding to the grid point (i, k) is  Aik = m2A + p2i + p2⊥k (mγ = 0, m± = m) . (12.15) In this finite difference representation the DF has a Klimontovich form and can be seen as a sum of Dirac deltas centered on the grid points (i, k) and multiplied by the energy density of particles on the same grid point FAik : FA (p , p⊥ ) = δ(p − pi ) δ(p⊥ − p⊥k ) FAik , (12.16) ik



I

K

where i,k = i=1 k=1 . From the preceding definition and from eq. (12.11), the energy and number densities of particles of sort A are given by ρA = FAik (12.17) i,k

nA =



nAik ,

(12.18)

i,k

where nAik = FAik /Aik . Then the mean parallel momentum, its mean squared value, and the mean squared value of the orthogonal momentum are 1 nAik pi , nA i,k 1

p2 A = nAik (pi − p A )2 , nA i,k 1

p2⊥ A = nAik p2⊥k . nA i,k

p A =

(12.19) (12.20) (12.21)

Owing to axial symmetry, the mean orthogonal momentum must be null identically

p⊥ A = 0 . Once electrons and positrons are produced, they are accelerated by the electric field in opposite directions. The time derivative of the electron or positron parallel momentum d p± in the presence of an electric field E is given by the equation of motion d p± = ±qE . dt

(12.22)

190

Kinetics of Particles in Strong Fields

Numerically, particles are moved from one cell to another such that the number of particles is conserved and eq. (12.22) is satisfied. Acceleration causes the changing with time of F±ik , which can be written as follows:   ∂F±ik = ± αi−1k F±i−1k + αik F±ik + αi+1k F±i+1k , (12.23) ∂ p where the coefficients α−,0,+ are defined as αi−1k = αik = αi+1k =

±ik

1 , pi − pi−1

(12.24)

1 1 − , pi − pi−1 pi+1 − pi

(12.25)

±i−1k

±ik ±i+1k

1 . pi+1 − pi

(12.26)

Also the electric field evolves according to the Maxwell equations (5.4). Once the currents of the moving pairs are computed, the time derivative of the electric field is known. Consequently, a new ordinary differential equation must be added to the system of eqs. (12.12) and (12.14). However, owing to the uniformity and homogeneity of the physical space, one can describe the electric field simply using the energy conservation law. Emission and absorption coefficients are computed in the same way as done in Chapter 11.

12.2.2 Evolution of Pairs Created in a Strong Electric Field Vlasov-Boltzmann equations (12.12) and (12.14) must be supplied with the boundary conditions: the initial electric field E0 and the initial DF FA0 (p , p⊥ ). Computations are performed for different initial electric fields but always with no particles at the beginning. When, in addition to external electric field, particles are also present from the beginning, oscillations still occur, but with higher frequency, as given by the plasma frequency [83]. So the computations are performed with the following initial conditions:  E0 = ξ Ec , ξ = {1, 3, 10, 30, 100} , (12.27) FA0 (p , p⊥ ) = 0 , p⊥ ∈ [0, +∞) , p ∈ (−∞, +∞) . Consequently, electrons and positrons are produced exclusively by the Schwinger process. To interpret the results, introduce some useful quantities. Initially, the energy is stored in the electric field, and it fixes the energy budget available as given by the

12.2 Creation and Thermalization of Pairs in Strong Electric Fields

191

energy density ρ0 =

E02 . 8π

(12.28)

The final state of the equilibrated thermal electron-positron plasma to be characterized by the temperature A 8 ρ E0 0  1.7 MeV , (12.29) Teq = 4 4σSB Ec where σSB is the Stefan-Boltzmann constant. The total energy densities of pairs ρ± and photons ργ are related to the actual and initial electric fields, E and E0 , by the energy conservation law ρ± = ρ+ + ρ− =

E02 − E 2 − ργ . 8π

(12.30)

Following [83], define the maximum achievable pair number density nmax =

E02 , 8 π mc2

(12.31)

which corresponds to the case of conversion of the whole initial energy density into electron-positron rest energy density ρ±rest = (n− + n+ ) mc2 ,

(12.32)

where n− and n+ are the electron and positron number densities, respectively. From the electron and positron DFs, one can extrapolate their bulk parallel momentum p as defined in eq. (12.19), and the symmetry of the problem implies that

p− = − p+ . One can use this identity to define the kinetic energy density of pairs: ⎡A ⎤  

p± 2 ρ±kin = ρ±rest ⎣ + 1 − 1⎦ . (12.33) mc Therefore ρ±kin is the energy density as if all particles are put together in the momentum state with p = p and p⊥ = 0 while their rest energy density is ρ±rest . The difference between the total energy density and all the others defined here is denoted as internal energy density ρ±in = ρ± − ρ±rest − ρ±kin .

(12.34)

The term internal refers here to the dispersion of the DF around a given point with coordinates ( p , p⊥ ) in the momentum space.

192

Kinetics of Particles in Strong Fields

150

E Ec p

100

me c

50 0 50 100 0.01

0.1

1

10

100

1000

t tc

Figure 12.1 Evolution of electric field E (dotted) and pairs bulk parallel momentum p ± (solid) obtained from the numerical solution of eq. (12.12) setting E0 = 30 Ec . Reprinted from [289]. © 2013, with permission from Elsevier.

Since interactions with photons operate on much larger timescales than the pair creation by vacuum breakdown, first the results obtained solving the relativistic ∗q Boltzmann equation (12.12) for electrons and positrons with χ±q = η± = 0 are presented. With these assumptions, one expects the results to be closely related to those reported in [83]. For all the explored initial conditions, there are important analogies between the approach adopted in [83] and the one discussed here. For each initial field the first half-period of the oscillation t1 is nearly equal to the corresponding one obtained in [83]. Also the evolution with time of p ± during this time lapse is very similar to the result given by their analytic method. The time evolution of electric field E and p ± in Compton units with tc = h/(mc2 ) is shown in Figure 12.1 for E0 = 30 Ec . In addition to these similarities, some new important features emerge. The manifestation of these new aspects is represented in Figure 12.2, where it is shown how the various forms of energy defined earlier evolve with time. These energy densities are normalized to the total initial energy density ρ0 defined by eq. (12.28). Some of the most important evidence for of this figure is that the rest energy density of pairs ρ±kin saturates to a small fraction of the maximum achievable one. This is in contrast with the result presented in [83], where the value given in eq. (12.31) was reached asymptotically.

12.2 Creation and Thermalization of Pairs in Strong Electric Fields

193

1.0 0.8 0.6 0.4

R

kin

0.2 0.0 0.1

1

10

100

1000

100

1000

t tc 1.0 0.8 R

in

0.6 0.4

rest

0.2 0.0 0.1

1

10 t tc

Figure 12.2 Evolution with time of the pair energies as defined by eqs. (12.30) and (12.32) (upper figure) and those defined by eqs. (12.33) and (12.34) (lower figure) for the collisionless case E0 = 30 Ec . All of them are normalized by the total initial energy ρ0 given by eq. (12.28). Reprinted from [289]. © 2013, with permission from Elsevier.

As a consequence, the energy is mainly converted into other forms, namely, the kinetic ρ±kin and internal ρ±int ones. Both these quantities oscillate with the same frequency but with shifted phase. Relative maxima and minima of ρ±kin correspond to the peaks of the bulk parallel momentum shown in Figure 12.1, as can be grasped from its definition in eq. (12.33). Looking at Figure 12.2, one can see their relative importance changing progressively with time. Even if they oscillate, the internal component dominates over the kinetic one as time advances. This trend points out that all the initial energy will be converted mostly into internal energy, while the contribution of the kinetic one will eventually be small.

194

Kinetics of Particles in Strong Fields

Table 12.1 Square root of the mean squared value of orthogonal p2⊥ ± and parallel p2 ± momentum, parallel momentum p1 , in units of m, and number density n1 of pairs at the first zero of the electric field, saturation number density ns normalized by the maximum achievable one given by eq. (12.31) for different initial electric fields E Ec



p2⊥ ±

1 3 10 30 100

0.4 0.8 1.3 2.0 3.5



p2 ±

75 37 35 87 127

p 1

n1 nmax

ns nmax

160 82 77 192 284

0.006 0.018 0.013 0.005 0.003

0.018 0.037 0.041 0.016 0.011

In this respect, from the electron and positron DFs, one obtains the mean squared values of the parallel and orthogonal momenta as defined by eqs. (12.20) and (12.21). These quantities give some insight into the spreading of the DF along the parallel and orthogonal components of the momentum. In Table 12.1, the values of these quantities are reported at the end of runs with different initial fields. It is clear that the larger the initial electric field is, the larger is p2⊥ ± . This is a direct consequence of the rate of pair production given by eq. (12.13) that already distributes particles along the orthogonal direction in the momentum space. The mean squared value of the parallel momentum p2 ± reaches a minimum value between 3 and 10 critical electric fields. This minimum was first found in [333]; see Figure 3 in that paper. In Table 12.1, also p ±1 , namely, the peak value of the bulk parallel momentum at the moment when the electric field vanishes for the first time, is given. It can be seen from the table that also this quantity has a minimum in the same range of initial fields as p2 ± . Both these minima are linked to the combined effects of pair creation and acceleration processes. However, it is important to compare p2 ± and p2⊥ ± for different initial fields, which gives quantitative information about the anisotropy of the DFs in the phase space. Looking at the numerical values, one observes how this anisotropy decreases with the increase of the initial electric field, which points out how an eventual approach toward isotropy, and therefore thermalization, would be much more difficult for lower initial fields. In Table 12.1, also two different number densities n1 and ns normalized to the maximum achievable one are reported. The first one is the number density of pairs at the moment when the electric field vanishes in the first oscillation t1 . The second is the saturation number density of pairs at the end of the run. It is found that the values of n1 are very close to the same densities computed in [83], with a significant

12.2 Creation and Thermalization of Pairs in Strong Electric Fields

195

amount of pairs produced already with a very small time lapse. Let us note that there are maxima of both n1 and ns in the range between 1 and 10 Ec in correspondence with minima of p2 ± and p2⊥ ± . Now turn to the dynamics of the system on much larger timescales. As discussed earlier, in the long run, interactions between created pairs become important. Consider two-particle interactions listed in Table 11.1, which are described by the collision integrals in eqs. (12.12) and (12.14) using the same range of initial fields used for the collisionless systems. Clearly a more sparse computational grid is used as calculations of collision terms imply performing multidimensional integrals in the phase space. The larger the electric field is, the higher the rate of pair production and consequently their number density is. Since the interaction rate is proportional to particle number densities, one expects them to be important sooner for a higher initial field. In this respect, it is worth mentioning that in [83] the time tγ was estimated at which the optical depth for electron-positron annihilation equals unity, τ (tγ ) = 1. There it was found that tγ decreases when the initial electric field increases. Besides, the order of magnitude of their estimations is in agreement with the time at which the photon number density is around a few percent of the pair number density. Interactions redistribute particles in the phase space and tend to isotropize their distributions, so the orthogonal grid must be extended to values comparable to the kinetic equilibrium temperature. The extension of the parallel grid remains essentially the same as in the collisionless case. To correctly describe the pair acceleration process, the time step of the computation must be a small fraction of their oscillation period. This constraint prevents studying the evolution up to the kinetic equilibrium within a reasonable time. After hundreds of oscillations, the energy density carried by the electric field is a small fraction of the pair and photon energy densities. In other words, most energy has already been converted into electron-positron plasma. Because of this, the acceleration of electrons and positrons does not affect their DFs appreciably. This allows one to neglect the presence of the electric field hereafter. To do that, the DF is used at this instant as the initial condition for a new computation in which the condition E = 0 is imposed. By neglecting oscillations induced by the electric field, the constraint on the time step of the numerical calculation is released, and it is now determined by the rate of the interactions. In Figure 12.3 the time evolution of the pair and photon energy densities for E0 = 100 Ec is shown. From this figure one can understand the hierarchy of timescales associated with the distinct physical phenomena. In the presence of an overcritical electric field, electron-positron pairs start to be produced in the shortest time, according to eq. (12.13). As soon as they are created, electrons and positrons are accelerated in opposite directions as the back-reaction effect on the external

196

Kinetics of Particles in Strong Fields 1030

erg/cm 3

1029 1028 1027

-+

1026 1025 1024 0.01

1

100

10 4

106

t /t c

Figure 12.3 Energy densities of pairs and photons obtained from the numerical solution of eqs. (12.12) and (12.14) with initial field E0 = 100 Ec . Reprinted from [289]. © 2013, with permission from Elsevier.

field. The characteristic duration of this back-reaction corresponds approximately to the first-half oscillation period. At early times, even after many oscillations, the energy density of photons is negligible compared to that of pairs, meaning that interactions do not play any role. Such a starting period, during which the system can be considered truly collisionless, exists independently of the initial electric field, even if its duration depends on it. From Figure 12.3 it is clear that the photon energy density increases with time as a power law approaching the pair energy density. Only when hundreds of oscillations have taken place, interactions start to affect the evolution of the system appreciably and cannot be neglected any further. The slope of the photon curve in 12.3 reduces, indicating that pair annihilation has become less efficient than the photon annihilation process. Now the evolution of the system is mostly governed by interactions. Möller, Bhabha, and Compton scatterings give rise to momentum and energy exchange between electron, positron, and photon populations. Besides, the same collisions have the tendency to distribute particles more isotropically in the momentum space. After some time, the photon energy density becomes equal and then overcomes the pair energy density. This growth continues until the equilibrium between pair annihilation and creation processes is established, e− e+ ↔ γ γ . For this reason, both pair and photon curves are flat on the right of Figure 12.3 (see also Chapter 11). However, at this point the DF is not yet isotropic in the momentum space, indicating that the kinetic equilibrium condition is not yet satisfied. In fact, kinetic equilibrium is achieved only at later times when also Möller, Bhabha, and Compton scatterings are in detailed balance. At that time the electron-positron plasma can be identified by a common

12.2 Creation and Thermalization of Pairs in Strong Electric Fields Log f

Log fg

80

26

60 p me

p me

22

80

60

40

20

40 20

21 0

400

200

0 200 p me

17 0

400

400

Log f 27

0 200 p me

400

26

80

60

60 p me

p me

200

Log fg

80

40

20

40 20 21

22 0

400

200

0 200 p me

0

400

400

Log f

200

0 200 p me

400

Log fg

80

33

33

80

60

60 p me

p me

197

40

20

40 20

28 0

400

200

0 200 p me

400

28 0

400

200

0 200 p me

400

Figure 12.4 Phase space distributions of electrons (left column) and photons (right column) for the initial condition E0 = 100 Ec . (top) 2.3 × 102 tc ; (middle) 2.3 × 102 tc ; (bottom) 4.6 × 106 tc . Reprinted from [289]. © 2013, with permission from Elsevier.

temperature and nonzero chemical potential. Thermalization is expected to occur after kinetic equilibrium is established; see Chapter 11 for details. As an example, in Figure 12.4, density plots of f− and fγ are shown in the left and right columns, respectively, for the initial condition E0 = 100. Their time

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evolution starts from the top line to the bottom one, corresponding to three different times.√After 2.3 tc both DFs are highly anisotropic as is well established by the ratio R = p2 ± / p2 ± = 0.06. At this stage, the electric field is highly overcritical, and a very small fraction of initial energy has been converted into rest mass energy of electrons and positrons. For this reason, electrons and positrons are easily accelerated up to relativistic velocities, explaining why the electron DF is shifted on the right side of the phase space plane characterized by p > 0. At this instant, electrons are characterized by a relativistic bulk velocity corresponding to a Lorentz factor γ  170. On the second line the time is 2.3 × 102 tc and the DFs are still anisotropic if one looks at the parameter R introduced earlier. However, the situation is different with respect to the previous stage because the electric field is only slightly overcritical and many more pairs and photons have been generated. As a consequence, interaction rates are much larger than before and an efficient momentum exchange between electron and positron populations occurs. Both small electric field and collisions prevent particles from reaching ultra-relativistic velocities, and for this reason the electron DF is now symmetric with respect to the plane p = 0. Only later on, at 4.6 × 106 tc for the bottom line, do collisions dominate the evolution of the system, whereas the presence of the electric field can be safely neglected. The pictures show a prominent DF widening toward higher orthogonal momenta, which is confirmed by the value R  0.23. This allows one to predict the forthcoming fate of the system to be an electron-positron plasma in thermal equilibrium. The DF isotropization in the momentum space indicates not only that the kinetic equilibrium condition is approached but also that the system is going to lose information about the initial preferential direction of the electric field. In the case of isotropic DF, the timescale on which thermal equilibrium is achieved can be estimated as τth  1/(nσT c). For the anisotropic DF the thermalization timescale is remarkably longer.

12.3 Emission from Hot Bare Quark Stars The idea that the ground state of nuclear matter is quarks in a deconfined state was proposed in [334, 335]; for review, see [336]. It was realized [310] that such a ground state may be relevant in astrophysics and that compact objects referred to as quark stars [310] may be composed of such strange quark matter. Such hypothetical objects are described within the standard Tolman-Oppenheimer-Volkoff equation with the equation of state provided by the MIT bag model [337] P = (ρ − 4B)/3, where B is a constant. They may represent an alternative to neutron stars with the following differences. Their mass-radius relation is opposite compared to the case of neutron stars: with increasing radius the mass increases as well. Maximum

12.3 Emission from Hot Bare Quark Stars

199

masses and radii of neutron and quark stars are similar. Unlike neutron stars, the mass of the crust of quark stars is only Mcrust ∼ 10−5 M [338]. The surface of quark stars is composed of deconfined quarks in statistical equilibrium, subject to short-range nuclear forces. At the same time, electrons obey long-range electromagnetic interactions, and their spatial distribution does not coincide with quarks. This fact results in the presence of strong electric fields, referred to as the electrosphere, at the surface of a quark star [339]. If the surface of a quark star is exposed to the vacuum, it is called a bare quark star [336]. The electric field on the surface of a bare quark star exceeds the Schwinger critical electric field for pair production [223]. At absolute zero temperature, pair production is blocked by the Pauli principle. If the temperature increases, the star can produce a wind of electron-positron pairs, but due to large density, its radiation in photons is suppressed [310] for temperatures below 1010 K. For the typical radius of a quark star (106 cm), a temperature of TS  109 –1010 K gives an energy injection rate in pairs E˙  1043 –1049 ergs s−1 [312]. For such powerful winds the pair density near the surface is very high, the wind is opaque for photons, and the pairs and photons are in thermal equilibrium almost up to the wind photosphere (see Section 13.6). The outflow may then be described fairly well by relativistic hydrodynamics [340, 341, 218, 342]. The emerging emission consists ˙ The photon spectrum is roughly Planckian with mostly of photons, so Lγ  E. 49 ˙ a temperature of ∼ 1010 (E/10 ergs s−1 )1/4 K. The emerging luminosity in e± 49 ˙ ˙ pairs is very small, Le = E − Lγ  10−7 (E/10 ergs s−1 )−1/4 Lγ . All this applies roughly down to E˙ ∼ 1042 ergs s−1 . In contrast, for E˙ < 1042 ergs s−1 (TS < 9 × 108 K), the thermalization time for the pairs and photons is longer than the escape time tesc  a few times R/c, and pairs and photons are not in thermal equilibrium. The results of numerical calculations of the characteristics of the emerging emission in pairs and photons in stationary winds with energy injection rates E˙ = 1035 –1042 ergs s−1 are reported below. Consider an electron-positron pair wind that flows away from a hot, bare, unmagnetized quark star with a radius of R = 106 cm. Pairs are injected from time t = 0, at a constant rate into the wind, which is assumed spherical. This results in a timedependent wind that becomes stationary after a time ∼ 100 ms. The relativistic Boltzmann equations for the pairs and photons (2.41) are solved numerically in [343, 269] within special relativity and in [344] within general relativity. The nonstationary pair plasma wind in the region outside the quark star is considered if the quark star is considered as an internal boundary with the constant energy flux of pairs, so one expects to obtain the steady solution at sufficiently large times. At the internal boundary, r = R, the input pair number flux depends on

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Kinetics of Particles in Strong Fields

the temperature TS at the stellar surface alone and is taken as [312] #   3 −1 $ T T S S exp −11.9 F˙± = 1039 109 K 109 K # $ ζ 3 ln(1 + 2ζ −1 ) π 5ζ 4 × + cm−2 s−1 , 3(1 + 0.074ζ )3 6(13.9 + ζ )4

(12.35)

where ζ = 20(TS /109 K)−1 . Their energy spectrum is thermal with temperature TS , and their angular distribution is isotropic. The energy injection rate in e± pairs is then E˙ = 4π R2 [me c2 + (3/2)kB TS ]F˙± .

(12.36)

The surface of the quark star is assumed to be a perfect mirror for both e± pairs and photons. At the external boundary (r = rext ), the pairs and photons escape freely. Although the injected pair plasma contains no radiation, as the plasma moves outward, photons are produced by pair interactions. The two-particle and threeparticle interactions between pairs and photons listed in Tables 11.1 and 11.2 are considered. The results for the properties of the emerging radiation after stationarity is achieved are given, so the total wind luminosity is equal to the energy injection ˙ The results are presented for different values of L, which rate: L = Le + Lγ = E. is the only free parameter. The corresponding surface temperature TS is found from eqs. (12.35) and (12.36). For L > L∗  2 × 1035 ergs s−1 , the emerging emission consists mostly of photons, while pairs dominate for L < L∗ . This simply reflects the fact that for L < L∗ the pair annihilation time is longer than the escape time and the injected pairs remain mostly intact. At higher luminosities, most pairs annihilate before escape, and reconversion into pairs is inefficient, as the mean energy of photons at the photosphere is rather below the pair-creation threshold. At low luminosities, L ∼ 1035 –1037 ergs s−1 , photons formed in the wind escape relatively freely, and the photon spectra resemble a very wide annihilation line. The small decrease in mean photon energy γ from ∼ 500 keV at L  1035 ergs s−1 to ∼ 400 keV at L  1037 ergs s−1 occurs because of the energy transfer from annihilation photons to pairs via Compton scattering (see Figure 12.5). As a result of this transfer, the emerging pairs are heated up to the mean energy e  400 keV at L  1037 ergs s−1 . For L > 1037 ergs s−1 , changes in the particle number due to three-body processes are essential, and their role in thermalization of the outflowing plasma increases with the increase of L. For L = 1042 ergs s−1 , the photon spectrum is almost Planckian, except for the presence of a high-energy tail at γ > 100 keV.

201

γ

εγ

12.3 Emission from Hot Bare Quark Stars

εγ

Figure 12.5 The energy spectra of emerging photons for different values of the total luminosity (measured in erg/s), as marked on the curves. Also shown, for L = 1042 erg · s−1 s, is the Planck spectrum with the same energy density as that of the photons at the photosphere of the outflowing wind (dashed line). This figure is based on improved calculations with the correct Jacobi matrix important for thermalization. The original calculations were presented in [269].

At this luminosity, the mean energy of the emerging photons is ∼ 40 keV, while the mean energy of the black body photons is ∼ 30 keV. For L  1042 ergs s−1 the emerging pair energy spectrum is close to a Maxwellian (4.43), while for L  1040 ergs s−1 it deviates significantly from it. All details of the steady wind for the calculated luminosities can be found in [269]. Later, in [344], two important effects were considered: influence of the general relativistic effects on the photon spectrum and the admixture of a small amount of nonthermal radiation from the surface of the quark star. It was shown that accounting for general relativity leads to redshift of the annihilation line. In addition, the admixture on the nonthermal photons does not remove the annihilation line at the luminosities 1038 –1040 erg/s. It is appropriate to make some remarks. The annihilation line and high-energy tail in the photon spectrum can be a diagnostic tool for the super-Eddington luminosities  1038 erg/s erg. These features can be affected by possible existence of some amount of usual matter around the bare quark star. The results reported here show that photon spectra become Planckian at luminosities  1042 erg/s. In principle, the bare quark star can support very large luminosities  1050 erg/s. Hence a candidate quark star is a compact object with the super-Eddington luminosity. In particular, soft γ -ray repeaters (SGRs), which are the sources of short bursts of hard X-rays with super-Eddington luminosities up to ∼ 1042 –1045 erg/s, are reasonable candidates for quark stars (see, e.g., [310, 345, 312]). The bursting activity

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Kinetics of Particles in Strong Fields

of SGRs may be explained by fast heating of the stellar surface up to the temperature of ∼ (1 − 2) × 109 K and its subsequent thermal emission [346, 347]. The heating mechanism may be either impact of comets into bare quark stars [348] or fast decay of overcritical (∼ 1015 G) magnetic fields [349]. For typical luminosities of SGRs L ∼ 1041 –1042 ergs s−1 , the mean photon energy is ∼ 40 keV, which is consistent with observations of SGRs [350]. Another important idiosyncrasy found is a strong anticorrelation between spectral hardness and luminosity. While at very high luminosities (L > 1042 – 1043 erg/s) the spectral temperature increases with luminosity as in thermal radiation, in the range of luminosities studied, where thermal equilibrium is not achieved, the expected correlation is opposite. Such anticorrelations were, indeed, observed for SGR 1806-20 and SGR 1900+14 where the burst statistics is good enough [351, 352, 353]. The recent progress in the theory of quark stars concerns photon emission. In particular, it was confirmed that for low luminosities L ≤ 1040 erg/s, the radiation in electron-positron pairs dominates that of photons [354]. It was also shown that bremsstrahlung in a strong electric field of the electrosphere can be comparable to the output in pairs [355]. This effect has been discussed, but for details, see [344]. For a review of astrophysical constraints on quark stars, see [356].

13 Compton Scattering in Astrophysics and Cosmology

One of the most simple quantum electrodynamical processes is Compton scattering of photons off electrons. This process plays a vital role in many astrophysical phenomena. In particular, Compton scattering is essential in blazars [357], in accretion discs in binary X-ray sources and AGN [358, 359], in hot coronae of accretion discs [360], in GRBs [361], and in cascades of very high energy cosmic rays [362]. In cosmology, Compton scattering is crucial in the formation of the cosmic microwave background (CMB) radiation [363, 364, 365] and for the propagation of this radiation in clusters of galaxies [366, 367, 368]. Basic properties of Compton scattering are described in almost any textbook on astrophysics (see, e.g., [43, 369, 370]) or cosmology (see, e.g., [371, 372]). In this chapter, some essential aspects of Compton scattering are discussed and various processes where Compton scattering plays an important role are illustrated. In particular, one of the most important astrophysical implications, the Sunyaev-Zeldovich effect, is discussed. The theory of comptonization in static and relativistically moving media and the Kompaneets equation are reviewed. 13.1 The Boltzmann Equation for Compton Scattering In this chapter the focus is on the scattering process (see relation (2.10)), where interacting particles are electron (or positron) and photon: e + γ −→ e + γ .

(13.1)

The energy-momentum conservation is described by eq. (2.11) and can be written as μ pμe + pμγ = p μ e + pγ ,

pμγ

(13.2)

pμe

and are energy-momentum four-vectors of photons and electrons, where respectively. Since photons are bosons and electrons are fermions, generally speaking, the Boltzmann equation (2.14) should contain Pauli blocking and Bose enhancement factors, as in eq. (2.29). In practice, electron degeneracy is weak, and 203

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Compton Scattering in Astrophysics and Cosmology

it is negligible in the nonrelativistic limit, where most applications of Compton scattering take place. Instead, the Bose enhancement cannot be neglected, since otherwise the equilibrium photon spectrum would be given by Wien law instead of Planck law. Then, in what follows, only the Bose enhancement factor is kept, and Pauli blocking is neglected. The collision integral due to Compton scattering of photons off electrons is   (St f )C = d 3 p e d 3 pγ d 3 p γ V wk ,p ;k,p fγ (p γ , x)[1 + fγ (pγ , x)] fe (p e , x)    − fγ (pγ , x) 1 + fγ (p γ , x) fe (pe , x) , (13.3) where the differential rate and the corresponding matrix element are given in Appendix B in eqs. (B.3) and (B.4). In principle, energy and momentum conservation expressed by the δ-function in differential probability wk ,p ;k,p in eq. (13.3) (see Section 11.3) can be used to reduce this 9D integral to the 5D one. Before discussing the Boltzmann equation for Compton scattering further, it is appropriate to discuss how the Compton scattering affects the spectrum of photons. 13.2 Mean Number of Scatterings The key parameter that determines the efficiency of Compton scattering in a medium is called the Compton y parameter, defined as the product of the average fractional energy change in a single scattering ε/ε and the mean number ¯ of scatterings

N ! " ε  ¯  y= N . (13.4) ε For y  1, multiple Compton scatterings alter the initial spectrum of photons significantly, while for y  1, the spectrum is only perturbed to a small degree. The average energy change in single Compton scattering can be computed by averaging over a given distribution of electrons, e.g., Maxwellian distribution (4.43) with a given temperature. This can be done in both nonrelativistic and ultrarelativistic cases [43] with the result ! "  ε 4θ , θ  1, ∼ (13.5) 16θ 2 , θ  1. ε The well-known result [43] for the finite static medium is that the mean number of scatterings is    τ, τ  1, ¯ N stat ∼ (13.6) τ 2 , τ  1,

13.2 Mean Number of Scatterings

where τ is the optical depth of the medium, defined as  1 τ= σ jμ dxμ , c L

205

(13.7)

where σ is the scattering cross section, jμ is the four-current of particles, and dxμ is the element of the world line. It is clear from eq. (13.7) that the optical depth is Lorentz invariant. For the finite medium, which is moving with relativistic velocity v in the direction of photon propagation, the mean number of scatterings is different from the one given in eq. (13.6). Particularly interesting is the case of spherically symmetric relativistic outflows with density decreasing as n ∝ r−2 [285, 279, 277]. Such outflows are likely produced in AGNs and GRBs.1 The outflows can be classified as “photon thick” if the number density in the outflow decreases significantly along the light-like world line connecting the origin and the observer, or “photon thin” otherwise [279, 277]. Roughly speaking, in the former case, such outflow can be represented as a steady wind, while in the latter, it can be considered as a thin shell. The mean number of scatterings in optically thick relativistic outflows can be computed in the reference frame co-moving with the medium. Denote quantities measured in this frame with subscript c, so that nc is the co-moving density, λc = 1/(σT nc ) is the co-moving mean free path of photons and tc is the co-moving time. The mean number of scatterings is defined as  t2   ¯ N = λ−1 (13.8) c cdtc , t1

where t1 is the co-moving time where the photon is injected in the medium which has the optical depth τ , and t2 is the co-moving time when the photon leaves the medium. The integral (13.8) is taken along the average photon path. The mean number of scatterings is shown in Figure 13.1, together with the results of MC simulations [285]. When the outflow can be approximated as a steady rela¯ ∝ τ . This result is in contrast with a tivistic wind, or it is photon thick, one has N static optically thick finite medium; see the second line in eq. (13.6). In the outflows that can be approximated as a thin shell, or photon thin, the number of scatterings ¯ ∝ τ 1/2 for most of the photons. For those photons that scatter at sufficiently is N ¯ ∝ τ 2 , as in the static medium. The transition between these two large distances, N regimes occurs at the characteristic diffusion radius RD = (τ0 η2 R0 l 2 )1/3 , where τ0 is initial optical depth of the outflow launched at the radius R0 , η is dimensionless entropy, and l is the outflow width. At the radius RD , most photons leave the outflow by the diffusion process [279]. 1

Jets are observed in AGNs and are expected in some models of GRBs. The jet with an opening angle ϑ can be considered spherically symmetric when its bulk Lorentz factor is γ > 1/ϑ.

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106

N

104

100

1

0.01 0.01

1

10 4

100

10 6

ti

Figure 13.1 Average number of scatterings as a function of initial optical depth τ . Results of MC simulations are shown with points, while analytic results are shown by the corresponding lines. In the optically thin regime the number of scatterings is proportional to τ . For τ < τ (RD ) it is proportional to τ 2 . For τ > τ (RD ) it is proportional to τ 1/2 . For even larger τ , corresponding to the photon thick case, the number of scatterings is again proportional to τ . Reproduced from [285] with permission. © AAS 2013.

13.3 Kompaneets Equation Most of the studies on Compton scattering are related to the famous Kompaneets equation, which describes the interaction of photons with nonrelativistic electrons. Kompaneets derived [373] his equation in 1949 within the work done for the Soviet Atomic project. For this reason its publication was delayed for several years [374]. In essence, this equation is a Fokker-Planck approximation to the Boltzmann equation with the collision integral for Compton scattering. The derivation here follows [375] (see also [376]). Two main approximations are adopted: r electrons are assumed nonrelativistic with kinetic energy εk ≡ e /me c2 − 1  1; r photons have energies much less than electron rest mass energy γ  me c2 .

It implies that the energy transfer in a single scattering is small, namely, (γ − γ )  γ . This allows expansion in Taylor series of the δ-function in energy in eq. (13.3) in the small parameter εk , as follows: $ #        ∂  δ p γ − pγ δ pe + pγ − pe − pγ = δ pγ − pγ + G pγ , pγ , pe pγ ∂ p γ $ #  2 ∂2   1 2 + G pγ , pγ , pe pγ δ pγ − pγ + O(pe )3 , 2 2 ∂ pγ (13.9)

13.3 Kompaneets Equation

207

where here and in the following, pe ≡ p0e and pγ ≡ p0γ are zero components of the momentum four-vectors of electrons and photons, respectively, and    2   1  G pγ , p γ , pe = pγ − p γ · pe + pγ − p γ . (13.10) me cpγ Electron DF follows from eq. (4.27) in the limit v/c  1, and it is nonrelativistic Maxwellian:

−3/2  (pe − me v)2 2 4 fe (x, pe ) = ne (x) 2π me c θe , (13.11) exp − 2m2e c2 θe where θe = kB Te /(me c2 ) is dimensionless electron temperature and ne (x) is their number density. The first two moments of the DF (13.11) are   3 ne (x) = fe (x, pe )d pe , ne (x)me ve = fe (x, pe )pe d 3 pe . (13.12) In the electron rest frame the matrix element (B.4) summed over polarizations is

# |M f i |2 = 25 π 2 α 2

$ p˜ γ + − sin2 ϑ˜ , p˜ γ p˜ γ p˜ γ

(13.13)

where a tilde denotes quantities in the electron rest frame and ϑ˜ is the scattering angle of the photon in that reference frame. In the reference frame, where radiation is isotropic, the matrix element can also be expanded in Taylor series in the small parameter εk , as follows: |M f i |2 = 25 π 2 α 2

4

Ii + O(εk )3 ,

i=0

I0 = 1 + μ2 , I2 = μ(1 − μ)εk2 ,

(13.14)

$ pγ · pe p γ · pe , I1 = −2μ(1 − μ) + me cpγ me cp γ #

#

pγ · pe p γ · pe I3 = (1 − μ)(1 − 3μ) + me cpγ me cp γ

$2 + 2μ(1 − μ)

(pγ · pe )(p γ · pe ) m2e c2 pγ p γ

,

I4 = (1 − μ)2 εk2 , where μ = cos ϑ. The collision integral (13.3) can be integrated over electron momentum, with the use of relations (13.12), which gives 3 σT ne (St f )C = 4 c



d p γ

p γ pγ



4 do Ci . 4π i=0

The coefficients Ci in expression (13.15) represent the following:

(13.15)

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r Thompson scattering

    C0 = δ(pγ − p γ )(1 + μ2 ) fγ x, p γ , t − fγ (x, pγ , t ) ,

(13.16)

r linear Doppler effect

#

$   ∂ 2 C1 = δ(p − p ) (1 + μ )v − p p γ e γ γ γ ∂ p γ  < · p v v · p e γ e γ − δ(pγ − p γ )2μ(1 − μ) + cpγ cp γ     × fγ x, p γ , t − fγ (x, pγ , t ) ,

(13.17)

r quadratic Doppler effect

 # $ 2   1 ∂2 C2 = δ(pγ − p γ ) (1 + μ2 ) ve · pγ − p γ (13.18) 2 2 ∂ pγ # $   · p v   v · p ∂ e γ e γ − δ(pγ − p γ ) 2μ(1 − μ) + ve · pγ − p γ ∂ pγ cpγ cpγ #    pγ · ve p γ · ve 2 2 + δ(pγ − pγ ) −(1 − 2μ + 3μ )ve + 2μ(1 − μ) c2 pγ p γ  2 ⎤⎫ ⎬ ve · p γ v e · pγ ⎦ + (1 − μ)(1 − 3μ) + ⎭ cpγ cp γ     × fγ x, p γ , t − fγ (x, pγ , t ) ,

r thermal Doppler effect

 # $ 2  1 ∂2 C3 = δ(pγ − pγ ) (1 + μ2 ) pγ − p γ (13.19) 2 2 ∂ pγ # $ ∂ 2 − 2μ(1 − μ ) δ(pγ − pγ ) (pγ − p γ ) (13.20) ∂ p γ <     + δ(pγ − p γ )(4μ3 − 9μ2 − 1) θe fγ x, p γ , t − fγ (x, pγ , t ) ,

r and recoil effect

$   2 − p p ∂ γ γ δ(pγ − p γ ) (1 + μ2 ) (13.21) C4 = − ∂ p γ 2me

    2 fγ (x, pγ , t ) fγ x, pγ , t + fγ x, pγ , t . × fγ (x, pγ , t ) + (2π h¯ )3 #

13.3 Kompaneets Equation

209

Assuming further that the distribution of photons is isotropic and close to equilibrium, integration over photon momentum can be performed resulting in a more tractable collision integral:    2   ∂ fγ 3 2 11 2 2 ∂ 2 fγ 2 (St f )C = σT ne ς + ve − ς pγ v + ς pγ 2 + (13.22) ∂ pγ 20 e 20 ∂ pγ 

   ∂ 4 2 ∂ fγ pγ me c θe , + fγ 1 + fγ + σT ne ∂ pγ ∂ pγ where ς = pγ · ve /(pγ c). This result generalizes the collision term obtained by Kompaneets [373] to the case where the medium has nonzero, but nonrelativistic, velocity. When the medium is at rest, only the last term in expression (13.22) survives. This gives the familiar form of the Kompaneets equation written for dimensionless quantities. It can be obtained using invariant definition of the Compton parameter (13.4), which is  1 y= θe σ jμ dxμ . (13.23) c L The DF can be expressed through the occupation number (see eq. (2.29)), defined as n(x, ¯ p) = (2π h¯ )3 f (x, p)

(13.24)

and the variable x = γ /kB Te = εγ /θe expressing the dimensionless photon energy. As a result [373, 377] one has 

∂ n¯ 1 ∂ ∂ n¯ = 2 x4 + n( ¯ n¯ + 1) . (13.25) ∂y x ∂x ∂x Interestingly, Kompaneets derived his equation based on physical arguments and not on formal calculation of all terms (13.16)–(13.21) in expansion of the collision integral for the Compton scattering. Indeed, the fact that the number of photons is conserved, as is the case of Compton scattering, allows one to fix the form of the kinetic equation uniquely. In fact, it has to be in a conservative form, ∂ n/∂y ¯ = −2 2 ¯ ¯ εγ ∂/∂εγ (εγ jγ ), where jγ is the photon flux in the energy space. Considering that the average fractional energy change of the photon is εγ /εγ ∼ θ ∼ εγ (see eq. (13.5)), expanding the collision integral (13.3) in terms of εγ , and using the fact that the Planck function n¯ 0 (x) = [exp(x) − 1]−1

(13.26)

with θγ = θe is a solution of the kinetic equation, it is possible to recover the function j¯γ and thus derive eq. (13.25) (see [373, 374]). Since the total photon number is conserved, the spectrum redistribution is such that the reduction in the

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Compton Scattering in Astrophysics and Cosmology

number of photons in one energy range results in the increase of photon number in another range. It is easy to verify that also the Bose-Einstein distribution n(x) ¯ = [exp(x + ν ) − 1]−1 , where ν = φγ /kB Te is dimensionless chemical potential of photons, is a solution of eq. (13.25). It implies that Compton scattering may lead to the so-called μ-type distortions [365], inducing nonzero chemical potential. From the mathematical point of view, exact solutions of the full partial differential equation (13.25) were found only recently [378]. When the quadratic term n¯ 2 can be neglected, the equation becomes linear and its exact solutions are well known [373] (see also [379]). It was also shown [364, 374] that when the term n¯ 2 dominates, SWs may form in the spectrum, leading to appearance of quasilines. In the next section the case with θγ  θe , most relevant for the Sunyaev-Zeldovich effect, is considered. 13.4 Sunyaev-Zeldovich Effect Just after the discovery of the CMB [380], Sunyaev and Zeldovich [366, 367] predicted that the CMB spectrum should be modified in the direction of a cluster of galaxies, the famous Sunyaev-Zeldovich (SZ) effect. Simple considerations on hydrostatic equilibrium can be used for estimation of electron temperature Te . The characteristic or virial radius, Rv (see eq. (14.42)), of a cluster is defined from the theory of structure collapse in an expanding universe. For clusters of galaxies Rv ∼ 1 Mpc, the intercluster gas, which constitutes approximately 13% of the total mass of the cluster of galaxies, is ionized and heated by the gravitational infall to temperatures close to the virial temperature kB Te ∼ GMm p/Rv (see eq. (14.42)), which ranges in clusters from 1 to 15 keV. Heated plasma must emit, and thermal bremsstrahlung emission from the intercluster medium is indeed observed [381]. Such nonrelativistic temperatures justify the use of Thomson scattering cross section σT as a first approximation. Considering the electron density in clusters of galaxies ne , the scattering optical depth is approximately τ  ne σT Rv ∼ 10−2 . At every scattering the photon energy change is ε/ε ∼ θe ∼ 10−2 . It implies that the overall change in the brightness of the CMB in a cluster of galaxies is approximately 10−4 , which is an order of magnitude larger than the primordial temperature fluctuations in the CMB. Unlike primordial fluctuations in the CMB, the SZ effect is local. Moreover, the SZ signature is redshift-independent, which makes the SZ effect a useful cosmological tool [382]. For instance, the distance to clusters can be determined from the analysis of the SZ effect and X-ray data, providing independent estimates of the Hubble constant. Large observational campaigns with

13.4 Sunyaev-Zeldovich Effect

211

dedicated instruments, such as South Pole Telescope,2 are designed to utilize the SZ effect to survey galaxy clusters [383] and produce catalogs of clusters of galaxies [384]. There are many excellent reviews, including [385, 386, 376, 387, 388], where this subject is treated.

13.4.1 Spectral Distortions The spectral distortion due to the SZ effect is best illustrated in the nonrelativistic approximation, from the analysis of eq. (13.25). Assuming θe  θγ , it is useful to transform to a new variable ξ = εγ /θe . In the limit of small ξ one has ∂ n/∂ξ ¯  2 n, ¯ n¯ , so it is safe to neglect the second term in the RHS of eq. (13.25), which reduces to  ∂ n¯ ∂ n¯ 1 ∂ = 2 ξ4 , (13.27) ∂y ξ ∂ξ ∂ξ where the Compton parameter (13.4) can be represented as  y = θe dτ.

(13.28)

Equation (13.27) can be reduced to the standard diffusion equation. Considering that for y  1 the spectral distortion is a small perturbation of the Planck spectrum (13.26), eq. (13.27) can be solved by noting that ∂ n/∂y ¯  n/y. ¯ Then the spectral distortion is x  xy exp(x)  n¯ = x coth − 4 . (13.29) [exp(x) − 1]2 2 In the limit x  1 it implies n/ ¯ n¯ = −2y. For arbitrary x and y the spectral distortion can be represented in the integral form [367]

 ∞ dξ (ln x − ln ξ + 3y)2 1 n¯ 0 (ξ ) exp − . (13.30) n¯ = √ 4y 4π y 0 ξ The energy density of radiation can be obtained by multiplying n¯ by x3 and integrating over all x with the result ρ(y) = σSB Tγ4 exp(4y), where Tγ is the unperturbed photon temperature. 2

https://pole.uchicago.edu/

(13.31)

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13.4.2 Relativistic Corrections There are two main concerns about the validity of the results reported here. First, the optical depth of clusters of galaxies is low τ  1, which means that most photons are not scattered at all, and few are scattered only once. This is in apparent contradiction with the Fokker-Planck approximation to the Boltzmann equation used to derive eq. (13.25). Besides, as the temperature in some clusters is as high as 15 keV, the approximation that electrons are nonrelativistic appears to be insufficient (see Figure 4.2). This requires more accurate treatment, which is discussed subsequently following [389, 387]. The distortion of photon spectrum can be calculated as follows. The probability of scattering of a photon [40] in electron rest frame is

3 1 P(μ, μ ) = 1 + μ2 μ 2 + (1 − μ2 )(1 − μ 2 ) , (13.32) 8 2 where the prime denotes quantities after scattering. The photon energy change is     εγ 1 + βμ ς = ln . (13.33) = ln εγ 1 − βμ The probability of a single scattering of a photon is obtained by the integration over the angles before the scattering as  (1 − β 2 )2 1 1 + βμ P(ς , β ) = P(μ, μ )dμ. (13.34) 3 2β (1 − βμ) −1 The distortion of photon spectrum is then obtained by integration of this probability with the electron distribution, given by eq. (4.44). For arbitrary optical depth the final spectrum is a weighted sum over the probabilities for any number of scatterings. However, since for clusters of galaxies, τ  1, it is sufficient to write P(ς ) = (1 − τ )δ(ς ) + τ P1 (ς ),

(13.35)

where the first term describes the scattering without energy change, while the second terms correspond to a single scattering. As a result, the spectral energy density of thermal radiation   I0 (x) = 2(kB Tγ )3 /(hc)2 x3 [exp(x) − 1]−1 (13.36) gets modified due to single scattering as I = τ [&(x, θe ) − 1], I0

(13.37)

13.4 Sunyaev-Zeldovich Effect

213

where &(x, θe ) = A(θe ) [ϕ1 (x, θe ) + ϕ2 (x, θe )] , (13.38)  1   1  1 tdt[exp(x) − 1] γ −1 ϕ1 (x, θe ) = γ dβ exp − q(t, μ, β )dμ, exp(xt ) − 1 θe 0 βm μm  1   μM   1  dt exp(x) − 1 γ −1 γ dβ exp − q(t, μ, β )dμ, ϕ2 (x, θe ) = 3 θe −1 0 t exp(x/t ) − 1 βm β −2 (3μ2 − 1) [(1 − βμ/t ) − 1]2 + (3 − μ2 ) , (1 − βμ)2  1

−1 3 γ −1 2 5 A(θe ) = β γ exp(− )dβ , 32 0 θe   1−t t 1 t −1+β , μm = − 1 − β , μM = . βm = 1+t β t tβ

q(t, μ, β ) =

These equations generalize the result in eq. (13.29) to a relativistic case, still under assumption τ  1. In clusters of galaxies, also a population of nonthermal electrons is present. Photons from the CMB can also scatter these electrons. Clearly the spectral distortions due to this nonthermal SZ effect should be analyzed based on the relativistic formalism discussed in this section. The convolution of the scattering probability in this case is done with the appropriate nonthermal electron distribution, instead of the relativistic Maxwellian given by eq. (4.44). It is important to stress that the thermal SZ effect is dominant with respect to the nonthermal one [388].

13.4.3 Kinematic SZ Effect Observations of the large-scale structure of the universe reveal that clusters of galaxies are not at rest with respect to the reference frame in which CMB is isotropic. It implies that spectral distortions in the CMB may occur not only due to the thermal SZ effect but also due to the bulk motion of clusters of galaxies, which is referred to as the kinematic SZ effect. The fact that radiation temperature decreases due to the bulk motion of clusters with temperature variation in the direction of the cluster given by Tγ vr τ , Tγ c

(13.39)

where vr is the velocity component along the line of sight, has been known for a long time [368]. The spectrum distortion due to this effect was first computed in [390] using the Boltzmann equation for photons; see also [388] for the derivation

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from the radiative transfer equation. The specific intensity change is found: I vr x exp(x) = −τ . I c exp(x) − 1

(13.40)

This result is valid for arbitrary velocity vr but for small optical depth τ  1. In principle, polarization measurement can supply the information on the tangential velocities as well [391]. Currently the SZ effect is not only observed in clusters and superclusters of galaxies but also used for detection of clusters [383]. It is also used together with other observations for constraining cosmological parameters [392, 393].

13.5 Comptonization in Static Media Many astrophysical sources, such as X-ray binaries [394] and AGNs [395], show power law spectra in X-rays. Such power law spectra naturally arise when soft photons are scattered off hot electrons in a cloud located near the source [396, 397, 358]. The process of formation of photon spectrum under multiple Compton scatterings is called comptonization. Usually, comptonization is understood as upscattering of photons with the final spectrum being harder than the initial one. In most studies of comptonization, electrons are assumed nonrelativistic. Here, following [398], the general case of relativistic electrons and arbitrary optical depth of the cloud is considered. Recall that the problem of X-ray spectrum formation in a hot plasma is connected to the number of scatterings that photons experience before they escape the cloud. Historically, this problem has been treated numerically applying the MC technique (see Section 8.2). Assume for simplicity that the cloud is spherical and uniformly filled with plasma; it has optical depth τ0  1 and temperature θe . The number of scatterings at the time t given in eq. (13.8) reduces to u = σT nct. For the impulsive source located at the center of the homogeneous sphere the distribution of the number of scatterings is given in [358] as # $ √   3τ02 3 3 τ02 π 2u 8π 2 u3/2 P(u) = √ 3/2 1 + −  exp − (13.41) 2 . 4u 4 πu 9τ04 3 τ0 + 23 The average number of scatterings is u¯ = τ02 /2, which agrees with the second line in eq. (13.6), while the average photon escape time is t¯ = τ0 R/(2c). In particular, it follows from eq. (13.41) that the distribution of the number of scatterings in the regime u  u¯ is P(u) = A(u, ¯ τ0 ) exp(−β ∗ u),

(13.42)

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215

  2  where β ∗ = π 2 / 3 τ0 + 23 . This regime is well described in the diffusion approximation, and the coefficient β ∗ represents the first eigenvalue of the differential operator L = (1/3)d 2 /dτ 2 . Consider now a disk with Thomson optical depth τ0 filled with free electrons at the temperature θe . The specific intensity of radiation in the plane geometry [359] is determined from the equation  ∞  1 ∂I(ν, μ, τ ) = dν d (13.43) μ ∂τ σT n 0 4π  ν       × σs ν → ν,  ·  , θe I ν , μ , τ − σs ν → ν ,  ·  , θe I(ν, μ, τ ) , ν where ν and ν are the photon frequencies before and after the scattering, respectively, μ = cos ϑ is the angle with respect to the normal to the plane,  and  are photon unit direction vectors before and after scattering, respectively, and τ is the Thomson optical depth. The scattering kernel σs [399] depends on the plasma temperature, the photon frequencies, and the cosine of the angles between the two photon directions. Assume that photons are injected with energies hν  me c2 θe , Considering the low-energy part of the spectrum hν < min(me c2 θe , me c2 ) only Doppler effect is taken into account, so the scattering kernel is # $     1 −  ·  2 3σT n 1 D D 3 f (β ) σs = d β 1+ 1− , − δ γ 16π νz γ DD z z (13.44) where z = hν/(me c2 ); the electron DF is the relativistic Maxwellian, given by eq. (4.44); and D = 1 − μβ is the Doppler factor. The crucial observation made in [398] is that the energy integral operator in eq. (13.43) satisfies the self-similarity property L(z1 , z) ≡ L(uz1 , uz) for any u = 0, so its eigenfunction is a power law. Owing to this property, the solution of eq. (13.43) can be factorized as I(ν, μ, τ ) = J(μ, τ )z−α p ,

(13.45)

where α p is the power law index. Then the problem is reduced to the solution of equations μ

dI(ν, μ, τ ) = −J(μ, τ ) + B(μ, τ ), dτ   1 1  B(μ, τ ) = R  ·  J(μ , τ )dμ , 2 −1  α p +2    3   1  D 3 f (β ) ˜ · ˜ 2 , R · = d β 2 1 +  4 γ D D

(13.46) (13.47) (13.48)

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˜ and  ˜ are measured in the electron rest frame. These equations imply where  that the radiative transfer in photon energy space and in configuration space can be treated separately. The solution of eqs. (13.46)–(13.48) gives ⎧  β∗ 9 ⎪ + θe , θe  1, ⎨ 4 αp = (13.49) %  & ⎪ 3[(α p +3)α p +4] (2α p +2) ∗ ⎩ 1 − ln  1. β , θ e (α p +3)(α p +2)2 ln (4θ 2 ) e

The spectral index α p is a function of just two parameters: electron temperature θe and the optical depth τ0 of the cloud. These results also imply that power law spectra are the exact solutions of the radiative transfer equations in the case of low-energy photon injection hν  me c2 θe . Clearly, at the high-temperature branch, θe  1 of the solution (13.49) is of limited interest, since at these temperatures in the optically thick plasma, all binary and triple interactions discussed in Chapter 11 are important.

13.6 Comptonization in Relativistic Outflows It is well known that the last Compton scattering of CMB photons produces spectral distortions [363]. In particular, because the baryons are nonrelativistic with an adiabatic index = 5/3, compared to the adiabatic index = 4/3 for the photon gas, the photons must transfer energy to the baryons during expansion of the universe, as the former cool adiabatically slower than the latter. A similar effect is present in emission from the photosphere of relativistic outflows. In particular, the low-energy part of the Planck spectrum gets modified significantly by this effect [287]. Photospheric emission is considered among the leading mechanisms generating gamma rays from relativistically expanding plasma in GRBs. It is worth discussing such emission in more detail (for reviews, see [277, 400]). Consider a spherically symmetric steady (time-independent) ultra-relativistic medium moving with velocity βc. In the ultra-relativistic limit the radiative transfer equation (2.39) becomes ill defined (see, e.g., [282]). It is convenient to write down the radiative transfer equation in the reference frame co-moving with the medium, with the only exceptions for radius r and Lorentz factor γ being measured in the laboratory frame: μ + β ∂ n(r, ¯ , μ) 1 − μ2 ∂ n¯ 1 − μ2 ∂ n¯ 1 η − χn + − β = , 1 + βμ ∂r r ∂μ r(1 + βμ) ∂ γ (1 + βμ) 1 + βμ (13.50) where n(r, ¯ , μ) = h3 f (r, , μ) is photon occupation number and  is photon energy.

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217

The collision integral takes into account that photons are scattered by the moving medium [401, 402] and consequently their distribution is anisotropic in the comoving reference frame:      3ne σT  2 η − χn = (1 − 2εe ) do (1 + x )n − n + 2εe do (x3 + x)n 16π   3 2 + 2θe do (2x − 3x − 2x + 1)n + do (1 + x2 )(1 − x)   1 ∂ 4 ∂ 1 ∂ 4 ∂ 2 (13.51) ε + 2 ε + 2n ε n , × θe 2 εγ ∂εγ γ ∂εγ εγ ∂εγ γ ∂εγ γ where θe is electron co-moving temperature, ne iselectronco-moving density, n¯ = n(, ¯ μ), n¯ = n(, ¯ μ ), do = dμ dφ , and x = 1 − μ2 1 − μ 2 cos(φ − φ ) + μμ . Here prime denotes quantities after the scattering. In the isotropic case the integration of collision integral (13.51) over angles gives the Kompaneets equation (13.25). Equation (13.50) with collision integral (13.51) can be integrated numerically by introducing the computational grid for angles and energy and applying the method of lines for evolutionary equations with continuous coordinate r. The scheme for the classical Kompaneets equation is similar to the one used in [379]. In numerical calculations one has to assume the dependence of temperature and number density of electrons on radius. Following [279], the co-moving density profile is assumed to be  2 R0 ne = n 0 B , (13.52) R ˙ 2 /L is the baryonic loading parameter, L is luminosity at the base of where B = Mc the wind at radius R0 , and M˙ is mass ejection rate. For the co-moving temperature it is assumed that  k R0 Te = T0 B , (13.53) BR where T0 = [L/(16π σSB R20 )]1/4 and k is a constant. This model with k = 2/3 describes steady adiabatically expanding relativistic wind with constant velocity. The electron co-moving density decreases with increasing laboratory radius, and so does the optical depth. Near the photosphere, where the optical depth decreases to unity, the coupling between photons and electrons weakens. It is possible to assume that electron co-moving temperature on the path of photons propagating outward is decreasing following eq. (13.53), even when the outflow becomes optically thin. This effect is parametrized by the coefficient k ≥ 2/3. Electrons are

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Figure 13.2 Anisotropy of photon distribution in the co-moving reference frame developing near the photosphere. When the outflow has high optical depth, the distribution is isotropic, but near the photosphere, it becomes increasingly anisotropic. The direction of medium motion is to the right. The optical depth for these curves is τ = 100, 10, 5, 1.4, 0.7, 0.3 with increasing anisotropy. Reproduced from [287] with permission.

described by the Maxwellian DF (4.43) in the co-moving reference frame with the temperature given by the relation (13.53). Since initially the outflow is highly opaque, photons have Planck spectrum and isotropic distribution in the co-moving reference frame. Near the photosphere, the coupling of photons to the medium is due to Compton scattering on electrons. Near the photosphere, the photon distribution in the co-moving reference frame becomes anisotropic [282]. This effect, illustrated in Figure 13.2, can be explained from the geometric point of view. Since collisions tend to isotropize photons, only those photons that are undergoing scatterings have nearly isotropic distribution. In contrast, photons that already experienced their last scattering have increasingly anisotropic distribution in the co-moving reference frame due to relativistic aberration. Hence the local photon field at the photosphere that contains all photons, those which continue to scatter and those which already propagate freely, becomes more and more anisotropic. The numerical solution of eq. (13.50) is obtained in the co-moving reference frame. To find the photon spectrum, which a distant observer can detect, one has to transform the spectrum into the laboratory reference frame using the following transformations:  1 μL − β N(L ) = 2π dμL r2 μL n(, μ))L2 ,  = γ (1 − βμL )L , μ = , 1 − βμL 0 (13.54) where the subscript L denotes quantities in the laboratory reference frame.

13.6 Comptonization in Relativistic Outflows

219

Figure 13.3 The spectrum of photons Fν transformed to the laboratory reference frame shown for selected values of radii: 5 × 1010 , 1011 , 1012 , 1013 , and 5 × 1014 cm. Solid curves show the observed spectra of photons integrated over angles, and dashed curves show the spectra of photons arriving to a given radius with Planck spectrum in the co-moving reference frame, transformed to the laboratory reference frame. Reproduced from [287] with permission.

The results are illustrated as follows with the model of steady relativistic wind characterized by the following parameters: luminosity L = 1054 erg/s, Lorentz factor γ = 500, and k = 2/3. Given these values, the photospheric radius is R ph =

σT L = 4.4 × 1012 cm. 8π m pc3 γ 3

(13.55)

The computation starts at large optical depth τi ∼ 102 , proceeds through the photosphere, and ends up at small optical depth τ ∼ 10−2 . While the spectrum in the co-moving reference frame initially keeps the Planck shape, it becomes distorted with decreasing optical depth. It is also the consequence of interaction with electrons, which on average have smaller energy than photons do (see Figure 13.3). In fact, near the photosphere, the average energy of photons saturates [282], but the one of electrons continues to decrease. When this spectrum is transformed to the laboratory reference frame using eq. (13.54), increasing deviations from Planck spectrum are found, essentially due to contributions from different angles (see Figure 13.3). Clearly this effect becomes less and less prominent with decreasing optical depth. These results agree quantitatively with those obtained by different methods: MC simulations [285] and “fuzzy photosphere” approximation [279]. The power law index k parametrizes the temperature profile of electrons in the radial direction. Generally speaking, this index can take different values, not only k = 2/3 as in the steady relativistic wind with constant velocity. It is interesting that the number of photons in the low-energy part of the spectrum increases with increasing k. This low-energy part of the spectrum can be fit with a power law. Performing the fit in the energy range between 1 and 10 keV, it is found [287] that

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the power law index changes as follows: α  1 − k.

(13.56)

This power law index α actually coincides with the one adopted in the Band phenomenological spectrum [403] and used frequently in GRB studies. In particular, for the temperature profile T ∝ r−2 , the typical low-energy photon index observed in GRBs α  −1 is recovered. This result indicates that the photospheric emission [277] can play a crucial role in the GRB phenomenon. 13.7 Monte Carlo Simulations of the Photospheric Emission from Relativistic Outflows In this section, MC simulations are used to study the decoupling of photons from ultra-relativistic spherically symmetric outflows [285]. Depending on the duration of the outflow, one of two basic approximations is generally adopted: steady relativistic wind [340, 341] or thin shell [218, 404, 273]. It is shown [279] that in both wind and shell models, when outflows with finite duration are considered, some important aspects of the optical depth behavior are overlooked. A new physically motivated classification was introduced [279]: photon thick and photon thin outflows with, respectively, R ph < Rt and R ph > Rt , with R ph being the photospheric radius, Rt = 2γ 2 l the radius of transition from acceleration to coasting expansion and l the laboratory width of the outflow; for a review, see [277]. In this section the probability of last scattering in finite relativistic outflows is discussed. The light curves and spectra from photon thick outflows are obtained and compared with results obtained by different methods. Following [279], consider a spherical relativistic wind of finite width l with laboratory number density of electrons being zero everywhere, except in the region R(t ) < r < R(t ) + l, where it is given by eq. (13.52). This outflow expands with constant Lorentz factor γ , so that R(t ) = R0 + βct, and accelerating outflows are not discussed. Then the optical depth is given by τ (r, ϑ, t ) = τ0 R0

#

ϑ − tan−1



r sin ϑ ct+r cos ϑ

r sin(ϑ )



 −β

$ 1 1 − , r (ct + r cos ϑ )2 + (r sin ϑ )2 (13.57)

where τ0 = σT n0 R0 =

σ Ll , 4π m pc3 R20 γ

(13.58)

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221

and r is the position of photon emission, ϑ is the angle between momentum of photon and the radius vector, t is the time the photon remains within the outflow, and L is outflow luminosity. Time t is found via the equations of motion of the outflow and that of the photon. The optical depth in finite relativistic outflows has two different asymptotics, depending on initial conditions: in the photon thick case, τ is almost constant within the outflow; instead, in the photon thin case, it is linearly increasing with depth from the outer boundary at R(t ) + l. It is found [279] that diffusion of photons within the outflow plays a crucial role in the photon thin case: the radius at which photons effectively diffuse out of the outflow is smaller than the photospheric radius R ph . The latter is defined by equating expression (13.57) to unity for emission with ϑ = 0 and t corresponding to the inner boundary. Hence, when the photon thin outflow reaches the photospheric radius, few photons are left in it. A spherically symmetric MC simulation of photon scattering within the outflow described by eq. (13.52) is employed in [285]. In this simulation, each photon is followed as it interacts with electrons until it decouples from the outflow. Photons are injected in the outflow at the optical depth of the inner boundary τi given in Table 13.1. It is also assumed that their initial DF is isotropic and thermal. The radial position of each photon inside the outflow is chosen randomly. The MC code consists mainly of a loop computing each scattering. One has to proceed in two steps. First, an infinite and steady wind also treated by [282] with l → ∞ and R → 0 is considered. For a given position characterized by r and ϑ of photon in the outflow, a maximal value for the optical depth τmax is computed using (13.57) with t → ∞. The probability for the photon to decouple from the outflow is exp(−τmax ). Then a random number X ∈ [0, 1] is chosen. On the one hand, if X < exp(−τmax ), the photon is considered decoupled. Afterward, the photon remains in the outflow but does not scatter. This case corresponds to decoupling in the photon thick case, for which the presence of boundaries is not essential. On the other hand, if X ≥ exp(−τmax ), the position rs of next scattering is computed from the optical depth   1 1 ϑ − ϑs −β − , (13.59) τ (r, rs , ϑ ) = τ0 R0 r sin(ϑ ) r rs where ϑs is the angle between photon momentum and the radius vector at the position of scattering. The new radial position rs such that exp[−τ (r, rs , ϑ )] ≡ X is found by iterations. The second step is to take into account the finite width of the outflow. To do so, rs is compared with the radii of the inner and outer boundaries of the outflow. If rs < R(ts ) or rs > R(ts ) + l, the scattering does not take place and the photon is

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Table 13.1 Parameters for the simulations

R0 (cm) l (cm) τ0 E0 (erg) γ R ph (cm) RD (cm) τi

Photon thick

Photon thin

108 4.5 × 108 2.3 × 1010 1.5 × 1052 500 4.6 × 1012 3.5 × 1013 102

108 108 1.2 × 1013 1054 300 3.3 × 1014 1.0 × 1014 1.3 × 103

considered decoupled. Such decoupling occuring at the boundaries corresponds to the photon thin case. In the opposite case, R(ts ) ≤ rs ≤ R(ts ) + l, the scattering is assumed to occur and the loop is repeated until the photon decouples. Two models of scattering are considered: coherent isotropic scattering and Compton scattering. The former is treated for comparison with results in the literature, such as [282]. Such a model is also interesting per se since it preserves Planck spectrum and traces only geometrical effects. Instead, when Compton scattering is considered, the equilibrium spectrum is the Wien one because stimulated emission is not taken into account by linear MC simulation. For initially large optical depth the spectrum indeed first evolves to the one described by the Wien law, and only at the photosphere does it change shape again. The coherent scattering is computed by Lorentz transformation to the reference frame co-moving with the outflow. In addition, when Compton scattering is considered, another Lorentz transformation to the rest frame of the electron is performed. The electron is chosen randomly from the Maxwell distribution (4.43) at a given co-moving temperature defined as  2/3   3E0 1/4 −1/3 R0 T = T0 γ T0  , (13.60) r 4π aR30 where a = 4σSB /c (see, e.g., [405]). The results for two specific cases are presented below. The set of parameters for these simulations are given in Table 13.1. The left (right) column corresponds to the photon thick (thin) case. Consider first the probability density of photon last scattering position as a function of depth (see Figure 13.4, top). Photon decoupling from photon thick outflows is expected to be local and the presence of boundaries should not change this probability substantially; indeed, the probability distribution function of last scattering is found to be almost independent of the depth. On the contrary, in photon thin

13.7 Monte Carlo Simulations of the Photospheric Emission 4

223

Photon thick Photon thin

3.5 3 Inner boundary

dN/d

2.5

Outer boundary

2 1.5 1 0.5 0 0

0.2

0.4

0.6

0.8

1

/l a

b

c

d

dP/d(log r)

1

0.1

4

3

2

1

0.01 0.1

10

1

100

r/Rph

Figure 13.4 (top) Probability density of last scattering as a function of normalized depth for photon thick and photon thin outflows (in the latter case decreased by a factor of 10 for a better presentation). (bottom) Probability density function for the position of last scattering in the following cases: infinite and steady wind (1), photon thick case of Table 13.1 (2), photon thick case of Table 13.1 with a different Lorentz factor = 300 (3), and photon thin outflow (4). The vertical line (a) represents the diffusion radius in the photon thin case, line (b) represents the photospheric radius, while lines (c) and (d) show the transition radius Rt for curves (2) and (3), respectively. Both coherent and Compton scattering models give the same results. Reproduced from [285].

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F (arbitrary units)

1

0.1

5

0.01

3

4 2

1

0.001 0.1

1

10

h /kTLOS

Figure 13.5 Time-integrated spectrum of photospheric emission from photon thick outflow computed within different approximations: Planck spectrum (1), sharp photosphere approximation (2), fuzzy photosphere approximation (3), MC simulation for Compton scattering (4), and MC simulation for coherent scattering (5). Spectra (1), (2), and (3) are shifted in energy by a factor 1.58 (see text for details). TLOS is the laboratory temperature of the outflow at the photospheric radius. Reproduced from [285].

outflow, there is enough time for photons to be transported at the boundaries by diffusion, as discussed in [279]. As a result, the probability density is peaked at the boundaries. The difference between photon thin and photon thick outflows is also reflected in the probability density of last scattering as a function of the radius, shown at the bottom of Figure 13.4. In the photon thin outflow, most photons escape from the outflow well before the photospheric radius, namely, near diffusion radius [279] RD = R0 (τ0 γ 2 l 2 )1/3 . Instead, for photon thick outflows the probability density function of last scattering is found to be close to the one of infinite and steady wind, found in [282]. The finite extension of the outflow results in the exponential cutoff for the probability density function at radii larger than Rt . It is remarkable how different the probability densities for photon thick and photon thin cases are. Not only the positions of the maximum but also the shapes are different. In the photon thin case at small radii the number of photons that diffuse out is determined by the change in the diffusion coefficient, i.e., the electron density, which follows a power law. In expanding plasma, while the density decreases, the mean free path for photons increases, which makes the diffusion faster. At large

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225

radii, almost all photons have already diffused out, and the probability has an exponential cutoff. As for the photon thick case, the probability density dependence on the radius is opposite. For small radii the probability for last scattering is exponentially suppressed because of large optical depth, while at large radii it follows a power law, as discussed in [406]. At larger radii an exponential decrease due to the finiteness of the outflow is found (see Figure 13.4). In Figure 13.5, time-integrated spectra of photon thick outflow obtained with different models are computed. Each of them involves 5 × 107 photons. The results obtained in fuzzy approximation are in good agreement with MC simulations. The latter gives the low-energy spectra index α = 0.24 for Compton scattering and α = 0.19 for coherent scattering, to be compared with α = 0 found in [279]. For spectra and light curves from photon thin outflows, see [285].

14 Self-Gravitating Systems

Gravity is one of the four fundamental interactions, and it is the weakest one. Nevertheless, on largest scales, the structure of the universe is governed by gravity alone. This is essentially due to the long range and unscreened nature of gravitational interaction. These properties make self-gravitating systems with large numbers of particles very different from other systems with short-range interactions, such as gases, or systems with long-range screened interactions, such as neutral plasmas. In this chapter, kinetic properties of self-gravitating systems are discussed and contrasted with kinetic properties of gases and plasmas discussed in previous chapters. While before the focus has been always on relativistic aspects of KT, most essential properties of self-gravitating systems can be understood within a nonrelativistic context. From the statistical physics point of view, there are fundamental differences between systems like gases and plasmas and systems with unscreened long-range interactions, in particular, self-gravitating systems.1 First, extensivity of energy does not hold for self-gravitating systems. Extensivity is a property of thermodynamic variables such as energy entropy and free energy, that is, their proportionality to the size of the system. In contrast, intensive variables, such as entropy or temperature, do not depend on system size. Generally speaking, extensivity does not hold for any system with long-range interactions. In principle, the extensivity property can be restored if the interaction energy scales with the inverse of particle number [407]. However, systems with long-range interactions also lack additivity of energy, namely, such a property that the system can be divided into parts with total energy being the sum of energies of these parts, with the energy of interaction between parts vanishing in the thermodynamic limit. These unusual properties have a consequence, which is ensemble inequivalence 1

Long-range interaction is assumed to be described by a potential V (r) ∝ r−a , where a < d and d is the dimensionality of the space.

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227

[408], namely, that statistical descriptions based on canonical and microcanonical distributions give different results. Recall that a microcanonical ensemble is a statistical ensemble with specified and constant energy. In contrast, a canonical ensemble is a statistical ensemble of particles in thermal equilibrium with a heat bath at a given temperature. Second, systems possessing only short-range interactions usually exist in different phases, e.g., gas, liquid, or solid. For a system in equilibrium at high temperature, such as gas, the kinetic energy of particles dominates their potential energy. As the temperature decreases, the kinetic energy decreases as well, and eventually becomes comparable to the potential energy. At this point a phase transition to a more condensed phase, say, a liquid, occurs. Remarkably, for self-gravitating systems in virial equilibrium, kinetic energy of particles is comparable to their potential energy. For this reason, virialized self-gravitating systems are similar to other systems with short-range interactions, such as gases at the verge of a phase transition [409]. In addition, it is well known that self-gravitating systems in virial equilibrium are characterized by negative heat capacity. While for systems described by a canonical ensemble, this is not possible, the microcanonical distribution admits this possibility. Negative specific heat is an unusual property of a medium that respond to increasing energy by lowering its temperature. Such a property is observed experimentally in mesoscopic systems [410] near the phase transition and is not uncommon [411] in nature. For discussion of these issues, see an excellent recent monograph [412]. Third, a peculiar property of self-gravitating systems is that the interaction potential is bounded from above. This leads to the possibility of particle evaporation from a self-gravitating system. Such a property makes self-gravitating systems special, as the total number of particles is not conserved. In addition, particle evaporation clearly makes impossible establishment of any steady state. To avoid difficulty with particle evaporation from open systems, usually they are considered to be confined in a (e.g., spherical) box. Fourth, systems with short-range interactions can be divided into noninteracting subsystems. This fact has a consequence that equilibrium systems in the absence of external fields are homogeneous. In previous chapters, it was shown that the only equilibrium distribution of particles in gas or plasma is Jüttner distribution, which reduces to the usual Maxwellian in the nonrelativistic case. Once the system is put in an external field, the equilibrium distribution is the Maxwell-Boltzmann one. It is clear that equilibrium configurations of self-gravitating systems are necessarily inhomogeneous. If thermal equilibrium for self-gravitating systems can in principle be established, it is described by a Maxwell-Boltzmann distribution in a given field. This field is the mean field, established by the self-gravitating system itself.

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14.1 Kinetic Theory of Self-Gravitating Systems One can derive the nonrelativistic kinetic equations for the self-gravitating system in two ways. One way is to construct moments of the Klimontovich random function (see, e.g., [75, 413]). The other way (see, e.g., [414, 415, 412]) is to start with Hamiltonian equations and arrive at the Liouville equation. Then one can define the N-particle DF, construct the BBGKY hierarchy, and derive the kinetic equations from it. In this section, basic kinetic equations for self-gravitating system are derived following [416]. 14.1.1 BBGKY Hierarchy Assume the system consists of N particles having equal mass m, so that total mass is M = mN. Particle dynamics is described by the Hamiltonian equations dri ∂H m = , dt ∂vi

∂H dvi =− m , dt ∂ri

H=

N mv2 i

i=1

2

− Gm2

i< j

1 . |ri − r j |

(14.1)

The N-particle DF, defined on the 6N-dimensional phase space, satisfies the Liouville equation  N  ∂PN ∂PN ∂PN + vi · =0 (14.2) + Fi · ∂t ∂r ∂v i i i=1 ri − r j Fi = −Gm F ( j → i) , (14.3) 3 = − r r i j j=i j=i where PN (w1 , ...wN ) is the probability density of finding at the time moment t the ith particle in the phase space with coordinates wi = (ri , vi ), and F( j → i) denotes the force exerted on the ith particle by the jth particle. Defining the jth DF as    (14.4) Pj w1 , ...w j = PN (w1 , ...wN ) dw j+1 ...dwN , one arrives at the BBGKY hierarchy ∂Pj ∂Pj ∂Pj + vi · + F (k → i) · ∂t ∂ri ∂vi i=1 i=1 k=i j

= − (N − j)

j

j  i=1

j

F ( j + 1 → i) ·

(14.5)

∂Pj+1 dw j+1 . ∂vi

At this point, all three descriptions are complete and the systems of equations (14.1), (14.2), and (14.5) are equivalent. The last system contains N coupled integro-differential equations.

14.1 Kinetic Theory of Self-Gravitating Systems

229

In the case of plasma considered in Chapter 5, the relativistic BBGKY hierarchy has been truncated assuming weak correlations (see eq. (5.32)). The small parameter has been the plasma parameter g p. Similarly, for the self-gravitating system [416], one can either adopt the interaction strength scaling G ∝ 1/N [407] or consider expansion in terms of the inverse of the number of particles in the Jeans sphere ggr = (nλ3J )−1  1, where the Jeans length λJ is defined in eq. (14.81). With the goal to truncate the system of kinetic equations at the order 1/N, one can write down explicitly the first two equations of the hierarchy:  ∂P1 (w, t ) ∂P1 (w, t ) ∂P1 (w, t ) +v· + (N − 1) · F (2 → 1) P1 (w2 , t ) dw2 ∂t ∂r ∂v  (14.6) ∂ = − (N − 1) · F (2 → 1) P2 (w, w2 , t ) dw2 ∂v and ∂P2 (w, w2 , t ) ∂P2 (w, w2 , t ) 1 ∂P2 (w, w2 , t ) +v· + F (2 → 1) · (14.7) 2 ∂t ∂r ∂v ∂P2 (w, w2 , t ) + (N − 2) · F (3 → 1) P1 (w3 , t ) dw3 ∂v    ∂P1 (w, t ) P1 (w2 , t ) · F (2 → 1) − F (3 → 1) P1 (w3 , t ) dw3 + ∂v  ∂P1 (w, t ) · F (3 → 1) P2 (w2 , w3 , t ) dw3 + (N − 2) ∂v  ∂ · F (3 → 1) P2 (w, w3 , t ) P1 (w2 , t ) dw3 − ∂v  ∂ · F (3 → 1) P3 (w, w2 , w3 , t ) dw3 + (1 ←→ 2) = 0, + (N − 2) ∂v where P2 (w, w2 , t ) = P1 (w, t ) P1 (w2 , t ) + P2 (w, w2 , t )

(14.8)

P3 (w, w2 , w3 , t ) = P1 (w, t ) P1 (w2 , t ) P1 (w3 , t ) + P2 (w, w2 , t ) P1 (w3 , t ) + P2 (w2 , w3 , t ) P1 (w, t ) + P2 (w3 , w, t ) P1 (w2 , t ) + P3 (w, w2 , w3 , t ) , (14.9) and where the notation (1 ←→ 2) implies that additional terms with the interchanging particles 1 and 2 are also present. Note that the index for the first particle is dropped for brevity. The first observation in eq. (14.7) is that the last term in this equation, containing the three-particle correlation P3 (w, w2 , w3 , t ), is of order 1/N 2 and can be neglected. This observation is crucial since it allows one to close the hierarchy [417]. The different approximations adopted further result in different kinetic equations.

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The RHS in eq. (14.6) is of the order 1/N, and it should be retained. The fourth term in eq. (14.7) is of the order 1/N 2 and can be neglected. This results in neglecting close encounters (weak coupling approximation). The sixth term corresponds to collective effects, namely, dynamical polarization created by the field of a given particle onto neighbor particles. This term is crucial in plasma physics and leads to regularization of kinetic equations at small angles of scattering, which results in replacement of the Landau equation by the Lenard-Balescu one. This term can be properly accounted for [418, 419]. However, this term is less crucial than others since regularization in a self-gravitating system, unlike neutral plasma, can be obtained by developing inhomogeneity. Also, the penultimate term is of the order 1/N 2 and can be neglected. Introducing the distribution function f (w, t ) = NP1 (w, t ) and two-particle correlation function g(w, w2 , t ) = N 2 P2 (w, w2 , t ), one finally2 obtains  ∂f ∂f N−1 ∂f N−1 ∂

F · +v· + =− · F (2 → 1) g (w, w2 , t ) dw2 ∂t ∂r N ∂v N ∂v (14.10) 1 ∂g ∂g ∂g +v· + F (1) · 2 ∂t ∂r ∂v ∂f ∂g f (w2 , t ) + δF (2 → 1) · + δF (2 → 1) · ∂v  ∂v N−2∂f · F (3 → 1) g (w2 , w3 , t ) dw3 + (1 ←→ 2) = 0, + (14.11) N ∂v where F is the mean force exerted by all particles on a given one and δF is fluctuating force exerted by particle 2 on particle 1, which are defined as  ∂

F = − = F (2 → 1) f (w2 , t ) dw2 (14.12) ∂x 1 (14.13) δF (2 → 1) = F (2 → 1) − F . N 14.1.2 The Vlasov-Poisson Equations When particle correlations can be neglected, setting g = 0 in eq. (14.10), in the limit N → ∞, one recovers the Vlasov-Poisson equations ∂f ∂f ∂ ∂ f + v· − · =0 ∂t ∂x ∂x ∂v  = 4π Gρ, 2

(14.14) (14.15)

Note that in [416] the “mass distribution function” is used, f˜(w, t ) = MP1 (w, t ), which implies f˜ = m f .

14.1 Kinetic Theory of Self-Gravitating Systems

where

 ρ = m f d3 p

231

(14.16)

is the mass density of particles, is the Laplace operator, and  is Newtonian gravitational potential. Since the term on the RHS of eq. (14.10) is of the order 1/N, the relaxation time is expected to scale linearly with N; see eq. (2.5.1). In principle, KT considers systems with N  1, therefore the range of validity of these equations is large. The dynamics in the collisionless regime occurs on the dynamical time tD ∝ (Gρ )−1/2 [420] (see Section 14.3). The relaxation by encounters (collisions) is expected to occur on the timescale (2.52), which can be estimated as t∗ ∼ NtD . 14.1.3 Gravitational Correlations in an Expanding Universe Unlike the electromagnetic interactions, there is no Debye screening in the gravitational interactions since there is no negative mass. Nevertheless, in an expanding universe, it is possible to introduce the gravitational correlation radius. Following [21], transform the Vlasov-Poisson equations (14.14) and (14.15) in the co-moving coordinates  ∂f u ∂f ∂ ∂ f 4π Gm + 2 − = 0  = f d 3 u − a3 ρ0 , (14.17) ∂t a ∂q ∂q ∂u a where ρ0 is the average density and  is gravitational potential. Here co-moving coordinates q and velocities u are related to the physical ones, as usual: x q= u = a (t ) [v − H (t ) x] , (14.18) a (t ) where a (t ) is cosmological scale factor and H = d ln a/da is the Hubble parameter. By comparison of eqs. (14.17) and (6.6), one finds that the average density in eq. (14.17) plays the role of opposite charge particles. By analogy with eq. (6.8), considering a gravitating particle in a uniform medium, insert in eq. (14.17) instead of a3 ρ0 a new density:   m a3 ρ = a3 ρ0 exp − . (14.19) kB T In the physical coordinates one obtains  

m  = 4π Gρ exp − −1 , kB T which is similar to eq. (6.8). For large distances it reduces to   1 d 4π Gρ0 m 2 d r +  = 0, 2 r dr dr kB T

(14.20)

(14.21)

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Self-Gravitating Systems

which gives   r 1  ∝ cos r rg

kB T , 4π Gρ0 m

rg2 ≡

(14.22)

where rg is called the gravitational Debye radius. It is important that this dependence, being introduced in the correlation function    (r1 , r2 ) gab = fab − fa fb ∝ exp −1 , (14.23) kB T leads to finite integrals at infinity.

14.1.4 Generalized Landau Equation Neglecting further collective effects in eq. (14.11), one arrives at the equation for the two-particle correlation function: 1 ∂g ∂g ∂g ∂f +v· + F · + δF (2 → 1) · f (w2 , t ) + (1 ←→ 2) = 0. 2 ∂t ∂r ∂v ∂v (14.24) The first three terms in this equation represent “Vlasov” terms ∂t∂ + L, where ∂ ∂ L =v · ∂r + F · ∂v . The last term can be moved to the RHS and viewed as a source expressed in terms of the one-particle DFs. This equation then can be solved by the method of characteristics [421] with the result

 t   ∂ ∂ g (w, w2 , t ) = − + δF (1 → 2) · dt G t, t − t δF (2 → 1) · ∂v ∂v2 0     × f w, t − t f w2 , t − t , (14.25) where the Green’s function G(t, t − t ) is    G t, t − t = exp −

t

t−t

˜ ˜ L (t ) dt

(14.26)

and it is assumed that g(w, w2 , t = 0) = 0. Substituting the solution (14.25) into eq. (14.10) results in a non-Markovian integro-differential equation for the function f (w, t ). One can further simplify this equation by assuming that in the limit N → ∞ the DFs in eq. (14.25) are frozen in time. This corresponds to the rapid attenuation of correlations principle adopted by Bogolyubov [11, 73] (see also Chapter 5). Replacing t − t → t in eq. (14.25) and extending the integral to +∞,

14.1 Kinetic Theory of Self-Gravitating Systems

233

one finally gets the generalized Landau equation  +∞  +∞   ∂f ∂f ∂ ∂f +v· + F · = · dt F (2 → 1) G t, t − t ∂t ∂r ∂v ∂v 0 0

∂ ∂ + δF (1 → 2) · f (w, t ) f (w2 , t ) dw. (14.27) × δF (2 → 1) · ∂v ∂v2 This equation was first derived in [421] using the projection operator formalism. Later it was rederived using the Klimontovich approach [413]. Owing to proper accounting for spatial inhomogeneity, this equation avoids divergence at large scales. However, as (rare) strong encounters have been neglected, there is a logarithmic divergence at small scales. This divergence can in principle be avoided by preserving the last term in eq. (14.11) at the expense of obtaining a very complicated kinetic equation, as done in [418, 419].

14.1.5 Vlasov-Landau Equation Integro-differential equation (14.27) is nonlocal and still very complicated to be solved. By making the local approximation on the RHS in eq. (14.27) with f (r2 , v2 , t ) → f (r, v2 , t ) and δF(i → j) → F(i → j) with F(2 → 1) = −F(1 → 2), and making the integrals over t and r, this equation can be reduced [416] to the Landau equation; i.e., one gets ∂f ∂f ∂f ∂sa +v + F · =− (14.28) ∂t ∂r ∂p ∂va       (v − v )2 δab − va − va vb − vb 3 ∂ f 2 ∂f sa = 2π mG  f −f d v, ∂ pb ∂ pb (|v − v |)3 One can easily verify that with the substitution Gm2 → −q2 the corresponding equation for plasma (5.40) is recovered. The inhomogeneity of the system is only accounted for in the “Vlasov” term L, since the collision integral is local. In plasma physics,  = log(χmax /χmin ) ∝ log(λD /λL ), where λL = q2 /(kB T ). Since λL ∝ 1/(nλ2D ), it follows that  ∝ log(nλ3D ) ∝ log(g−1 p ), where in the last passage, eq. (6.21) is used. In other words, Coulomb logarithm is a logarithm of the number of particles in the Debye sphere. Analogously, in a self-gravitating system, the quantity  ∝ log(nR3 ) is a logarithm of the number of particles in the system confined within linear size R. Turning back to eq. (14.10) and recalling that the RHS is of the order 1/N, one concludes more precisely that the relaxation time (2.52) is tD N/ log N. This result was first noted by Chandrasekhar [422]. In self-gravitating systems with large amounts of objects with similar masses, such as globular clusters, the relaxation time is very long and is comparable to the

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Self-Gravitating Systems

age of the universe. Indeed, with a typical number of stars N ∼ 105 and densities in the core of 103 star in cubic parsecs, the relaxation time is approximately 4 × 109 years. Therefore, globular clusters are in the collisional regime. The same holds for open stellar clusters, galaxy clusters, and center regions of galaxies [414]. Instead, if one considers elliptical galaxies with the number of stars as large as N ∼ 1011 , the relaxation time turns out to be much larger than the age of the universe. Yet, elliptical galaxies appear to be well-relaxed objects. This observation formed the basis of the Lyndel-Bell arguments, introducing the idea of violent relaxation [420], discussed later on. One should be careful, however, when attempting to estimate the relaxation time of galaxies. In the preceding considerations, both the dark matter and supermassive black hole [423] contributions are neglected. In estimating relaxation time, one has to keep in mind all the history of galaxy formation, including assembly and merging.

14.1.6 Jeans Equations Useful information can be inferred from macroscopic quantities constructed from the DF, in a similar way to gases and plasmas discussed in Part I. Consider the Vlasov-Poisson equations (14.14) and (14.15). They conserve the total mass M, as a consequence of the continuity equation for particle mass density. Indeed, taking moments of the Vlasov equations, one gets the Jeans equations3 ∂n ∂ (n¯vi ) + =0 ∂t ∂xi   ∂ (n¯vi ) ∂ nvi v j ∂ + +n = 0, ∂t ∂xi ∂xi

(14.29) (14.30)

 where v¯ i =n−1 f vi d 3 v is the average particle velocity and n is particle density. Using eq. (14.29) and the definition vi v j = n−1 f vi v j d 3 v the second equation can be rewritten as     ∂ v¯ j ∂ v¯ j ∂ ∂ ρσi j + n¯vi = −n − , (14.31) n ∂t ∂xi ∂x j ∂xi where σi j = vi v j − v¯ i v¯ j is the velocity dispersion tensor. Equations (14.29) and (14.30) are called Jeans equations . Equation (14.30) has the form of a Euler equation in fluid dynamics with the fluid velocity replaced by v¯ i and with the stress tensor replaced by nσi j . They have wide astrophysical applications and allow one to 3

From this section through the whole chapter, vector notation is replaced for clarity by notation in components. Components are numbered by Latin indices i, j, and so on.

14.1 Kinetic Theory of Self-Gravitating Systems

235

establish relations between observed density profiles n(r) and velocity distributions characterized by v¯ i and σi j . In particular, these equations reduce to the hydrostatic equilibrium equation for an isotropic in velocity space and spherically symmetric configurations: dP ∂ = −nm , dr ∂r

(14.32)

where P = nm¯vr2 and v¯ r is the radial component of the mean velocity. The symmetric tensor σi j in this case reduces to σi j = σ 2 δi j . 14.1.7 Virial Theorem Equations (14.29) and (14.30) are obtained from the Vlasov equation by taking its moments. These are partial differential equations for functions of spatial coordinates and time. These equations can be further integrated over physical coordinates to obtain information on the global properties of the system (see, e.g., [414]). Multiplying eq. (14.31) by mxk and integrating over spatial variables, one has      ∂ ρvi v j ∂ 3 ∂ (ρ v¯ i ) 3 3 xk d x = − xk d x − ρxk d x, (14.33) ∂t ∂xi ∂xi where the last term is the potential tensor  ∂ 3 Wik = − ρxk d x. ∂ri

(14.34)

Assuming that the density vanishes at large distances from the origin, the second term can be rewritten with the help of the Gauss theorem,     ∂ ρvi v j 3 xk d x = − δki ρvi v j d 3 x = −2Kk j , (14.35) ∂xi where Kk j is the kinetic energy tensor, which can be split into two terms corresponding to bulk and random motion, respectively: 1 Kik = Tik + "ik ,  2 1 ρ v¯ i v¯ k d 3 x, Tik = 2

"ik =

1 2

(14.36)

 ρσik2 d 3 x.

(14.37)

Finally, adding to eq. (14.33) the same equation, obtained by interchanging indices i ↔ k, one obtains the tensor virial theorem  1 d 2 Iik 1 ρ (xi v¯ k + xk v¯ i ) d 3 x = 2Tik + "ik + Wik , = (14.38) 2 dt 2 2

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Self-Gravitating Systems

where Iik is the moment of inertia tensor. Taking a trace of eq. (14.38), a scalar version of this theorem is obtained: 1 d2I = 2T + W. 2 dt 2

(14.39)

If the system is in steady state, it follows 2K + W = 0,

(14.40)

where T and W are, respectively, total kinetic and potential energies of the system. In particular, from eq. (14.40), it follows  = K + W = −K < 0;

(14.41)

namely, the total energy of the virialized self-gravitating system is negative. Consider the spherical collapse of a uniform sphere of collisionless particles of total mass M. As a collisionless system does not dissipate energy, during the collapse the gravitational potential energy has to be converted into kinetic energy of the particles involved in the collapse. The sphere should eventually relax to a structure supported by random motions. Assuming that the final state is uniform as well, its potential energy is W = −(3/5)GM 2 /Rv , where Rv is the virial radius. Considering that the kinetic energy is K = (1/2)M v¯ 2 , where v¯ is the average velocity of particles from eq. (14.40), one has, dropping the numerical factors, 1 2 GM kB T v¯ ∼ ∼ , 2 Rv mp

(14.42)

where the last relation holds if particles are in thermal equilibrium. It gives the relation between the temperature of the system and its global parameters. Instead of the average velocity, the velocity dispersion σ is frquently used in eq. (14.42). In practice, galaxies and clusters of galaxies are extended objects, and their virial radius is difficult to determine observationally. The operational definition of the virial radius of galactic halo is the radius at which the density exceeds the critical density of the universe, ρc = 3H 2 /(8π G), by a factor 200. 14.2 Gravitational Instability The matter in the present-day observable universe is distributed inhomogeneously with stars, gas, dust, and systems composed thereof, such as galaxies and their clusters. This fact is in sharp contrast with two critical observations: of cosmic microwave background (CMB) radiation and of nearly homogeneous matter distribution on largest cosmological scales. CMB is almost perfectly isotropic with temperature relative variations of the order T/T ∼ 10−5 . The largest gravitationally

14.2 Gravitational Instability

237

bound objects in the universe are superclusters of galaxies, separated by giant voids. When the averaging is performed on even larger scales, the matter appears to be distributed homogeneously. In addition, superclusters are not completely relaxed yet. All this indicates that matter in the early universe was distributed homogeneously and clustering observed now is a consequence of some interaction in the matter. This interaction is gravity, and the process of clustering originates from the instability of systems ruled by gravity. Numerous observations, including CMB, point out that most matter in the universe is dark, i.e., not participating in electromagnetic interaction, and has a nonbaryonic nature. For this reason, both stellar systems and systems of dark matter particles can be considered as self-gravitating collisionless systems of pointlike particles. In fact, the gravitational instability is considered as the basic mechanism of structure formation in the universe (see, e.g., [371, 217]). It is believed that small perturbations were already present at some initial time in the early universe. In particular, within the inflationary paradigm, these perturbations originated from quantum fluctuations of the inflaton field [424]. Such small perturbations grow due to gravitational attraction, because overdense regions accrete matter from the neighboring regions, raising a density contrast. One of the simplest examples showing the process of gravitational instability is a perfect fluid model. If density distribution in self-gravitating fluid is slightly nonuniform, i.e., small density perturbations exist, they tend to grow. When the density contrast is small, linear approximation can be used. The main advantage of linear theory is that perturbations on different scales evolve independently. A crucial result of this theory is that the growth of perturbations is damped by the cosmic expansion. It leads to a power law for the time dependence of the amplitude of density perturbations, in contrast with the exponential growth in Minkowski space. This is also different from plasma instabilities, which initially grow exponentially (see Chapter 10). Only during the nonlinear stage with large density contrast does the evolution become faster. It is at the nonlinear stage where formation of gravitationally bound structures occurs. The theory of linear density perturbations in a homogeneous medium initiated by Jeans [425] is greatly developed. His study was motivated by the intention to explain the mechanism of star formation. It is well known that the matter content of the universe includes relativistic species and the expansion regime was once dominated by radiation. Hence realistic cosmological perturbation theory should be necessarily relativistic. The linear perturbations in the expanding homogeneous and isotropic universe were analyzed by Lifshitz [426] within general relativity. Relativistic theory is quite cumbersome and interpretation of results on scales larger than the cosmological horizon requires usage of the gauge-independent formalism [427, 428, 429]. Relativistic treatment is required when the scale of perturbations

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Self-Gravitating Systems

lies outside the horizon or when the matter is relativistic, which is the case for neutrinos in the early universe as well as for photons. In the simplest cases, such as perturbations in dark matter well inside the horizon at the matter-dominated stage of expansion, it is sufficient to consider nonrelativistic theory, based on Newtonian gravity (see, e.g., [430, 431, 217]). Bonnor [430] was the first to study evolution of perturbations in Newtonian cosmology. For a comparison between Newtonian and relativistic cosmologies with respect to the background solutions and small perturbations, see, e.g., [432, 433]. The study of perturbations in multicomponent media including baryons, photons, dark matter, and neutrinos has proven to be computationally demanding [434, 435]. But these results can be confronted with observations, especially those of CMB. In fact, owing to coupling between plasma and photons in the expanding universe in early epochs, when radiation finally decouples, its anisotropy traces density perturbations in plasma, hence CMB is a crucial cosmological probe. Dedicated experiments to observe CMB anisotropies, such as COBE [436], BOOMERanG [437], and later WMAP [438] and PLANCK [439] satellites, required high-accuracy predictions and detailed exploration of parameter space of the cosmological models. Such computations requested development of new numerical techniques. The novel approach implementing integration along the photon past light cone [440] and the associated code CMBFast, later superseded by CAMB,4 allowed for strong improvement in computational time. Progress in observations of the large-scale structure within 2dF [441] and SDSS [442, 443] projects as well as observation of baryon acoustic oscillations [444] required development of efficient cosmological structure formation codes. The implementation of the fast Fourier transform algorithm, as well as integration of SPH and N-body codes (see Section 9.4), with feedback from supernovae (SNe) and active galactic nuclei (AGNs), greatly improved the quality of modeling of the large-scale structure of the universe. Currently a large collection of codes for precision calculations in a cosmological framework is available.5

14.2.1 General Relativistic Treatment The theory of linear density perturbations is developed in great detail in reviews [445, 446, 447, 448, 449] and is presented in many textbooks [217, 415, 450]. The monograph [371] presents one of the most direct and comprehensive uses of kinetic equations in cosmology (see also [451]). Gauge-invariant perturbation theory is introduced in [427] (see also [445, 428, 446, 452]). In this section the cosmological 4 5

http://camb.info/ http://lambda.gsfc.nasa.gov/toolbox/

14.2 Gravitational Instability

239

perturbation theory and the use of kinetic equations are illustrated,6 following the classic work [453] and the review [454]. Cosmological perturbations are separated into scalar, vector, and tensor modes [426]. As mentioned earlier, in linear approximation these modes are uncoupled and can be studied independently. Only scalar perturbations in cold collisionless dark matter components are considered in what follows. This regime is of primary importance for large-scale structure formation. The gauge issue, already encountered in electrodynamics in Chapter 10, is a more complicated concept in general relativity. Equations of general relativity are invariant under diffeomorphisms. Within linear perturbation theory a decomposition gμν = g¯ μν + δgμν into background spacetime g¯ μν and a small perturbation δgμν is introduced. Since this decomposition is not unique, perturbations may have different dynamics when another particular choice of the decomposition is made. On one hand, it results in the freedom to choose a gauge that fixes the decomposition; on the other hand, it may add spurious unphysical solutions [427]. Most of the works on perturbations nowadays adopt conformal Newtonian, or longitudinal, gauge. It is most suited for the study of scalar perturbations [446]. The metric tensor in this case can be written as   gμν = a (η)2 (1 + 2) dη2 − (1 − 2' ) δi j dxi dx j , (14.43) where the quantity

 η=

t

a−1 dt

(14.44)

0

is called conformal time, δi j is the Kronecker delta symbol,   1 is the Newtonian potential, and '  1 is the perturbation of the spatial curvature. This metric reduces to the Friedmann-Robertson-Walker (FRW) one, g¯ μν = a(η)2 δμν , for the background when ' =  = 0. The starting point in analysis of cosmological perturbations are Einstein field equations 1 Rμν − gμν R = 8π GTμν , (14.45) 2 where Rμν is Ricci tensor and R is scalar curvature. The energy-momentum conservation ∂T μν μ ρ ∇ν T μν = + ρσ T ρσ + ρσ T μσ = 0, (14.46) ∂xν where T μν is the energy-momentum tensor, and where ∇ν stands for the covariant derivative, is a consequence of Einstein’s equations. The background in the RHS 6

In this section the units with c = h¯ = kB = 1 are used.

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Self-Gravitating Systems

of Einstein equations is represented in the fluid approximation ¯ μν , T¯μν = ρ U ¯ μUν − Pg

(14.47)

where Uμ is the contravariant four-velocity, ρ¯ is energy density, P¯ is pressure, and ¯ ρ¯ is the adiabatic index (see eq. (4.41)). For ultra-relativistic fluid, e.g., = 1 + P/ photons, one has = 4/3. In the dust approximation, the pressure can be neglected, resulting in = 1. For Einstein’s cosmological term, = 0. In this approximation, eq. (14.46) results in ρ¯ ∝ a−3 .

(14.48)

The FRW metric ansatz for the background leaves nontrivial only the 00component of the Einstein equations, which results in the Friedmann equation 8π G H2 = ρ, ¯ ρ¯ = ρ¯i , (14.49) 3 i where H = d ln a/dt = a−2 da/dη is the Hubble parameter and the index i refers to various matter content of the universe, including baryons b, photons γ , neutrinos ν, dark matter d, and dark energy (cosmological term) . Spatially flat metric is assumed. Given this mixed content, eq. (14.48) has a solution 2

2

a ∝ t 3 ∝ η 3 −2 .

(14.50)

Given current values of energy density of different components ρ¯γ  ρ¯d < ρ (see, e.g., [439]), eq. (14.50) implies that the evolution of the universe can be divided into several epochs, in chronological order: radiation-dominance ρ¯  ρ¯γ ∝ a−4 and a ∝ η ∝ t 1/2 , matter-dominance ρ¯  ρ¯d ∝ a−3 and a ∝ η2 ∝ t 2/3 , and dark energy dominance ρ¯  ρ = const with a ∝ η−1 ∝ exp(t ). Assume that also the energy-momentum tensor is perturbed, Tμν = T¯μν + δTμν . One can identify the components δT00 = δρ, ¯ i, δTi0 = ρ v δT ji

=

−δPδ ij

(14.51) +

ij ,

where δρ and δP are density and pressure perturbations, vi is velocity perturbation, and ij is anisotropic shear perturbation. The last two quantities are usually expressed as = ∂ i vi (14.52) i

ρ ∂ ¯ σ =− 2

 i, j

where ∂i ≡ ∂/∂xi and ∂ 2 ≡ ∂i ∂i .

 1 2 ∂i ∂ j − ∂ δi j ij , 3

(14.53)

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241

As usual for linear perturbations, it is convenient to solve equations in the Fourier space. Given the existence of the preferred coordinate, the cosmic time, the Fourier transformation7 is performed only for spatial coordinates as      i  d3k (14.54) G x = exp iki xi Gˆ ki . 3 (2π ) Finally, introducing the density contrast δ = δρ/ρ, the first-order perturbed Einstein equations in the Fourier space give   a˙ a˙ ˙ 2 −k  − 3 ' +  = 4π Ga2 ρδ, ¯ (14.55) a a   ˙ + a˙  = 4π Ga2 ρ , −k2 ' ¯ (14.56) a #  2 $   a ˙ a ¨ k2 ¨ + ˙ + 2' ˙ + 2 − a˙ '  (14.57)  − ( − ' ) = 4π Ga2 δP, a a a 3 k2 ( − ' ) = 12π Ga2 ρ σ, ¯

(14.58)

where a dot denotes differentiation with respect to η. Also the energy-momentum conservation (14.46) implies, respectively, the continuity and Euler equations for the perturbations   ˙ δ˙ =  + 3' (14.59)     ˙ ˙ = 3 a˙ − 4  −  − k2  + σ + − 1 δ ,  (14.60) a 3 where = 1 + P/ρ. Equations (14.55)–(14.58) together with (14.59) and (14.60) form the system of six ODEs for six functions δ, δP, , σ , , '. Each of the matter components in the universe contributes both to the background quantities and to perturbations. In principle, the energy-momentum tensor does not need to be assumed in the form (14.47). It can be derived from the kinetic equations. Moreover, interactions between the components, such as baryons and photons, might imply that the energy-momentum tensor of individual components does not conserve, although the conservation (14.46) holds for the sum of the components. Since the wave number k is arbitrary, perturbation wavelength can exceed the particle horizon of the universe, which is equal to the conformal time η in the units with c = 1. Perturbations with kη  1 are referred to as super-horizon perturbations, while those with kη  1 are called subhorizon perturbations. According to the standard cosmological model, initial perturbations are generated at the inflationary stage. Inflation occurred when the energy content of the 7

Since only the spatial coordinates are Fourier transformed, the quantity k in this section refers to the absolute value of the wave vector ki .

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universe was presumably dominated by scalar fields, with  0, resulting in quasiexponential expansion. The presence of such a rapid expansion phase in the evolution of the early universe offers a solution to several problems of the standard cosmological model [455, 456, 217]. It is quantum fluctuations of the scalar field that gives origin to cosmological perturbations. In the simplest scenarios, perturbations are adiabatic and have a Harrison-Zeldovich spectrum. Such a spectrum is also called scale invariant since perturbations have equal amplitude when crossing the particle horizon. 14.2.2 Linearized Vlasov Equation The kinetic equation for one-particle DF in general relativity is given by eq. (2.25). Using the definitions from Part I, in particular, eqs. (2.18) and (2.17), one can write the Boltzmann equation (2.25) for the DF f (xμ , pi ) as ∂ f d pi 1 ∂ f dxμ + = 0 St [ f ] . μ i ∂x dt ∂ p dt p

(14.61)

It is clear that on the LHS of eq. (14.61), there is a full derivative of the DF df /dt with respect to time. It is also true for conformal time, defined in eq. (14.44). In most of the applications of KT in cosmology, the RHS of this equation is zero. This is supposed to be true for dark matter particles; this is also true for neutrinos after their decoupling at a temperature of approximately 1 MeV, so in the following it is assumed that St [ f ] = 0. The components in the four-momentum are not independent and are related through the mass shell condition gμν pμ pν = m2 c2 .

(14.62)

The three-momentum pi is a canonical conjugate of the co-moving coordinate xi , so the quantity f (xi , p j , η)d 3 xd 3 p gives the number of particles in the differential volume d 3 xd 3 p of the phase space. The three-momentum pi is related to the proper momentum p˜ i measured by a co-moving observer with fixed coordinates xi as pi = a (1 − ' ) p˜ i .

(14.63)

In the unperturbed FRW metric p˜ i ∝ a−1 . From eqs. (14.62) and (14.63) it follows that p0 = (1 + ) .

(14.64)

To eliminate the metric perturbations from the definition of the momenta, it is convenient [435] to replace the three-momenta pi by the co-moving ones8 qi = a p˜ i . 8

Not to be confused with elementary charge q.

14.2 Gravitational Instability

243

The co-moving three-momentum is further decomposed into its magnitude q and directions ni as qi = qni , where the latter are normalized by the conditions δi j ni n j = 1. Using these new variables, one may rewrite eq. (14.61) as df ∂f ∂ f dxi ∂ f dq ∂ f dni = + i + + = 0. dη ∂η ∂x dη ∂q dη ∂ni dt

(14.65)

Since the background spacetime is homogeneous and isotropic, the corresponding DF should be isotropic as well. Therefore, it can be represented as      f xi , η, p j = f0 (q) 1 + ψ xi , η, q j , (14.66) where p j = (1 − ')q j . Recall the definition of the energy-momentum tensor (1.14). Using the generalization rule (2.28), one has   d3 p μ ν Tμν = det gμν p p f, (14.67) p0  where det gμν = a−4 (1 −  + 3') and d 3 p = (1 − 3')q2 dqd, where d is the solid angle associated with the direction q j . Then using eq. (14.66), the background energy-momentum tensor can be represented as  4π 0 ¯ T0 = ρ¯ = 4 (14.68) q2 dq f0 (q) a  4π q2 T¯ii = −P¯ = 4 q2 dq f0 (q) , (14.69) 3a  where in the last equation there is no summation over the index i, while from definitions of perturbations (14.51) it follows that  1 δ = 4 q2 dqd f0 (q) ψ, (14.70) ρa ¯  1 q2 δP = 4 q2 dqd f0 (q) ψ, (14.71) a   1 0 (14.72) δTi = 4 q2 dqdqni f0 (q) ψ, a    1 q2 1 i 2 (14.73) ni n j − δi j f0 (q) ψ.

j = − 4 q dqd a  3 The equation for the perturbation of the DF ψ (xi , η, q j ) follows from eq. (14.65). With the definitions adopted, one finds that dxi /dη = −qi /. From the geodesic ˙ + ni ∂i . It turns out that the last equation (2.17), it also follows that dq/dη = q' term in eq. (14.65) is of second order and can be dropped. Then the perturbation of

244

Self-Gravitating Systems

the DF satisfies the equation in the Fourier space:     i ∂ ln f0 ˙ q  i ˙ ' + i ki n  . ψ − i ki n ψ =  ∂ ln q q

(14.74)

If the thermal motion of particles is nonrelativistic, then perturbation evolution is well described within the fluid approximation. This is completely analogous to the case of plasma discussed in Chapter 10. Indeed, one can take moments of eq. (14.65) and, using eqs. (14.68)–(14.73), reduce it to equations (14.59) and (14.60); for details, see, e.g., [371].

14.2.3 Ultra-Relativistic Case Assume particles are ultra-relativistic (massless). It is convenient to integrate out the q-dependence and expand the angular dependence in a series of Legendre polynomials Pl . One can define the new function  3 ∞  i    q dq f0 (q) ψ (−i)l (2l + 1) Fl (ki , η) Pl ki ni . (14.75) F x , η, n j =  3 = q dq f0 (q) l=0 Equation (14.74) for F reduces to       ˙ + i k i ni  . F˙ − i ki ni F = 4 '

(14.76)

It follows that the monopole in eq. (14.75) corresponds to the density contrast (see eq. (14.70)), i.e., δ = F0 . The dipole corresponds to the velocity contrast  = (3/4)kF1 , and the anisotropic stress is related to the quadrupole σ = −(1/2)F2 . So in the ultra-relativistic case, perturbation equations reduce to the continuity and Euler equations (14.59) and (14.60) with = 4/3 for the first two moments: 4 ˙ δ˙ =  + 4' 3 

(14.77)



˙ = −k2  + σ + 1 δ ,  4

(14.78)

while higher moments are related by the infinite hierarchy Fl =

k [(l + 1) Fl+1 − lFl−1 ] , 2l + 1

l ≥ 2.

(14.79)

To solve these equations, one has to specify the background evolution. Solutions of eqs. (14.77)–(14.79) for the radiation-dominated background indicate that perturbations remain static on super-horizon scales, while on sub-horizon scales, perturbations decay [371, 454]. The latter corresponds to the short-wavelength regime, which is discussed in the following sections in the nonrelativistic framework.

14.2 Gravitational Instability

245

14.2.4 Jeans Length Before discussing collisionless systems further, it is instructive to recall the nonrelativistic limit of equations for perturbations in Newtonian cosmology in the fluid approximation. From eqs. (14.55)–(14.58), (14.59), and (14.60), it follows that   a˙ δ¨ − 3 δ˙ − 4π Ga2 ρ¯ − vs2 k2 = 0, (14.80) a ¯ ρ¯ )1/2 is the sound speed. One can obtain the critical (co-moving) where vs = (d P/d wave number and the corresponding critical Jeans length 1/2    4π Ga2 ρ¯ π 1/2 2π a kJ = , λJ = = vs . (14.81) vs kJ Gρ¯ It is well known that perturbations with wavelengths larger than λJ are unstable, while those with shorter wavelengths oscillate. Recall that in the absence of expansion, a˙ = 0 and eq. (14.80) reduces to the original Jeans equation [425]. From that equation with vs → 0, the exponential growth of perturbations, similar to the case of plasma, is recovered. 14.2.5 Free Streaming Following [457, 458], consider Vlasov-Poisson equations (14.17) for collisionless particles with equal mass m within Newtonian cosmology with zero spatial curvature. Both DF and gravitational potential can be separated into a background and a perturbation as f (x, v, t ) = f0 (v, t ) + f1 (x, v, t )

 (x, t ) = 0 (x, t ) + 1 (x, t ) ,

(14.82)

with f1  f0 and 1  0 . One has to take into account cosmic expansion, which results in the fact that every scale changes with time as a(t ) ∝ t 2/3 . The background solution for a Newtonian universe is well known [430, 431, 217, 459]: 2x 1 2 ∂0 2 (t ) π Gρ x = ,  = , . (14.83) 0 6π Gt 2 3 ∂x 9 t2 Usually in cosmology at this point, co-moving coordinates are introduced, which take explicitly into account cosmic expansion. Alternatively, equations (14.14) and (14.15) can be solved by the method of integration along trajectories [460], which does not request the change of coordinates. The characteristics of eq. (14.14) in Cartesian coordinates with xi = (x, y, z) are ρ0 =

dt =

dvi dxi = − 2 xi . vi 9 t2

(14.84)

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Self-Gravitating Systems

The integrals of the system of equations (14.84) are 2 ui = vit 2/3 − vit −1/3 . (14.85) 3 Any function of ci and ui satisfies eq. (14.14), but additionally it should also satisfy the condition (14.16). The unperturbed DF is assumed to be nonrelativistic Maxwellian in the form    2 u 1 i f0 = T˜ −3/2 exp − i , (14.86) 6π 5/2 Gm T˜ ci = 3vit 1/3 − xit −2/3

where the effective temperature T ∝ T˜ /a2 decreases with expansion. Equations for perturbations follow from eqs. (14.14) and (14.15) as  ∂ f1 ∂ f1 ∂0 ∂ f1 ∂1 ∂ f0 + v· − · − · = 0 1 = 4π Gm f1 d 3 v. ∂t ∂x ∂x ∂v ∂x ∂v (14.87) The equations for trajectories coincide with equations for characteristics of the Vlasov equation. Coordinates and velocities of a particle at a given time moment can be expressed from their values at a different time moment via eq. (14.85), as follows:    2/3  −1/3   1/3 x = 3vxt 1/3 − xt −2/3 t + 2xt − 3vxt 2/3 t (14.88)   −1/3 1  −1/3   −2/3 2 3vxt 1/3 − xt −2/3 t 2xt + − 3vxt 2/3 t . vx = 3 3 As usual, perturbations are considered in the form of plane waves 1 = exp (iκξ ) ϕ (t )

f1 = exp (iκξ ) fˆ (t ) ,

where ξ = x/t 2/3 , so that dimensionally κ is not a usual wave number. The solution to the linearized equations is  t ∂ ∂ f0 f1 = dt f1 (0) = 0. 0 ∂x ∂v

(14.89)

(14.90)

Substituting eq. (14.89) into eq. (14.87), using eqs. (14.14), (14.15), and (14.88), and integrating by parts, one has ∂ f0 f1 = iκ ∂u1

t

     −1/3  ϕ t exp 3iκu1 t −1/3 − t dt .

(14.91)

0

The Poisson equation gives  = −κ 2 ϕ (t ) t −4/3 exp (iκξ ) = exp (iκξ )

 f d 3 v.

(14.92)

14.2 Gravitational Instability

247

Integrating eq. (14.91) and substituting it into eq. (14.92), one finally has    t    −1/3 9 2˜ 2 −2/3 ϕ (t ) = 2t ϕ t t˜ exp − κ T t˜ dt t˜ = t −1/3 − t . (14.93) 4 0 For long waves with 9κ 2 T˜ t˜2 /4  1, the exponential is substituted by unity and the equation is reduced to  t   ϕ (t ) = 2t −2/3 (14.94) ϕ t t˜. 0

This equation can be differentiated, and a simple second-order differential equation follows: t2

d 2 ϕ 8 dϕ = 0. + t dt 2 3 dt

(14.95)

This equation describes the well-known hydrodynamic solution ϕ (t ) ∝ t −5/3

fˆ (t ) ∝ t 2/3 ,

(14.96)

which is the usual result of gravitational instability in the matter-dominated epoch of the universe (see, e.g., [217]). What is remarkable is that the solution for short waves is strikingly different. It is well known that in the hydrodynamic picture the interplay between gravity and pressure results in oscillations of the amplitude of perturbations at small scales (see, e.g., [459]). The solution of eq. (14.93) can be found in the limit 94 κ 2 T˜ t −2/3  1 ˜ = 9 κ 2 T˜ and assume that using the method of steepest descents. One can define λ 4 9 2 ϕ(t ) is a slowly varying function, while t˜ exp(− 4 κ T˜ t˜2 ) quickly decreases with ˜ The following differential equation is obtained: increasing λ. 8 π ˜ −3/2 2 dϕ λ + ϕ = 0. (14.97) 9 t 2e dt It has the following solution: ⎤ ⎡ A 3 ˜ 1 2eλ 1 ⎦ ϕ (t ) ∝ exp ⎣ , 9 π t

(14.98)

which means that perturbations are damped with time. The damping time is on the order of 8 π 2 t . (14.99) tfs = 9 ˜3 0 2eλ

248

Self-Gravitating Systems

This phenomenon is analogous to the Landau damping in plasma discussed in Section 10.4 and is called the gravitational Landau damping or simply free streaming. Interestingly, unlike plasma, the presence of two counterstreaming beams does not lead to additional instability [457]. Free streaming within relativistic cosmology was first addressed in [435]. Later a multicomponent model including cold dark matter, neutrinos, baryons, and photons, as well as a cosmological constant was developed [453], on which most numerical codes dealing with CMB and large-scale structure are based.

14.2.6 Nonrelativistic Case Now assume particles are nonrelativistic. In that case perturbations are described by the continuity and Euler equations (14.59) and (14.60) with = 1 for the first two moments: ˙ δ˙ =  + 3'

˙ = −k2  − a˙ δ,  a

(14.100) (14.101)

which can be combined into a single second-order differential equation,   a ˙ a ˙ 2 ˙ = 0. ¨ + ' δ¨ + δ˙ + k  − 3 ' (14.102) a a For the matter-dominated background, using eqs. (14.55)–(14.58), at subhorizon scales, this reduces to the Poisson equation (14.87) in Newtonian cosmology in the long wavelength limit. As, in that case,  = ' = const, the solution of eq. (14.102) is δ ∝ a ∝ η2 ∝ t 2/3 . It implies that density perturbations grow with time. Such growth in fact is supposed to be responsible for all large-scale structure observed in the present universe. From the same equation one can also see that matter perturbations grow only logarithmically with time during the radiation-dominated stage.

14.2.7 The Role of Neutrinos Light neutrinos (electron, muon, and tau) are the second most abundant particle species in the universe after photons [461], with a density approximately 112 cm−3 today. They were among the favorite candidates for dark matter in the beginning of the 1980s (see reviews [461, 459, 454]). It was soon realized that they cannot be the dominant component of dark matter, essentially since free streaming strongly damps perturbations and because structures at galactic scales observed at present do not have enough time to form [462, 463]. The discovery of neutrino

14.2 Gravitational Instability

249

oscillations [464] implies a lower limit9 mν > 0.056 eV. Owing to this fact, the interest in cosmological effects of massive neutrinos is revived within the currently favored CDM cosmological model [217]. The free streaming phenomenon in the neutrino component is strongly affecting cosmological structure formation on large scales [465]. One can introduce the following characteristic scale (see, e.g., [454]), by which perturbations in the neutrino component are damped:  λFS =

4π Gρa ¯ 2 vν

1/2 ,

(14.103)

where vν  3Tν /mν  150(a0 /a)(1 eV/mν ) km/s is thermal velocity of neutrinos. It implies that today λFS  11 Mpc, the scale of large superclusters and voids. Constraints on neutrino mass and possible lepton asymmetry can be obtained from the analysis of CMB alone [466]. But the strongest constraints follow from combination with other cosmological data sets, such as galaxy surveys, baryon acoustic oscillations, and lensing, and currently result in the bound [467] mν < 0.23 eV,

(14.104)

which is stronger than the constraint obtained in the laboratory measurements of tritium beta decay, mν < 2.3 eV [468, 469]. One of the unresolved puzzles in cosmology is baryon asymmetry, which results today in large entropy (the number of photons exceeds the number of baryons). A mechanism involving neutrinos was proposed to resolve this puzzle, with baryon asymmetry generated from lepton asymmetry (with heavy neutrinos) in the early universe [470, 471]. The role of lepton asymmetry in cosmology has been discussed in [461]. Strong limits on electron neutrino asymmetry follow from observations of light element abundances, confronted with predictions of BBN theory. The existence of neutrino oscillations implies also a strong bound on leptonic asymmetry [472]. Another important issue concerning neutrinos is the description of neutrino decoupling in the early universe. Recently an extensive study has been performed with numerical integration of relativistic Boltzmann equations [473, 474]. These results are essential for correct parametrization of neutrinos in terms of masses and effective degrees of freedom.

9

Neutrinos are observed in flavor states: νe , νμ , and ντ . Each of these states is a combination of mass states. The neutrino oscillation experiments are only sensitive to the squared difference of neutrino masses. The absolute values of masses m1 , m2 , and m3 are unknown. If m2 < m3 , the mass hierarchy is said to be “normal”; in the opposite case it is called “inverted.” This limit is valid for a normal hierarchy; a similar limit exists for an inverse hierarchy.

250

Self-Gravitating Systems

14.3 Collisionless (Violent) Relaxation Stellar systems such as globular clusters and elliptical galaxies appear to be virialized and relaxed objects. Estimation of the relaxation time for globular clusters point out that they have enough time for relaxation through binary encounters (see Section 14.1). Similar estimation for elliptical galaxies gives a relaxation time that exceeds the age of the universe. It is clear that, unlike gases or plasmas, “collisions” (or binary encounters) cannot be responsible for relaxation of these systems. Hence another relaxation mechanism works for systems that underwent gravitational instability. The Vlasov-Poisson equations (14.14) and (14.15) are the exact equations for the one-particle DF in the limit N → ∞. The mathematical structure of the Vlasov equation is similar to the Liouville equation, and it represents the evolution of the one-particle DF in the phase space as the density of an incompressible fluid. There is an infinite number of associated conserved quantities, or “Casimir invariants” [475, 476], defined as  C [ f ] = g ( f ) d 3 pd 3 x, (14.105) where g( f ) is an arbitrary local function of the DF. Boltzmann entropy (1.15) is a particular case of the Casimir invariant. It implies that systems with longrange interactions, and, in particular, self-gravitating systems with N → ∞, cannot reach true statistical equilibrium. However, they can relax to quasi-stationary states (qSS), whose lifetime diverges as N → ∞, due to phase mixing. The phenomenon of phase mixing is well known in plasma physics [85]. For illustration of this phenomenon, consider an example [477]. Assume particles are placed in a rectangular potential well and each one moves with constant velocity (see Figure 14.1). When particles hit the wall of the well, they bounce, and the direction of their velocity changes to the opposite. While in the beginning, only the lower half of the phase space is filled, in the course of time, the DF tends to fill all the phase space. Clearly the system is time reversible; the DF preserves the memory of the initial state. Indeed, the DF can be determined only with finite resolution (see Chapter 3). For this reason the DF    1 f¯ (x, p) = f x , p d 3 p d 3 x , (14.106) 3 ( x p) x, p

obtained with finite resolution on the phase space, is called coarse-grained DF. In contrast, the exact one-particle DF is called fine-grained DF. While the fine-grained DF keeps evolving with time according to the Vlasov equation, the

1.0

1.0

0.5

0.5

0.0

0.0

v

v

14.3 Collisionless (Violent) Relaxation

-0.5

-0.5 -1.0 0.0

251

0.2

0.4

0.6

0.8

-1.0 0.0

1.0

0.2

0.4

1.0

1.0

0.5

0.5

0.0

0.0

v

v

0.8

1.0

0.6

0.8

1.0

x

x

-0.5

-0.5 -1.0 0.0

0.6

0.2

0.4

0.6

0.8

1.0

-1.0 0.0

0.2

0.4 x

x

Figure 14.1 Phase space of particles moving in a rectangular potential well. Time increases from left to right and from top to bottom. Initially, particles occupy only half of the phase space. At late times, particles, when viewed with finite resolution, appear to fill the entire available phase space.

coarse-grained one reaches some stationary state. Another example of phase mixing with independent pendula is given in [414]. In general, the procedure of coarse graining is arbitrary and the results obtained by coarse graining depend on it [478, 476]. Independent of the particular coarsegraining procedure, it can be shown [479] that the coarse-grained entropy      ¯ ¯ S = S (t ) − S (t0 ) = s f¯, t − s ( f , t0 ) d 3 pd 3 x, (14.107) where the entropy density s( f , t ) ∝ − f ln f , increases with time. It should be noted, however, that unlike gases where the H-theorem is proved (see Chapter 4), the Boltzmann entropy for a self-gravitating system does not have a maximum [478]. A relaxation mechanism related to phase mixing was discovered by Linden-Bell [420]. It operates in a newly formed gravitationally bound collisionless system such as a galactic halo or cluster of galaxies. When a particle moves in a fixed potential with ∂/∂t = 0, its specific energy 1  = v2 +  2

(14.108)

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Self-Gravitating Systems

does not change in time. When the potential is time varying, (x, t ), the energy is no longer constant:   d 1 dv 2 d dv ∂ ∂ = + = v· + ∇ + = . (14.109) dt 2 dt dt dt ∂t ∂t x(t ) This is a mechanism of redistributing particles in the phase space, i.e., relaxation. It differs from particle collisions, because the energy change does not depend on mass. Since the energy fluctuation is stochastic, the relaxation timescale can be estimated [420] as #   $1/2 1 d 2 trel  2 . (14.110)  dt Using the virial theorem with 2T ∼ W or W ∼ 2E, one has (1/2)mv 2 ∼ −(1/4)m or  ∼ (3/4). It implies trel

3  4

#

1 2



d dt

2 $1/2 .

(14.111)

Defining the effective radius from W = −GM 2 /R and writing the moment of inertia I = ι2 MR2 with ι  const from the nonequilibrium virial theorem, eq. (14.39), one obtains #  2 $ 2 R d 2 GM dR ι2 R 2 + + . (14.112) = dt dt M R Assuming now R(t ) = R¯ + δR(t ) and expanding this differential equation for R(t ) around its equilibrium value R¯ = −GM 2 /(2) gives d 2 δR GM = − 2 3 δR, 2 dt 2ι R¯

(14.113)

which is a harmonic oscillator equation with the frequency ω2 =

GM 2π G = ρ, ¯ 2 3 ¯ 3ι2 2ι R

(14.114)

where ρ¯ = M/(4π R30 /3) is the mean density of the system. This implies the following estimates: −

GM NR

d GM dR − 2 , dt NR dt

(14.115)

14.4 Quasi-Stationary States

253

and hence trel

3 = 4

#

1 R2



dR dt

2 $1/2

3  tf f , ι

(14.116)

√ where t f f = 3/(32π Gρ¯ ) is the free fall time, namely, the timescale on which the cold uniform mass collapses to a point. Since the free fall time of both globular clusters (105 years) and elliptical galaxies (108 years) is well shorter than the age of the universe, both systems should be presently relaxed. Lynden-Bell has developed a statistical theory of collisionless relaxation [420] which was further refined in [480, 481]. The main advantage of this statistical approach is that it predicts the equilibrium, which does not depend on initial state of the system. However, it is well known that this theory, having internal inconsistencies, is neither confirmed by numerical simulation nor supported by observations [482]. One of the reasons for the failure of this theory, noted by Lynden-Bell himself [420], is the incompleteness of the process of violent relaxation, due to the fact that the timescale on which fluctuations of the gravitational potential decay is shorter than the timescale of relaxation of the DF. Numerical simulations [483, 484, 485] have shown that relaxation is not complete and the resulting qSS strongly depends on initial conditions.

14.4 Quasi-Stationary States As discussed in the introduction to this chapter, systems with unscreened longrange interactions have very different properties from systems such as gases or neutral plasmas. Such systems, when formed, relax to a long-lived qSS due to phase mixing and violent relaxation. Among systems with long-range interactions, the self-gravitating systems have peculiar properties. In particular, it has been recalled that the gravitational potential is bounded from above. It leads to the possibility that some particles may become unbounded as their kinetic energy exceeds the potential energy. Such particles leave the system, and the phenomenon is referred to as particle evaporation. Due to this peculiar property, many studies of self-gravitating systems focused on simplified models with reduced spatial dimensions: one (1D) and two (2D) dimensional cases, e.g., [486, 487, 488, 489]. A 1D self-gravitating system consists of N sheets of mass m uniformly distributed in the y-z plane and free to move on the x-axis [490]. The particles can cross each other. The total mass of the system is M = mN. The Poisson equation for this system reads  (x, t ) = 4π Gρ1 (x, t ) ,

(14.117)

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Self-Gravitating Systems



where ρ1 = m f dv is the mass density and is a 1D Laplace operator. One can rescale the quantities in eq. (14.117) by introducing an arbitrary length scale L0 and making them dimensionless. This is equivalent to setting M = G = 1 and defining the dynamical time tD = (4π Gρ1 )−1/2 . For an isolated particle at the origin with the density given by 1D δ-function ρ1 (x) = δ (x) ,

(14.118)

one can recover from eq. (14.117) the potential 1D = |x| .

(14.119)

Thus the potential is unbounded and particle evaporation is not possible, unlike in the 3D case. Similarly, a 2D self-gravitating system consists of N rods of mass m [491] with azimuthal symmetry. One can rescale the quantities and make them dimensionless with the corresponding dynamical time tD = L0 (2GM)−1/2 in a way analogous to the 1D case. Considering the density for an isolated particle given by eq. (14.118), the potential is recovered: 2D = ln (x) .

(14.120)

Thus the potential is also unbounded. The analysis of these systems has been performed in the cosmological context [492, 493, 491]. Numerical study of 1D and 2D cases was performed with a one-level waterbag DF , f0 (x, v) ∝  (xmax − |x|)  (vmax − |v|) ,

(14.121)

where xmax and vmax are some values of coordinate and velocity, respectively (see [479] for a review). It was shown that Lynden-Bell theory gives quantitatively accurate predictions for this class of DF when initially the system is close to the virial condition (see Section 14.1.7). For a general interacting potential (xi − x j ) one can define the average poten ¯ = 1 i, j (xi − x j ). If  ¯ ¯ satisfies the condition (x ¯ i ) = λ−u (λx tial  i ), so that 2 ¯  is a homogeneous function of order u, the virial condition (14.40) can be written [479] as 

2K = uW,

(14.122)

where K = N −1 i p2i /(2m) and W = N −1 V¯ . A similar procedure is possible for a function (14.120) (see [494]). To consider the deviation from the virial condition, the virial ratio R=

2K uW

(14.123)

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255

is defined. For the 1D system it is R1D ≡ 2K/W , while for the 2D system it is R2D ≡ 2¯v 2 , where v¯ is the average velocity, and for the usual 3D gravity it is R3D ≡ −2K/W . It was found that when the system is far from the virial condition R = 1, it experiences strong density oscillations due to imbalance of kinetic and potential energies. The interactions between particle density and waves allow some particles to move in the regions of phase space, which are highly improbable from the point of view of Boltzmann or Lynden-Bell statistics. As the oscillations decay, the system relaxes to a characteristic core-halo configuration, a qSS. The explanation for these results [479] is that during violent relaxation the system develops quasi-periodic oscillations [495]. Some particles entering in dynamical resonance with these oscillations gain energy at the expense of collective motion [496], a sort of nonlinear Landau damping. A similar study for the 3D case performed in [497, 498] has shown that a selfgravitating system confined within a spherical region relaxes to a qSS that can be described reasonably well by a Lynden-Bell distribution with a cutoff. Generally speaking, there is no reason for a self-gravitating system to be formed with the condition R ≈ 1. That is why Lynden-Bell theory is not universal and the qSS of such a system should depend on initial conditions at its formation. 14.5 Self-Gravitating Systems in Equilibrium In this section thermalized self-gravitating configurations, called isothermal spheres, are considered. Their thermodynamic stability is discussed. 14.5.1 Isothermal Sphere There is a correspondence between the gravitating collisional gas and collisionless self-gravitating system in equilibrium [414]. One can assume that the DF is given10 by   ρ0 f () = exp , (14.124) (2π σ 2 )3/2 σ2 where ρ0 is a constant and particle energy  is given by eq. (14.108). Integrating this DF over velocities and taking into account eq. (14.108), one obtains ρ = ρ0 exp(/σ 2 ). Hence the Poisson equation (14.15) in the spherically symmetric case reduces to   1 d 2 d r = 4π Gρ, (14.125) r2 dr dr 10

This DF can be also obtained using the standard approach of statistical mechanics by variation of the Boltzmann entropy (cf. (14.53)) with fixed total energy and mass.

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Self-Gravitating Systems

which gives for the mass density ρ = nm   d 4π Gρ 2 2 d ln ρ r =− r . dr dr σ2

(14.126)

The gravitating gas in equilibrium also satisfies the hydrostatic equation (14.126), which can be rewritten as kB T dρ ∂ GM (r) dP = =ρ = −ρ , (14.127) dr m dr ∂r r2 where T is gas temperature and P is its pressure. The interior mass is defined by integrating the differential mass dM(r) = 4π ρr2 dr. After multiplication by r2 m/(ρkB T ) and differentiation with respect to r, one has   4π Gρm 2 d 2 d ln ρ r =− r . (14.128) dr dr kB T It follows that eqs. (14.126) and (14.128) are identical if kB T . (14.129) m It implies that the structure of an isothermal self-gravitating sphere of gas is identical to the structure of a collisionless system with DF given by eq. (14.124). The solution of eq. (14.126) gives σ2 =

σ2 , (14.130) 2π Gr2 which for r → ∞ results in infinite mass M(r) → ∞. In practice, however, selfgravitating systems are not isolated, and the density cannot decrease much beyond the average cosmological density ρ. ¯ The solution (14.130) is referred to as a singular isothermal sphere, since the density (but not mass) diverges as r → 0. Both these limitations are overcome for a regular isothermal sphere confined within a spherical region with radius R. Using eq. (14.124) and eq. (14.129), one can obtain an equation for the gravitational potential, instead of density,     d  − 0 2 d 2 r = 4π Gρ0 r exp − , (14.131) dr dr kB T ρ (r) =

with ρ0 and 0 being density and potential in the center, which is subject to the boundary condition d(r = 0)/dr = 0. There is no analytic solution of this equation. One can, however, reduce this second-order differential equation to the firstorder one [499, 409]. First, dimensionless variables are introduced:     4π Gρ0 1/2 ρ 4π Gρ0 3/2 M (r)  − 0 $= , r, ν = , μ = , χ= kB T ρ0 kB T 4π ρ0 kB T (14.132)

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257

3.0 2.5

v

2.0 1.5 1.0 0.5 0.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

u

Figure 14.2 Solution for isothermal sphere in variables u and v defined in eq. (14.134). The dotted line corresponds to λ = λc  −0.335, the dashed line represents λ > λc , and the dash-dotted line represents λ < λc .

which are used to rewrite eq. (14.131) as   1 d 2 dy $ = exp (−χ ) = n, $ 2 d$ d$

(14.133)

which is subject to the boundary conditions dχ ($ = 0)/d$ = χ ($ = 0) = 0. Second, another set of new variables is introduced as v ($ ) =

μ $

u ($ ) =

ν$ 3 , μ

(14.134)

so one arrives at $ dv $ du =u−1 = 3 − v − u, (14.135) v d$ u d$ with boundary conditions v(0) = 0 and u(0) = 3. Dividing the first of these equations by the second one finally gets u dv u−1 =− , (14.136) v du v+u−3 with the boundary conditions dv/du |v(u=3)=0 = −5/3 and v(u = 3) = 0. Numerical solution of this equation is shown in Figure 14.2. The solution starts at the origin r = 0 corresponding to the point (u, v) = (3, 0) and spirals indefinitely around the point (u, v) → (1, 2), which corresponds to the large distance limit r → ∞.

258

Self-Gravitating Systems 0.4

l

0.2 0.0 0.2 0.4 1

10

100

1000

104

105

x

Figure 14.3 The function λ(x) decreases for small x, has an absolute minimum at λc  −0.335, and then tends to the asymptotic value −0.25.

The total energy of the isothermal sphere is given as the sum of kinetic and potential ones: 3 K = MkB T, 2



R

W =− 0

GM (r) dM dr. r dr

(14.137)

When expressed in variables defined in eqs. (14.132) and (14.134), it turns out that one can define a dimensionless parameter

R 1 3 λ= , = u (0) − GM 2 v (0) 2

(14.138)

which depends only on the variables u and v. One can fix the value of the parameter λ by specifying the values of the global parameters of the isothermal sphere: its energy, mass, and size. For a given parameter λ, eq. (14.138) gives a line −1

v=λ



 3 . u− 2

(14.139)

The intersection between the line (14.139) and the solution of eqs. (14.136) obviously specifies the isothermal sphere with fixed λ. One can check that there is a critical parameter λc  −0.335 below which there is no intersection. The parameter λ as a function of the variable x is shown in Figure 14.3. One can see that this is exactly the first minimum in the function that corresponds to the value λc . Moreover, this is the absolute minimum as λ(x) > λc . It implies that self-gravitating systems with λ  λc cannot be isothermal.

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259

14.5.2 Instability of the Isothermal Sphere, Negative Heat Capacity, and Gravothermal Catastrophe Antonov [500, 501] first realized that an isothermal sphere might be unstable under certain conditions. He considered an isothermal self-gravitating gas with energy  and total mass M confined to a spherical region with radius R. By considering density perturbations at constant energy and mass, and the associated entropy perturbation, he found that under certain conditions, small density perturbation leads to an increase in the entropy. For sufficiently small radii the system is nearly homogeneous, as its kinetic energy exceeds its potential energy. Hence the system reduces to an ideal gas confined in a spherical container, which is obviously thermodynamically stable. As the radius increases (with constant mass and total energy), the system develops inhomogeneity and turns unstable. The analysis [500, 502, 409] shows that instability onset corresponds to the dimensionless radius $c  34.2, which gives for the density contrast between the center and the edge the value ρ0  708.61ρ($c ). For larger density contrast the system is thermodynamically unstable, since the local entropy maximum for the self-gravitating system with ρ0  708.61ρ($c ) transforms to the saddle point at ρ0  708.61ρ($c ). This property is referred to as Antonov instability. This can be understood by considering heat capacity of the self-gravitating gas in thermal equilibrium [411, 414]. Its kinetic energy is 3 K = NkB T¯ , 2

(14.140)

where T¯ is the mass-weighted mean temperature. From the virial theorem, eq. (14.41), it then follows that CV =

3 d = − NkB . 2 d T¯

(14.141)

It means that the self-gravitating gas in virial equilibrium has negative specific heat. This property is not only restricted to self-gravitating systems [411, 479] but is common in other systems with long-range interactions, such as nonneutral plasmas, two-dimensional vortices, and the Hamiltonian mean field model [412]. It is well known that the system with the negative specific heat in thermal contact with the heat bath is thermodynamically unstable. The analysis shows [502] that the specific heat in the isothermal sphere becomes negative already with ρ0  32.2ρ($c ). The fact that there is no need for the contact with thermal bath for the instability to occur with ρ0  708.61ρ($c ) makes this instability particularly important. The term gravothermal catastrophe was coined for this instability by Lynden-Bell and Wood [502].

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Self-Gravitating Systems

From the physical point of view the onset of instability can be understood as follows. The self-gravitating system confined in a box can be separated into a central core and an extended halo. The latter is dominated by kinetic energy and has positive specific heat. The former is dominated by the potential energy and has negative specific heat. The thermal contact between the core and a halo provides the condition for the runaway process. The study of the gravothermal instabilities in general relativity has been performed recently [503]. The study is based on the analysis of isothermal spheres in general relativity, which satisfy the Tolman-Oppenheimer-Volkoff equation √ [504, 505] as well as Tolman conditions [506, 507] on the temperature g00 T (r) = √ const, and Klein conditions [508] on the chemical potential g00 μ(r) = const. The instability criterion is formulated using the dimensionless parameter ξ=

2GM , Rc2

(14.142)

where M is the total rest mass of the system. In the limiting case ξ → 0 for low temperature, θ  1, the Antonov instability is recovered, while for high temperature, θ  1, instability of radiation is recovered. The upper bound ξ < 0.35 is obtained in [503], implying that there exists an upper bound on the number of noninteracting particles contained in a sphere of radius R that can thermalize without collapsing.

14.6 Cosmic Structure Formation The processes discussed in this chapter provide a mechanism that can explain the large-scale structure formation in the universe. Almost homogeneous matter initially has small density fluctuations subject to gravitational instability. At a given moment of cosmic time, there is a certain characteristic mass scale M ∗ (t ). Structures with M < M ∗ are already formed. Structures with M ∼ M ∗ become gravitationally bound, detach from the Hubble flow, relax by phase mixing and violent relaxation, and end up as virialized systems in a qSS. Structures with M > M ∗ are still subject to gravitational instability in the linear regime. The mass scale M ∗ (t ) increases with time, so the process of structure formation repeats on larger and larger scales. At present, M ∗ ∼ 1015 M , which corresponds to the scale of superclusters of galaxies. This bottom-up picture of structure formation is called hierarchical clustering, and it is supported by numerical simulations [415]. Galaxies are assumed to be tracers of the underlying matter distributions, so the large-scale structure is inferred from galaxy surveys [442]. Since baryons and dark matter behave in a different way, information obtained in this way suffers a

14.6 Cosmic Structure Formation

261

bias [509], called galaxy bias. Nevertheless, dark matter distribution can be probed directly with gravitational lensing [510]. In addition to this simple scenario, many physical effects influence structure formation. First, accretion of surrounding material and merging of already formed structures result in increase of the mass of relaxed objects. Second, the structure formation process depends on the environment: large galaxies are formed in the centers of clusters and superclusters, while they do not form in voids or low-density regions. Third, the baryonic component evolves in a different manner than dark matter does. Baryons are able to cool and condense, and they form stars, which later explode as supernovae; some galaxies contain supermassive black holes in their centers. If such black holes are fed by surrounding baryonic material, they produce an AGN. Such effects back-react on the structure formation and are accounted for in most recent cosmological codes. The largest success of the numerical N-body simulations resulted in the so-called Navarro-Frenk-While (NFW) profile of dark matter halos [511]. The dark matter halo density profile is inferred from numerical simulations and has a universal shape with mass density profile ρ δc = , ρc (r/rs ) (1 + r/rs )2

(14.143)

where ρc is the critical density, δc is the characteristic density, and rs is the scale radius. Subsequent numerical simulations with increased resolution showed some deviations from the NFW profile. The spherically averaged halo is shown to be better described by the Einasto [512] profile or Burkert profile [513]. It is crucial that dispersion velocities of stars in galaxies depend on the total gravitational potential, created by dark matter and baryonic matter. Since the baryonic component can be separated using the luminosity data, the dark matter profile can be recovered from such observations. For a review of different phenomenological profiles as well as comparisons with observations, see [514].

15 Neutrinos, Gravitational Collapse, and Supernovae

Most energy generated in the gravitational collapse to a supernova is radiated in neutrinos, hence the role of these particles in a supernova explosion is crucial. Current models of core collapse supernovae focus on multidimensional hydrodynamics and nuclear burning and treat neutrino transport in a simplified manner. In this last chapter an example of accurate neutrino treatment in a spherically symmetric collapse is given. The role of multidimensional effects is discussed. These results are of interest for the multidimensional models with large-scale convection as well as for the ongoing experimental search for neutrinos from supernovae. 15.1 Supernova Models and Neutrinos Supernovae (SNe) are produced by stars that end their late evolution in a catastrophic explosive process. The name supernova was introduced and the difference between SNe and novae in terms of their estimated explosive powers was described in [515]. The luminosity of a SN at its maximum, which lasts for several days, is comparable to the total luminosity of its host galaxy. It was first hypothesized in [516] that SN explosions should be accompanied by the formation of neutron stars; the neutron had been discovered just two years earlier. The total energy involved into explosion is approximately 1053 erg, mostly released in the form of neutrinos. Approximately 1% of this energy, namely, 1051 erg, is released in the form of kinetic energy of the SN ejecta. Only approximately 1% of that kinetic energy, i.e., 1049 erg, is emitted in the form of photons, which are detected as the SN event [517, 518]. The relevant timescales are as follows: collapse of the core occurs on a ≤ 0.1 s timescale, the SW propagation inside the collapsing core takes ∼10 ms, and the neutrino cooling time of the hot neutron star is 10 s. There are two general types of a SN, classified according to the presence of hydrogen absorption lines in observed spectra. Absorption lines are present in the spectra of Type II SNe and absent in the spectra of Type I SNe. In addition, Type II 262

15.1 Supernova Models and Neutrinos

263

SNe normally contain compact remnants, although such a remnant was not found in the nearby SN 1987A [519]. In this chapter, mainly Type II SNe, or core-collapse SNe, are discussed. A progenitor of a Type II SN is a star with a zero age main sequence (ZAMS) mass exceeding 10M in which thermonuclear burning is approaching its end, yielding an iron core at its center. As the density of the matter at the center of the star becomes high enough, electrons become degenerate and relativistic, and their energy becomes sufficient for reactions in which electrons are captured by atomic nuclei, giving rise to β-unstable elements, as was fist indicated in [520]. This process is called neutronization [521]. The first review of neutrino processes in a stellar core was given in [522]. Treatment of the explosion of core-collapse SNe can be divided into two parts with different characteristic timescales: the gravitational collapse and calculation of light curves. Light curve calculation is the best-studied part of SN theory: extensive observational data on visible and X-ray light curves and well-developed numerical models for the radiation hydrodynamics enable the determination of the explosion energy at the SN center and the chemical composition of the progenitor. At the same time, the gravitational collapse part remains incompletely understood. The only direct probe of the dynamics at the center of a SN is offered by neutrino. So far few neutrino events associated with SNe were observed only once [523, 524, 525], during the SN 1987A explosion. Overall, around 20 events were detected in a 13 s interval at a time consistent with the estimated time of the core collapse. Owing to poor statistics the determination of SN and neutrino parameters was not possible. The lack of neutrino detection [526] represents the major difficulty in understanding the gravitational collapse and the explosion of SNe. For a recent review on predictions for neutrinos from SNe, see [527]. Models of progenitors for core-collapse SNe are based on stellar evolution codes, such as KEPLER1 or MESA;2 for a review, see [528]. At the end of its evolution, a star with a ZAMS mass of 10–25M exhausts its nuclear fuel reserves, and its iron core with a mass of 1.2–1.6M begins to collapse. Neutronization initiates the photodissociation of iron nuclei and provokes instability. The collapse of a low-mass core begins at a high temperature, when the mean adiabatic index is still > 4/3, while the collapse of a massive core begins at the stability boundary = 4/3 [529, 530]. The collapse is accompanied by a loss of energy via neutrino emission, the URCA process [531, 520]. The density and temperature increase during the collapse, and neutrinos carry away an energy comparable to the gravitational energy of the stellar core in its final stationary state. 1 2

http://2sn.org/kepler/doc/Introduction.html http://mesa.sourceforge.net/

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Neutrinos, Gravitational Collapse, and Supernovae

Hence one of the first SN explosion models is based on the neutrino-driven mechanism [532, 533]. In this model, to explain the SN explosion, it is necessary to understand how less than 1% of the energy carried away by neutrinos is deposited in the envelope of the collapsing star [527]. Despite the clarity of the problem, the decadelong efforts are still far from success [534, 535]. Recent simulations [517] indicate that weak Type II SNe with kinetic energy up to Ek ∼ 1051 erg can be produced by the neutrino-driven mechanism, while luminous SNe with Ek > 1051 erg and hypernovae need an alternative explanation, possibly a magnetorotational mechanism [536, 537, 538]. In the latter mechanism, rotational energy of the newly formed proto-neutron star is converted into magnetic energy. Owing to large electric conductivity, magnetic fields are frozen in plasma and the role of magnetic energy increases during the collapse. Hence the magnetic pressure or dissipative heating via magnetorotational instability can drive the explosion [539, 540, 541, 542]. Such a mechanism gives asymmetric explosions. In what follows the focus is on neutrino-driven mechanisms. The first computations carried out in spherical symmetry [532] did not solve the full physical problem: the model was not correct due to the assumption that the material was transparent to neutrinos. However, the correct conclusion was reached that, provided the envelope would absorb some fraction of the neutrinos, it would be ejected [543]. Then, it was realized that the central region of the iron core is opaque to neutrinos [544, 545]. Hence the core is divided into two regions: opaque and transparent, separated by the neutrinosphere, a surface analogous to the photosphere in the problem of radiative transfer. The opacity of the neutrinos during the core collapse was first taken into account using the equations describing neutrino thermal conductivity. The limitation of such an approach is that the spectrum of neutrinos depends only on two parameters: chemical potential and temperature. Besides, neutrinosphere radius depends on the type of neutrino and on energy. Such an approach based on thermal conductivity is relevant during the prolonged cooling of the hot, newly formed neutron stars [546]. A more accurate treatment of neutrino transport in spherical symmetric collapse was later developed in [131]. This scheme is based on flux limiter diffusion adopted for photon transport in neutron stars [547]. The method for multidimensional, multitemperature hydrodynamics presented in Chapter 9 generalizes this approach to the multidimensional case. Later the group in the Oak Ridge National Laboratory developed an approach based on the Boltzmann equation for the spherical symmetry with all weak interactions taken into account (see Appendix C [548, 118, 549, 550, 551]). The approach adopted in Chapter 11 is similar: opaque and transparent regions have to be considered separately. In comparison with the heat conduction models, the models with Boltzmann transport give larger average

15.2 Spherically Symmetric Collapse of a Stellar Iron Core

265

neutrino energy. It is important to stress that all spherically symmetric models of SN explosions cannot explain the energy  1051 erg of expanding ejecta [534]. The reason for failure of one-dimensional models could be that convection remains unaccounted for. In the spherically symmetric problem, there are two unstable convection regions with the negative gradient of the entropy per nucleon.3 One region is formed for  10 ms in the center of the forming proto-neutron star, and the second region is formed at the accretion shock. This indicates possible importance of convection for the explosion of the core-collapse SN. Convection provides two effects. First, the neutrino energy flux through the envelope can be increased. Second, the average neutrino energy is increased as well. This effect obviously requires a multidimensional approach. Multidimensional models based on neutrino diffusion with the flux limiters or simplified transport along the radius were developed in [554, 555, 556]. Simulations in 2D (see, e.g., [557, 558]) and 3D (see, e.g., [559, 560]) apparently successfully generate large-scale convection, leading to explosion. Careful examination of the resolution effects in 3D simulations shows that the dynamics of the turbulent cascade of energy from large to small scale is severely affected by numerical viscosity [561, 562], and hence successful explosions in 3D models might be numerical artifact. As summarized earlier, 1D models do not account for turbulence and convection and other multidimensional effects, such as standing accretion shock instability [563] and the lepton-emission self-sustained asymmetry [564], and do not show explosions in a self-consistent way. Nevertheless, spherically symmetric simulations are a very efficient way for detailed study of the neutrino light curves and spectra, in particular for the longtime cooling evolution of the newly formed neutron star over tens of second. Accurate hydrodynamic simulations [561, 562] indicate that convection develops on small scales only, hence such study is relevant for observations of neutrino signals from core-collapse SNe. Therefore, even the development of 1D models that include sufficiently complete physics remains very topical [550, 565, 566]. In this chapter, spherically symmetric collapse with a full account of neutrino interactions is considered using the finite difference approach described in Chapter 8. 15.2 Spherically Symmetric Collapse of a Stellar Iron Core with Neutrino Transport As recalled from the previous section, evolution of massive stars with masses in the range from 15M to 25M ends with the formation of iron cores with masses 3

Generally speaking, the Ledoux criterion [552] should be used due to the conservation of leptonic number [553].

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Neutrinos, Gravitational Collapse, and Supernovae

1.2–1.6M [567, 568]. In this section, gravitational collapse of an iron core with mass 1.4M is considered. In the previous sections in Part III, only electromagnetic interactions were analyzed. All components were considered at the kinetic level and corresponding transport equations were solved. In these sections, in addition to electromagnetic interactions, neutrino processes are taken into account. The computations are reduced to the solution of the hydrodynamic equations for the matter interacting electromagnetically. The transport equation is applied for the neutrinos, and also the kinetics equations for the difference between the numbers of electrons and positrons per nucleon Ye [120] are included. The kinetics of neutrino production and capture by free nucleons and nuclei are taken into account, as well as neutrino scattering on electrons and positrons and other weak interactions. All these interactions conserve lepton number. On one hand, only two-particle interactions are treated, unlike in previous sections. On the other hand, there are no numerical methods that conserve exactly the lepton number. Also, compared to the previous sections, degeneracy of electrons should be taken into account. The hydrodynamics and transport equations are solved in spherical symmetry in a co-moving reference frame, with terms of first order in v/c in a Newtonian gravity. The method of lines is used for numerical integration of hydrodynamic and transport equations. The thermodynamic functions of the electron-positron gas are computed using the code described in [569]. In comparison with previous treatments [548, 118, 565] the adopted scheme is fully implicit and uses high-order Gear’s method with the automatic selection of the order from 1 to 5 to optimize time steps. In addition, unlike the treatment in [549], the described approach is based on exact evaluation of the matrix elements for the neutrino scattering on electrons. Neutrino annihilation is computed assuming isotropic distribution of both neutrinos and antineutrinos. r In the Lagrangian variables m = 0 (r )2 ρ(r , t )dr , and t, the hydrodynamic equations can be written in the following form [570]: ∂r = v, ∂t

(15.1)

     ∂v ∂ (r2 v) 4π Gm 1 2 ∂ +r P−ζ =− 2 + dμdν μ χνq Eν − ηνq , ∂t ∂m ∂V r ρc νq (15.2)    2 2   ∂ (r v) ∂ (r v) 1 ∂ + P−ζ = dμdν χνq Eν − ηνq , (15.3) ∂t ∂V ∂m ρ ν where  is the specific internal energy of the matter, ζ is the coefficient of artificial viscosity, and V = r3 /3. The following transport equation is valid for the spectral

15.2 Spherically Symmetric Collapse of a Stellar Iron Core

267

energy density Eν (r, μ, ν , t ) of neutrinos of sort ν, in the direction whose angle with the radius vector corresponds to the cosine μ:    1 ∂Eν ∂r2 Eν 1 ∂ 3v r ∂r2 v 2 +μ + (1 − μ ) 1 + − μ Eν c ∂t ∂V r ∂μ c c ∂V  

r ∂r2 v v ∂ν Eν 1 2 3v μ − − + r c c ∂V c ∂ν    2 r ∂r2 v 1 v r ∂r v 2 3v + −μ − Eν = −χνq Eν + ηνq . + (15.4) r c c ∂V c c ∂V In addition, one should solve the kinetics equation for the difference in the numbers of electrons and positrons per nucleon Ye : ∂Ye = Y˙e (ρ, T, Ye , Iν ). (15.5) ∂t To close the system (15.1)–(15.4), one should have an expression for the emission and absorption coefficients on the RHS of (15.4) and should specify the equation of state, P = P(ρ, T, Ye ). A computational grid in the phase space is introduced, (ω , μk , r j ). The radii r j+1/2 and velocities v j + 1/2 are determined at the cell boundaries, while the specific energy of the matter  j , number of electrons per nucleon Ye, j , and spectral energy density of the neutrinos Eν, j,k,ω are determined as the mean values within the phase volume. The spatial derivatives (15.1), (15.2), and (15.3) are approximated using the central differences [120] with artificial viscosity. Introducing the notation ( f ) j ≡ f j+1/2 − f j−1/2 and the analogous notation for the subscript k, the spatial derivatives in the Boltzmann equation for (15.4) can be approximated [120]: (r2 μ¯ k Eν,k,ω ) j 1 ∂ Eν, j,k,ω + c ∂t V j    ¯  r¯ j (r2 v) j 3¯v j 1 1 2 − (1 − μ ) 1 + μ Eν, j,ω + r j μk 2c 2c V j k  

¯ μ3k /3 3¯v j v¯ j (ν Eν, j )ω r¯ j (r2 v) j 1 − − + r j μk c c V j c ν,ω   ¯  μ2k /2 3¯v j r¯ j (r2 v) j r¯ j (r2 v) j v¯ j 1 + − Eν, j,k,ω + − r j c c V j μk c c V j = −χν, j,ω Eν, j,k,ω + ην, j,ω . (15.6) r2 /2

Here, Eν, j,k,ω = ν,ω Eν, j,k,ω , (1/r) j = r3j /3 , r¯ j = (r j−1/2 + r j+1/2 )/2, v¯ j = j (v j−1/2 + v j+1/2 )/2, μ¯ k = (μk−1/2 + μk+1/2 )/2.

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The approximation for the flux along the direction r can be written as    μ¯ + 1  μ¯ k − 1 k Eν, j,k + Eν, j+1,k (μ¯ k Eν,k,ω ) j+ 1 = 1 − ην, j+ 1 ,ω 2 2 2 2 Eν, j,k + Eν, j+1,k (15.7) + ην, j+ 1 ,ω 2 2 7  χν, j,ω r j kν, j+1,ω r j+1 χν, j,ω r j χν, j+1,ω r j+1 ην, j+ 1 ,ω = 1+ . 2 χν, j,ω r j + χν, j+1,ω r j+1 χν, j,ω r j + χν, j+1,ω r j+1 (15.8) The approximation for the flux along the direction μ using a second-order approximation and integrating along the characteristic can be written as ⎧ μ ( Eν, j,k,ω − Eν, j,k−1,ω ) ⎪ , Eν, j,k,ω + k ( μ ⎪ k + μk−1 ) ⎪ ⎪   ⎪ ⎪ vj r¯ (r2 v) ⎪ ⎨ for 1 + 3¯ − 2cj V j j μk ≥ 0, 2c Eν, j,k+ 1 ,ω = (15.9) μ ( E j,k+2,ω − Eν, j,k+1,ω ) 2 ⎪ ⎪ Eν, j,k+1,ω − k+1 ( μν,k+1 , ⎪ + μk+2 ) ⎪ ⎪   ⎪ 2 ⎪ 3¯ v r ¯ v) (r j ⎩ for 1 + − 2cj V j j μk < 0. 2c With these expressions for (μ¯ k Eν,k,ω ) j+ 1 Eν, j,k+ 1 ,ω , in the opaque region, 2 2 where χ r j is small and the transport dominates, the spatial derivatives in (15.6) against the flow can be approximated, in accordance with the direction of propagation of the perturbations. False oscillations are absent in the numerical solution, but the scheme is first order in the derivatives in r. In the opaque region, the local energy exchange between the neutrinos and matter dominates, and approximating the derivatives in r using the central differences does not lead to false oscillations. The resulting second-order approximation in the spatial derivatives is important for the correct behavior of the numerical solution in the region with high optical depth. There is a transition to the equations of neutrino thermal conductivity. A similar approximation for the spatial derivatives in the equation for the neutrino DF occupation number is adopted in [118]. The system of ODEs y˙ = fi (y)

(15.10)

for the unknowns y = r j+1/2 , v j+1/2 ,  j , Eν, j,k,ω , Ye, j is solved numerically. Since the rate at which matter flows through and the rate at which neutrinos are redistributed can differ substantially, the eigenvalues of the Jacobi matrix can differ appreciably. Hence Gear’s method is used again. The equation of state takes into account the equilibrium radiation of photons, the electron-positron gas, and a mixture of nuclei in so-called nuclear statistical

15.2 Spherically Symmetric Collapse of a Stellar Iron Core

269

Table 15.1 Characteristics of the stellar core at the beginning and end of the computations Time, s

Etotal , erg

Egr , erg

Ekin , erg

ρc , gcm−3

Tc , K

Rstar , cm

t=0 t = 40

−5.1 × 1050 −8.3 × 1052

−3.7 × 1051 −1.8 × 1053

0 3.4 × 1050

4.0 × 109 2.8 × 1014

7.2 × 109 5.2 × 1010

1.9 × 108 2.7 × 106 M = 1.353M

equilibrium with free nucleons [571]: (Ai , Zi )  (Ai − Zi )n + Z p.

(15.11)

The neutrons and protons have statistical weights ωn,p = 2. The following nuclei are considered: 2 He4 nuclei with ωHe = 1 and binding energy per nucleon QHe = 28.296 MeV and 26 Fe56 nuclei with ωFe = 1 and QFe = 492.322 MeV, where Qi = c2 [Zi m p + (Ai − Zi )mn − mi ]. Fermi-Dirac statistics is assumed for the nucleons in the equation of state in a nonrelativistic approximation (for details, see [572]). The dependence of the pressure on the density in the initial state is a polytropic equation of state P = P(ρ ) = Kρ 1+1/n , with the polytropic index n = 3 corresponding to the pressure of a degenerate, relativistic electron gas is adopted. The central density ρc was chosen so that the star was near the boundary of instability to collapse (more precisely, the approximate boundary of the region of stability): 7  dm dm 4

≡ P P= , (15.12) ρ ρ 3  P where = ∂∂ ln is the adiabatic index. The collapse of a massive stellar core ln ρ s  2M occurs due to photodissociation of iron nuclei precisely at the stability boundary, = 4/3. For less massive stars with iron-core masses ≈ 1.4M , the collapse occurs at higher central densities and temperatures, when the neutrino energy losses during the neutronization of the core becomes important, but

> 4/3. The global parameters for the initial state are presented 15.1.  r in2 Table The 100 × 8 × 12 grid for the mass coordinate m = 0 (r ) ρ(r , t )dr , the cosine of the angle between the neutrino velocity and the radius vector μ, and the energies of the neutrinos and antineutrinos ν , ν˜ are used for the computations of the collapse. The numerical results are summarized in Table 15.1. The collapse begins after roughly 4 s, when the density and temperature at the center of the star and the kinetic energy grow sharply. The stellar radius takes on its minimum value 5 × 107 cm, at time 4.2 s (see Figure 15.1). The time for the transition to

270

Neutrinos, Gravitational Collapse, and Supernovae

Figure 15.1 Luminosity of neutrinos (thin curve) and antineutrinos (thick curve), mean energy of neutrinos (thin curve) and antineutrinos (thick curve), and the total energy of the star (dotted curve) at various times. Calculations confirm the results in [572] (see their Figure 4).

the collapse depends on the energy losses to neutrino emission and is determined by the initial density and temperature. This time is  much longer than the characteristic timescale for the initial state, which is 1/ GM/R3s = 0.8 s. At the initial times, the energy of the neutrino emission is modest compared to the energy of the matter; the power released in the form of neutrinos and antineutrinos is −dEt /dt ∼ 1048 erg/s (see the light curve in Figure 15.1). Owing to the slow rate at which the parameters of the matter change, the neutrino energy flux, which is

15.2 Spherically Symmetric Collapse of a Stellar Iron Core

271

Figure 15.2 Dependence of the velocity of the matter in the star on the mass coordinate at time 4.19 s. Calculations confirm the results in [572] (see their Figure 5).

primarily emitted from the inner regions of the star, rapidly reaches the stellar surface and is a constant along the mass coordinate m for the outer layers of the star. With increase in the density and temperature at the center of the star (ρc , Tc ), the flux of energy carried away by neutrinos grows sharply, accelerating the collapse. However, when the value ρc ∼ 1011 g/cm3 is reached, a neutrino-opaque region appears near the center, bounded by the neutrinosphere. In this opaque region inside the neutrinosphere, the neutrino transport is slowed by absorption. Matter is accreted from outside, giving rise to a shock. The transition to collapse occurs very sharply: the central density increases from 3 × 1010 g/cm3 to 2 × 1011 g/cm3 (when the neutrinosphere forms) over only 100 ms and reaches values 1013 g/cm3 after only another 15 ms. This time is comparable to the time for a neutrino to travel from the center to the surface of the star, Rs /c = 18 ms (during which time the stellar radius changes only slightly). At approximately t1 = 4.19 s, the maximum energy flux in neutrinos is reached at the neutrinosphere. At later times, the flux from it is lower, and a growth of the flux is observed outward from the neutrinosphere, which then decreases toward the stellar surface. This nonmonotonic behavior is associated with the finite velocity of the neutrinos as well as the motion and compression of the matter. The maximum energy flux at the neutrinosphere is reached at the time when the shock has its maximum intensity (see Figure 15.2). A large growth in flux near it by a factor of a few is accompanied by strongly nonequilibrium kinetics. The degree of neutronization Ye in the accreting flow decreases by nearly a factor of 2 in the transition through the shock inside the neutrinosphere (see Figure 15.3). This is neutronization on the

272

Neutrinos, Gravitational Collapse, and Supernovae

Figure 15.3 Dependence of Ye in the star on the mass coordinate at time 4.19 s. Calculations confirm the results in [572] (see their Figure 6).

SW. A maximum of Ye (m) is observed in front of the shock, with Ye > 0.47, obviously due to the absorption of neutrinos by free neutrons, which form due to the dissociation of iron at the enhanced temperature. The time of neutrino luminosity L(t ), the total energy E(t ) =  t dependence E(0) − 0 L(t )dt , and the mean energy of the neutrinos are presented in Figure 15.1. The total energy radiated over the entire collapse is 1.2 × 1053 erg. The light curve maximum due to neutrinos reaches 1.8 × 1054 erg/s at time t = 4.195 s and has a half-width of 10 ms. During this time, the total energy of the star is decreased by ≈ 8 × 1051 erg, which comprises a few percent of the total radiated energy, and the kinetic energy is strongly reduced (by several orders of magnitude). The maximum kinetic energy, 2.9 × 1051 erg, is reached at time t = 4.188 ms. This is 10 ms earlier than the light curve maximum, which corresponds to the time for neutrinos to travel from the neutrinosphere to the stellar surface. The appearance of a narrow maximum in the light curve is due to the reprocessing of kinetic energy and the release of the gravitational energy of the infalling flow of matter in the region of the neutrinosphere and shock. The characteristic mean energies of the particles are 20 MeV in the region of the radiation maximum and 10–17 MeV at later times for neutrinos and antineutrinos, respectively. Narrow maxima with values (2–6) × 1053 erg/s in light curves with characteristic widths of tens of milliseconds have not appeared in the solutions obtained in other studies [573, 574] as is true in the present work roughly at the time of

15.2 Spherically Symmetric Collapse of a Stellar Iron Core

273

formation of the shock in the star. Time-resolved measurements of this narrow maximum in the light curve could be used to estimate the mass of the neutrinos [575]. The relationship between energy and velocity for particles with nonzero mass is 2 given by ν = √ mν c2 2 . For high energies and ultra-relativistic velocities, the delay 1−v /c

time for the arrival of the neutrinos is 1 τ1 = 2



mν c2 ν

2 t0 ,

(15.13)

where t0 is the travel time for light from the star to the observer. For a collapse in our galaxy (at a distance of 10 kpc), t0 = 1.0 × 1012 s. Owing to the difference in the arrival times, an observer on Earth sees the light curve peak to be spread out in time:  ∞ ν dL , (15.14) LE (τ ) = dt d 2(τ − t ) 8

0

ν τ −t +t1

t0 . If the neutrino peak were observed and found to have 2(τ − t ) a width ≈10 ms, this would constrain the neutrino mass to be 4 eV: at higher masses, the broadening of the light curve would become appreciable for a source at a distance of 10 kpc, i.e., a source in our galaxy. This constraint on the neutrino mass is of little practical interest. However, according to eq. (15.13), the delay is proportional to the distance to the observer and inversely proportional to the square of the particle energy. In other words, for a collapse at a distance a factor of k2 larger than our previously chosen 10 kpc, or with particle energies a factor of k lower than the mean energy at the maximum, it follows a constraint on the neutrino mass that is a factor of k more stringent. High energies (10 MeV) are favorable for detection with neutrino observatories, such as Super-Kamiokande. A second notable result is the absorption of part of the energy of the neutrinos and antineutrinos in outer layers of the stellar core (see Table 15.1). This same effect was manifest in the computations [576, 577] and increases with decreasing core mass. This effect is believed to be due to inaccuracies in the initial model, the simplified equation of state used, and the assumption of spherical symmetry. However, it is important to bear this effect in mind in further studies of SN mechanisms. First, specially conducted computations with a zero neutrino-absorption coefficient do not give rise to expansion of the envelope; i.e., the obtained weak explosion is associated with neutrino absorption, not a hydrodynamical outflow. Second, neutrinos have substantially higher energies compared to the case where the approximation of neutrino thermal conductivity is adopted. This effect can occur in multidimensional computations with large-scale convection, when neutrinos from deep

where ν = mν c2

274

Neutrinos, Gravitational Collapse, and Supernovae

layers with higher energies rise toward the surface due to large-scale motions in the region of convection. The cross section for scattering on electrons increases with increasing neutrino energy [578]: σeν = 1.7 × 10−40 cm2

ν EF , for EF  ν me c2 me c2

(15.15)

where the Fermi energy of the electrons is EF = (3π 2 )1/3h¯ cn1/3 e . According to eq. (C.4), in a collision with a stationary electron, e = EF , the neutrino transfers a fraction of its energy: ν − ν ν = ≈ 1, for EF  ν . ν EF + ν

(15.16)

Thus a shell with thickness lsh and with a density of degenerate electrons nsh absorbs a fraction of the neutrino energy: Ek = σeν lsh nsh . Etotal (t = 0) − Etotal (tfin )

(15.17)

For an estimate, take lsh = Rstar,min = Rstar (t = 4.3648 s) = 5 × 107 cm (Figure 15.4), and to estimate the number density, take the mean density for the ejected mass 0.047M from Table 15.1. Thus, the number of electrons in the shell is Nsh = 2.8 × 1055 , ne = 5.3 × 1031 cm−3 , EF /(me c2 ) = 4.5. For an absorbed fraction of the neutrino energy Ek /(Etotal (t = 0) − Etotal (tfin )) = 0.0041, one should obtain a mean energy for the neutrinos of ∼10 MeV, in agreement with the computations; i.e., the ejection of the envelope is associated with the transfer of some part of the neutrino energy. At the end of the computations, a stationary, cool, degenerate neutron star is obtained. Kinetic energy of the neutron star is close to zero; the difference of the temperature from zero only insignificantly influences the pressure and energy. Because of the inclusion of the gas pressure due to interacting baryons in the equation of state, the equation of state for the final models of the star are not polytropic. As the temperature decreases, general relativistic effects, neglected here, become dominant, and also a more realistic equation of state for nuclear densities is required. 15.3 Supernova Explosion Mechanism with Large-Scale Convection and Neutrino Transport The large-scale instability (convection) of a hot, collapsing, proto-neutron-star core was first considered in [579]. The equation of state included the relativistic electronpositron gas and an ideal ion gas. An equilibrium configuration with a central

15.3 Supernova Explosion Mechanism with Large-Scale Convection

275

Figure 15.4 Dependence of the star radius from time. Calculations confirm the results in [572] (see their Figure 3).

density of 2 × 1013 g × cm−3 was adopted for the initial state. A constant specific entropy was chosen, except for the central region, where a high temperature and entropy were established due to neutronization behind the accretion shock. Threedimensional modeling indicated the development of convection, manifest as the upward rising of a hot bubble. Convection can arise when g∇s > 0. Here the study connected with large-scale convection considered the evolution of the neutrino spectrum in a hot bubble [580] is reported. The neutrinos were initially assumed to be captured in the hot bubble. The evolution of their spectrum was studied until the expansion of the homogeneous bubble [579], taking into account the density variations with time in accordance with the solution obtained in [579]. Therefore there was no dependence on spatial variables or angles in the model, the energy and time dependence of the neutrino DF was retained in the homogeneous and isotropic treatment, and emission, absorption, and scattering reactions with the participation of electrons and nuclei were taken into account. This yielded an important result: the mean neutrino energy decreased to  40 MeV during the expansion. This implies that the means used to redistribute the neutrino spectrum to higher energies adopted in the previous section are completely justified. The interactions between high-energy neutrinos and the envelope of a collapsing stellar core were also taken into account in [581], where the Euler gas dynamics of the SN envelope was considered, allowing for neutrino absorption. The dependence of the neutrino emission on the coordinates (r, θ ) during the development of

276

Neutrinos, Gravitational Collapse, and Supernovae

Figure 15.5 Dependence of the specific entropy density per nucleon s/kB in the star on the mass coordinate at times t = 4.1759 (1), 4.18360 (2), 4.1866 (3), 4.1873 (4), 4.1882 (5), and 4.1893 (6) s. The entropy in the first mass interval, m ≈ 0, exceeds the real physical value, and it is the numerical artifact.

convective instability was specified. The total neutrino luminosity was assumed to be constant at 5 × 1052 erg/s, and the neutrino energy spanned in the range 30–60 MeV. The kinetic energy of the expanding envelope was determined to be (1.5 − 50) × 1051 erg, depending on the initial accretion rate of the envelope material. In Figure 15.5 the entropy profiles are shown at different time moments. One can see two unstable regions ds/dr < 0: one region near the center during 10 ms and another region on the accretion SW during long time interval. A more accurate criterion for the convection, called the Ledoux criterion, is [552]     ∂P ds dYl ∂P + < 0, (15.18) ∂s ρYl dr ∂Yl ρs dr where l is the number of leptons. In the central region inside the neutrinosphere the neutrinos are trapped, so the approximate Schwarzschild criterion for specific entropy is appropriate if one recalculates entropy after the exclusion of Yl from the equation of state. In the recent work [582] the convection in the center is analyzed by numerical solution of the hydrodynamic equations in three dimensions with the entropy profile in the initial stationary solution corresponding to the gravitational collapse with neutrino transport (see Figure 15.5). An important result was obtained: the convection in the center develops during the time less than 10 ms; its scale is large. So this

15.3 Supernova Explosion Mechanism with Large-Scale Convection

277

Figure 15.6 Dependence of the electron chemical potential μe in the star on the mass coordinate at the same times as those in Figure 15.5.

effect should be important for the collapse. Another interesting result is the profile of chemical potential of electrons, shown in Figure 15.6. While the temperatures of degenerate electrons are ∼10 MeV, they are highly degenerate. The average energy of electrons and neutrinos is approximately 100 MeV. Then a cold, unstable bubble formed as the result of convection in the center contains energetic neutrinos with energy approximately 100 MeV. Such energetic neutrinos might be detected in neutrino observatories.

Appendix A Hydrodynamic Equations in Orthogonal Curvilinear Coordinates

In orthogonal curvilinear coordinates xi the metric tensor is   ∂r ∂r −1 −1 gi j = , j = diag(g11 , g22 , g33 ), gi j = diag(g−1 11 , g22 , g33 ). (A.1) i ∂x ∂x Christoffel symbols are     ∂gi j ∂gi j 1 ∂gik ∂g jk 1 kk ∂gik ∂g jk k i j = g + − k = + − k , (A.2) 2 ∂x j ∂xi ∂x 2gkk ∂x j ∂xi ∂x and the covariant derivative of the tensor is i ··· j

i ··· j ∇l T j11··· jqp

=

∂T j11··· jqp ∂xl

i ···i

i ···i

+ lsi1 T j1 ··· jpq + lsp T ji11······sjq − lsj1 Ts···1 jq p − lsjq T j11···s p . s···i

i

(A.3)

One uses the norm of units vectors for Riemannian geometry. The preceding definitions are enough to derive all other derivatives. The gradient of the scalar gi j ∇ j T = the gradient of the vector is ∂ui ∂ui 1 ∇ j ui = j + ijs us = j + ∂x ∂x 2gii



1 ∂T , gii ∂xi  ∂g ∂g j j ii us + j ui − uj , ∂xs ∂x ∂xi

∂gi j s

(A.4)

(A.5)

the divergence of the vector is ∂ui ∂ui 1 ∂gkk i k j + u = + u k j ∂xi ∂xi 2gkk ∂xi √ √ ∂ui ∂ ln gkk i 1 ∂ |g|ui = i + u =√ , ∂x ∂xi |g| ∂xi k

∇ i ui =

278

(A.6)

Hydrodynamic Equations in Orthogonal Curvilinear Coordinates

and the divergence of the tensor is   ∂σ i j j is ij i sj + is σ + is σ ∇i σ = ∂xi i   1 ∂ √|g|σ i j ∂ ln g j j i j 1 ∂gii ii = + σ − σ . √ ∂xi 2g j j ∂x j |g| ∂xi i

279

(A.7)

Hydrodynamical conservations laws (for the one component ideal fluid without viscosity) are ∂ρ + ∇i (ρui ) = 0, ∂t ∂ρu j + ∇i σ i j = 0, ∂t and

  ∂ρE ∂T + ∇i (ρE + P)ui = ∇i κgii j + ρQ, ∂t ∂x

(A.8) (A.9)

(A.10)

where E =  + ui gik uk /2, κ = κ (ρ, T ), Q = Q(ρ, T ), σ i j = ρui u j + Pgi j . Let us introduce new “physical” “vectors” and “tensors” in physical units for all components: v2 √ vi ≡ gii ui , E =  + i , 2 √ "i j ≡ gii g j j σ i j = ρvi v j + Pδi j . Then 1 ∂ ∇m u = √ |g| i ∂xi m

√  |g|vi ≡ divv. √ gii

(A.11) (A.12)

(A.13)

The mass conservation law is ∂ρ + div(ρv) = 0. ∂t

(A.14)

The momentum conservation law is √   σi j v ∂ρ √gj j j ∂ ln g j j σi j 1 ∂ |g| √gii g j j 1 ∂gii σii + = 0, + − √ √ √ ∂xi ∂xi gii g j j 2g j j ∂x j gii gii ∂t |g| i (A.15) or ∂ρv j + (Div") j = 0, ∂t

(A.16)

280

where

Hydrodynamic Equations in Orthogonal Curvilinear Coordinates

 A   √ ∂ ln gii |g|g j j 1 1 ∂ "i j − "ii . (A.17) Div" j ≡ √ √ gjj i gii ∂x j |g| ∂xi

The energy conservation law is ∂ρE + div((ρE + P)v − κgradT ) = ρQ, ∂t

(A.18)

1 ∂T (gradT )i ≡ √ . gii ∂xi

(A.19)

where

Appendix B Collision Integrals in Electron-Positron Plasma

B.1 Collision Integrals for Binary Interactions B.1.1 Compton Scattering γ e± → γ e± Consider scattering (2.10) process between electron with momentum p and photon with momentum k. After scattering, these quantities are denoted with primes. The time evolution of the distribution functions of photons and pair particles due to Compton scattering may be described by [252, 31]   ∂ fγ (k, t ) (B.1) ∂t γ e± →γ e±  = dk dpdp V wk ,p ;k,p [ fγ (k , t ) f± (p , t ) − fγ (k, t ) f± (p, t )],   ∂ f± (p, t ) (B.2) ∂t γ e± →γ e±  = dkdk dp V wk ,p ;k,p [ fγ (k , t ) f± (p , t ) − fγ (k, t ) f± (p, t )], where wk ,p ;k,p =

|M f i |2 h¯ 2 c6 δ( +  −  −  )δ(k + p − k − p ) (B.3) γ ± γ ± (2π )2V 16γ ± γ ±

is the probability of the process, # 2  m2 c2 m2 c2 m2 c2 m2 c2 2 6 2 2 |M f i | = 2 π α + + + s − m2 c2 u − m2 c2 s − m2 c2 u − m2 c2   1 s − m2 c2 u − m2 c2 , (B.4) − + 4 u − m2 c2 s − m2 c2 281

282

Collision Integrals in Electron-Positron Plasma

is the square of the matrix element, s = (p + k)2 and u = (p − k )2 are invariants, k = (γ /c)(1, eγ ) and p = (± /c)(1, β± e± ) are energy-momentum four-vectors of photons and electrons, respectively, dp = d± do±2 β± /c3 , dk = dγ γ 2 do γ /c3 , and do = dμdφ. The energies of photon and positron (electron) after the scattering are γ =

± γ (1 − β± b± ·bγ ) , ± (1 − β± b± ·b γ ) + γ (1 − bγ ·b γ )

± = ± + γ − γ ,

(B.5)

bi = pi /p, b i = p i /p , b ± = (β± ± b± + γ bγ − γ b γ )/(β± ± ). For photons, the absorption coefficient (11.16) in the Boltzmann equations (11.3) is    γ |M f i |2h¯ 2 c2 1 ∂ fγ abs 1 γ e± →γ e± χγ dn± doγ Jcs fγ = − = fγ , c ∂t γ e± →γ e± (2π )2 16± γ ± (B.6) where dni = di doi i2 βi fi /c3 = di doi Ei /(4π i ). From equations (B.1) and (B.6), one can write the absorption coefficient for photon energy density Eγ averaged over the , μ-grid with zone numbers ω and k as  1 ± ± γ e± →γ e± (χ E )γ ,ω ≡ dγ dμγ (χ E )γγ e →γ e γ ,ω γ ∈ γ ,ω  γ |M f i |2h¯ 2 c2 1 1 = dnγ dn± doγ Jcs , (B.7) (2π )2 γ ,ω γ ∈ γ ,ω 16± ± where the Jacobi matrix of the transformation is γ ±  . Jcs = γ ± 1 − β± bγ ·b±

(B.8)

Similar integrations can be performed for the other terms of equations (B.1) and (B.2), and one obtains  γ 2 |M f i |2h¯ 2 c2 1 1 γ e± →γ e± ηγ ,ω = dn dn do J , (B.9) γ ± cs γ (2π )2 γ ,ω γ ∈ γ ,ω 16± γ ±  γ |M f i |2h¯ 2 c2 1 1 γ e± →γ e± = dn dn do J , (B.10) η±,ω γ ± cs γ (2π )2 ±,ω ± ∈ ±,ω 16± γ  γ |M f i |2h¯ 2 c2 1 1 γ e± →γ e± = dnγ dn± doγ Jcs . (B.11) (χ E )±,ω (2π )2 ±,ω ± ∈ ±,ω 16γ ± To perform integrals (B.7)–(B.11) numerically over φ (0 ≤ φ ≤ 2π ), one can introduce a uniform grid φl∓1/2 with 1 ≤ l ≤ lmax and φl = (φl+1/2 −

B.1 Collision Integrals for Binary Interactions

283

φl−1/2 )/2 = 2π /lmax . It is assumed that any function of φ in equations (B.7)–(B.9) in the interval φ j is equal to its value at φ = φ j = (φl−1/2 + φl+1/2 )/2. It is necessary to integrate over φ only once at the beginning of calculations. The number of intervals of the φ-grid depends on the average energy of particles and is typically taken as lmax = 2kmax = 64. B.1.2 Pair Creation and Annihilation γ1 γ2  e− e+ Now consider the process of electron-positron pair creation and annihilation. Electrons and positrons have momenta p− and p+ , respectively, while photons have momenta k1 and k2 . The rates of change of the distribution function due to pair creation and annihilation are    ∂ fγ j (ki , t ) = − dk j dp− dp+V wp− ,p+ ;k1 ,k2 fγ1 (k1 , t ) fγ2 (k2 , t ), ∂t γ1 γ2 →e− e+ (B.12)    ∂ fγi (ki , t ) = dk j dp− dp+V wk1 ,k2 ;p− ,p+ f− (p− , t ) f+ (p+ , t ), ∂t e− e+ →γ1 γ2 (B.13) for i = 1, j = 2, and for j = 1, i = 2, and    ∂ f± (p± , t ) = dp∓ dk1 dk2V wp− ,p+ ;k1 ,k2 fγ (k1 , t ) fγ (k2 , t ), ∂t γ1 γ2 →e− e+ (B.14)    ∂ f± (p± , t ) = − dp∓ dk1 dk2V wk1 ,k2 ;p− ,p+ f− (p− , t ) f+ (p+ , t ), ∂t e− e+ →γ1 γ2 (B.15) where |M f i |2 h¯ 2 c6 δ( +  −  −  )δ(p + p − k − k ) . − + 1 2 − + 1 2 (2π )2V 16− + 1 2 (B.16) Here the matrix element |M f i |2 is given by eq. (B.4) with the new invariants s = (p− − k1 )2 and u = (p− − k2 )2 (see [48]). The energies of photons created via annihilation of a e± pair are wp− ,p+ ;k1 ,k2 =

1 (b1 ) =

m2 c4 + − + (1 − β− β+ b− ·b+ ) , − (1 − β− b− ·b1 ) + + (1 − β+ b+ ·b1 )

2 (b1 ) = − + + − 1 , (B.17)

284

Collision Integrals in Electron-Positron Plasma

while the energies of pair particles created by two photons are found from √ B ∓ B2 − AC , + (b− ) = 1 + 2 − − , (B.18) − (b− ) = A where A = (1 + 2 )2 − [(1 b1 + 2 b2 )·b− ]2 , B = (1 + 2 )1 2 (1 − b1 ·b2 ), C = m2 c4 [(1 b1 + 2 b2 )·b− ]2 + 12 22 (1 − b1 ·b2 )2 . Only one root in eq. (B.18) has to be chosen. Energy-momentum conservation gives k 1 + k 2 − p − = p+ ;

(B.19)

taking the square from the energy part, one has 12 + 22 + −2 + 21 2 − 21 − − 22 − = +2 ,

(B.20)

and taking the square from the momentum part, one gets 12 + 22 + −2 β−2 + 21 2 b1 ·b2 − 21 − β− b1 ·b− − 22 − β− b2 ·b− = (+ β+ )2 . (B.21) There are no additional roots because of the arbitrary e+ : 1 2 (1 − b1 ·b2 ) − 1 − (1 − β− b1 ·b− ) − 2 − (1 − βb2 ·b− ) = 0

(B.22)

− β− (1 b1 + 2 b2 )·b− = − (1 + 2 ) − 1 2 (1 − b1 ·b2 ). Eliminating β gives 12 22 (1 − b1 ·b2 )2 − 21 2 (1 − b1 ·b2 )(1 + 2 )−   + (1 + 2 )2 − [(1 b1 + 2 b2 )·b− ]2 −2

(B.23)

= [(1 b1 + 2 b2 )·b− ] (−m ). 2

Therefore, the condition to be checked reads − β− [(1 b1 + 2 b2 )·b− ]2

(B.24)

= [− (1 + 2 ) − (1 2 )(1 − b1 ·b2 )] [(1 b1 + 2 b2 )·b− ] ≥ 0. Finally, integration of equations (B.12)–(B.15) yields  12 |M f i |2h¯ 2 c2 1 1 − + 2 ηγe ,ωe →γ1 γ2 = d n J (B.25) ± ca (2π )2 γ ,ω 16− + 2 1 ∈ γ ,ω   1 |M f i |2h¯ 2 c2 2 + d n± Jca , 16− + 2 ∈ γ ,ω  1 |M f i |2h¯ 2 c2 1 1 e− e+ →γ1 γ2 2 = d n J (B.26) (χ E )e,ω ± ca (2π )2 e,ω 16+ 2 − ∈ e,ω   1 |M f i |2h¯ 2 c2 , d 2 n± Jca + 16− 2 + ∈ e,ω

B.1 Collision Integrals for Binary Interactions − +

γ2 →e (χ E )γγ1,ω

e

− +

γ1 γ2 →e ηe,ω

e

=

=

1 1 2 (2π ) γ ,ω

1 1 2 (2π ) e,ω

 1 ∈ γ ,ω

d 2 nγ Jca

(B.27)

 2 2 2  β |M | h ¯ c − − f i + d 2 nγ Jca , 161 + 2 ∈ γ ,ω   2 β− |M f i |2h¯ 2 c2 d 2 nγ Jca − (B.28) 161 2 + − ∈ e,ω   − β− |M f i |2h¯ 2 c2 2 , d nγ Jca + 161 2 + ∈ e,ω 

where d 2 n± = dn− dn+ do1 , d 2 nγ = dnγ1 dnγ2 do− , 2 dnγ1,2 = d1,2 do1,2 1,2 fγ1,2 and the Jacobi matrix is Jca =

− β− |M f i |2h¯ 2 c2 162 +

285

dn± = d± do± ±2 β± f± ,

+ β− . (+ + − ) β− − (1 b1 + 2 b2 ) ·b−

(B.29)

± ± ± Møller Scattering of Electrons and Positrons e± 1 e2 → e1 e2

Consider now the scattering (2.10) process between electron (positron) with momenta p1 and p2 . After scattering these quantities are denoted with primes. The time evolution of the distribution functions of electrons (or positrons) is described by    ∂ fi (pi , t ) = dp j dp 1 dp 2V wp 1 ,p 2 ;p1 ,p2 (B.30) ∂t e1 e2 →e e 1 2

× [ f1 (p 1 , t ) f2 (p 2 , t ) − f1 (p1 , t ) f2 (p2 , t )], with i = 1, j = 2, and with j = 1, i = 2, and where |M f i |2 h¯ 2 c6 δ( +  −  −  )δ(p + p − p − p ) 1 2 1 2 1 2 1 2 (2π )2V 161 2 1 2 (B.31)  2

2 1 s +u + 4m2 c2 (t − m2 c2 ) |M f i |2 = 26 π 2 α 2 2 t 2

1 s2 + t 2 + 4m2 c2 (u − m2 c2 ) + 2 u 2 s   4 s (B.32) − m2 c2 − 3m2 c2 , + tu 2 2

wp 1 ,p 2 ;p1 ,p2 =

with s = (p1 + p2 )2 = 2(m2 c2 + p1 p2 ), t = (p1 − p 1 )2 = 2(m2 c2 − p1 p 1 ), and u = (p1 − p 2 )2 = 2(m2 c2 − p1 p 2 ) [48]. The energies of final state particles are given by (B.18) with new coefficients A˜ = (1 +2 )2 − (1 β1 b1 ·b 1 + 2 β2 b2 ·b 1 )2 , B˜ = (1 +2 )[m2 c4 +1 2 (1 − β1 β2 b1 b2 )],

286

Collision Integrals in Electron-Positron Plasma

and C˜ = m2 c4 (1 β1 b1 ·b 1 + 2 β2 b2 ·b 1 )2 + [m2 c4 + 1 2 (1 − β1 β2 b1 ·b2 )]2 . The condition to be checked is    1 (1 + 2 ) − m2 c4 − (1 2 )(1 − β1 β2 b1 ·b2 ) (1 β1 b1 + 2 β2 b2 )·b 1 ≥ 0. (B.33) Integration of eqs. (B.30), similar to the case of Compton scattering in Section B.1.1, yields  1 2 β1 |M f i |2h¯ 2 c2 1 1 e1 e2 →e 1 e 2 2 ηe,ω = d nJ (B.34) ms (2π )2 e,ω 161 2 2 1 ∈ e,ω   1 β1 |M f i |2h¯ 2 c2 2 + d nJms 161 2 2 ∈ e,ω  1 β1 |M f i |2h¯ 2 c2 1 1 e1 e2 →e 1 e 2 2 = d nJ (B.35) (χ E )e,ω ms (2π )2 e,ω 162 2 1 ∈ e,ω    β |M f i |2h¯ 2 c2 , d 2 nJms 1 1 + 161 2 2 ∈ e,ω 2 β1,2 f1,2 , and the Jacobi matrix is where d 2 n = dn1 dn2 do 1 , dn1,2 = d1,2 do1,2 1,2

Jms =

2 β2 . (1 + 2 )β1 − (1 β1 b1 + 2 β2 b2 )·b 1

(B.36)

B.1.3 Bhaba Scattering of Electrons on Positrons e− e+ → e− e+ The time evolution of the distribution functions of electrons and positrons due to Bhaba scattering is described by    ∂ f± (p± , t ) = dp∓ dp − dp +V wp − ,p + ;p− ,p+ (B.37) ∂t e− e+ →e− e+ × [ f− (p − , t ) f+ (p + , t ) − f− (p− , t ) f+ (p+ , t )],

where wp − ,p + ;p− ,p+ =

|M f i |2 h¯ 2 c6 δ( +  −  −  )δ(p + p − p − p ) , (B.38) − + − + − + − + (2π )2V 16− + − +

and |M f i | is given by eq. (B.32), but the invariants are s = (p− − p + )2 , t = (p+ − p + )2 , and u = (p− + p+ )2 . The final energies − , + are functions of the outgoing particle directions in a way similar to that in Section B.1.2 (see also [48]).

B.2 Collision Integrals for Binary Reactions with Protons

287

Integration of eqs. (B.37) yields − +

− +

e e →e η±,ω

− +

e

− +

(χ E )e±,ωe →e

e

1 1 = 2 (2π ) ±,ω

=

1 1 2 (2π ) ±,ω

 ∈  − e,ω

d 2 n ± Jbs

− 2 β− |M f i |2h¯ 2 c2 16− + +

(B.39)

 2 2 2  β |M | h ¯ c f i + d 2 n ± Jbs − − ∈  16− + + e,ω   β |M f i |2h¯ 2 c2 d 2 n ± Jbs − − (B.40) 16+ + − ∈ e,ω   − β− |M f i |2h¯ 2 c2 2 + d n± Jbs , 16− + + ∈ e,ω 

where d 2 n ± = dn− dn+ do − , dn± = d± do± ±2 β± f± , and the Jacobi matrix is Jbs =

+ β+ . (− + + )β− − (− β− b− + + β+ b+ )·b −

(B.41)

Analogously to the case of pair creation and annihilation in Section B.1.2, the energies of final state particles are given by (B.18) with the coefficients A˘ = (− + + )2 − (− β− b− ·b − + + β+ b+ ·b − )2 ,   (B.42) B˘ = (− + + ) m2 c4 + − + (1 − β− β+ b− ·b+ ) ,  2 4  2 2 C˘ = m c + − + (1 − β− β+ b− ·b+ ) + m2 c4 − β− b− ·b − + + β+ b+ ·b − . To select the correct root, one has to check the condition (B.33), changing the subscripts 1 → −, 2 → +.

B.2 Collision Integrals for Binary Reactions with Protons B.2.1 Compton Scattering on Protons γ p → γ p The rate for this process tγ−1p , compared to the rate of Compton scattering of electrons tγ−1 e , is much longer: tγ−1p =

n p  ± 2 −1 tγ e n± Mc2

 ≥ mc2 .

(B.43)

Moreover, it is longer than any timescale for binary and triple reactions considered, and thus this reaction can be excluded from the computations.

288

Collision Integrals in Electron-Positron Plasma

B.2.2 Electron-Proton and Positron-Proton Scattering e± p → e ± p The time evolution of the distribution functions of electrons due to ep → e p is described by    ∂ f± (p, t ) = dqdp dq V wp ,q ;p,q [ f± (p , t ) f p (q , t ) − f± (p, t ) f p (q, t )] ∂t ep→e p (B.44)    ∂ f p (q, t ) = dpdp dq V wp ,q ;p,q [ f± (p , t ) f p (q , t ) − f± (p, t ) f p (q, t )], ∂t ep→e p (B.45) where |M f i |2 h¯ 2 c6 δ( +  −  −  )δ(p + q − p − q ) , (B.46) e p e p (2π )2V 16e  pe  p  1 2 2 6 2 21 (s + u2 ) + (m2 c2 + M 2 c2 )(2t − m2 c2 − M 2 c2 ) , |M f i | = 2 π α 2 t 2 (B.47)

wp ,q ;p,q =

the invariants are s = (p + q)2 = m2 c2 + M 2 c2 + 2p · q, t = (p − p )2 = 2(m2 c2 − p · p ) = 2(M 2 c2 − q · q ), and u = (p − q )2 = m2 c2 + M 2 c2 − 2p · q , s + t + u = 2(m2 c2 + M 2 c2 ). The energies of particles after interaction are given by (B.18) with 2  A¯ = (± +  p )2 − (± β± b± +  pβ pb p )·b ± , (B.48) B¯ = (± +  p )[m2 c4 + ±  p (1 − β± β pb± ·b p )],  2 C¯ = m2 c4 (± β± b± ·b ± +  pβ pb p·b ± )2 + [m2 c4 + ±  p (1 − β± β pb± ·b p )] . The correct root is selected by the condition (B.33) with the substitution 1 → ±, 2 → p. Absorption and emission coefficients for this reaction are  ± 2 β± ± |M f i |2h¯ 2 c2 1 1 ep (χ E )±,ω = dn dn do J , (B.49) ± p ep ± (2π )2 ±,ω ± ∈ ±,ω 16±  p±  p  ± 2 β±  p|M f i |2h¯ 2 c2 1 1 (χ E )ep = dn dn do J , (B.50) ± p p,ω ± ep (2π )2  p,ω  p ∈  p,ω 16±  p±  p  ± 2 β± ± |M f i |2h¯ 2 c2 1 1 ep η±,ω = dn dn do J , (B.51) ± p ep ± (2π )2 ±,ω ± ∈ ±,ω 16±  p±  p  ± 2 β±  p |M f i |2h¯ 2 c2 1 1 ep dn± dn pdo± Jep , (B.52) η p,ω = (2π )2  p,ω  p ∈  p,ω 16±  p±  p

B.3 Collision Integrals for Triple Interactions

289

where dni = di doi i2 βi fi , i = ±, p, and the Jacobi matrix is Jep =

 p β p (± +  p )β± − ( pβ pb p + ± β± b± )·b ±

.

(B.53)

The rate for proton-electron (proton-positron) scattering is −1 tep ≈

 −1 t , Mc2 ee

±   p.

(B.54)

B.2.3 Proton-Proton Scattering p1 p2 → p 1 p 2 This reaction is similar to e1 e2 → e 1 e 2 , described in Section B.1.2. The time evolution of the distribution functions of electrons is described by    ∂ fi (pi , t ) = dq j dq 1 dq 2V wq 1 ,q 2 ;q1 ,q2 (B.55) ∂t p1 p2 →p p 1 2

× [ f1 (q 1 , t ) f2 (q 2 , t ) − f1 (q1 , t ) f2 (q2 , t )], with j = 3 − i, and where |M f i |2 h¯ 2 c6 δ( +  −  −  )δ(q + q − q − q ) 1 2 1 2 1 2 1 2 (2π )2V 161 2 1 2 (B.56)  2

2 1 s +u + 4M 2 c2 (t − M 2 c2 ) |M f i |2 = 26 π 2 α 2 2 t 2

1 s2 + t 2 2 2 2 2 + 4M c (u − M c ) + 2 u 2 s  4 s (B.57) − M 2 c2 − 3M 2 c2 , + tu 2 2

wq 1 ,q 2 ;q1 ,q2 =

and the invariants are s = (q1 + q2 )2 = 2(M 2 c2 + q1 · q2 ), t = (q1 − q 1 )2 = 2(M 2 c2 − q1 · q 1 ), and u = (q1 − q 2 )2 = 2(M 2 c2 − q1 q 2 ). For the rate one has 8 8 m n p −1 m −1 v± , tee , vp ≈ v± ≈ c. (B.58) t pp ≈ M n± M B.3 Collision Integrals for Triple Interactions The emission coefficients for triple interactions are adopted from [251].

290

Collision Integrals in Electron-Positron Plasma

The bremsstrahlung is ∓ ∓ ∓ ∓ ηγe e →e e γ

=

(n2+

+

αc 3 ε

16 n2− )



e2 mc2

2

3√ 2θ + 2θ 2 θ 5 , × ln 4ξ (11.2 + 10.4θ ) ε exp(1/θ )K2 (1/θ ) 2  16 2αc e2 = n+ n− 3 ε mc2

√ 2 + 2θ + 2θ 2 2 θ , × ln 4ξ (1 + 10.4θ ) ε exp(1/θ )K2 (1/θ )  2 16 αc e2 = (n+ + n− )n p 3 ε mc2

1 + 2θ + 2θ 2 θ , × ln 4ξ (1 + 3.42θ ) ε exp(1/θ )K2 (1/θ )

2

− +

ηγe

e →e− e+ γ

±

ηγpe

→p e± γ

(B.59)

(B.60)

(B.61)

where ξ = e−0.5772 and K2 (1/θ ) is the modified Bessel function of the second kind of order 2. The double Compton scattering is ± ± ηγe γ →e γ γ

128 αc = (n+ + n− )nγ 3 ε



e2 mc2

2

θ2 . 1 + 13.91θ + 11.05θ 2 + 19.92θ 3 (B.62)

The three-photon annihilation is ± ∓

ηγe

e →γ γ γ

 = n+ n− αc

2

e mc2

2

 2 π2 1 2ξ θ + − 2 ln 6 2 1  , ε 4θ + 1 2 ln2 2ξ θ + π 2 − 1 6 2 θ2 4 θ



(B.63)

where the two limiting approximations given by [251] are joined together. The radiative pair production is

n2γ K2 (1/θ ) 2 γ γ →γ e± e∓ e± e∓ →γ γ γ ηe = ηγ . (B.64) n+ n− 2θ 2 The electron-photon pair production is ⎧  2 2  2 ⎪ e ⎨ (n+ + n− )nγ αc mc exp − θ 16.1θ 0.541 , θ ≤ 2, 2 ± ± ± ∓ e1 γ →e1 e e ηγ =  2 2   1 ⎪ 56 8 ⎩ (n+ + n− )nγ αc e 2 ln 2ξ θ − 27 , θ > 2. 9 1+0.5/θ mc (B.65)

B.4 Mass Scaling for the Proton-Electron/Positron Reaction

The proton-photon pair production is ⎧  2 2   1 ⎪ e ⎨ n αc exp − θ2 1+0.9θ , n p γ 2 mc ± ∓ ηγpγ →p e e =  2   ⎪ 28 92 ⎩ n pnγ αc e22 (ln 2ξ θ + 1.7) − , 9 27 mc

291

θ ≤ 1.25277,

θ > 1.25277. (B.66) The absorption coefficient for three-body processes can be written as χγ3p = ηγ3p /Eγeq ,

(B.67)

where ηγ3p is the sum of the emission coefficients of photons in the three particle processes, Eγeq = 2π  3 fγeq /c3 , where fγeq is given by (11.24). From eq. (11.58), the law of energy conservation in the three-body processes is  (ηi3p − χi3p Ei )dμd = 0. (B.68) i

For exact conservation of energy in these processes the following coefficients of emission and absorption for electrons are introduced:  χe3p

=

(ηγ3p − χγ3p Eγ )ddμ  , Ee ddμ

 ηe3p

= 0,

(ηγ3p − χγ3p Eγ )ddμ > 0, (B.69)

or  3p (ηγ − χγ3p Eγ )ddμ ηe3p  =− , Ee Ee ddμ

 χe3p

= 0,

(ηγ3p − χγ3p Eγ )ddμ < 0. (B.70)

B.4 Mass Scaling for the Proton-Electron/Positron Reaction The mass ration m/M  1/1836 presents serious difficulties in numerical simulation of plasmas (see, e.g., [583],[211]). This complication is avoided by making use of the mass scaling in matrix elements. Since proton mass is larger than electron mass-energy M  m,  then for the CM frame, p1 + p2 , ≈ 1, J1 ≈ 1, M   1 1 − 1 ≈ V e 01 − e01 p0 ∝ , M

V≈

(B.71) (B.72)

292

Collision Integrals in Electron-Positron Plasma

and also s2 ≈ M 4 + 4mM 3 + 6m2 M 2 , c4 u2 ≈ M 4 − 4mM 3 + 6m2 M 2 , c4 2   M f i ∝ 1 6m2 − 2t M 2 , 2 t while

  2 −2m2 βe0 1 − ee0 e e0 t= 2 1 − βe0    −1  −2m2 βe2 1 − ee e e  1 + O M = 1 − βe2

for small angles. This leads to the following scaling for the reaction rate:  2   e − e M f i 1 ep ep ∝ ηeω − (χ E )eω ∝ e  pe  p M.

(B.73) (B.74) (B.75)

(B.76)

(B.77)

ep0 0 One can therefore compute ηeω , (χ E )ep eω for a pseudo-particle with mass M0  m,  instead of M and obtain

M0 ep0 η (B.78) M eω M0 0 (χ E )ep (χ E )ep (B.79) eω ≈ eω . M For such purposes, the mass of this pseudo-particle is taken as M0 = 20m. ep ηeω ≈

Appendix C Collision Integrals for Weak Interactions

C.1 Scattering of Neutrinos on Electrons Taking into account the filled final electron states, one has     ∂ fν (q, t) V dpV dq V dp = wq ,p ;q,p (1 − Fe (p, t )) fν (q , t ) fe (p , t ) 6 ∂t (2π h¯ ) νe  − (1 − Fe (p , t )) fν (q, t ) fe (p, t ) , (C.1) where Fe ≡

(2π h¯ c)3 fe , 2

the probability of the process is

wq ,p ;q,p = c(2π h¯ )4

|M f i |2 (¯hc)4 δ(q + p − q − p ) , V3 16ν e ν e

the square of the matrix element is [131, 584]   GF 2 2  2 (CV + CA )2 (pq)(p q ) c |M f i | = 32 (¯hc)3  + (CV − CA )2 (p q)(pq ) − (CV2 − CA2 )m2e c2 (qq ) , m2

(C.2)

(C.3)

m2

where CV = 1 + 2ma 2 gW2 = 12 + 2 sin2 θW = 1.2, CA = 1 − 2mb 2 gW2 = 12 , the mass of Z Z the intermediate bosons is mW = 37.3/ sin θW GeV, mZ = 74.6/ sin(2θW ) GeV, the tangent of the Weinberg mixing angle is tan θW = g /g, and the Fermi constant GF is √ −5 GF 2 g2 given by (¯hc)3 = 8 m2 c4 = 1.015·10 . In the computations, it is used q = cν (1, eν ), m2 c4 W

p

p = ce (1, βee ), dq = dν dων ν2 /c3 , dp = de dωe e2 β/c3 , do = dμdφ. The conservation of energy-momentum yields the energies of the product particles as a function of their initial momenta and the final angle of the neutrino trajectory: ν =

ν e (1 − βe be · bν ) , e (1 − βe be · b ν ) + ν (1 − βe bν · b ν ) e = e + ν − ν ,

(C.4) (C.5) 293

294

Collision Integrals for Weak Interactions

where the unit vector bi ≡ pi /pi coincides with the direction of the particle momentum i. The direction of the final electron trajectory is b e = (βe e be + ν bν − ν b ν )/(βe e ). Consider the absorption process in more detail. Using a delta function for the momenta, one can integrate the reaction rate over dp . In the following integration over dq = dν (ν )2 dων /c3 , one should use e from the previous momentum conservation relation, p = q + p − q :   −1 ∂ (ν + e ) dν δ(ν + e − ν − e ) = d(ν + e ) δ(ν + e − ν − e ) ∂ν = |Jνe |,

(C.6)

where −1 = Jνe

∂ (ν + e ) ∂ (e ) = 1 + . ∂e ∂ν

(C.7)

Using the momentum conservation, βe e = (ν bν + βe e be − ν b ν )b e ,

(C.8)

∂ (βe e ) 1 ∂ (e ) ∂ (ν ) = = − b b = −b ν b e ∂ν βe ∂ν ∂ν ν e

(C.9)

one obtains

Jνe =

1 . 1 − βe b ν b e

(C.10)

The absorption coefficient for the neutrino for the considered reaction is   1 ∂ fν abs νe χν f ν = − c ∂t νe  ( )2 |M f i |2h¯ 2 c2 = dnν dne do ν Jνe (1 − Fe (p , t )) ν fν , (C.11) 16ν e ν e where dni = di doi i2 βi fi /c3 = di dωi Ei /(2π i ). One can obtain the remaining numerical coefficients in similar fashion:  1 νe (χ E )ν,ω,k (T, μe ) = dnν dne do ν ν,ω μk ν ∈ ν,ω ,μν ∈ μk × Jνe (1 − Fe (p , t ))

ν 2 ν |M f i |2h¯ 2 c2 , 16ν e ν e

(C.12)

C.2 Absorption of Neutrinos by Neutrons νe ην,ω,k (T, μe ) =

1 ν,ω μk

 ν ∈ ν,ω ,μ ν ∈ μk

× Jνe (1 − Fe (p , t ))

295

dnν dne do ν

ν 2 ν |M f i |2h¯ 2 c2 . 16ν e ν e

(C.13)

The quantities Eν do not depend on φ, and the Ee values do not depend on μ or φ. One can calculate the corresponding integrals once [269], using the relations   d dμ 2β dp 2 8π 2 dne = → , (C.14) e e (2π h¯ )3 (2π h¯ c)3 ddμ,φ=0 exp( −μ exp( −μ )+1 )+1 kT kT 1 − Fe (p) = 1 −

1 e exp( −μ )+1 kT

=

e ) exp( −μ kT

e exp( −μ )+1 kT

.

(C.15)

In the case of antineutrino scattering on electrons, one makes the substitution q ↔ −q (equivalent to the substitution CA → −CA ) in the matrix element above. C.2 Absorption of Neutrinos by Neutrons The change in the neutrino distribution function due to absorption of neutrinos by neutrons is      ∂ fν (q, t) V dPV dQV dp − f = w (q, t ) f (P, t ) , p,Q;q,P ν n ∂t (2π h¯ )6 νn where the probability of this process is wp,Q;q,P = c(2π h¯ )4

|M f i |2 (¯hc)4 δ(q + P − p − Q) , V3 16ν n e  p

(C.16)

and the square of the matrix element |M f i |2 is presented in [131, 584]:   GF 2 2  2 c |M f i | = 32 (ν + 1)2 (Pq)(Qp) + (ν − 1)2 (Pp)(Qq) (¯hc3 )  (C.17) + (ν 2 − 1)mn m pc2 (pq) , with ν = 1.21. The conservation of energy-momentum yields the energy of the product particles as a function of their initial momenta and the angle for the final trajectory of the electron. Denote the energy-momentum four-vectors q for ν, P for n, p for e, and Q for p. The nonstandard selection of Q instead P for the protons makes it possible to avoid the use of primes and simplifies our treatment of the reverse reaction. One should first exclude from the system ν + n =  p + e ,

(C.18)

ν bν + βn n bn = β p pb p + βe e be ,

(C.19)

296

Collision Integrals for Weak Interactions

the energy and velocity of p. All terms relating to p on the right-hand sides of the two equations are retained. Taking squares on both sides of the two equations and subtracting the first from the second one yields the equality (−m2n − m2e + m2p )c4 − 2ν n (1 − βn bn bν ) + 2e (ν + n ) = 2e βe be (βn bn + ν bν ).

(C.20)

Squaring this last equation, one obtains a quadratic equation for the final energy e as a function of the angles be be and the initial states. The required root satisfies conservation of energy-momentum. Further more, using a delta function for the momenta, one can integrate the reaction rate over dQ. In the following integration over dp, one should use e from the conservation of momentum relation, p = q + P − Q:   ∂ (e +  p ) −1 δ(ν + n −  p − e ) de δ(ν + n −  p − e ) = d(e +  p ) ∂e (C.21) = |Jνn |, where −1 = Jνn

∂ (e +  p ) ∂ ( p ) =1+ , ∂e ∂e

(C.22)

with β p p = (ν bν + βn n bn − βe e be )b p,

(C.23)

∂ (β p p ) 1 ∂ ( p ) ∂ (βe e ) 1 = =− be b p = − be b p. ∂e β p ∂e ∂e βe

(C.24)

The Jacobi matrix transformation is Jνn =

1 . 1 − β pbe b p/βe

(C.25)

The neutrino absorption coefficient in this reaction is  1 νn (χ E )ν,ω,k = dnν dnn doe ν,ω μk ν ∈ ν,ω ,μν ∈ μk × Jνn (1 − Fe (p))(1 − Fn (Q))

e2 βe ν |M f i |2h¯ 2 c2 . 16ν n e  p

(C.26)

Since the distribution functions for the neutrons, protons, and electrons are independent of angle (in a co-moving system), the absorption coefficient likewise does not depend on μ: μk νn νn χν,ω,k χν,ω,k = . (C.27) 2 k

C.3 Creation of Neutrinos

297

This reaction changes the numbers of electrons and positrons and is included in the equation for Y˙e .

C.3 Creation of Neutrinos The change in the neutrino distribution function due to this process, taking into account the neutron degeneracy, is    ∂ fν (q, t) V dPV dQV dp = wq,P;p,Q (1 − Fn (P, t )) fe (p, t ) f p (Q, t ). ∂t (2π h¯ )6 νn (C.28) The probability wp,Q;q,P was already determined in the previous section. One can use the conservation of energy-momentum to obtain the energies of the product particles as functions of the initial momenta and the angle of the final trajectory of the neutron. Carrying out the procedure described above yields an equation for ν as a function of the angle of the product neutrino bν : (m2p + m2e − m2n )c4 + 2e  p (1 − βe be β pb p ) = 2ν ( p (1 − β pb pbν ) + e (1 − βe be bν )).

(C.29)

One can remove the extra solution from the conservation of momentum. Further more, using a delta function, one can integrate the rate over dP. In the following integration over dq, one should use  p from the previous momentum conservation relation Q = P + q − p, which leads to the Jacobi matrix transformation −1 Jep = 1 − βn bν bn . One obtains for the emission coefficient for the neutrino via this reaction  1 νn ην,ω,k = dne dn pdoν ν,ω μk ν ∈ ν,ω ,μν ∈ μk × |Jep|(1 − Fn (P, t ))

(ν )2 ν |M f i |2h¯ 2 c2 . 16e  pν n

(C.30)

The absorption coefficient likewise does not depend on μ. This reaction should be taken into account in the equation for Y˙e . The absorption and emission coefficients for antineutrino reaction with nucleons can be considered in the same way. Moreover, the computations include reactions for the absorption and emission of neutrinos by nuclei [585, 118]. The production of neutrinos via the annihilation of electron-positron pairs also was included in the computations. Isotropic distribution of neutrinos is assumed in the calculations of their annihilation (in exact calculations the rate of the inverse

298

Collision Integrals for Weak Interactions

reaction depends on the two angles for the two neutrinos). The role of neutrino annihilation in the shell ejection is similar to neutrino scattering on electrons, because number of antineutrinos is less in comparison with neutrinos at the neutralization of the matter (matter is converting in neutrons). The scattering of neutrinos on nucleons (this can be treated analogously to the scattering of neutrinos on electrons) is also included.

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Index

A-cascade, 184, 185 A-stable method, 85, 87 absorption coefficient, 22, 23, 96, 157–159, 179, 190, 267 active galactic nuclei, 152, 203, 205, 238 additivity of energy, 226 adiabatic index, 35, 216, 263, 269 advection equation, 107 AGN, 214, 261 Antonov instability, 259 artificial viscosity, 98, 99, 112, 121, 127, 266, 267 B-splines, 132 backward-lightcone, 28 baryon acoustic oscillations, 249 baryon asymmetry, 249 baryonic loading, 162, 171, 175–177, 217 BBGKY equations, 42, 44 BBGKY hierarchy, 10, 41, 43, 44, 48, 186, 228, 229 beam-laser interaction, 182 Belyaev-Budker equation, 47, 48 Bogolyubov method, 10 Bose enhancement, 164, 203, 204 Bose-Einstein distribution, 210 Burkert profile, 261 canonical ensemble, 227 canonical form of the second-order PDE, 67 cardinality, 61 Cauchy problem, 64, 69, 83, 85 causal Green’s function, 47 center of momentum, 24, 52, 158 central difference, 71, 75, 97, 98, 111, 267, 268 central limit theorem, 101 characteristic, 69, 70, 72, 98, 113–115, 232, 245, 246, 269 charge conservation, 51, 132, 165, 166 Chebyshev set of variable time steps, 81 chemical potential, 36, 120, 155, 159, 160, 163–166, 168–173, 181, 210, 260, 264, 277

326

chirped pulse amplification, 182 Christoffel symbols, 19, 278 classicality parameter, 154 cloud-in-cell, 132 coarse-grained distribution function, 250 coarse graining, 26, 251 coherent isotropic scattering, 222 collision integral, 17, 19, 21, 31, 34, 38, 48, 155, 158, 159, 167, 178, 186, 188, 195, 204, 207, 209, 217, 233 collisionless shocks, 144, 148, 149 comoving coordinates, 108, 231, 245 comoving reference frame, 151, 178, 205, 216–219, 222, 266 Compton parameter, 204, 209, 211 comptonization, 203, 214 computational grid, 64, 70, 91, 94, 97, 108, 109, 126, 129, 171, 174, 195, 217, 267 conformal gauge, 239 conservation laws, 30, 31, 108, 111, 112, 121, 123, 124, 163 conservative form, 96, 187, 209 conservative method, 96, 167 contact discontinuity, 76, 108, 109, 115–118 core-collapse supernova, 263 core-halo configuration, 255 correlation radius, 52, 53, 231 cosmic microwave background, 100 cosmological horizon, 237 cosmological perturbation theory, 237 cosmological perturbations, 239 cosmological term, 240 Coulomb logarithm, 47, 53, 119, 154, 157, 173, 233 counterpropagating laser beams, 185, 186 Courant condition, 107, 109, 132 covariant derivative, 20, 239 covariant derivative of the tensor, 278 Crank-Nicolson method, 106 Crankicolson method, 81

Index critical density, 236, 261 critical electric field, 199 critical field, 182 cross section, 24, 25, 52–54, 104, 154, 156, 161, 205, 274 cutoff frequency, 140 cyclic reduction method, 91, 94, 99, 112 cycling reduction, 98 d’Alembert operator, 41 dark energy, 240 dark matter, 234, 237–240, 242, 248, 260, 261 Debye length, 41, 48, 52, 140, 152, 154 degeneracy factor, 21 degeneracy parameter, 57, 155 degeneracy temperature, 57 derivative on the grid, 63 detailed balance condition, 24, 164, 181 diffeomorphism, 239 differential cross section, 24, 52, 155, 156 differential optical depth, 23 diffusion approximation, 215 diffusion equation, 68, 211 diffusion radius, 205, 224 diffusion with flux limiter, 106, 107 dilute gas, 53, 54 dilute plasma, 48, 52, 54, 55 dimensional splitting, 81, 108, 111 Dirichlet problem, 83 discrete Fourier transform, 62, 63, 82 discriminant of the second-order PDE, 67 dispersion curve, 141 dispersion of the random value, 100 dispersion relation, 135–137, 139–142, 145–148, 150 divergence of the tensor, 279 divergence of the vector, 278 domain of dependence, 72 Doppler factor, 215 Einasto profile, 261 Einstein-Vlasov equations, 11, 48 electron-positron plasma, 56, 86, 95, 140, 143, 150, 152, 153, 155, 164–167, 171, 173–175, 177, 181, 183, 186, 191, 195, 196, 198 electroneutrality condition, 120 electrosphere, 199, 202 elliptic equation, 67, 68, 81 elliptical galaxy, 234, 250, 253 emission coefficient, 23, 96, 179 energy conservation law, 280 energy density, 34, 35, 111, 124, 146, 149, 155, 162, 167, 168, 170–172, 175, 177, 180, 187, 189, 191, 192, 195, 196, 211, 212, 240 energy-momentum conservation, 31, 38, 156, 203, 239, 241 ensemble averaging, 27, 42, 43 ensemble inequivalence, 226 enthalpy, 35, 124

327

entropy density, 35, 251, 276 entropy flux, 15 equation of state, 94, 106, 109, 110, 112, 115, 118–120, 124, 128, 162, 198, 267–269, 273, 274 equidistant grid, 63 ergodic systems, 29 ergodicity, 29, 44 Eulerian approach, 126 explicit Euler method, 84 explicit method, 84, 85, 87, 96, 107 explicit scheme, 64, 70, 72, 74–81, 109 extensivity of energy, 226 extinction coefficient, 23 Faraday rotation, 152 fast Fourier transform, 82, 130, 152, 238 Fermi energy, 56, 274 Fermi-Dirac statistics, 269 filamentation instability, 144 fine-grained distribution function, 250 fine-grained phase space, 128 finite difference, 62, 64, 66, 81, 83, 95, 97, 104, 130 finite difference equation, 64, 69, 70, 72, 94 finite difference method, 61, 81 finite difference scheme, 70, 73, 75, 81, 104, 108 finite difference time domain method, 131 first moment of distribution function, 15 first-order approximation, 98, 109 first-order explicit Euler method, 65, 83 flux limiter multigroup diffusion, 107 Fokker-Planck approximation, 48, 53, 157, 158, 178, 206, 212 forward difference scheme, 71, 72 four-flux, 15, 38 Fourier stability analysis, 73 Fourier transform, 62, 63, 82 free electron laser, 182 free fall time, 253 free streaming, 248, 249 freeze-out, 39, 180 Friedmann equation, 240 FRW metric, 239, 240, 242 fugacity, 159, 174 fuzzy photosphere, 219 galaxy bias, 261 gamma-ray burst, 144, 148, 150, 153, 177, 179, 203, 205, 216, 220, 221 Gauss distribution, 101 Gauss elimination, 89 Gauss theorem, 17, 235 Gauss-Seidel method, 92 Gear’s method, 88–90, 95, 99, 112, 266, 268 geodesic equation, 243 geometric progression, 62 global equilibrium, 34 globular cluster, 233, 250, 253 Godunov high-order methods, 108

328

Index

gradient of the scalar, 278 gradient of the vector, 278 gravitational collapse, 95, 106, 107, 110, 123, 183, 262, 263, 266, 276 gravitational instability, 143, 237, 247, 250, 260 gravitational lensing, 261 gravothermal catastrophe, 259 Green’s function, 41, 232 growth factor, 85 growth rate, 145–148 H-theorem, 33, 45, 251 Hamiltonian mean field model, 259 hierarchical clustering, 260 hierarchical tree algorithm, 128 Hubble parameter, 240 hydrodynamic instability, 143 hydrostatic equilibrium, 210, 235 hyperbolic equation, 67–69, 70–72, 74–76, 78 impact parameter, 52, 158 implicit method, 85, 87, 95 implicit scheme, 72, 74, 75, 77, 81, 91, 99, 106–108, 161 induced current, 135, 136 instability to collapse, 269 integration along the characteristic, 268 ionization potential, 120 isothermal sphere, 256–260 Jüttner distribution, 34, 227 Jacobi matrix, 84, 85, 89, 99, 124, 125, 157, 201, 268 Jacobi method, 91 Jeans equations, 234 Jeans length, 229, 245 kinetic equilibrium, 155, 159–161, 163–166, 170–172, 176, 181, 195, 196, 198 kinetic instability, 143 Klein conditions, 260 Klimontovich random function, 11, 14, 16, 19, 26, 40–42, 48, 130, 228 Kompaneets equation, 178, 206, 209, 217 Kramers-Kronig relations, 136 Lagrangian approach, 108, 126 Landau damping, 142–145, 248, 255 Landau equation, 48, 230, 233 Langmuir wave, 140, 141 Laplace equation, 83 large-scale structure, 100, 213, 238, 239, 248, 260 large-particle representation, 103 large-scale convection, 265, 273, 275 Larmor frequency, 147 leapfrog method, 128, 129, 131 Ledoux criterion, 276 left-eigenvectors, 125 Legendre polynomials, 244

lepton asymmetry, 249 lepton era, 152 lepton-emission self-sustained asymmetry, 265 light curve, 178, 225, 263, 265, 270, 272, 273 linear congruent generator, 103 linear interpolation, 62, 118 linear Monte Carlo method, 104 linear perturbation, 237, 241 linear perturbation theory, 239 linear response tensor, 136, 138, 139 linearized Riemann problem, 109 Liouville equation, 9, 10, 228 Lipschitz condition, 83 local equilibrium, 34 local model for the equation of state, 109, 110, 115, 118, 119 longitudinal wave, 139–141 Lorentz factor, 37, 124, 149, 158, 177, 183, 185, 216, 219, 220, 223 Lorentz reference frame, 15, 16, 20, 34, 138–140, 144, 147 Lynden-Bell distribution, 255 Mach speed, 121 macroscopic gravity, 11 macroscopic Maxwell equations, 27 macroscopic quantity, 9, 14, 26, 34, 130, 234 magnetorotational instability, 264 Mandelstam variables, 24 Markov chain, 100 mass conservation law, 279 mass ejection rate, 217 material relations, 27 mathematical expectation, 100, 101 matrix inversion, 89 mean free path, 53, 54, 105, 154, 157, 205, 224 mean number of scatterings, 204, 205 mesh-free method, 126 mesoscopic system, 227 method of lines, 83, 95, 99, 107, 124, 178, 217, 266 metric tensor, 19, 239, 240, 278 microcanonical ensemble, 227 microscopic gravity, 48 Minkowski space, 17, 28, 237 MIT bag model, 198 mixed congruential generator, 103 molecular chaos hypothesis, 44 momentum conservation law, 279 monoatomic gas, 120 monotonic scheme, 76, 78 monotonic solution, 76, 108, 109 Monte Carlo simulations, 103, 178, 206, 219, 221, 225 multicomponent gas, 109, 111 multiplicative congruential generator, 103 N-body, 40, 128, 130, 238, 261 Navarro-Frenk-While profile, 261 negative heat capacity, 227

Index negative specific heat, 227, 259, 260 neutrino luminosity, 272, 276 neutrino mass, 249, 273 neutrino spectrum, 265, 275 neutrino-driven mechanism, 264 neutrinosphere, 264, 271, 272, 276 neutron star, 95, 183, 198, 262, 264, 265, 274 neutronization, 263, 269, 271, 275 Newton’s iteration, 88, 94, 117 Newtonian cosmology, 238, 245, 248 nuclear statistical equilibrium, 269 numerical relativity, 106, 123 occupation number, 21, 155, 209, 216, 268 ODE system with decay, 86 one-particle distribution function, 12, 13, 19, 22, 23, 26, 29, 34, 55, 110, 154, 232, 242, 250 optical depth, 21, 23, 95, 96, 105, 153, 154, 160, 161, 177–180, 195, 205, 206, 210, 212, 214–222, 225, 268 optically thick region, 99 optically thin region, 99, 108 order of accuracy, 63, 109 parabolic equation, 67, 68, 78, 79 Parseval’s identity, 73 particle evaporation, 227, 253, 254 particle horizon, 241 particle number conservation, 31, 130, 163, 165, 167 particle-in-cell, 41 Pauli blocking, 164, 203 phase mixing, 250, 251, 253, 260 phase space, 9, 12–14, 17, 19, 83, 95–97, 104, 106, 107, 129, 153, 167, 178, 187, 188, 195, 198, 228, 242, 250–252, 255, 267 phase velocity, 140, 142–144 photon thick outflow, 205, 220, 222, 224, 225 photon thin outflow, 205, 220, 221, 223–225 photospheric emission, 178, 181, 216, 220, 224 Planck spectrum, 104, 181, 201, 211, 216, 218, 219, 222, 224 plasma frequency, 51, 140, 145, 147, 190 plasma instability, 143, 237 plasma parameter, 41, 54, 154, 229 Poisson equation, 68, 94, 246, 248, 253, 255 predictor-corrector method, 88 pressure, 34, 35, 76, 108, 110, 115, 116, 118–122, 124, 128, 146, 162, 177, 240, 247, 256, 264, 269, 274 probability, 100 probability density, 101 proper time, 14 pseudo-random numbers, 103 pulsar magnetosphere, 144 quark star, 152, 183, 198–202 quarks, 198 quasi-particle, 129, 130

329

quasi-stationary state, 159, 250, 253, 255, 260 quasilinear form, 112, 113 radiative transfer, 21–23, 178, 264 radiative transfer equation, 23, 214, 216 random value, 100 Rankine-Hugoniot conditions, 117 rapid attenuation of correlations, 44, 232 rarefaction wave, 116–119 relative velocity, 24, 25, 38, 53, 104, 154 relativistic jet, 123, 144 relativistic Maxwellian distribution, 36, 37, 104, 150, 213, 215 relativistic outflow, 205, 216, 220, 221 relativistic transport equation, 17 relaxation, 45, 55, 81, 152, 161, 167, 174, 175, 231, 250–253 relaxation factor, 93 relaxation time, 25, 231, 233, 234, 250 Ricci tensor, 239 Riemann problem, 109, 110, 112, 114–116 Riemann problem solver, 110, 123, 124, 126 Riemann problem solver for the linearized system, 124 right-eigenvectors, 125 Ritz-Galerkin method, 83 Runge-Kutta method, 84, 86, 124 Rutherford cross section, 52, 153 S-cascade, 184, 185 Saha ionization equation, 120 scalar curvature, 239 scalar perturbations, 239 scale-invariant spectrum, 242 scattering coefficient, 23 Schwarzschild criterion, 276 second moment of distribution function, 15 second-order approximation, 97, 98 second-order linear partial differential equations, 66 self-gravitating system, 29, 44, 128, 143, 226–230, 233, 236, 237, 250, 251, 253–256, 258–260 shape function, 130, 132 shock wave, 76, 108–110, 112, 115–122, 149, 210, 262, 272, 276 Simpson quadrature, 85 singular isothermal sphere, 256 singularity, 42, 49 smoothed-particle hydrodynamics, 126, 128, 238 smoothing kernel, 127 sound speed, 107, 124, 146, 245 spacelike surface, 16 spacetime averaging, 28 sparse matrix, 94, 99 spatial correlation function, 52 spectral energy density, 96, 166, 187, 267 spectral method, 75, 81 square matrix, 89 stability of the scheme, 64, 81, 132

330 standard deviation, 101 standing accretion shock instability, 265 statistical equilibrium, 29, 199, 250 steady relativistic wind, 217, 219, 220 stiff equation, 86, 95, 99, 104 stiff stability, 88 stochastic process, 99, 100, 252 strange quark matter, 198 subluminal wave, 141, 150 successive overrelaxation method, 93 summational invariant, 31, 34 Sunyaev-Zeldovich effect, 210, 211, 213, 214 super-Eddington luminosity, 201 superluminal wave, 141, 142 supernova, 95, 144, 238, 261–265, 275 tangential discontinuity, 117 Taylor series, 63, 83, 85, 206, 207 thermal equilibrium, 39, 53, 54, 57, 153, 154, 159–162, 166, 167, 169, 170, 172–175, 180, 181, 198, 199, 202, 227, 236, 259 thermal radiation, 212 thin shell, 205, 220 tokamak technology, 143 Tolman conditions, 260 Tolman-Oppenheimer-Volkoff equation, 198, 260 transfer factor, 65 transition rate, 18 transverse wave, 139–141

Index trapezoidal rule, 87, 88 two-particle distribution function, 27, 43, 44, 55 two-stream instability, 144, 147 ultra-intense lasers, 152, 177, 179 upwind derivatives approximation, 98 upwind difference, 98 upwind scheme, 72, 74 URCA process, 263 variational method, 83 violent relaxation, 234, 253, 255, 260 virial condition, 254, 255 virial equilibrium, 227, 259 virial radius, 236 virial theorem, 235, 252 Vlasov-Maxwell equations, 41, 45, 129, 132, 135, 137 Vlasov-Poisson equations, 46, 230, 234, 245, 250 von Neumann stability analysis, 73 waterbag distribution function, 145, 254 Weibel instability, 144, 146, 147 Wien law, 204, 222 Wien spectrum, 105 world lines, 16, 17, 47 X-ray binaries, 214 ZAMS mass, 263