# Elements of Nonequilibrium Statistical Mechanics 3030622320, 9783030622329

This book deals with the basic principles and techniques of nonequilibrium statistical mechanics. The importance of this

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English Pages 326 [332] Year 2021

Preface
Acknowledgments
Contents
Prologue
Chapter 1 Introduction
1.1 Fluctuations
1.2 Irreversibility of macroscopic systems
Chapter 2 The Langevin equation
2.1 Introduction
2.2 The Maxwellian distribution of velocities
2.3 Equation of motion of the tagged particle
2.4 Exercises
2.4.1 Moments of the speed in thermal equilibrium
2.4.2 Energy distribution of the tagged particle
2.4.3 Distribution of the relative velocity between two particles
2.4.4 Generalization to the case of many particles
Chapter 3 The fluctuation-dissipation relation
3.1 Conditional and complete ensemble averages
3.2 Conditional mean squared velocity as a function
of time
3.3 Exercises
3.3.1 The contradiction that arises if dissipation is neglected
3.3.2 Is the white noise assumption responsible for the contradiction?
3.3.3 The generality of the role of dissipation
Chapter 4 The velocity autocorrelation function
4.1 Velocity correlation time
4.2 Stationarity of a random process
4.3 Velocity autocorrelation in three dimensions
4.3.1 The free-particle case
4.3.2 Charged particle in a constant magnetic field
4.4 Time reversal property of the correlation matrix
4.5 Exercises
4.5.1 Higher-order correlations of the velocity
4.5.2 Calculation of a matrix exponential
4.5.3 Correlations of the scalar and vector products
Chapter 5 Markov processes
5.1 Continuous Markov processes
5.2 Master equations for the conditional density
5.2.1 The Chapman-Kolmogorov equation
5.2.2 The Kramers-Moyal expansion
5.2.3 The forward Kolmogorov equation
5.2.4 The backward Kolmogorov equation
5.3 Discrete Markov processes
5.3.1 The master equation and its solution
5.3.2 The stationary distribution
5.3.3 Detailed balance
5.4 Autocorrelation function
5.5 Remarks on ergodicity, mixing, and chaos
5.6 Exercises
5.6.1 The dichotomous Markov process
5.6.2 The Kubo-Anderson process
5.6.3 The kangaroo process
Chapter 6 The Fokker-Planck equation and the Ornstein-Uhlenbeck distribution
6.1 Diffusion processes
6.2 The SDE–FPE correspondence
6.3 The Ornstein-Uhlenbeck distribution
6.4 The uctuation-dissipation relation again
6.5 Exercises
6.5.1 Verification
6.5.2 Green function for the Fokker-Planck operator
6.5.3 Joint distribution of the velocity
Chapter 7 The diffusion equation
7.1 Introduction
7.2 The mean squared displacement
7.3 The fundamental Gaussian solution
7.4 Diffusion in three dimensions
7.5 Exercises
7.5.1 Derivation of the fundamental solution
7.5.2 Green function for the diffusion operator
7.5.3 Solution for a rectangular initial distribution
7.5.4 Distributions involving two independent particles
7.5.5 Moments of the radial distance
7.5.6 Stable distributions related to the Gaussian
Chapter 8 Diffusion in a finite region
8.1 Diffusion on a line with reecting boundaries
8.2 Diffusion on a line with absorbing boundaries
8.3 Solution by the method of images
8.4 Exercises
8.4.1 Solution by separation of variables
8.4.2 Diffusion on a semi-in nite line
8.4.3 Application of Poisson's summation formula
Chapter 9
Brownian motion
9.1 The Wiener process (Standard Brownian motion)
9.2 Properties of Brownian paths
9.3 Khinchin's law of the iterated logarithm
9.4 Brownian trails in d dimensions: recurrence properties
9.5 The radial distance in d dimensions
9.6 Sample paths of diffusion processes
9.7 Relationship between the OU and Wiener processes
9.8 Exercises
9.8.1 r2(t) in d-dimensional Brownian motion
9.8.2 The nth power of Brownian motion
9.8.3 Geometric Brownian motion
9.9 Brief remarks on the It^o calculus
Chapter 10 First-passage time
10.1 First-passage time distribution from a renewal equation
10.2 Survival probability and first passage
10.3 Exercises
10.3.1 Divergence of the mean first-passage time
10.3.2 Distribution of the time to reach a specified distance
10.3.3 Yet another aspect of the x2 ˘ t scaling
Chapter 11 The displacement of the tagged particle
11.1 Mean squared displacement in equilibrium
11.2 Time scales in the Langevin model
11.3 Equilibrium autocorrelation function of the displacement
11.4 Conditional autocorrelation function
11.5 Fluctuations in the displacement
11.6 Cross-correlation of the velocity and the displacement
11.7 Exercises
11.7.1 Verification
11.7.2 Variance of X from its mean squared value
11.7.3 Velocity-position equal-time correlation
11.7.4 Velocity-position unequal-time correlation
Chapter 12 The Fokker-Planck equation in phase space
12.1 Recapitulation
12.2 Two-component Langevin and Fokker-Planck equations
12.3 Solution of the Langevin and Fokker-Planck equations
12.4 PDFs of the velocity and position individually
12.5 The long-time or diffusion regime
12.6 Exercises
12.6.1 Velocity and position PDFs from the joint density
12.6.2 Phase space density for three-dimensional motion
Chapter 13 Diffusion in an external potential
13.1 Langev in equation in an external potential
13.2 General SDE—FPE correspondence
13.3 The Kramers equation
13.4 The Brownian oscillator
13.5 The Smoluchowski equation
13.6 Smoluchowski equation for the oscillator
13.7 Escape over a potential barrier: Kramers' escape rate formula
13.8 Exercises
13.8.1 Phase space PDF for the overdamped oscillator
13.8.3 Diffusion in a constant force field: Sedimentation
Chapter 14 Diffusion in a magnetic field
14.1 The PDF of the velocity
14.1.1 The Fokker-Planck equation
14.1.2 Detailed balance
14.1.3 The modified OU distribution
14.2 Diffusion in position space
14.2.1 The diffusion equation
14.2.2 The diffusion tensor
14.3 Phase space distribution
14.4 Exercises
14.4.1 Velocity space FPE in vector form
14.4.3 Conditional mean velocity and displacement
14.4.4 Calculation of Dij
14.4.5 Phase space FPE in vector form
Chapter 15 Kubo-Green formulas
15.1 Relation between D and the velocity autocorrelation
15.2 Generalization to three dimensions
15.3 The mobility
15.3.1 Relation between D and the static mobility
15.3.2 The dynamic mobility
15.3.3 Kubo-Green formula for the dynamic mobility
15.4 Exercises
15.4.1 Application to the Brownian oscillator
15.4.2 Application to a particle in a magnetic field
15.5 Further remarks on causality and stationarity
Chapter 16 Mobility as a generalized susceptibility
16.1 The power spectral density
16.1.1 Definition of the power spectrum
16.1.2 The Wiener-Khinchin theorem
16.1.3 White noise; Debye spectrum
16.2 Fluctuation-dissipation theorems
16.3 Analyticity of the mobility
16.4 Dispersion relations
16.5 Exercises
16.5.1 Position autocorrelation of the Brownian oscillator
16.5.2 Power spectrum of a multi-component process
16.5.3 The mobility tensor
16.5.4 Particle in a magnetic eld: Hall mobility
16.5.5 Simplifying the dispersion relations
16.5.6 Subtracted dispersion relations
Chapter 17 The generalized Langevin equation
17.1 Shortcomings of the Langevin model
17.1.1 Short-time behavior
17.1.2 The power spectrum at high frequencies
17.2 The memory kernel and the GLE
17.3 Frequency-dependent friction
17.4 Fluctuation-dissipation theorems for the GLE
17.4.1 The first FD theorem from the GLE
17.4.2 The second FD theorem from the GLE
17.5 Velocity correlation time
17.6 Exercises
17.6.1 Exponentially decaying memory kernel
17.6.2 Verification of the first FD theorem
17.6.3 Equality of noise correlation functions
Epilogue
Appendix A Gaussian integrals
A.1 The Gaussian integral
A.2 The error function
A.3 The multi-dimensional Gaussian integral
A.4 Gaussian approximation for integrals
Appendix B The gamma function
Appendix C Moments, cumulants and characteristic functions
C.1 Moments
C.2 Cumulants
C.3 The characteristic function
Appendix D The Gaussian or normal distribution
D.1 The probability density function
D.2 The moments and cumulants
D.3 The cumulative distribution function
D.4 Linear combinations of Gaussian random variables
D.5 The Central Limit Theorem
D.6 Gaussian distribution in several variables
D.7 The two-dimensional case
Appendix E From random walks to diffusion
E.1 A simple random walk
E.2 The characteristic function
E.3 The diffusion limit
E.4 Important generalizations
Appendix F The exponential of a (2 x 2) matrix
Appendix G Velocity distribution in a magnetic field
Appendix H The Wiener-Khinchin Theorem
Appendix I Classical linear response theory
I.1 Mean values
I.2 The Liouville equation
I.3 The response function
I.4 The generalized susceptibility
Appendix J Power spectrum of a random pulse sequence
J.1 The transfer function
J.2 Random pulse sequences
J.3 Shot noise
J.4 Barkhausen noise
Appendix K Stable distributions
K.1 The family of stable distributions
K.2 The characteristic function
K.3 The three important special cases
K.4 Asymptotic behavior: `heavy-tailed' distributions
K.5 Generalized Central Limit Theorem
K.6 Infinite divisibility