Trails in Kinetic Theory: Foundational Aspects and Numerical Methods (SEMA SIMAI Springer Series, 25) 3030671038, 9783030671037

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Trails in Kinetic Theory: Foundational Aspects and Numerical Methods (SEMA SIMAI Springer Series, 25)
 3030671038, 9783030671037

Table of contents :
Preface
Acknowledgements
Contents
Contributors
Recent Development in Kinetic Theory of Granular Materials: Analysis and Numerical Methods
1 The Boltzmann Equation for Granular Gases
1.1 Cauchy Theory of the Granular Gases Equation
1.2 Large Time Behavior
1.3 Compressible Hydrodynamic Limits
2 Fourier Spectral Methods for the Inelastic Boltzmann Collision Operator
2.1 The Direct Fourier Spectral Method
2.2 The Fast Fourier Spectral Method
3 Numerical Experiments and Results
3.1 GPU Parallelized Implementation
3.2 Numerical Results
4 Conclusion
References
Asymptotic Methods for Kinetic and Hyperbolic Evolution Equations on Networks
1 Introduction
2 Coupling Conditions for Linear Problems
2.1 Equations
2.2 Boundary and Coupling Conditions for Linear Equations via Kinetic Layer Analysis
2.3 Approximate Solution of the Half Space Problem via Half-Fluxes
2.4 Half-Moment Coupling Conditions
2.5 Numerical Results
3 Coupling Conditions for Nonlinear Problems
3.1 Equations
3.2 Boundary Conditions via Layer Analysis
3.3 Coupling Conditions
3.4 Numerical Results
4 Outlook
References
Coagulation Equations for Aerosol Dynamics
1 Introduction
2 Preliminaries
2.1 Conservation of Mass and Continuity Equation
2.2 Coagulation Kernels for Aerosols in the Atmosphere
2.3 From Particle Models to Smoluchowski's Coagulation Equation
2.4 Notation
3 One-Component Equation
3.1 Main Results
3.2 Well-Posedness for the Time-Dependent Problem
3.3 Stationary Solutions with Injection
3.3.1 Existence of Stationary Solutions
3.3.2 Nonexistence of Stationary Solutions
4 Discrete Multi-Component Coagulation Equation with Constant Kernel
4.1 Mass Localization in Time-Dependent Solutions
4.2 Mass Localization in Stationary Solutions
5 Perspectives and Open Problems
References
Multibody and Macroscopic Impact Laws: A Convex Analysis Standpoint
1 Introduction
2 A Closer Look to Micro and Macro Impact Laws
2.1 Saddle Point Formulation of the Microscopic Impact Law
2.2 Saddle Point Formulation of the Macroscopic Impact Law
3 Micro-Macro Issues
4 Anisotropic Macroscopic Collision Laws
5 Homogenization Issues
5.1 General Procedure
5.2 Homogenization for Structured Configurations
6 Evolution Models
Appendix
Linear Constraints
Unilateral Constraints
References
An Introduction to Uncertainty Quantification for Kinetic Equations and Related Problems
1 Introduction
2 Uncertainty Quantification for PDEs
2.1 PDEs with Random Inputs
2.2 Overview of Techniques
2.2.1 Monte Carlo (MC) Sampling Methods
2.2.2 Stochastic Galerkin (SG) Methods
3 Uncertainty in Kinetic Equations
3.1 The Boltzmann Equation with Random Inputs
3.2 Numerical Methods for UQ in Kinetic Equations
4 Single Control Variate (bi-Fidelity) Methods
4.1 Space Homogeneous Case
4.1.1 Local Equilibrium Control Variate
4.1.2 Time Dependent Control Variate
4.2 Non Homogeneous Case
5 Multiple Control Variate (Multi-Fidelity) Methods
5.1 Hierarchical Methods
5.2 Multi-Level Monte Carlo Methods
6 Structure Preserving Stochastic-Galerkin (SG) Methods
6.1 Equilibrium Preserving SG Methods for the Boltzmann Equation
6.2 Generalizations for Nonlinear Fokker–Planck Problems
7 Hybrid Particle Monte Carlo SG Methods
7.1 Particle SG Methods for Fokker–Planck Equations
7.2 Direct Simulation Monte Carlo SG Methods
References
A Brief Introduction to the Scaling Limits and Effective Equations in Kinetic Theory
1 The Foundations of Kinetic Theory
2 Low-Density Limit and Boltzmann Equation
2.1 Hard-Sphere Hierarchies
2.2 Lanford's Theorem
3 Weak-Coupling Limit and Landau Equation
3.1 Remarks on the Scaling Limits
3.2 Weak-Coupling Limit for Classical Systems
References
Statistical Description of Human Addiction Phenomena
1 Introduction
2 Kinetic Description of Addiction Phenomena
3 Fokker–Planck Description and Equilibria
4 Relaxation to Equilibrium
5 Conclusions
References
Boltzmann-Type Description with Cutoff of Follow-the-Leader Traffic Models
1 Introduction
2 FTL-Inspired Binary Interactions
2.1 The Case n=1
2.2 The Case n=2
3 Boltzmann-Type Kinetic Description with Cutoff
4 Fokker–Planck Asymptotics
4.1 The Case n=1
4.2 The Case n=2
4.2.1 The Case δ=1
5 Numerical Tests
5.1 Log-Normal Equilibrium n=1
5.2 Gamma Equilibrium (n=2)
6 Conclusions
References

Citation preview

SEMA SIMAI Springer series  25

Giacomo Albi · Sara Merino-Aceituno Alessia Nota · Mattia Zanella  Eds.

Trails in Kinetic Theory Foundational Aspects and Numerical Methods

SEMA SIMAI Springer Series Volume 25

Editors-in-Chief Luca Formaggia, MOX–Department of Mathematics, Politecnico di Milano, Milano, Italy Pablo Pedregal, ETSI Industriales, University of Castilla–La Mancha, Ciudad Real, Spain Series Editors Mats G. Larson, Department of Mathematics, Umeå University, Umeå, Sweden Tere Martínez-Seara Alonso, Departament de Matemàtiques, Universitat Politècnica de Catalunya, Barcelona, Spain Carlos Parés, Facultad de Ciencias, Universidad de Málaga, Málaga, Spain Lorenzo Pareschi, Dipartimento di Matematica e Informatica, Università degli Studi di Ferrara, Ferrara, Italy Andrea Tosin, Dipartimento di Scienze Matematiche “G. L. Lagrange”, Politecnico di Torino, Torino, Italy Elena Vázquez-Cendón, Departamento de Matemática Aplicada, Universidade de Santiago de Compostela, A Coruña, Spain Paolo Zunino, Dipartimento di Matemática, Politecnico di Milano, Milano, Italy

As of 2013, the SIMAI Springer Series opens to SEMA in order to publish a joint series aiming to publish advanced textbooks, research-level monographs and collected works that focus on applications of mathematics to social and industrial problems, including biology, medicine, engineering, environment and finance. Mathematical and numerical modeling is playing a crucial role in the solution of the complex and interrelated problems faced nowadays not only by researchers operating in the field of basic sciences, but also in more directly applied and industrial sectors. This series is meant to host selected contributions focusing on the relevance of mathematics in real life applications and to provide useful reference material to students, academic and industrial researchers at an international level. Interdisciplinary contributions, showing a fruitful collaboration of mathematicians with researchers of other fields to address complex applications, are welcomed in this series. THE SERIES IS INDEXED IN SCOPUS

More information about this series at http://www.springer.com/series/10532

Giacomo Albi • Sara Merino-Aceituno • Alessia Nota • Mattia Zanella Editors

Trails in Kinetic Theory Foundational Aspects and Numerical Methods

Editors Giacomo Albi Department of Computer Science University of Verona Verona, Italy Alessia Nota Department of Information Engineering Computer Science and Mathematics University of L’Aquila L’Aquila, Italy

Sara Merino-Aceituno Faculty of Mathematics University of Vienna Vienna, Austria Mattia Zanella Department of Mathematics University of Pavia Pavia, Italy

ISSN 2199-3041 ISSN 2199-305X (electronic) SEMA SIMAI Springer Series ISBN 978-3-030-67103-7 ISBN 978-3-030-67104-4 (eBook) https://doi.org/10.1007/978-3-030-67104-4 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

In the last decades, kinetic theory has emerged as one of the most prominent fields of modern mathematics. It was originally developed as a field of Mathematical Physics to investigate interacting particle systems and their corresponding continuum descriptions. Yet, in the recent years, there has been an explosion of applications of kinetic theory to other areas of research such as biology and social sciences. Kinetic-type equations currently represent a common ground for the crossfertilization between a heterogeneity of communities that include both pure and applied disciplines. The source for the broad applicability of kinetic theory lies on the omnipresence of emergent phenomena in real-life applications. Emergent phenomena correspond to the appearance of large-scale (observable) structures from the underlying microscopic, discrete dynamics. Kinetic theory provides precisely the mathematical framework to link these discrete dynamics with their corresponding continuum equations at the macro-scale. Nowadays, countless applications, ranging from plasma physics to socio-economic and soft matter, have roots in kinetic theory. At the same time, the investigation of emerging phenomena in these new fields of applications presents many mathematical challenges at the level of modelling, mathematical analysis and numerics. All these aspects have been presented in the School Trails in Kinetic Theory: Foundational Aspects and Numerical Methods organized by the editors of the present book during the Trimester Program on Kinetic Theory at the Hausdorff Institute for Mathematics of Bonn in June 2019. During this event, four eminent lecturers and eleven invited speakers of the highest profile presented research advances and cutting-edge results to broadest scientific community. In the following, we briefly describe the main topics of this event, which will constitute the backbone for the book. Theoretical Aspects The problem of deriving macroscopic evolution equations from the microscopic description based on the fundamental laws of mechanics, through suitable scaling limits, is a central problem of non-equilibrium statistical mechanics. The resulting v

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Preface

kinetic equations are essential to describe the relevant physical properties of the system and their time evolution. Classical examples of kinetic equations are the Boltzmann equation, the Landau equation and the Vlasov equation. During the last couple of decades, several important results have been achieved, both in the derivation and in the analysis of the kinetic equations, which have increased the current understanding of kinetic theory. Most of the rigorous results available in this direction have been obtained for dilute gases or for systems with weak interactions. The rigorous analysis of the mean-field limit and the Vlasov equation is a wellunderstood subject in the case of smooth potentials, but the interesting case of the Coulomb interaction is still open. Much more subtle are the limiting physical situations leading to the Boltzmann equation and the Landau equation. After the fundamental Lanford’s result on the rigorous derivation of the Boltzmann equation from a gas of hard spheres even though only for a short time interval, the case of smooth short-range interaction potentials has been studied, but the validity (or non-validity) of the Boltzmann equation in the case of long-range potentials is still open and challenging, as well as the short-time validity limitation. Concerning the derivation of the Landau equation, a rigorous proof is still missing, even for short time. Recently, some partial results on the extension to long-range interactions have been proposed in the simplified case of the Lorentz gas, which consists of a single particle moving through infinitely heavy, randomly distributed scatterers. Significant progress has been made in the derivation of kinetic equations for quantum particles, despite that the understanding of interacting particle systems is much more partial than the classical ones. Another well-established research direction in kinetic theory is the qualitative behaviour of the solutions of kinetic equations, and the analysis of their longtime asymptotics, closely related to the problem of hydrodynamics. It is worth to mention that also in this direction several hard problems still need further investigation, as, for instance, the proof of global well-posedness of classical solutions of Boltzmann equation, which is still elusive. Kinetic equations are also used in various other settings. One example is, for instance, the Smoluchowski equation arising from problems of polymerization, particle aggregation in aerosols or drop formation in rain. The analysis and validation from particle systems of the Smoluchowski equation is much less developed, but there has recently been progress in understanding the properties of self-similar solutions and a first step in understanding the derivation of coagulation equations from mechanical particle systems. Applications in Socio-Economic and Life Sciences As mentioned before, emergent phenomena are ubiquitous in nature. Being able to link phenomena at the different scales is crucial to giving answers to questions in the experimental sciences: How do we explain the self-organization of a tissue from its underlying constituents? How crowds of pedestrian self-organize into lanes? How does opinion dynamics evolve over time from local and macroscopic interactions? Kinetic theory has found in the recent years multiple applications in biology and social sciences, especially in the fields of collective motion and opinion dynamics.

Preface

vii

This poses many new challenges: first, at the level of the modelling, we need to consider simple models that are tractable enough but that, at the same time, capture the phenomena under investigation; second, new coarse-graining tools need to be developed to obtain macroscopic equations since the classical tools cannot be applied (this can be due, for example, to the lack of conserved quantities or to the appearance of phase transitions); third, new numerical methods need to be developed (for example, for hyperbolic non-conservative equations); finally, some systems exhibit violation of the propagation of chaos, making it impossible to derive kinetic equations with classical methods. In conclusion, the applications of kinetic theory to emergent phenomena in biology and social sciences open new fascinating questions that will push further the borders of our mathematical understanding and methodologies. Numerical Methods and Uncertainty Quantification The development of numerical methods for kinetic equations has been the subject of extremely active investigations in the past decades, especially in relation to efficient approximations of equilibrium states and in connection to the multi-scale limits of collisional equations. Significant progress has been obtained in a variety of problems in kinetic theory, e.g. granular gases, kinetic methods for soft matter physics, optimal control of kinetic equations. Furthermore, in recent years, significant efforts have been dedicated to incorporate possible deviations from the systems’ prescribed deterministic behaviour. Our analytical understanding of the structural randomness seems to be crucial to provide reliable descriptions of real-world models. A step towards realistic modelling demands a quantification of the possible deviations of a model-driven approach measuring errors and uncertainties. In a kinetic setting, the general strategy to take into consideration the realistic lack in information due, for example, to empirical assumptions or incomplete knowledge of boundary terms or initial data relies on an increased dimensionality of the particles’ distribution. In the recent literature, authors deal with this challenge through uncertainty quantification (UQ) methods whose approach provides accurate algorithms for the a priori estimation of the impact of uncertainties in terms of statistical moments. All these methods seem to be particularly appropriate also in connection to socio-economic and life sciences phenomena since most of the models are generally not derived from first principles. Verona, Italy Vienna, Austria L’Aquila, Italy Pavia, Italy

Giacomo Albi Sara Merino-Aceituno Alessia Nota Mattia Zanella

Acknowledgements

The editors of this book wish to express their gratitude to the Hausdorff Institute for Mathematics (HIM) of Bonn, Germany, for their financial support in organizing the School “Trails in Kinetic Theory: Foundational Aspects and Numerical Methods”, May 20–24, 2019. We would like to thank all the speakers of this exciting event for their inspiring presentations that nourished fruitful discussions among the participants.

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Contents

Recent Development in Kinetic Theory of Granular Materials: Analysis and Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . José Antonio Carrillo, Jingwei Hu, Zheng Ma, and Thomas Rey

1

Asymptotic Methods for Kinetic and Hyperbolic Evolution Equations on Networks .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Raul Borsche and Axel Klar

37

Coagulation Equations for Aerosol Dynamics . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Marina A. Ferreira Multibody and Macroscopic Impact Laws: A Convex Analysis Standpoint . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Félicien Bourdin and Bertrand Maury

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97

An Introduction to Uncertainty Quantification for Kinetic Equations and Related Problems .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 141 Lorenzo Pareschi A Brief Introduction to the Scaling Limits and Effective Equations in Kinetic Theory .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 183 Mario Pulvirenti and Sergio Simonella Statistical Description of Human Addiction Phenomena .. . . . . . . . . . . . . . . . . . . 209 Giuseppe Toscani Boltzmann-Type Description with Cutoff of Follow-the-Leader Traffic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 227 Andrea Tosin and Mattia Zanella

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Contributors

Raul Borsche Department of Mathematics, TU Kaiserslautern, Kaiserslautern, Germany Félicien Bourdin Laboratoire de Mathématiques d’Orsay, Université Paris-Sud, Orsay, France José Antonio Carrillo Mathematical Institute, University of Oxford, Oxford, UK Marina A. Ferreira Department of Mathematics and Statistics, University of Helsinki, Helsingin yliopisto, Finland Jingwei Hu Department of Mathematics, Purdue University, West Lafayette, IN, USA Axel Klar Department of Mathematics, TU Kaiserslautern, Germany

Kaiserslautern,

Zheng Ma Department of Mathematics, Purdue University, West Lafayette, IN, USA Bertrand Maury Département de Mathématiques et Applications, Ecole Normale Supérieure, PSL University, Paris, France Lorenzo Pareschi Department of Mathematics and Computer Science, University of Ferrara, Ferrara, Italy Mario Pulvirenti Department of Mathematics, University of Roma “La Sapienza”, Roma, Italy Thomas Rey Laboratoire Paul Painlevé, Batiment M2 Bureau 202, Université Lille, Lille, France Sergio Simonella ENS de Lyon, UMPA UMR 5669 CNRS, Lyon Cedex 07, France Giuseppe Toscani Department of Mathematics “F. Casorati”, University of Pavia, Pavia, Italy xiii

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Contributors

Andrea Tosin Department of Mathematical Sciences “G. L. Lagrange”, Politecnico di Torino, Torino, Italy Mattia Zanella Department of Mathematics “F. Casorati”, University of Pavia, Pavia, Italy

Recent Development in Kinetic Theory of Granular Materials: Analysis and Numerical Methods José Antonio Carrillo, Jingwei Hu, Zheng Ma, and Thomas Rey

Abstract Over the past decades, kinetic description of granular materials has received a lot of attention in mathematical community and applied fields such as physics and engineering. This article aims to review recent mathematical results in kinetic granular materials, especially for those which arose since the last review Villani (J Stat Phys 124(2):781–822, 2006) by Villani on the same subject. We will discuss both theoretical and numerical developments. We will finally showcase some important open problems and conjectures by means of numerical experiments based on spectral methods. Keywords Granular gases equation · Inelastic Boltzmann equation · Fast spectral method · Asymptotic behavior · Review · Hydrodynamic equations · Granular flows · GPU

1 The Boltzmann Equation for Granular Gases Granular gases have been initially introduced to describe the nonequilibrium behavior of materials composed of a large number of unnecessarily microscopic particles, such as grains or sand. These particles form a gas, interacting via energy dissipating inelastic collisions. Statistical mechanics description of particle systems through inelastic collisions faces basic derivation problems such as the inelastic collapse [72], i.e. infinite many collisions in finite time. Nevertheless, the kinetic

J. A. Carrillo () Mathematical Institute, University of Oxford, Oxford, UK e-mail: [email protected]; [email protected] J. Hu · Z. Ma Department of Mathematics, Purdue University, West Lafayette, IN, USA e-mail: [email protected]; [email protected] T. Rey Univ. Lille, CNRS, UMR 8524, Inria – Laboratoire Paul Painlevé, Lille, France e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 G. Albi et al. (eds.), Trails in Kinetic Theory, SEMA SIMAI Springer Series 25, https://doi.org/10.1007/978-3-030-67104-4_1

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description of rapid granular flows [57, 65, 66] has been able to compute transport coefficients for hydrodynamic descriptions successfully used in situations that are a long way from their supposed limits of validity, to describe, for instance, shock waves in granular gases [28, 82], clustering [32, 37, 61], and the Faraday instability for vibrating thin granular layers [27, 28, 36, 46, 73, 88–90]. A large amount of practical systems can be described as a granular gas, such as for example spaceship reentry in a dusty atmosphere (Mars for instance), planetary rings [7, 70] and sorting behavior in vibrating layers of mixtures. A lot of other examples can be found in the thesis manuscript [40], and in the seminal book of Brilliantov and Pöschel [31]. Usually, a granular gas is composed of 106–1016 particles. The study of such a system will then be impossible with a direct approach, and we shall adopt a kinetic point of view, studying the behavior of a one-particle distribution function f , depending on time t ≥ 0, space x ∈  ⊂ Rdx and velocity v ∈ Rd , for dx ≤ d ∈ {1, 2, 3}. The statistical mechanics description of the system has been then admitted in the physical community as the tool to connect the microscopic description to macroscopic system of balance laws in rapid granular flows [30, 57, 59, 65, 66] as in the classical rarefied gases [39]. In this first section, we shall review some basics on the inelastic Boltzmann equation, and present the mathematical state of the art since the previous review paper on the subject [91]. Microscopic Dynamics The microscopic dynamics can be summarized with the following hypotheses: 1. The particles interact via binary collisions. More precisely, the gas is rarefied enough so that collisions between 3 or more particles can be neglected. 2. These binary collisions are localized in space and time. In particular, all the particles are considered as point particles, even if they describe macroscopic objects. 3. Collisions preserve mass and momentum, but dissipate a fraction 1 − e of the kinetic energy in the impact direction, where the inelasticity parameter e ∈ [0, 1] is called restitution coefficient: ⎧   ⎪ ⎨ v + v∗ = v + v∗ , 1−e ⎪ ⎩ |v  |2 + |v∗ |2 − |v|2 − |v∗ |2 = − |(v − v∗ ) · ω|2 ≤ 0, 2 2

(1)

with ω ∈ Sd−1 being the impact direction. Using these conservation, one has the following two possible parametrizations (see also Fig. 2) of the post-collisional velocities, as a function of the pre-collisional ones: • The ω-representation or reflection map, given for ω ∈ Sd−1 by 1+e ((v − v∗ ) · ω) ω, 2 1+e v∗ = v∗ + ((v − v∗ ) · ω) ω. 2 v = v −

(2)

Granular Materials

3

Fig. 1 Geometry of the inelastic collision in the physical space (green is elastic, red is inelastic)

v∗

v ω

v∗

v

• The σ -representation or swapping map, given for σ ∈ Sd−1 by v + v∗ 1+e 1−e + (v − v∗ ) + |v − v∗ |σ, 2 4 4 1−e v + v∗ 1+e − (v − v∗ ) − |v − v∗ |σ. v∗ = 2 4 4 v =

(3)

Remark 1 Taking e = 1 in both (2) and (3) yields the classical energy-conservative elastic collision dynamics, as illustrated in Fig. 1. The geometry of collisions is more complex than the classical elastic one. Indeed, fixing v, v∗ ∈ Rd , denote by ± :=

1−e v + v∗ ± (v∗ − v), 2 4

O :=

v  + v∗ v + v∗ = . 2 2

Then if u := v − v∗ is the relative velocity, one has |+ − v  | = |− − v∗ | =

1+e |u|, 4

namely v  ∈ S (+ , |u|(1 + e)/4) and v∗ ∈ S (− , |u|(1 + e)/4), where S(x, r) is the sphere centered in x and of radius r (see also Fig. 2). Restitution Coefficient The physics literature is quite divided on the question of whether the restitution coefficient e should be a constant or not [31]. Although most of the early mathematical results on the topic consider a constant e [91], it seems that this case is only realistic in dimension 1 of velocity (the so-called “collisional cannon” described in the chapter 4 of [31] is a famous counter-example). The true realistic case considers that e depends on the relative velocity |v−v∗ | of the colliding particles. Even more precisely, it must be close to the elastic case 1 for small relative velocities (namely no dissipation, elastic case), and decay towards 0 when this relative velocity is large. The first mathematical result on this direction can be found in [85], where e(|v − v∗ |) =

1 , 1 + c |v − v∗ |γ

(4)

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v

v ω

v

θ

σ O

−

v∗

+ h v∗

v∗

Fig. 2 Geometry of the inelastic collision in the phase space (dashed lines represent the elastic case)

for a nonnegative constant c characterizing the inelasticity strength (c = 0 being elastic), and γ ∈ R. Another important case is the so-called viscoelastic hard spheres one, thoroughly studied mathematically in a series of papers [4–6, 11], where e is given by the implicit relation e(|v − v∗ |) + a|v − v∗ |1/5e(|v − v∗ |)3/5 = 1,

(5)

for a > 0. More details on the derivation of this expression can be found in [31, Chapter 3]. Another quite rigorous study has been made in [81], with a threshold-dependent restitution coefficient:  1 if r < r∗ , e(r) = e¯ if r ≥ r∗ , e¯ < 1 and r∗ > 0 being fixed. Finally, the case e = 0 describes sticky collisions: the normal component of the kinetic energy being completely dissipated during impact, the particles stick and travel together in the tangent direction after impact. A derivation of the model from the microscopic dynamics on the line can be found in [29, 42].

Granular Materials

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Remark 2 This model is meaningful even in dimension 1, which is not the case for elastic collisions. Indeed, such monodimensional collisions are only {v  , v∗ } = {v, v∗ }, meaning that the particle velocities are either swapped or preserved. The particles being indistinguishable, nothing happens.1 In the 1d inelastic case, the collisions are given using (3) by {v  , v∗ } = {v, v∗ }

 or

v + v∗ e ± (v − v∗ ) 2 2



depending on the value of σ ∈ {±1}. The Granular Gases Operator: Weak Form Using the microscopic hypotheses (1–1–1), one can derive the granular gases collision operator QI , by following the usual elastic procedure (see [91] for more details). Its weak form in the σ representation is given by Rd

QI (f, f )(v) ψ(v) dv =



1 f∗ f ψ  + ψ∗ − ψ − ψ∗ 2 Rd ×Rd ×Sd−1 · B(|v − v∗ |, cos θ, E(f )) dσ dv dv∗ ,

(6) where the collision kernel is typically of the form B(|u|, cos θ, E(f )) = (|u|)b(cos θ, E(f )), and E(f ) is the kinetic energy of f , namely its second moment in velocity, the postcollisional velocities are computed by (3), and θ is the angle between σ and u. We shall assume in all the following of this section that the collision kernel is of generalized hard sphere type, namely B(|u|, cos θ, E)) = (|u|) b(cos θ, E) = |u|λ b(cos θ ) E γ ,

(7)

where λ ∈ [0, 1] (λ = 0 being the simplified Maxwellian pseudo-molecules case and λ = 1 the more relevant hard sphere case), γ ∈ R and the angular cross section b verifies 0 < β1 ≤ b(x) ≤ β2 < ∞,

∀x ∈ [−1, 1].

(8)

Remark 3 Note that we assumed that the collision kernel B in (6) depends on the relative velocity, the angle of collision, and on E(f ). These former dependencies are quite classical, but the latter is not. Nevertheless, it makes a lot of sense physically speaking, as one can see in [83]. 1 Because of that, the elastic collision operator is simply equal to 0 for a one-dimensional velocity space, the Boltzmann equation reducing only to the free transport equation.

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The weak form in the ω-representation can be written analogously as Rd

QI (f, f )(v) ψ(v) dv =

1 2

Rd ×Rd ×Sd−1



f∗ f ψ  + ψ∗ − ψ − ψ∗

· B(|u|, cos θ, E(f )) dω dv dv∗ , (9)

where the postcollisional velocities are computed by (2), θ is the angle between ω and u, and

B(|u|, cos θ, E) = |u|λ b(cos θ )E γ with b(t) = 3|t|b(1 − 2t 2 ) for −1 ≤ t ≤ 1 by the change of variables between the σ - and the ω-representation, see [34] for details. The Granular Gases Operator: Strong Form Deriving a strong form of QI with the reflection map in the ω-representation is a matter of a change of variables. However, deriving a strong form of QI is not as easy as in the elastic case in the σ -representation since the collisional transform (v, v∗ , σ ) → (v  , v∗ , σ ) is not an involution and we have to go through the ω-representation, see [34] for details. More precisely, given the restitution coefficient e = e(|u|) depending on the relative velocity of the particles u = v − v∗ , we assume the collisional transform’s Jacobian for (2) is J (|u|, cos θ ) = 0 for all z. Notice J = e in the constant restitution case. It is in general a complicated expression of the relative speed r = |u| and s = cos θ involving e and its derivative. Then, the precollisional velocities read as 1+e ((v − v∗ ) · ω) ω, 2e 1+e  v∗ = v∗ + ((v − v∗ ) · ω) ω. 2e 

v=v−

(10)

The final strong from of the operator is QI (f, f )(v) = Q+ I (f, f )(v) − f (v) L(f )(v) with the loss part of the operator given by L(f )(v) =

Rd ×Sd−1

− v∗ |, cos θ, E(f ))f∗ dω dv∗ B(|v

and the gain part of the operator in strong form written as Q+ I (f, f )(v) =

Rd ×Sd−1

+

+  e (|u|, cos θ )be (cos θ )

Eγ   f f∗ dω dv∗ , J (|u|, cos θ ) (11)

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7

+

+ with  e (r, s) and be (s) given by 

be+ (s)

= b 



s e2 + (1 − e2 )s 2

(12)

,

and

+  e (r, s) = 

r 

 r  λ e2 + (1 − e2 )s 2 = e2 + (1 − e2 )s 2 . e e

(13)

We can derive now the following strong form of the collision operator also in the σ -representation by just changing variable in the operator from ω to σ , see [34], to find the expressions of the loss and the gain terms in the σ -representation: L(f )(v) =

Rd ×Sd−1

B(|v − v∗ |, cos θ, E(f ))f∗ dσ dv∗

and Q+ I (f, f )(v) =

Rd ×Sd−1

+ + e (|u|, cos θ )be (cos θ )

Eγ   f f∗ dσ dv∗ , J (|u|, cos θ ) (14)

+ with + e (r, s) and be (s) given by

be+ (s)



(1 + e2 )s − (1 − e2 ) =b (1 + e2 ) − (1 − e2 )s

√ 2

 

(1 + e2 ) − (1 − e2 )s

,

(15)

and    λ r  r  2 ) − (1 − e 2 )s = √ 2 ) − (1 − e 2 )s (r, s) =  √ (1 + e (1 + e . + e 2e 2e (16) In these expressions, the precollisional velocities are given in the σ -representation by v + v∗ 1−e + (v − v∗ ) + 2 4e 1−e v + v∗  − (v − v∗ ) − v∗ = 2 4e 

v=

1+e |v − v∗ |σ, 4e 1+e |v − v∗ |σ. 4e

(17)

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J. A. Carrillo et al.

The granular gases collision operator has then the same structure of the elastic Boltzmann operator under Grad’s cutoff assumption, namely it can be seen as the difference between the inelastic gain term Q+ I (f, f ) and the loss term f L(f ), which depends only on the chosen collision kernel, but not on the inelasticity. We shall call the following granular gases equation, or the inelastic Boltzmann equation: ∂f + v · ∇x f = QI (f, f ). ∂t

(18)

We shall denote its first fluid moments (resp. density, mean momentum, and kinetic energy) by (ρ(f ), ρ(f )u(f ), E(f )) :=

Rd

  1, v, |v|2 /2 f (v) dv.

Remark 4 There is another popular approach to describe granular gases, which uses an Enskog-type collision operator. It is more relevant physically because it allows to keep the particles’ radii δ positive, hence delocalizing the collision.2 The microscopic hypothesis 1 is in particular not valid anymore. The strong form of the collision operator in the constant restitution coefficient case is given by QE (f, f )(x, v) = δ d−1

Rd ×Sd−1

+

+ ( e (|u|, cos θ )be (cos θ )

G(ρ+ )   f+ f∗ e

−G(ρ− )f− f∗ ) dω dv∗ , (19) where ρ is the local density of f , ± denotes for a given function g = g(x) the shorthand notation g± (x) := g(x ± δ ω), and G is the local collision rate (also known as the correlation rate, see [91]). The global existence of renormalized solutions for the full granular gases equation (18) with the collision operator (19) has been established for both elastic and inelastic collisions in [45]. Existence and L1 (dx dv) stability of such solutions has been proved in [95], for close to vacuum initial datum.

2 Note that using a BBGKY approach [50] to derive (11) is not expected to succeed, because among other problems the macroscopic size of the particles composing a granular gas is incompatible with the Boltzmann-Grad scaling assumption.

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1.1 Cauchy Theory of the Granular Gases Equation The Space Homogeneous Setting Most of the rigorous mathematical results concerning the granular gases equation are obtained in the space homogeneous setting, where f = f (t, v) is the solution to ⎧ ⎪ ⎨ ∂t f = QI (f, f ) , ε ⎪ ⎩ f (0, v) = f (v),

(20)

in

for a given scaling parameter ε > 0. The first existence results for solution to (20) can be found in [12, 13, 15, 16, 21, 38]. These works deal with the generalized Maxwellian pseudo-molecule kernel (7) λ = 0, b ≡ 1 and γ = 1/2, with a velocity dependent restitution coefficient e = e(|v − v∗ |). Such a model allows to use some Fourier techniques to deal with the collision operator, altogether with the correct large time behavior for the kinetic energy, the so-called Haff’s cooling Law (26), and the correct hydrodynamic limit (31). The main result of these works is the global wellposedness of the solutions to (20) in L12 (R3 ), the convergence towards equilibrium, and the contraction in different metrics for the equation. The proof relies in the careful study of the self-similar solutions to (20). Some extensions of these results, using the same collision kernel, can be found in the works [17–19], where in particular the uniqueness is obtained. The physically relevant case of the hard sphere kernel λ = 1, γ = 0 was first considered in [85] in the unidimensional case. This work establishes the global existence of measure solutions with finite kinetic energy for this problem. The proof relies on a priori estimates of the solution to (20) in the Monge-KantorovichWasserstein metrics W2 . This work also investigates the quasi-elastic 1 − e2 ∼ ε → 0 limit of the model, a nonlinear McNamara-Young-like friction equation. This friction model was later investigated in [71], where its global wellposedness was shown in the space of measure of finite energy. The tail behavior of the equilibrium solution to the granular gases equation with a thermal bath v f was investigated in many papers, the main ones being [20, 44, 51]. They all proved the existence of non-Gaussian, overpopulated tails for diffusively excited granular gases, namely: Theorem 1 (From [20] and [44]) Let F (v) ≥ 0 for v ∈ Rd be a solution to the stationary equation QI (F, F ) + v F = 0

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J. A. Carrillo et al.

with all polynomial moments in velocity. Then,

F (v) ∼|v|→∞ exp −|v|α , with α = 1 in the Maxwellian molecules case and α = 3/2 in the hard spheres case. Indeed, the thermal bath gives an input of kinetic energy, preventing the appearance of trivial Dirac delta equilibria. The propagation of the Sobolev norms of the space dependent version of this equation was then established in [51]. It uses a careful estimate of the inelastic entropy production (23), and a fixed point argument for the existence and uniqueness of solutions. Finally, the work [77] establishes the global well posedness of the granular gases equation without a thermal bath, for a general case of collision kernel (which contains (7)) and velocity dependent restitution coefficients: Theorem 2 (Theorem 1.4 of [77]) Let 0 ≤ fin ∈ L13 ∩ ∈ BV4 . Then for any T ∈ (0, Tc ), where Tc := sup {T > 0 : E(f )(t) > 0, ∀ t < T } is the so-called blowup time, there exists an unique nonnegative solution f ∈ C(0, T ; L12 ) ∩ L∞ (0, T ; L13 ) of (20). It preserves mass and momentum, and converges in the weak-* topology of measures towards a Dirac delta. Their proof relies on careful estimates of the collision operator QI in Orlicz space (specially the L log L space of finite entropy measures). Remark 5 The related (but still mostly open) problem of the propagation of chaos was considered in [78] for a very simplified inelastic collision operator with a thermal bath. Cauchy Problem in the Space Dependent Setting The case of the space inhomogeneous setting3 has been much less investigated. The first result can be found in [9] for the model introduced in [85] with a restitution coefficient given by (4), in one dimension of space and velocity. This work establishes the existence and uniqueness of mild (perturbative) solutions, first for small L1 (dx dv) initial data, and then for compactly supported initial data. Their main argument is reminiscent from a work due to Bony in [23] concerning discrete velocity approximation of the Boltzmann equation in dimension 1. The global existence of mild solutions in the general R3x × R3v setting, for a large class of velocity-dependent restitution coefficient, but for initial data close to vacuum, was obtained in [6]. The proof is based on a Kaniel-Shinbrot iteration on a very small functional space. The stability in L1 (R3x × R3v ) under the same assumptions was established in [94]. Finally the existence and convergence to equilibrium in T3x × R3v for a weakly inhomogeneous granular gas4 with a thermal bath was proved in [87], using a perturbative approach. 3 Physically

more realistic, in part because of the spontaneous loss of space homogeneity that has been observed in [58]. 4 Namely, the initial condition is chosen with a lot of exponential moments in velocity, and close to a space homogeneous profile.

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1.2 Large Time Behavior Macroscopic Properties of the Granular Gases Operator Modeling-wise, the main microscopic difference between a granular gas and a perfect molecular gas is the dissipation of the kinetic energy. Using the weak form (6) among with the microscopic relations (1) of the inelastic collision operator, this yields ⎞ ⎛ ⎞ 1 0 QI (f, f )(v) ⎝ v ⎠ dv = ⎝ 0 ⎠ , |v|2 −D(f ) ⎛

Rd

where D(f ) ≥ 0 is the energy dissipation functional, which depends only on the collision kernel: f f∗  (|v − v∗ |, E(f )) dv dv∗ . (21) D(f ) := Rd ×Rd

The quantity  (|u|, E) is the so-called energy dissipation rate, given using (1) by  (|u|, E) :=

1 − e2 4

|u · ω|2 B(|u|, cos θ, E) dω ≥ 0,

Sd−1

∀e ∈ [0, 1]. (22)

This dissipation of kinetic energy has a major consequence on the behavior of the solutions to the granular gases equation. Indeed, combined with the conservation of mass and momentum, it implies (at least formally) an explosive behavior, namely convergence in the weak-* topology of solutions to (18) towards Dirac deltas, centered in the mean momentum u: f (t, ·)  δv=u ,

t → ∞.

As for the entropy, it is not possible to obtain any entropy dissipation for this equation, in order to precise this large time behavior. Indeed, as noticed in [51], taking ψ(v) = log f (v) in (6) yields Rd

QI (f, f )(v) log f (v) dv = =

1 2 1 2 +





Rd ×Rd ×Sd−1

Rd ×Rd ×Sd−1

1 2



 f  f∗ B dσ dv dv∗ f f∗      f f∗ f  f∗ f∗ f log + 1 B dσ dv dv∗ − f f∗ f f∗ f∗ f log

Rd ×Rd ×Sd−1

f∗ f  − f∗ f B dσ dv dv∗ .

(23)

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The first term in (23), the elastic contribution, is nonpositive because log λ−λ+1 ≤ 0 (this is Boltzmann’s celebrated H Theorem). Nevertheless, the second term has no sign a priori: it is 0 only in the elastic case (because of the involutive collisional transformation (v, v∗ , σ ) → (v  , v∗ , σ )). Boltzmann’s entropy H(f ) :=

R dv

f (v) log f (v) dv

is then not dissipated by the solution of the granular gases equation if e < 1. Some work has been done on that direction in the numerical side. Indeed, adding a drift term or a thermal bath in velocity can yield numerical entropy dissipation, as noticed in [53]. Kinetic Energy Dissipation and the Haff’s Cooling Law Let us assume in this subsection that the granular gas considered is space homogeneous, namely f is solution to (20). Having no known entropy, one has then to use other macroscopic quantities to study the large time behavior of solutions to (18). Because of its explicit dissipation functional, kinetic energy is a good candidate for this. Moreover, being related to the variance, it allows to measure the concentration in velocity of the solution. In order to have an explicit bound for the energy dissipation, let us assume that the collision kernel is of the general type (7). Using polar coordinates, it is straightforward to compute the dissipation rate (22):  (|u|, E) = b1

1 − e2 λ+2 γ |u| E , 4

(24)

where thanks to (8)     b1 = Sd−2 

π

cos2 (θ ) sind−3 (θ ) b(cos(θ )) dθ < ∞.

0

Using the conservation of mass and momentum, one can always assume that the initial condition is of unit mass and zero momentum. Plugging (24) into (21) then yields using Hölder and Jensen inequalities 1 − e2 d E(f )(t) = −b1 E(f )γ (t) dt 4



Rd ×Rd

≤ −b1

1 − e2 E(f )γ (t) 4

≤ −b1

1 − e2 E(f )1+γ +λ/2 (t). 4

Rd

f f∗ |v − v∗ |λ+2 dv dv∗

f (v) |v|λ+2 dv (25)

In particular, one will have the following large time behaviors: Setting Ce = b1 ρ (1 − e2 )/4 and α := γ + 1/2,

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13

• Maxwellian pseudo-molecules (λ = γ = 0) decays exponentially fast towards the Dirac delta : E(f )(t) = E (fin ) e−Ce t ; Notice that the inequality in (25) is an identity for this case. • Hard spheres (λ = 1, γ = 0) exhibits the seminal quadratic Haff’s cooling Law [60]: −2  . E(f )(t) ≤ E (fin )−1/2 + Ce t/2

(26)

• Anomalous gases (γ = 0) exhibits more general behaviors: ⎧

− 1 ⎪ E (fin )α + Ce α t α ⎪ ⎪ ⎪ ⎨ E(f )(t) ≤ E (fin ) e−Ce t ⎪ ⎪ ⎪

⎪ − 1 ⎩ E (fin )α − Ce α t α

if γ > −1/2 (α > 0, finite time extinction); if γ = −1/2; if γ < −1/2 (α < 0).

All of these formal results have been proven to be rigorous and sharp, with explicit lower bounds, in [13, 15] for the Maxwellian and hard sphere cases [74], and in [83] for the anomalous cases. Extension to the viscoelastic case can be found in [5, 6], where the energy is shown to behave as E(f )(t) ∼t →∞ C (1 + t)−5/3 . All these papers share a common approach of proof, using the fact that the space homogeneous granular gases equation admits a self-similar behavior. Hence, introducing some well chosen time-dependent scaling function ω and τ , the distribution f is written as f (t, v) = ω(t)d g (τ (t), ω(t) v) , to take into account the concentration in the velocity variables.5 The rescaled function g is then solution to the granular gases equation, with an anti-drift term in velocity: ∂t g + ∇v · (v g) = QI (g, g).

5 One can see the velocity scaling function ω as the inverse of the variance of the distribution f . This scaling is then a continuous “zoom” on the blowup, and can be used to develop numerical methods for solving the full granular gases equation, see [49].

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Using some regularity estimates of the gain term of QI “à la” Lions/BouchutDesvillettes [24] and some new Povzner-like estimates [3, 74] then obtains a lower bound for the energy of g, yielding the generalized Haff’s law by coming back to f . Remark 6 In the viscoelastic case, note that the rescaling in velocity induces a time dependency on the restitution coefficient, complicating the proof of the Haff’s cooling law [4]. It is also the case in the anomalous setting, where the rescaling function depends nonlinearly on the solution f . The question of the uniqueness, stability and exponential return to an universal equilibrium profile (hypocoercivity, see [92]) of the self-similar solutions has then been fully addressed in the series of work [75, 76], for a constant restitution coefficient, with and without a thermal bath. Extension of these results to the viscoelastic case has been done in [5]. Remark 7 These results are reminiscent of the works [8, 12, 13, 21, 22, 35], where the exponential convergence to equilibrium of the solution to the nonlocal granular media equation or the Maxwellian case has been shown in the W2 or Fourier metrics.

1.3 Compressible Hydrodynamic Limits Let us consider in this subsection the following hyperbolic scaling of the granular gases equation: ∂fε 1 + v · ∇x fε = QI (fε , fε ). ∂t ε

(27)

Determining the precise hyperbolic limit ε → 0 of Eq. (27) is a fundamental, yet very difficult question. Indeed, for the elastic case e = 1, one simply has to use the fact that the equilibria of the collision operator are at the thermodynamical equilibrium (gaussian distributions) and the conservation of mass, momentum and kinetic energy to obtain the classical compressible Euler–Fourier system [39]. Because of the trivial Dirac equilibria, this question is more intricate for the true inelastic case. Pressureless Euler Dynamics Adopting the same approach as in the elastic case, one can formally plug the “equilibrium” Dirac deltas in the pressure to obtain the following pressureless Euler system: ⎧ ∂ρ ⎪ ⎪ + ∇ · (ρ u) = 0, ⎨ ∂t ⎪ ⎪ ⎩ ∂(ρu) + ∇ · (ρ u ⊗ u) = 0. ∂t

(28)

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15

This system can describe various interesting physical situations, such as galactic clusters, but is notoriously difficult to study mathematically. Its solution are in general ill-posed, as classical solutions cannot exists for large times and weak solutions are not unique. In the unidimensional case, it is however possible to recover a well posed theory by imposing a semi-Lipschitz condition on u. This theory was introduced in [25], and later extended in [26] and [63]. We cite below the main result of [63], where M 1 (R) denotes the space of Radon measures on R and L2 (ρ) for ρ ≥ 0 in M 1 (R) denotes the space of functions which are square integrable against the density ρ. Theorem 3 (From [63]) For any ρ 0 ≥ 0 in M 1 (R) and any u0 ∈ L2 (ρ 0 ), there exists ρ ∈ L∞ (R+ , M 1 (R)) and u ∈ L∞ (R+ , L2 (ρ)) solution to (28) in the sense of distribution and satisfying the semi-Lipschitz Oleinik-type bound u(t, x) − u(t, y) ≤

x−y , t

for a.e. x > y.

(29)

Moreover the solution is unique if u0 is semi-Lipschitz or if the kinetic energy is continuous at t = 0 ρ(t, dx) |u(t, x)|2 −→ ρ 0 (dx) |u0 (x)|2 , as t → 0. R

R

The proof of Theorem 3 is quite delicate, relying on duality solutions. For this reason, we only explain the rational behind the bound (29), which can be seen very simply from the discrete sticky particles dynamics. We refer in particular to [29] for the limit of this sticky particles dynamics as N → ∞. Consider N particles on the real line. We describe the ith particle at time t > 0 by its position xi (t) and its velocity vi (t). Since we are dealing with a one dimensional dynamics, we can always assume the particles to be initially ordered in . x1in < x2in < . . . < xN

The dynamics is characterized by the following properties d xi (t) = vi (t). 1. The particle i moves with velocity vi (t): dt 2. The velocity of the ith particle is constant, as long as it does not collide with another particle: vi (t) is constant as long as xi (t) = xj (t) for all i = j . 3. The velocity jumps when a collision occurs: if at time t0 there exists j ∈ {1, . . . , N} such that xj (t0 ) = xi (t0 ) and xj (t) = xi (t) for any t < t0 , then all the particles with the same position take as new velocity the average of all the velocities

vi (t0 +) =

1 |j |xj (t0 ) = xi (t0 )|

 j |xj (t0 )=xi (t0 )

vj (t0 −).

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Note in particular that particles having the same position at a given time will then move together at the same velocity. Hence, only a finite number of collisions can occur, as the particles aggregates. This property also leads to the Oleinik regularity. Consider any two particles i and j with xi (t) > xj (t). Because they occupy different positions, they have never collided and hence xi (s) > xj (s) for any s ≤ t. If neither had undergone any collision then xi (0) = xi (t) − vi (t) t > xj (0) = xj (t) − vj (t) t or

vi − vj + 1

< , t xi − xj +

(30)

where x+ := max(x, 0). It is straightforward to check that (30) still holds if particles i and j had some collisions between time 0 and t. As one can see this bound is a purely dispersive estimate based on free transport and the exact equivalent of the traditional Oleinik regularization for Scalar Conservation Laws, see [80]. It obviously leads to the semi-Lipschitz bound (29) as N → ∞. This result was extended to the one dimensional (in space and velocity) granular gases equation (27) in [64]. The basic idea of the proof of this work is to compare the granular gases dynamics to the pressureless gas system (28). The main difficulty is to show that fε becomes monokinetic at the limit (see also the very recent paper [68]). This is intimately connected to the Oleinik property (29), just as this property is critical to pass to the limit from the discrete sticky particles dynamics. Unfortunately it is not possible to obtain (29) directly. Contrary to the sticky particles dynamics, this bound cannot hold for any finite ε (or for any distribution that is not monokinetic). This is the reason why it is very delicate to obtain the pressureless gas system from kinetic equations (no matter how natural it may seem). Indeed we are only aware of one other such example in [69]. One of the main contributions of [64] is a complete reworking of the Oleinik estimate, still based on dispersive properties of the free transport operator v ∂x but compatible with kinetic distributions that are not monokinetic, through the introduction of a new, global nonlinear energy. The main result in this paper is the following: Theorem 4 (from [64]) Consider a sequence of weak solutions fε ∈ L∞ ([0, T ], Lp (R2 )) for some p > 2 and with total mass 1 to the granular gases Eq. (27) such that all initial v-moments are uniformly bounded in ε, some moment in x is uniformly bounded, and fε0 is, uniformly in ε, in some Lp for p > 1. Then any weak-* limit f of fε is monokinetic, f = ρ(t, x) δ(v − u(t, x)) for a.e. t, where ρ, u are a solution in the sense of distributions to the pressureless system (28) while u has the Oleinik property (29). Quasi-Elastic Limit The physical community usually considers another approach, that is assuming that the granular gas is in a quasi-elastic 1 − e2 ∼ ε → 0 setting. This was first proposed in [65, 66], using an approach very similar to the

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17

seminal Grad’s 13 moments closure for rarefied gas dynamics. The difficulty of a hydrodynamic description of granular materials has been addressed in well reasoned terms in [55, 56], and as already discussed in the introduction, the hydrodynamic equations obtained with the kinetic theory of granular gases have been shown to be insightful well beyond their supposed limit of validity, i.e., away from the quasielastic limit assumption with external sources of energy. In fact, assuming that solutions of the kinetic problem do not deviate from being Gaussians, one can then obtain in the hard sphere case the following quasi-elastic compressible Euler system ⎧ ∂ρ ⎪ ⎪ + ∇ · (ρ u) = 0, ⎪ ⎪ ⎪ ∂t ⎪ ⎪ ⎨ ∂(ρu) + ∇ · (ρ (u ⊗ u) + ρ T I) = 0, ⎪ ∂t ⎪ ⎪ ⎪ ⎪ ⎪ ∂W ⎪ ⎩ + ∇ · (u (W + ρ T )) = −K ρ T 3/2, ∂t

(31)

representing the evolution of number density of particles ρ(t, x), velocity field u(t, x) and the total energy W , which is given by W = 32 ρT + 12 ρ|u|2 = ρ + 12 ρ|u|2 , with T being the granular temperature (3D). Here K = K(d, B) is an explicit nonnegative constant, see below. This is a compressible Euler-type system, which dissipates the kinetic energy thanks to its nonzero right hand side. The particular expression of this RHS allows, after integration in space, to recover the correct Haff’s cooling Law. The assumption that the solutions are not far from Gaussians obviously degenerates in a free cooling granular gas at some point leading to the so-called clustering instability studied by means of (31), see for instance [37] and the references therein. In fact, this assumption can be shown to be valid in the quasi-elastic limit, see [76] for a rigorous justification of this property. Physicists argue that this assumption is also generically true in practical experiments with external sources of energy such as the shock waves in granular flows under gravity [28, 82], clustering [32, 37, 61], the Faraday instability for vibrating thin granular layers [27, 28, 36, 46, 73, 88–90], and many other applications, see [36, 67] and the references therein. Passing from the granular gases equation (18) to (31) has not been established properly. It can still be done formally under the weak inelasticity hypothesis 1−e2 ∼ ε, see [86]. This particular scaling insures that the granular gases operator converges towards the elastic Boltzmann operator, as was shown rigorously in [75] in the space homogeneous setting. Moreover, it allows to characterize the equilibrium distribution of the limit operator, which is Gaussian. A first step towards a rigorous compressible hydrodynamic limit is available in [84], where the study of the spectrum of the heated granular gases operator QI + τ v , linearized with respect to the equilibrium described in [76], is done. For small inelasticity 1 − e2 ∼ 0, this work provides a spectral decomposition, and more importantly the existence and computation of eigenvalue branches. This work generalizes the seminal paper [43] on the spectrum of the linearized Boltzmann operator in L2 to the L1 and inelastic setting.

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Other types of fluid limits (such as viscous limits) of the granular gases equation has been described in the review paper [41] and in the recent survey [54] for many different physical regimes, but none has been rigorously established. To illustrate the kind of equations obtained through these procedure, we write the generalized Navier–Stokes compressible equations for granular media in conservative form, see [36], as ∂ρ + ∇ · (ρu) = 0 ∂t ∂(ρu) + ∇ · [ρ (u ⊗ u)] = ∇ · P + ρ F ∂t ∂W + ∇ · [uW ] = −∇ · q + P : E + u · (∇ · P ) − γ + ρu · F , ∂t

(32)

representing the evolution of number density of particles ρ(t, x), velocity field u(t, x) and the total energy W , which is given by W = 32 ρT + 12 ρ|u|2 = ρ + 12 ρ|u|2 , with T being the granular temperature (3D). The symbol F stands for external forces applied to the system. The constitutive relations for the momentum and heat fluxes write, as usual,     2 Pij = −p + λ − μ Eii δij + 2μEij 3 i



 ∂Uj i for the stress tensor, with Eij = 12 ∂U + ∂xj ∂xi . The thermal conductivity relates linearly the heat flux q to the temperature gradient, q = −χ∇T . The equation of state is relevant here: we use the expression G(ν) = [1 − 4 (ν/νmax ) 3 νmax ]−1 for the contact value of the pair correlation function G(ν), which accounts for high densities, and the equation of state p = ρT (1 + 2(1 + e)νG(ν)), where ν is the packing fraction and e the constant restitution coefficient. Random close-packing is achieved in 3D at νmax = 0.65; we do not allow any packing fraction higher than 99.99% of this value. The transport coefficients for constant restitution given in [14, 65] write, γ =

12 √ (1 − e2 )ρT 3/2 G(ν), σ π

for the cooling coefficient γ , which models energy dissipation through collisions. Other kinetic coefficients are the shear viscosity, √     5 4 12 πT μ= ρσ +1+ 1+ G(ν) , 6 16G(ν) 5 π

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the bulk viscosity, √ 8 λ = √ ρσ T G(ν) 3 π and the thermal conductivity    6G(ν) 5 15 √ 32 +1+ πT ρσ χ= 1+ , 16 24G(ν) 5 9π implemented directly with no further assumptions. A first attempt to derive equations of compressible Navier–Stokes type was done in the paper [33] using singular perturbations of the collision operator QI and a central manifold approach inspired from [47]. The fact is that transport coefficients for compressible Navier–Stokes like equations can be derived by moment closures under different assumptions and these equations are able to recover realistic phenomena in granular gases, see [54] for a very recent review. For instance, the Faraday instability for vertically oscillated granular layer was analysed in [2, 36] comparing molecular dynamics simulations, and different closures proposed in the literature. The conclusion, as the reader can check in the numerical experiments in [1], is that the qualitative behavior of the numerical experiments using Eqs. (32) fully recovers the physical expected outcome. Thus, this is a physical validation of these kind of approximations that are totally opened for mathematicians to be derived in full rigor.

2 Fourier Spectral Methods for the Inelastic Boltzmann Collision Operator Due to the complexity of the inelastic Boltzmann collision operator QI (f, f ) (a five-fold nonlinear integral operator), numerical simulation of granular gases is challenging and mostly done at the particle level (direct simulation Monte Carlo method [10] or its variants). Over the past decade, a class of deterministic numerical methods—the Fourier–Galerkin spectral method—has received a lot of popularity for its high accuracy and relatively low computational cost. The first attempt was made in [79] for the one-dimensional model. Later in [48, 49, 52], both two and three dimensional cases were considered. Although the implementation details may differ, the essential ideas in these works are the same, that is, utilizing the translational invariance of the collision operator and convolution property of the Fourier basis to write the collision operator as a weighted convolution in the Fourier space. In this way, the O(N 3d ) cost per evaluation of the collision operator in the Galerkin framework (since QI (f, f ) is quadratic) is readily reduced to O(N 2d ), where N is the number of basis used in each velocity dimension. Even though this reduction is dramatic compared to other spectral basis, numerical implementation of the

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“direct” Fourier spectral method is still computationally demanding; what makes it worse is that the method also requires O(N 2d ) memory to store the precomputed weights, which quickly becomes a bottleneck as N increases. Recently, a fast Fourier spectral method was introduced in [62], wherein the key idea is to shift some offline precomputed items to online computation so that the weighted convolution in the original method can be rendered into a few pure convolutions to be evaluated efficiently by the fast Fourier transform (FFT). As a result, both the computational complexity and the memory requirement in the direct Fourier method are reduced. In this section, we briefly review the original direct Fourier spectral method proposed in [48] and then its fast version introduced in [62]. To this end, let us introduce the following weak form of the inelastic Boltzmann collision operator QI (f, f ) using the σ -representation (3):

Rd

QI (f, f )(v)ψ(v)dv =



Rd

Rd

Sd−1



B(|v − v∗ |, σ · (v − v∗ ))ff∗ ψ  − ψ dσ dvdv∗ ,

(33) where σ · (v − v∗ ) = cos θ (uˆ denotes the unit vector along u), and for simplicity we assume B does not depend on the kinetic energy E (compare with (7)). From the numerical point of view, this does not impose any limitation since the E part can always be factored out from the integral sign.

2.1 The Direct Fourier Spectral Method We first perform a change of variable v∗ → g = v − v∗ in (33) to obtain

Rd

QI (f, f )(v)ψ(v)dv =

Rd

Rd

Sd−1

B(|g|, σ · g)f ˆ (v)f (v − g) ψ(v  ) − ψ(v) dσ dgdv,

where v = v −

1+e (g − |g|σ ). 4

We then assume that f has a compact support: Suppv (f ) ≈ BS , where BS is a ball centered at the origin with radius S. Hence it suffices to truncate the infinite integral in g to a larger ball BR with radius R = 2S:

Rd

QI (f, f )(v)ψ(v)dv =

Rd

BR

Sd−1

B(|g|, σ · g)f ˆ (v)f (v − g) ψ(v  ) − ψ(v) dσ dgdv.

(34)

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Next we restrict v to a cubic computational domain DL = [−L, L]d , and approximate f by a truncated Fourier series: −1 

N 2

f (v) ≈ fN (v) =

fˆk ei L k·v ,

fˆk =

π

k=− N2

1 (2L)d



f (v)e−i L k·v dv. π

DL

N/2−1

:= √ k1 ,...,kd =−N/2 . The choice of L should be chosen at least as L ≥ (3 + 2)S/2 to avoid aliasing, see [48] for more details. Now substituting fN into (34) π and choosing ψ(v) = e−i L k·v , we can obtain the k-th mode of the collision operator as Here k = N/2−1

(k1 , . . . , kd ) is a multidimensional index, and

N 2

−1 

Qˆ k =

G(l, m)fˆl fˆm ,

k=−N/2

(35)

l,m=− N2 l+m=k

where the weight G(l, m) is given by G(l, m) =

e BR

−i πL m·g

 Sd−1

 π 1+e   i L 4 (l+m)·(g−|g|σ ) B(|g|, σ · g) ˆ e − 1 dσ dg.

In the original spectral method [48], the weight G(l, m) is precomputed and stored since it is independent of the solution f which leads to a memory requirement of O(N 2d ). During the online computation, the weighted sum (35) is directly evaluated whose complexity is O(N 2d ).

2.2 The Fast Fourier Spectral Method To reduce the complexity of the direct spectral method as well as to alleviate its memory requirement, the key idea introduced in [62] is to render the weighted convolution (35) into a pure convolution so that it can be computed efficiently by the FFT. One way to achieve this is through a low-rank approximation of G(l, m), namely, G(l, m) ≈

Np  p=1

αp (l + m)βp (m),

(36)

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where αp and βp are some functions to be determined and the number of terms Np in the expansion is small. Then (35) becomes Np 

Qˆ k ≈

N 2 −1



αp (k)

p=1

  fˆl βp (m)fˆm ,

(37)

l,m=− N2 l+m=k

where the inner summation is a pure convolution of two functions fˆl and ˆ k (for all k) is brought down to βp (m)fˆm . Hence the total complexity to evaluate Q O(Np N d log N), i.e., a few number of FFTs. Specifically, we first split G(l, m) into a gain term and a loss term: G(l, m) = Ggain (l, m) − Gloss(m), where Ggain (l, m) :=

e



−i πL m·g

Sd−1

BR

B(|g|, σ · g)e ˆ

i πL

1+e 4 (l+m)·(g−|g|σ )

 dσ dg,

and

π

e−i L m·g

Gloss(m) :=



BR

Sd−1

 B(|g|, σ · g)dσ ˆ dg.

Note that the loss term is readily a function of m, hence no approximation/decomposition is actually needed. This suggests to evaluate the loss term of the collision operator by a precomputation of Gloss (m) and then compute Qˆ − k

N 2 −1

=



  fˆl G(m)fˆm

l,m=− N2 l+m=k

by FFT. For the gain term, to get a decomposition of form (36), we introduce a quadrature rule to discretize g, then Ggain (l, m) can be approximated as Ggain(l, m) ≈



wρ wgˆ ρ d−1 e−i L ρ m·gˆ F (l + m, ρ, g), ˆ π

(38)

ρ,gˆ

where ρ := |g| ∈ [0, R] is the radial part of g and gˆ ∈ Sd−1 is the angular part, and wρ and wgˆ are the corresponding quadrature weights. The function F is given by F (l + m, ρ, g) ˆ :=

Sd−1

π

B(ρ, σ · g)e ˆ iLρ

1+e ˆ ) 4 (l+m)·(g−σ

dσ.

(39)

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With this approximation, the gain term of the collision operator can be evaluated as Qˆ + k



 ρ,gˆ

N 2 −1

wρ wgˆ ρ

d−1

F (k, ρ, g) ˆ



! π fˆl e−i L ρ m·gˆ fˆm ,

l,m=− N2 l+m=k

which is in the same form as explained in (37). As for the quadratures, the radial direction ρ can be approximated by the GaussLegendre quadrature. Since the integrand in (38) is oscillatory on the scale of O(N), the number of quadrature points needed for ρ should be O(N). The angular direction in 2D can be discretized using simple rectangular rule which is expected to yield spectral accuracy due to the periodicity. While in 3D, we choose to use the spherical design [93] which is the near optimal quadrature on the sphere. To summarize, the total complexity to evaluate Qˆ k is O(MN d+1 log N), where M is the number of points used on Sd−1 and M  N d−1 . Furthermore, the only quantity that needs to be precomputed and stored is (39), which in the worst scenario only requires O(MN d+1 ) memory.

3 Numerical Experiments and Results The accuracy and efficiency of the fast spectral method has been validated in [62]. In this section, we perform some additional tests to demonstrate the potential of the method in predicting some mathematical theories. We also introduce a GPU implementation of the method that significantly improves the CPU version in [62]. This is critical especially for long time simulation. We consider the following spatially homogeneous equation with a heat bath: ∂t f = QI (f, f ) + τ v f,

(40)

where τ is the parameter describing the strength of the heat bath. Notice that it is not necessarily related to the inelasticity parameter e, contrarily to e.g. [76]. The heat bath v f will also be discretized using the Fourier spectral method and Runge– Kutta method is used for time marching. For the collision operator, we consider the simplified variable hard sphere kernel B(|g|, σ · g, ˆ E) = Cλ |g|λ ,

0 ≤ λ ≤ 1,

where Cλ > 0 is some constant (namely (7) with b = Cλ and γ = 0.

(41)

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For Maxwell molecules, given the initial condition f0 (v) whose macroscopic quantities are ρ0 , u0 and T0 , the density and velocity are conserved so ρ(t) = ρ0 , u(t) = u0 and the temperature will evolve as  T (t) = T0 −

8τ 1 − e2



  8τ ρ0 (1 − e2 ) t + , exp − 4 1 − e2

(42)

We could see lim T (t) =

t →∞

8τ . 1 − e2

As in [62], this analytical formula of temperature works as the reference solution to ensure the correctness of the numerical solution.

3.1 GPU Parallelized Implementation From the implementation perspective, we dramatically improve the efficiency of the fast spectral method by using GPU via Nivida’s CUDA. GPU-parallelized implementations are run on 2 Intel® Xeon® Silver 4110 2.10 GHz CPUs with 4 NVIDIA Geforce GTX 2080 Ti (Turing) GPUs accompanying CUDA driver 10.0 and CUDA runtime 10.0. The operating system used is 64-bit Unbuntu 18.04. The CPU has 8 cores, 16 threads with max turbo frequency 3.00 Ghz equiped with 128 GB DDR4 REG ECC memory. The GPU has 4352 CUDA cores, 11 GB device memory. The algorithm has been written in python with packages Numpy and Scipy. The CPU implementation is based on PyFFTW which is a python wrapper of the C library FFTW. The GPU implementation is based on CuPy which is an implementation of NumPy-compatible multi-dimensional array on CUDA. As shown in Table 1, GPU version is up to 15 times faster than CPU version depending on different Ns. Table 1 Average running time per evaluation of the collision operator in 3D. Comparison between the CPU and GPU-parallelized implementation for various Ns (# of Fourier basis in each velocity dimension) and fixed Nρ = 30, Msph = 32 (# of quadrature points used in radial and spherical direction, respectively) with 2 Xeon® Silver 4110 2.10 GHz CPUs with 4 NVIDIA Geforce GTX 2080 Ti (Turing) GPUs N 8 16 32 64

CPU 7.68 ms 61.2 ms 546 ms 5.38 s

GPU 5.89 ms 5.97 ms 12.1 ms 109 ms

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3.2 Numerical Results Test 1: Validation of Exponential Convergence to the Equilibrium in 2D As a first test, our goal is to verify the theoretical result of the exponential convergence to the equilibrium. We consider the Maxwell molecule by taking the collision kernel as B(|g|, σ · g, ˆ E) = C0 =

1 . 2π

(43)

The initial condition is chosen as 2

f0 (v) =

0) ρ0 − (v−u e 2T0 , 2πT0

(44)

with ρ0 = 1, u0 = (0, 0) and T0 = 8. The theoretical results from [20] states that if e = 1 − τ , then there exists a unique equilibrium solution f∞ to (40) and one has H(f |f∞ )(t) ∼ O(e−λt ),

(45)

where the relative entropy is defined as H(f |g) =

f ln

  f dv . g

In order to confirm this result, we first choose the following physical parameters e = 0.95,

τ = 1 − e = 0.05 .

The numerical parameters we use to compute the 2D collision operator are Nv2 = 64 × 64,

Nρ = 32,

Mgˆ = 16,

R = 20,

L = 5(3 +



2).

For time integrator, a 4th order Runge–Kutta method is used with t = 0.01 and we compute sufficient long time to get f∞ and in both cases the 2 difference of solutions between the last 2 time steps are of O(10−12). The results are shown in Fig. 3 where one can observe the exponential convergence of relative entropy (45) from the left figure. In the right figure we find a perfect match of the temperature evolution between numerical solution and the analytical solution (42). We also plot the profile of the equilibrium solution f∞ in Fig. 4 which shows a Gaussian-like density function.

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Fig. 3 Test 1. Convergence to equilibrium. e = 0.95 with heat bath τ = 0.05. Initial data is the Maxwellian (44). Left: Semi-log plot of the relative entropy of f and f∞ = f (t = 100, v). Right: numerical temperature (orange dots) with exact temperature (blue √ line) as (42). Numerical parameters: Nv2 = 64 × 64, Nρ = 32, Mgˆ = 16, R = 20, L = 5(3 + 2) and t = 0.01

Fig. 4 Test 1. The equilibrium profile of e = 0.95 with heat bath τ = 0.05, initial data is the Maxwellian (44). Numerical parameters: Nv2 = 64 × 64, Nρ = 32, Mgˆ = 16, R = 20, L = √ 5(3 + 2) and t = 0.01

Although the exponential convergence result is only available for the case that e = 1 − τ , we expect something similar happens even when e and τ are not related. In order to investigate this, we choose e = 0.5,

τ = 0.1,

and perform the test using the same initial data (44). The results are shown in Figs. 5 and 6 where we indeed observe the same exponential convergence to equilibrium.

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Fig. 5 Test 1. Convergence to equilibrium. e = 0.5 with heat bath τ = 0.1, initial data is the Maxwellian (44). Left: Semi-log plot of the relative entropy of f and f∞ = f (t = 55, v). Right: numerical temperature (orange dots) with exact temperature (blue √ line) as (42). Numerical parameters: Nv2 = 64 × 64, Nρ = 32, Mgˆ = 16, R = 20, L = 5(3 + 2) and t = 0.01

Fig. 6 Test 1. The equilibrium profile of e = 0.5 with heat bath τ = 0.1, initial data is the Maxwellian (44). Numerical parameters: Nv2 = 64 × 64, Nρ = 32, Mgˆ = 16, R = 20, L = √ 5(3 + 2) and t = 0.01

To confirm that we could always get Gaussian-like equilibrium solution regardless of the initial data, we also consider the following initial condition ⎧ ⎪ ⎨ 1 , f0 (v) = 4w02 ⎪ ⎩0,

for v ∈ [−w0 , w0 ] × [−w0 , w0 ], otherwise,

(46)

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Fig. 7 Test 1. Convergence to equilibrium. e = 0.5 with heat bath τ = 0.1, initial data is the flat function (46). Left: Semi-log plot of the relative entropy of f and f∞ = f (55, v). Right: numerical 2 temperature (orange dots) with exact temperature (blue √ line) as (42). Numerical parameters: Nv = 64 × 64, Nρ = 32, Ngˆ = 16, R = 20, L = 5(3 + 2) and t = 0.01

Fig. 8 Test 1. The equilibrium profile of e = 0.5 with heat bath τ = 0.1, initial data is the√flat function (46). Numerical parameters: Nv2 = 64 × 64, Nρ = 32, Mgˆ = 16, R = 20, L = 5(3 + 2) and t = 0.01.

√ with w0 = 2 6 such that ρ0 = 1, u0 = (0, 0) and T0 = 8. With restitution coefficient e = 0.5 and heat bath τ = 0.1, the results are shown in Figs. 7 and 8. Test 2: Investigation of Tail Behavior of the Equilibrium in 2D We now compare the different tail behaviors of the equilibrium solutions for the Maxwell molecules collision kernel (43) and for the hard spheres collision kernel B(|g|, σ · g) ˆ = |g|/(2π)

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Fig. 9 Test 2. The equilibrium profile of e = 0.3, 0.5, 0.7 with heat bath τ = 0.1, initial data is the flat function (46). Left: Semi-log plot of f∞ (v1 , 0.17) = f (t = 55, v1 , 0.17) for Maxwell molecules. Right: Semi-log plot of f∞ (v1 , 0.17) = f (t = 55, v1 , 0.17) for hard spheres. The red lines are the reference profiles. Numerical parameters: Nv2 = 128 × 128, Nρ = 32, Mgˆ = 16, √ R = 20, L = 5(3 + 2) and t = 0.01

in 2D. To see the tail we need higher resolution in velocity space so the velocity mesh is increased to Nv2 = 128 × 128. We plot the profile in vx (v1 ) by choosing a fixed vy (v2 ) for different es (0.3, 0.5 and 0.7). From Fig. 9, we see that the numerical scheme generates overpopulated equilibrium tails: the Maxwell molecules case behaves like f (v, t = ∞) ∼ e−α|v| , and the hard spheresones behaves like f (v, t = ∞) ∼ e−α|v|

3/2

.

These results corresponds accurately to what was predicted theoretically in [20, 44] (summarized in Theorem 1). Test 3: 3D Hard Sphere The last test is more related to physics in real world, by simulating the so-called “Haff’s cooling Law” (26). We use the following initial data which is a Maxwellian with nonzero bulk velocity: f0 (v) =

ρ0 2 e−(v−u0) , 3/2 (2πT0 )

(47)

where ρ0 = 1, T0 = 2 and u0 = (0.5, −0.5, 0)T . We consider the hard spheres collision kernel in 3D, namely B=

1 |g|. 4π

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In the first two tests, we consider a realistic set-up where the restitution coefficient e depends on the distance of the relative velocity, i.e., e is a function of ρ = |g| instead of a constant, e(ρ) =

e0 − 1 e0 + 1 tanh(ρ − 4) + , 2 2

where 0 < e0 < 1. This allows to mimics the physically relevant visco-elastic hard spheres case (see also (5)). We numerically evaluate the temperature and the results for e0 = 0.8 and e0 = 0.2 are shown in Figs. 10 and 11. Compared with the cases where e is constant, we observe a slight slower decay of the temperature. Another parameter that may affect the decay rate of temperature is the variable hard spheres exponent λ from (7). In Fig. 12 we show that, in the presence of heat bath, for e = 0.5 but with λ = 1 (hard spheres), λ = 0.5 and λ = 0 (Maxwellian

Fig. 10 Test 3. Haff’s cooling law with Maxwellian initial data (47). Left: plot of inhomogeneous e. Right: comparison of temperature between constant e = 0.8 (dash line) and e(|g| = ρ) = −0.1 tanh(ρ − 4) + 0.9. Numerical parameters: Nv3 = 32 × 32 × 32, Nρ = 30, Mgˆ = 32, R = 8, √ L = 5(3 + 2) and t = 0.01

Fig. 11 Test 3. Haff’s cooling law with Maxwellian initial data (47). Left: plot of inhomogeneous e. Right: comparison of temperature between constant e = 0.2 (dash line) and e(|g| = ρ) = −0.4 tanh(ρ − 4) + 0.6. Numerical parameters: Nv3 = 32 × 32 × 32, Nρ = 30, Mgˆ = 32, R = 8, √ L = 5(3 + 2) and t = 0.01

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Fig. 12 Test 3. Haff’s cooling law with heat bath for different variable hard spheres exponent λs 3 and Maxwellian initial data (47). The heat bath √ (τ = 0.1). Numerical parameters: Nv = 32 × 32 × 32, Nρ = 30, Mgˆ = 32, R = 8, L = 5(3 + 2) and t = 0.01

Fig. 13 Test 3. Heated Haff’s cooling law with Maxwellian initial data (47). Left: regular plot. Right: log-log plot. Numerical parameters: Nv3 = 32 × 32 × 32, Nρ = 30, Mgˆ = 32, R = 8, √ L = 5(3 + 2) and t = 0.01

molecules), the decay rate of temperature will decrease after certain time (notice the slopes after t = 5). Finally, with the heat bath τ = 0.1, we numerically evaluate the temperature up to time tfinal = 20 for various values of restitution coefficients. The time evolution of T is shown in Fig. 13 where one can observe the transition of decays from e = 0.5 to e = 0.95 (near elastic case).

4 Conclusion In this review article, we summarized recent theoretical and numerical development in kinetic theory of granular materials. The main focus is the inelastic Boltzmann equation in both spatially homogeneous and inhomogeneous setting. In the theoret-

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ical part, we discussed the Cauchy theory and large time behavior of the equation, as well as the hydrodynamic limits. In the numerical part, we discussed the Fourier Galerkin spectral method to approximate the inelastic collision operator and the fast version of the method. Leveraging on the fast algorithm and GPU parallelized architecture, we are able to conduct long time simulations. In the last section, the exponential convergence to equilibrium and tail behavior were numerically verified. We also simulated a physically relevant case with visco-elastic hard spheres, where the “near” Haff’s cooling law has been observed. Acknowledgments JAC acknowledges support by the Advanced Grant Nonlocal-CPD (Nonlocal PDEs for Complex Particle Dynamics: Phase Transitions, Patterns and Synchronization) of the European Research Council Executive Agency (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 883363), by the Engineering and Physical Sciences Research Council (EPSRC) under grant no. EP/P031587/1, and by the National Science Foundation (NSF) under grant no. RNMS11-07444 (KI-Net). JH was partially funded by NSF grant DMS-1620250 and NSF CAREER grant DMS-1654152. TR was partially funded by Labex CEMPI (ANR-11-LABX-0007-01) and ANR Project MoHyCon (ANR-17-CE40-0027-01).

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Asymptotic Methods for Kinetic and Hyperbolic Evolution Equations on Networks Raul Borsche and Axel Klar

Abstract We consider kinetic and associated macroscopic equations on networks. A general approach to derive coupling conditions for the macroscopic equations from coupling conditions of the underlying kinetic problem is presented using an asymptotic analysis near the nodes of the network. This analysis leads to the consideration of a fixpoint problem involving the coupled solutions of kinetic halfspace problems. The procedure is explained for two simplified situations. The linear case is discussed for a linear kinetic BGK-type model leading in the macroscopic limit to a linear hyperbolic problem. The nonlinear situation is investigated for a kinetic relaxation model and an associated macroscopic scalar nonlinear hyperbolic conservation law on a network. Numerical comparisons between the solutions of the macroscopic equation with different coupling conditions and the kinetic solution are presented for the case of tripod and more complicated networks. Keywords Kinetic and hyperbolic equations · Networks · Zero relaxation coupling conditions · Half-space problems

1 Introduction There have been many attempts to define coupling conditions for macroscopic partial differential equations on networks including, for example, drift-diffusion equations, scalar hyperbolic equations, or hyperbolic systems like the wave equation or Euler type models, see for example [4, 5, 5, 9, 10, 14, 16, 18–22, 26, 29, 33, 35]. In [18, 33] coupling conditions for scalar hyperbolic equations on networks are discussed and investigated. [26, 46] treat the wave equation and general nonlinear hyperbolic systems are considered in [4, 5, 9, 14, 19]. Nevertheless, for hyperbolic systems on networks there are still many unsolved problems, like finding suitable coupling conditions without restricting to subsonic situations. R. Borsche · A. Klar () Department of Mathematics, TU Kaiserslautern, Kaiserslautern, Germany e-mail: [email protected]; [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 G. Albi et al. (eds.), Trails in Kinetic Theory, SEMA SIMAI Springer Series 25, https://doi.org/10.1007/978-3-030-67104-4_2

37

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On the other hand, coupling conditions for kinetic equations on networks have been discussed in a much smaller number of publications, see [11, 27, 34]. In [11] a first attempt to derive a coupling condition for a macroscopic equation from the underlying kinetic model has been presented for the case of a kinetic equations for chemotaxis. Here, a more general and more accurate procedure to derive coupling conditions for macroscopic equations from the underlying kinetic ones is reviewed, see [12] and in [13] for more details. We consider linear and nonlinear systems and use an asymptotic analysis of the situation near the nodes. The procedure has been motivated by the classical procedure to find kinetic slip boundary conditions for macroscopic equations. They are derived from the underlying kinetic equations via an analysis of the kinetic layer. We refer to [7, 8, 30, 45] for a kinetic layer analysis or [39, 47–49] for the case of hyperbolic relaxation systems. To explain the procedure in more detail, we consider a scaled kinetic problem in one spacial dimension with spatial and velocity variables x ∈ R, v ∈ R and the distribution function f solving ∂t f + v∂x f =

1 Q(f ) 

(1)

with a scaling parameter . One obtains for  → 0 associated macroscopic equations for the moments of f as, for example density and flux ρ=



∞ −∞

f (v)dv ,

q=

∞ −∞

vf (v)dv .

Considering additionally boundary conditions on an interval [0, b] one uses for the kinetic model typically f (0, v), v > 0, f (b, v), v < 0 . For the macroscopic equations boundary conditions are usually given via conditions on the ingoing characteristic variables [6, 44]. If such equations are considered on a network, it is sufficient to study a single coupling point or node, where coupling conditions are required. We consider a node connecting n edges, which are, for simplicity, oriented away from the node. Each edge i = 1, . . . , n is parametrized by the interval [0, bi ] and the kinetic quantities are denoted by f i . A typical choice of coupling conditions for the kinetic problem is given by f i (0, v) =

n 

cij f j (0, −v), v > 0 ,

(2)

j =1

compare [11]. The total mass in the system is conserved, if

n

i=1 cij

= 1 holds.

Asymptotic Methods for Kinetic and Hyperbolic Evolution Equations on Networks

39

Before deriving macroscopic coupling conditions from the kinetic ones, we revisit the classical kinetic layer analysis to determine macroscopic boundary conditions from kinetic conditions, see [7, 17, 24]. Focussing on the left boundary of the interval [0, b], a rescaling of the spatial variable in the kinetic equation with  gives to first order in  a stationary kinetic half space problem for xˆ = x ∈ [0, ∞] v∂xˆ ϕ = Q(ϕ) .

(3)

At xˆ = 0 the boundary conditions for the half space problem are ϕ(0, v) = k(v) = f (0, v), v > 0 . On the right side, i.e. at xˆ = ∞, the kinetic half-space problem requires additional conditions on the distribution function ϕ. The number of additional conditions is expected to correspond to the number of outgoing characteristics of the macroscopic equations, see, for example, [24] for the case of linear layer problems. Solving the half space problem gives not only the value of ϕ and its moments at infinity and thereby the boundary conditions for the macroscopic problem, but also the outgoing distribution A[k](v) = f (0, v) = ϕ(0, v), v < 0 , where the notation A is used for the so called Albedo operator of the half space problem. On the network, we proceed as follows. Starting from the kinetic coupling conditions (2) we determine the coupling conditions for the macroscopic equations in the following way. We use the kinetic coupling conditions to obtain conditions on the in- and outgoing solutions of the half space problems on the different arcs. That means ϕ i (0, v) =

n 

cij ϕ j (0, −v), v > 0

j =1

or, if we denote the ingoing function of the half-space problem on arc i by k i (v), v > 0 and the outgoing solution by Ai [k i ](v), v < 0 k i (v) =

n 

cij Aj [k j ](−v), v > 0 .

(4)

j =1

This is a fix point equation for k i , i = 1, . . . , n. As before, additional conditions at infinity related to the values of the outgoing solutions of the macroscopic equations are needed to solve the half-space problems. The solutions of the halfspace problems at infinity finally yield the desired conditions for the macroscopic equations at each node. Using the above described procedure explicit coupling

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conditions have been derived for the linear wave equation from an underlying linear kinetic model in [12] and for a simple nonlinear situation in [13]. The survey is organized in the following way. We start with the linear case. In Sect. 2.1 we discuss the kinetic and macroscopic equations and the boundary and coupling conditions for the linear case. In Sect. 2.2 kinetic boundary layers are discussed, as well as an asymptotic analysis of the kinetic equations near the nodes. This leads to an abstract formulation of the coupling conditions for the macroscopic equations at the nodes based on a fix-point problem involving kinetic half-space equations. A refined method to determine the solution of the half space problems is derived, compared to previous approximate solution methods for halfspace problems and applied to the problem of finding accurate coupling conditions for the macroscopic equations. Moreover, the macroscopic equations on the network with the different coupling conditions are numerically compared to each other and to the full solutions of the kinetic equations on the network in Sect. 2.5. Then, the nonlinear case is investigated for an equation with a scalar limit problem. In Sect. 3.1 we state a relaxation model and the associated scalar conservation law. In Sect. 3.2 kinetic boundary layers are discussed, as well as the combination of these layer solutions with suitable Riemann solvers. This leads to classical boundary conditions for the Burgers problem depending on the kinetic boundary condition. In the following Sect. 3.3 a short outline is given, how coupling conditions for the scalar hyperbolic problem are derived from the kinetic coupling conditions. For the case of a node with three edges we state explicit coupling conditions for the macroscopic equation based on the kinetic coupling conditions. Finally, again, the solution of the nonlinear macroscopic equation on the network is numerically compared to the full solution of the kinetic equation in Sect. 3.4. An outline of further work is included at the end of the paper.

2 Coupling Conditions for Linear Problems 2.1 Equations We consider a linear kinetic BGK model in 1D with two collision invariants and x ∈ R, v ∈ [−1, 1] ∂t f + v∂x f = Q(f ) = − with a 2 =

1 3

  1 v 1 f − (ρ + 2 q)  2 a

and ρ=





−∞

f (v)dv,

q=

∞ −∞

vf (v)dv .

(5)

Asymptotic Methods for Kinetic and Hyperbolic Evolution Equations on Networks

41

The associated macroscopic equation for  → 0 is the wave equation ∂t ρ + ∂x q = 0

(6)

∂t q + ∂x (a 2 ρ) = 0 . Considering an interval x ∈ [0, b] we prescribe for the kinetic equation f (0, v), v > 0 and f (b, v), v < 0 .

For (6) the boundary conditions are given in characteristic variables [44]. The corresponding Riemann Invariants are r1/2 = q ∓ aρ .

(7)

As boundary data the value of q + aρ at the left boundary and q − aρ at the right boundary are prescribed. If these equations are considered on a network, it is sufficient to study a single coupling point. At each node coupling conditions are required. In the following we consider a node connecting n edges, which are oriented away from the node, as in Fig. 1. Each edge i is parametrized by the interval [0, bi ] and the kinetic and macroscopic quantities are denoted by f i and ρ i , q i respectively. A natural choice of coupling conditions for the kinetic problem is given by f i (0, v) =

n 

cij f j (0, −v), v > 0 ,

(8)

j =1

compare [11]. The total mass in the system is conserved, if n 

cij = 1

(9)

i=1

holds. In the following we use the notation f + = Cf − , v > 0 , where f + = (f 1 (0, v), . . . , f n (0, v))T and f − = (f 1 (0, −v), . . . , f n (0, −v))T and C is a n × n matrix. The coupling conditions for the macroscopic quantities are Fig. 1 Node connecting three edges and orientation of the edges

3 1 2

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conditions on the characteristic variables r2i (0) = q i (0) + aρ i (0) using the given values of r1i (0) = q i (0) − aρ i (0) . We refer to [19, 28] for systems of macroscopic equations on networks. In the following we will first review the boundary layer analysis for the kinetic eqautions under consideration and then derive macroscopic coupling conditions from the kinetic coupling conditions (8).

2.2 Boundary and Coupling Conditions for Linear Equations via Kinetic Layer Analysis In the classical procedure to determine boundary conditions for macroscopic equations, [7, 17, 24, 28], one rescales the spatial variable in Eq. (24) with . We consider the left boundary of the interval [0, b]. One obtains 1 1 ∂t f + v∂x f = Q(f ) .   This yields to first order in  the following stationary kinetic half space problem for x ∈ [0, ∞]   v  1 v∂x ϕ = − ϕ − ρ + 2q , 2 a

(10)

where ρ and q are here the zeroth and first moments of ϕ. At x = 0 the boundary conditions for the half space problem are ϕ(0, v) = k(v) = f (0, v), v > 0 . On the right side of the half-space, i.e. at x = ∞, prescribing an arbitrary linear combination of the invariants of the half-space problem < vϕ >" and < v 2 ϕ > is required. Here, and in the following we use the notation < ϕ >= ϕdv. We use the values of the first Riemann Invariant (7) r1 = q − aρ of the macroscopic system (6) to fix   v2 < v− ϕ >= r1 . a

Asymptotic Methods for Kinetic and Hyperbolic Evolution Equations on Networks

43

The boundary condition for (6) is obtained by determining r2 from the asymptotic solution of the half space problem setting r2 = q∞ + aρ∞ . The values ρ∞ and q∞ are the macroscopic quantities associated to the solution of the half-space problem at infinity, which has the form ϕ(∞, v) =

 1 v ρ∞ + 2 q∞ . 2 a

The solution of the half space problem is also used to determine the outgoing distribution A[k](v) = f (0, v) = ϕ(0, v), v < 0 . Coupling conditions for the macroscopic equations on the network are now found by the following procedure. We start from the kinetic coupling conditions f i (0, v) =

n 

cij f j (0, −v), v > 0.

j =1

As explained in the introduction, this leads for the half-space problems to ϕ (0, v) = i

n 

cij ϕ j (0, −v), v > 0

j =1

or, if we denote the ingoing function of the half-space problem on arc i by k i (v), v > 0 and the outgoing solution by Ai [k i ](v), v < 0 k i (v) =

n 

cij Aj [k j ](−v), v > 0 .

(11)

j =1

This is a fix point equation for k i , i = 1, . . . , n. Additional conditions are needed to solve the half-space problems, i.e.   v2 < v− ϕ i >= r1i a with r1i = q i − aρ i . The coupling conditions for the wave equation are conditions on the outgoing characteristic variables at x = 0. We define i i r2i (0) = q∞ [k i ] + aρ∞ [k i ] .

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In the following we do not consider an analytical investigation of the above fixpoint problem in general. Instead we discuss an approximation of the half space solution and solve the approximate problem explicitely. This yields explicit coupling conditions which are then numerically compared to the solution of the original kinetic problem on the network and to other approximate solution methods for the half-space problem.

2.3 Approximate Solution of the Half Space Problem via Half-Fluxes To solve kinetic half space problems approximately, a variety of different methods can be found in the literature. For approaches via a Galerkin method we refer to [23, 36, 37]. Approximate methods to determine only the asymptotic states and outgoing distributions have been developed in [31, 40, 41]. In the following we present a procedure to determine the solution of the halfspace problem via approximation of the BGK equation by half-moment equations and use it for the derivation of coupling conditions. To start with we determine a half moment approximation for the half-space solution, compare [11]. We define ρ− = −

q =



0

−1



ρ+ =

f (v)dv ,

1

f (v)dv , 0

0

−1



+

q =

vf (v)dv ,

(12)

1

vf (v)dv . 0

As closure assumption we use the following approximation of the distribution function f by affine linear functions in v to determine half-moment equations, see [11] and references therein. f (v) = a + + vb + , v ≥ 0

f (v) = a − + vb − , v ≤ 0 .

and

One obtains 1 ρ − = a − − b− , 2

1 ρ + = a + + b+ , 2

1 1 q − = − a − + b− , 2 3

q+ =

1 + 1 + a + b 2 3

and 0

1

1 v 2 f (v)dv = − ρ + + q + , 6



0

1 v 2 f (v)dv = − ρ − − q − . 6 −1

Asymptotic Methods for Kinetic and Hyperbolic Evolution Equations on Networks

45

Finally, integrating the kinetic equation with respect to the corresponding halfspaces, we get the half-moment approximation of the kinetic equation as ⎧ ⎪ + + ⎪ ⎪ ∂t ρ + ∂x q ⎪ ⎪ ⎪   ⎪ ⎪ ⎪ 1 + ⎪ + + ⎪ ∂t q + ∂x − ρ + q ⎨ 6 ⎪ ⎪ ∂ ρ− + ∂ q − ⎪ t x ⎪ ⎪ ⎪ ⎪   ⎪ ⎪ 1 ⎪ ⎪ ⎩ ∂t q − + ∂x − ρ − − q − 6

   3q 1 ρ ρ+ − +  2 4 q  1  + ρ + =− q −  4 2    3q 1 ρ − − =− ρ −  2 4 q  1 −  ρ . =− q − − +  4 2 =−

(13)

Introducing the even-odd variables ρ = ρ+ + ρ− ,

ρˆ = ρ + − ρ − ,

q = q+ + q− ,

qˆ = q + − q − ,

we can rewrite the system as ⎧ ∂t ρ + ∂x q =0 ⎪ ⎪ ⎪ ⎪   ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎨ ∂t q + ∂x − 6 ρ + qˆ = 0

  1 3 ⎪ ⎪ ∂ q ρ ˆ + ∂ q ˆ = − ρ ˆ − t x ⎪ ⎪  2 ⎪ ⎪   ⎪   ⎪ ⎪ 1 ρ 1 ⎪ ⎩ ∂t qˆ + ∂x − ρˆ + q = − qˆ − . 6  2 Obviously, the half-moment model has again the wave equation (6) as macroscopic limit as  goes to 0. Rescaling the spatial variable in the half-moment problem with  and rewriting the equations in terms of the even-odd variables, one obtains to zeroth order the following half-space problem for x ∈ R+ ⎧ ⎪ ⎪ ∂x q + ⎪ ⎪ ⎪ ⎪   ⎪ ⎪ ⎪ 1 + ⎪ + ⎪ ρ − + q ∂ ⎨ x 6 ⎪ ⎪ ⎪ ∂x q − ⎪ ⎪ ⎪ ⎪   ⎪ ⎪ 1 ⎪ ⎪ ⎩ ∂x − ρ − − q − 6

   3q ρ + + =− ρ − 2 4   ρ q + = − q+ − 4 2    3q ρ − − =− ρ − 2 4  ρ  q  = − q− − − + 4 2

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or ⎧ ∂x q =0 ⎪ ⎪ ⎪ ⎪   ⎪ ⎪ ⎪ 1 ⎪ ⎪ ∂ ρ + q ˆ =0 − ⎪ x ⎨ 6

  3 ⎪ ⎪ ∂x qˆ = − ρˆ − q ⎪ ⎪ 2 ⎪ ⎪   ⎪  ⎪ ⎪ ρ 1 ⎪ ⎩ ∂x − ρˆ + q = − qˆ − . 6 2

(14)

We have to provide boundary conditions for ρ + (0) and q + (0), as well as a condition at x = ∞ q∞ − aρ∞ = r1 (0) = C. Then, the half space problem can be solved explicitly. We determine a solution up to 3 constants which will be fixed with the above 3 conditions. First, we observe, that we have 2 invariants q = C1 −

ρ + qˆ = C2 6

(15)

with constants C1 and C2 . From the last equation in (14) we can deduce that at x=∞ qˆ∞ =

ρ∞ . 2

Combining this with (15) gives ρ∞ = 3C2 or qˆ∞ = 3C2 2 . From the third equation 3C1 of (14) we obtain ρˆ∞ = 3q 2 = 2 . This simplifies (14) to ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

q = C1 ρ = 6qˆ− 6C2  3 ∂x qˆ = − ρˆ − C1 ⎪ 2 ⎪   ⎪ ⎪

⎪ 1 ⎪ ⎩ ∂x − ρˆ = − −2qˆ + 3C2 . 6

Asymptotic Methods for Kinetic and Hyperbolic Evolution Equations on Networks

47

The ODEs for ρˆ and qˆ have the solutions 2x 2x 3C1 ) + γˆ exp( ) + a a 2 a 2x a 2x 3C2 qˆ = γ exp(− ) − γˆ exp( ) + . 2 a 2 a 2 ρˆ = γ exp(−

Since we are looking only for bounded solutions we are left with 2x )+ a 2x a qˆ = γ exp(− ) + 2 a ρˆ = γ exp(−

3C1 2 3C2 2

q = C1 ρ = 3aγ exp(−

2x ) + 3C2 . a

The three parameters are fixed with the 3 conditions mentioned above. At x = 0 inflow data is given 1

q(0) + q(0) ˆ = q+ (0) 2 1

ρ(0) + ρ(0) ˆ = ρ+ (0) 2 and the Riemann Invariant at x = ∞ gives q∞ − aρ∞ = C .

(16)

Inserting the above determined solution we obtain   3C2 a 1 C1 + γ + = q+ (0) 2 2 2   3C1 1 3aγ + 3C2 + γ + = ρ+ (0) , 2 2 which can be rewritten in terms of the asymptotic values

3q∞ 4

q∞ ρ∞ a + + γ = q+ (0) 2 4 4   ρ∞ 3a + 1 + + γ = ρ+ (0) . 2 2

(17)

48

R. Borsche and A. Klar

Together with the condition at infinity (16), this determines the asymptotic values q∞ , ρ∞ and γ . The outgoing quantities ρ− (0), q− (0) are then determined by



3q∞ 4

q∞ ρ∞ a − − γ = q− (0) 2 4 4   1 3a − 1 + ρ∞ + γ = ρ− (0) . 2 2

Remark 1 The so-called Maxwell approximation determines the asymptotic states by q+ (0) = q+ (∞) =

ρ∞ 1 q∞ (q∞ + qˆ∞ ) = + 2 2 4

and the condition at infinity (16). The outgoing quantities are q∞ ρ∞ + 4 2 3q∞ ρ∞ ρ− (0) = ρ− (∞) = − . 2 4 q− (0) = q− (∞) = −

To estimate the accuracy of our method, we consider the classical problem of determining the so-called extrapolation length [31, 40, 43]. For x ∈ R+ , v ∈ [−1, 1] we consider the half space equation   3 ρ v∂x f = − f − ( + vq) 2 2 with

"1

−1 vf dv

= q = 0 and f (0, v) = v, v > 0. Thus we obtain  ρ . v∂x f = − f − 2

The extrapolation length is the value of λ∞ = f (∞, v) = ρ2∞ . The Maxwell approximation gives λ∞ = 23 . The above half moment approximation gives ρ∞ a 1 + γ = q+ (0) = 4 4 3 3a + 1 1 ρ∞ +( )γ = ρ+ (0) = . 2 2 2 This leads to ρ∞ =

9a + 4 6a + 3

Asymptotic Methods for Kinetic and Hyperbolic Evolution Equations on Networks

and with a 2 =

1 3

49

we obtain ρ∞

√ 3 3+4 . = √ 2 3+3

Thus, the extrapolation length is approximated as λ∞ ∼ 0.7113. The exact value computed from a spectral method is 0.7104, see [23, 37]. This yields an error for the above half-moment method of approximately 0.1%. In contrast, the Maxwell approximation gives 0.6666, which is an error of 6.1%. The variational method in [31, 40] gives 0.7083, which is an error of 0.3%.

2.4 Half-Moment Coupling Conditions In this subsection we determine the coupling conditions on the basis of the half-moment approximation of the half-space problem. Multiplying with v and integrating the kinetic coupling conditions (8) with respect to the positive and negative half moments gives i (0) = − q+

n  j =1

j

(18)

cij q− (0) .

Inserting the half moment approximations (17) yields  j  n j i i q∞ ρ∞ a i  ρ∞ a j q∞ . + + γ = + + γ cij − 2 4 4 2 4 4 j =1

Again, a summation w.r.t. i = 1, . . . , n directly gives the equality of fluxes 

i q∞ =0.

(19)

i

For a uniform node with equal distribution cij = obtains using the balance of fluxes

1 n−1 , i

= j and 0 otherwise, one j

n−2 i a a ρi n−2 j ρ∞ q∞ + ∞ + γ i = q∞ + + γj . 2n 4 4 2n 4 4 which gives the invariance of ρ∞ +

2(n − 2) q∞ + aγ . n

(20)

50

R. Borsche and A. Klar

Further, integrating the kinetic coupling conditions (8) with respect to the positive and negative half moments, we obtain i ρ+ (0) =

n  j =1

j

cij ρ− (0) .

With the half moment approximations (17) this reads i ρi 3q∞ + ∞ + 4 2



     n j  3a + 1 1 3a − 1 3q ∞ j + ρ∞ + cij − γi = γj . 2 4 2 2 j =1

Summing these conditions for i = 1, . . . , n yields n 

γi = 0 .

(21)

i=1

Thus, in the case of a uniform node, we derive another coupling invariant ρ∞ +

  3(n − 2) n−2 q∞ + 3a + γ . 2n n

(22)

Alltogether (19), (20), (21) and (22) yield 2 + 2(n − 1) = 2n conditions at a node. In combination with the conditions at infinity we have 3n conditions for 3n i , qi . quantities γ i , ρ∞ ∞ Note that the invariants (22) and (20) can be combined such that γ is eliminated, which gives the invariance of  ρ∞ +

n−2 n

 9a + 4 4a + 2

 

n−2 n n−2 n

  q∞ .

(23)

This together with (19) and the conditions at infinity yields 1 + n − 1 + n = 2n i , q i . The resulting linear system is easily shown conditions for the 2n variables ρ∞ ∞ to have a unique solution. One can derive similiar conditions using the Marshak approach described above or an approach using a full moment approximation, see [12]. All these coupling conditions for the wave equation with uniform nodes are given by the conservation of mass and an invariant of the form ρ + Cq . They differ in the value of the factor C, see [12].

Asymptotic Methods for Kinetic and Hyperbolic Evolution Equations on Networks

51

With a = √1 and n = 3 the half moment approximation gives C = 0.7313 3 compared to 0.6666 for Maxwell. A numerical comparison of the resulting network solutions is presented in the next section. Remark 2 For numerical comparison of the different approaches we consider for the solution of the wave equation (6) the mathematical entropy   1 1 2 2 ρ + 2q . e= 2 a It evolves according to the conservation law ∂t e + ∂x (ρq) = 0 . Along one edge this entropy is conserved, but the total entropy in the network can change according to the entropy-fluxes at the nodes. Note that for all above models with the coupling invariant ρ + Cq = C˜ and C > 0 the total entropy decays, since n  i=1

ρq =

n  n n n      q −C q 2 = −C q2 < 0 . C˜ − Cq q = C˜ i=1

i=1

i=1

i=1

2.5 Numerical Results In this section we compare the numerical results of the different models on networks. The solutions of the kinetic equation are compared with the half-moment approximation (13) and the macroscopic wave equation (6) with the different coupling conditions. The networks are composed of coupled edges, each arc is given by an interval x ∈ [0, 1], which is discretized with 400 spatial cells if not otherwise stated. In the kinetic model the velocity domain [−1, 1] is discretized with 400 cells and we choose  = 0.001 if not otherwise stated. For the advective part of the equations we use an upwind scheme. The source term in the kinetic and half moment equations is approximated with the implicit Euler method. We note that for the wave equation the upwind scheme yields the exact solution by choosing the CFL number equal to 1. For comparison we consider in addition to the coupling conditions discussed above a coupling based on the assumption of equal density at the node, see [25, 32] ρi = ρj 3  i=1

qi = 0 .

i = j, i, j = 1, 2, 3

52

R. Borsche and A. Klar

In general, at the outer boundaries of the network, boundary conditions have to be imposed. For the kinetic problem the values for the ingoing velocities have to be prescribed as mentioned before. For the half-moment approximation the natural boundary conditions are given by integrating the kinetic conditions and using ρ+ and q+ at left boundaries and ρ− and q− at right boundaries. For the wave equations with Maxwell and half-moment conditions we use the corresponding approximations discussed above to provide boundary values for the macroscopic equations. Additionally a full moment coupling condition is shown, which is explained in detail in [12]. In a first example, we consider a tripod network with initial conditions f 1 (x, v) = 12 ,f 2 (x, v) = 13 and f 3 (x, v) = 0. The corresponding macroscopic states are (ρ 1 , q 1 ) = (1, 0), (ρ 2 , q 2 ) = ( 23 , 0) and (ρ 3 , q 3 ) = (0, 0). We use free boundary conditions at the exterior boundaries. The computational time is chosen such that the waves generated at the node do not reach the exterior boundaries. In Fig. 2 we compare the kinetic, the half-moment and the wave equation with coupling conditions given by the Maxwell, the half-moment and the full-moment approach and the assumption of equal density at time T = 1 at edge 1. We observe first, that the half moment model gives a very accurate approximation of the kinetic equation. Also the Maxwell approximation provides a good approximation in this case. Note that a boundary layer is appearing at the node in the kinetic model, see Fig. 3 for a magnification of the situation on edge 1. Moreover, we investigate the evolution of the total entropy in the network, i.e.   3  1 1 i 2 i 2 (ρ ) + 2 (q ) dx . 2 a i=1

half-moment wave-full

wave-Maxwell wave-equal

1

0

0.9

−0.05 −0.1

0.8

q

ρ

kinetic wave-half

0.7 0.6 0

0.2

0.4

0.6

x on edge 1

0.8

1

0

0.2

0.4

0.6

0.8

1

x on edge 1

Fig. 2 Kinetic equation, half-moment equation and wave equation with coupling conditions given by the Maxwell, the half-moment and the full-moment approach and the assumption of equal density at time T = 1

Asymptotic Methods for Kinetic and Hyperbolic Evolution Equations on Networks

kinetic Maxwell

53

half-moment wave-half

0.688

−0.178

0.686 −0.18 q

ρ

0.684 −0.182

0.682 0.68

−0.184

0.678 0

0.1 0.15 5 · 10−2 x on edge 1

0.2

0

0.1 0.15 5 · 10−2 x on edge 1

0.2

Fig. 3 Magnification of the situation from Fig. 2 on edge 1 at the node Table 1 Total entropy and entropy loss at time T = 0.1 for different coupling conditions Coupling conditions Wave equal density Wave full moment Wave Maxwell Wave half moment Half moment  = 10−6 Kinetic  = 10−6

Total entropy 7.2222139e − 01 7.1701785e − 01 7.1621400e − 01 7.1597274e − 01 7.1586300e − 01 7.1574793e − 01

Entropy loss −8.3353857e − 07 −5.2043681e − 03 −6.0082256e − 03 −6.2494790e − 03 −6.3592218e − 03 −6.4742970e − 03

Fig. 4 Diamond network E1

E5

E2 E4 E3

E7

E6

Initially, the total entropy at t = 0 is equal to 0.722222. In this case we use a very fine grid with 30000 spatial cells for all models and 400 cells in velocity space for the kinetic equation. In Table 1 the value of the total entropy at time T = 0.1 is shown for the different coupling conditions together with the half moment and the kinetic solution for comparison. One observes the very accurate approximation given by the wave equation with half moment coupling conditions. As a second example we consider a more complicated network, see Fig. 4, as, for example, studied in [26] for the wave equation. As initial conditions for the kinetic equation we choose f 1 (x, v) = 1,f 2 (x, v) = 5 1 j 6 and f (x, v) = 2 for j = 3, . . . , 7, which corresponds to macroscopic densities ρ 1 = 2, ρ 2 = 53 and ρ j = 1 for j = 3, . . . , 7 and fluxes q j = 0 j = 1, . . . , 7. These data are also prescribed at the two outer boundaries, i.e. k 1 (v) = 1, v ∈ [0, 1] and 7 (v) = 12 , v ∈ [−1, 0] as ingoing functions at the left and right end respectively.

54

R. Borsche and A. Klar kinetic wave - Maxwell wave - half

half-moment wave - full wave - equal 1.55

1.5

1.5 ρ

ρ

1.4 1.3

1.45

1.2

1.4

1.1 0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

x on edge 4

x on edge 4

Fig. 5 ρ on edge 4 at time t = 3 (left) and time t = 10 (right)

Boundary conditions for the wave equation with full moment, Maxwell and half moment conditions are derived as detailed above. In case of the equal density conditions, we determine the ingoing characteristic using ρ = 1, q = 0 at the E1 -boundary and ρ = 12 , q = 0 at the E7 -boundary. In Fig. 5 the density ρ 4 on edge 4 is displayed at time t = 3 and t = 10. As before, we observe a good agreement of the half moment coupling with the kinetic and half moment model. Also the Maxwell approximation is relatively close to the kinetic results. The states of the full moment coupling and the equal density coupling deviate remarkably from the kinetic results.

3 Coupling Conditions for Nonlinear Problems 3.1 Equations We consider the following relaxation model for x ∈ R, v > 0 and F = F (u).   1 vu − F (u) f1 −  2v   1 F (u) + vu f2 − ∂t f2 + v∂x f2 = −  2v

∂t f1 − v∂x f1 = −

with u = f1 + f2 . Defining the flux uˆ = v1 f1 + v2 f2 yields f1 =

vu − uˆ , 2v

f2 =

uˆ + vu . 2v

(24)

Asymptotic Methods for Kinetic and Hyperbolic Evolution Equations on Networks

55

The associated macroscopic equation for  → 0 is a conservation law for the quantity u given by ∂t u + ∂x F (u) = 0 .

(25)

Rewriting the kinetic problem in terms of u and uˆ yields ∂t u + ∂x uˆ = 0 ∂t uˆ + ∂x P (u, u) ˆ =−

1

uˆ − F (u) . 

(26)

with P (u, u) ˆ = v2 u .

(27)

Convergence of the kinetic equation as  → 0 is obtained under the subcharacteristic condition [38] −v ≤ F  (u) ≤ v .

(28)

We have to prescribe f2 at the left boundary and f1 at the right boundary. For the nonlinear hyperbolic limit problem, boundary conditions have been considered in many works [1, 2, 6, 15, 42]. For boundary conditions and layers of hyperbolic problems with stiff relaxation terms and for the derivation of conditions for the corresponding limit equations we refer for example to [39, 47–49]. In order to determine the boundary conditions for the limit equations we combine an analysis of the kinetic layer with the solution of a half-Riemann problem for the limit equation, compare for example [3, 47]. A similar procedure will then be used to find kinetic based coupling conditions for the Burgers equation on a network. To proceed, we first state the layer equations. We consider the left boundary of the domain located at x = 0. A rescaling of the spacial coordinate near the boundary with x˜ = x gives the layer problem on [0, ∞) as vu − F (u) − f1 2v F (u) + vu − f2 . v∂x˜ f2 = 2v

−v∂x˜ f1 =

In the macroscopic variables u, uˆ this is ∂x˜ uˆ = 0 ∂x˜ P (u, u) ˆ = F (u) − uˆ ,

(29)

56

R. Borsche and A. Klar

with P as in (27). This gives ∂x˜ uˆ = 0 v 2 ∂x˜ u = F (u) − uˆ and therefore uˆ = C = const

(30)

a∂x˜ u = F (u) − C with a = v 2 > 0. Remark 3 For a right boundary we obtain the layer problem as −a∂x˜ u = F (u) − C .

For the following computations we concentrate on the Burgers equations and choose F (u) = u2 .

3.2 Boundary Conditions via Layer Analysis In a first step we determine the boundary conditions for the Burgers equation from the kinetic boundary condition. We use the boundary layer equations and couple them with half-Riemann solvers. First we discuss the solution in the boundary layer. The boundary layer equation near a left boundary is given by a∂x u = u2 − C . √ √ For C √> 0 this problem has two fixpoints u = ± C, where C is instable and − √C is a stable fixpoint. The domain of attraction of the stable fix-point is (−∞, C). The explicit solution is given by u(x) =

√ √ Ctanh(− C(x + C2 )/a)

for

|u(0)|


√ C.

and

We determine C2 from u(0) =

√ √ Ctanh(− CC2 /a)

for

|u(0)|




C.

√ √ One observes that for u(0) < C the limit x → ∞ leads to u(x) → − C and for √ √ ). u(0) > C the layer solution diverges at x = −C2 = √a arcoth( u(0) C C For C = 0 we obtain u(x) = −

a a =− a x + C2 x − u(0)

and convergence to 0 for u(0) < 0 and divergence for u(0) > 0. The solutions are sketched in Fig. 6. √ In the following we use the notation (U ) for the unstable solution u(x) = C and the notation (S) for the (partially) stable solutions. The asymptotic states as x → ∞ are denoted by uK . The layer solution for the right boundary can be discussed analogously. Since the layer solution can not cover the full range of possible states at a boundary, we have to consider additionally a Riemann Problem for the Burgers equation connecting the state in the domain with the layer. In particular, for the left boundary we need to know, which asymptotic states uK from the kinetic layer can be connected to a given right side state from the Burgers equation uB using only waves with non-negative speeds. For the Burgers equation we have the following cases: RP1 uB ≥ 0 ⇒ uK ∈ [0, ∞), since there is either an arbitrary wave with positive speed, if uK > 0, or a rarefaction wave starting at u = 0.

u



u(0) u(0)

u

C

u (0) u =0

u=0 u(0)

√ − C

u(0) x=0

x=∞ (a)

u (0) u (0)

C =0 u (x )

u (0) x =0

x =∞ (b)

Fig. 6 Possible solutions to the layer equation. (a) Layer solutions for C > 0. (b) Layer solutions for C = 0

58

R. Borsche and A. Klar

RP2 uB < 0 ⇒ uK ∈ {uB } ∪ (−uB , ∞), since there is either no wave or a shock wave moving to the right. Thus, for a given uB we can select uK only from the above subsets. For a boundary on the right hand, we study the analogous cases. Finally, to find the macroscopic boundary conditions at a left boundary from the underlying kinetic problem, we combine the solution of the half space problem (30) on [0, ∞] with asymptotic solution uK and of a Riemann Problem with left state uK and right state uB . The details of the procedure are as follows. We consider a domain x ≥ 0 and determine macroscopic boundary conditions at x = 0 in the following way. For the kinetic problem we prescribe f2 (x = 0). Moreover, the actual macroscopic value at x = 0+ is denoted by uB . From these two values we have to determine a (potentially new) boundary value for the macroscopic solution uK and a value for f1 (0), the outgoing kinetic value. We consider different cases coupling stable or unstable layer solutions, denoted by (U ) or (S), and Riemann problem solutions RP 1 or RP 2. Case 1, RP1-U The flow is ingoing with uB > 0 and f2 (0) > 0. The layer solution is u(x) =

√ C >0.

Determine the value of C > 0 from √ C +v C uˆ + vu(0) = . f2 (0) = 2v 2v This equation has a positive solution C under the above condition on f2 (0). This gives 2  # 1 2 C= −v + v + 8vf2 (0) 4 √ and the new boundary condition uK = C given √ by the value of the layer solution C . This yields at ∞. The outgoing layer solution is f1 (0) = C+v 2v f1 (0) = and u(0) =

√ C − f2 (0)

√ C. See Fig. 7a.

Case 2, RP1-S The flow is transonic with uB > 0 and f2 (0) < 0. In this case C = 0 and u(0) is given by f2 (0) = C+vu(0) or 2v u(0) = 2f2 (0) < 0 .

Asymptotic Methods for Kinetic and Hyperbolic Evolution Equations on Networks

u

u u(0)

u(x)

uK

u

uB

u=0

59

u uB uK

u =0

u (x ) u (0)

x=∞

x=0

x =∞

x =0 (b)

(a)

Fig. 7 Boundary layer and Riemann problem solution for positive uB . (a) uB > 0 and u(0) > 0. (b) uB > 0 and u(0) < 0 a The layer solution is then u(x) = − x−a/u(0) . We do not need the exact form of the solution, but only the fact that for all u(0) < 0 the asymptotic value is 0. The new boundary condition is uK = 0 and

f1 (0) = f2 (0) . In this case u(0) and uB cannot be connected by an outgoing Burgers wave. The kinetic layer solution takes care for a part (from u(0) to 0) of the full jump from u(0) to uB , see Fig. 7b. Case 3, RP2-S The flow is outgoing with uB < 0 and f2 (0) ≤ case the value of the layer solution at infinity is given by uB . Thus,

u2B −vuB . 2v

In this

√ u(∞) = − C = uB . Therefore C = u2B . Determine u(0) from f2 (0) = the above assumption on f2 (0) we have u(0) = −

C+vu(0) 2v

=

u2B +u(0) , 2v

i.e. under

√ u2B + 2f2 (0) ≤ −uB = C . v

Then, the layer solution is given √ by the formulas in the last subsection and converges to the stable fixpoint − C. Moreover, f1 (0) = u(0) − f2 (0) = −

u2B + f2 (0) . v

In this case u(0) and uB cannot be connected by an outgoing Burgers wave. The kinetic layer solution handles the full jump, see Fig. 8a.

60

R. Borsche and A. Klar

u

u

u

u(0) u=0

u K uB

u (0)

u (x )

u =0

u uK uB

u(x) x=∞

x=0

x =∞

x =0 (b)

(a)

Fig. 8 Boundary layer and Riemann problem solution for negative uB . (a) uB < 0 and u(0) ≤ −uB . (b) uB < 0 and u(0) ≥ −uB

Case 4, RP2-U The flow is ingoing with uB < 0 and f2 (0) ≥ first case the layer solution is given by u(x) =



u2B −vuB . 2v

As in the

C.

The value of C > 0 is determined from f2 (0) =

√ C +v C 2v

or  2 # 1 2 C= −v + v + 8vf2 (0) . 4 √ The boundary condition uK = C and the outgoing layer solution are given as in case 1. In this case the full jump can be covered by a Riemann problem solution, see Fig. 8b. Remark 4 Thus, we obtain conditions similar to the ones obtained in [3]. For a proof of convergence of the kinetic BVP to the macroscopic one, we refer [47], where the transonic case has been excluded.

3.3 Coupling Conditions We consider the kinetic and Burgers problems on a network. In the present case the orientation of the edges is important. In the kinetic case we have to prescribe at the node for each ingoing edge the quantity f1 and for the outgoing edges f2 . The macroscopic coupling conditions are then derived from the kinetic conditions using the above procedures.

Asymptotic Methods for Kinetic and Hyperbolic Evolution Equations on Networks Fig. 9 1–2 node

i

61

j k

We concentrate on the case of a node with 3 edges and 1 ingoing and 2 outgoing edges. For other cases we refer to [13]. Arc i is oriented into the node and arc j and k are oriented out of the node (Fig. 9). We choose the symmetric kinetic coupling conditions similar to [11] j

f1i = 12 (f1 + f1k )

(31)

j

f2 = 12 (f2i + f1k )

(32)

j

f2k = 12 (f2i + f1 ) .

(33)

In these coupling conditions we assume that the value of v is identical on all edges. If the velocities are not symmetric the fluxes have to be weighted such that the conservation of mass is maintained. We reformulate the above conditions in terms of u and uˆ and obtain  1 j vu − uˆ j + vuk − uˆ k 2   1 uˆ j + vuj = uˆ i + vui + vuk − uˆ k 2   1 uˆ k + vuk = uˆ i + vui + vuj − uˆ j . 2 vui − uˆ i =

(34) (35) (36)

Remark 5 In general linear mass conserving kinetic coupling conditions in the 1-2 case have the form j

f1i = αf1 + (1 − β)f1k j

f2 = (1 − γ )f2i + βf1k j

f2k = γf2i + (1 − α)f1 with 3 free parameters α, β, γ .

The method to find the coupling conditions for the scalar limit equation proceeds along the following lines. Step 1: Combine Kinetic Layer and Coupling Conditions First the combination of the coupling conditions with the layer equations is considered. Each layer can have either a stable solution (S) or an unstable solution (U). Thus, for three edges

62

R. Borsche and A. Klar

we have eight possible combinations of stable and unstable layer solutions, which are combined via the coupling conditions. Step 2: Combine Kinetic Layer and Riemann Problems Assuming the states j j uiB , uB , ukB to be given, we have to determine the new states uiK , uK and ukK at the node. We have to consider eight different configurations of 1- and 2-Riemann problems. For each of them all possible combinations with stable or unstable layer solutions have to be discussed. Going through all possible cases one obtains the following cases for the macroscopic coupling conditions, we refer for details to [13]. j

Case 1, RP1-1-1 uiB < 0, uB > 0, ukB > 0. Then Ci = Cj = Ck = 0 and j

uiK = 0

uK = 0

ukK = 0 .

j

Case 2, RP1-1-2 uiB < 0, uB > 0, ukB < 0. Then Ci = Ck , Cj = 0 and j

uiK = ukB < 0

uK = 0

ukK = ukB < 0 .

j

Case 3, RP1-2-1 uiB < 0, uB < 0, ukB > 0. Then Ci = Cj , Ck = 0 and j

j

uiK = uB < 0

j

uK = uB < 0 j

ukK = 0 .

Case 4, RP2-1-1 uiB > 0, uB > 0, ukB > 0. Then Cj = 1 j uK = √ uiB > 0 2

uiK = uiB > 0

Ci 2 , Ck

=

Ci 2

and

1 ukK = √ uiB > 0 . 2

j

Case 5, RP1-2-2 uiB < 0, uB < 0, ukB < 0. Then Ci = Cj + Ck and uiK

#

= − (u2B )2 + (ukB )2 < 0

j

j

uK = uB < 0

ukK = ukB < 0 .

j

Case 6, RP2-1-2 uiB > 0, uB > 0, ukB < 0. Then we have 3 subcases. √ If uiB ≥ − 2ukB , then Cj = C2i , Ck = C2i and 1 j uK = √ uiB > 0 2

uiK = uiB > 0

1 ukK = √ uiB > 0 . 2

√ If − 2ukB > uiB ≥ −ukB , then Cj = Ci − Ck and uiK = uiB > 0

j

uK =

#

(uiB )2 − (ukB )2 > 0

ukK = ukB < 0 .

Asymptotic Methods for Kinetic and Hyperbolic Evolution Equations on Networks

63

If uiB < −ukB , then Ci = Ck , Cj = 0 and j

uiK = ukB < 0

uK = 0

ukK = ukB < 0 .

j

Case 7, RP2-2-1 uiB > 0, uB < 0, ukB > 0. We have again 3 subcases. √ j If uiB ≥ − 2uB , then Cj = C2i , Ck = C2i and 1 j uK = √ uiB > 0 2

uiK = uiB > 0

1 ukK = √ uiB > 0 . 2

√ j j If − 2uB > uiB ≥ −uB then Ck = Ci − Cj and uiK = uiB > 0

j

#

j

uK = uB < 0

ukK =

j

(uiB )2 − (uB )2 > 0 .

j

If uiB < −uB , then Ci = Cj , Ck = 0 and j

j

uiK = uB < 0

j

uK = uB < 0

ukK = 0 .

j

Case 8, RP2-2-2 uiB > 0, uB < 0, ukB < 0. We have 4 subcases. √ j √ If uiB ≥ − 2uB , uiB ≥ − 2ukB then Cj = C2i , Ck = C2i and 1 j uK = √ uiB > 0 2

uiK = uiB > 0

1 ukK = √ uiB > 0 . 2

# √ j j If − 2uB > uiB ≥ (uB )2 + (ukB )2 , then Ck = Ci − Cj and uiK = uiB > 0

j

#

j

uK = uB < 0

ukK =

j

(uiB )2 − (uB )2 > 0 .

# √ j If − 2ukB > uiB ≥ (uB )2 + (ukB )2 , then Cj = Ci − Ck and uiK = uiB > 0 If uiB ≤

j

uK =

#

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# j (uB )2 + (ukB )2 , then Ci = Cj + Ck and

# j uiK = − (uB )2 + (ukB )2 < 0

j

j

uK = uB < 0

ukK = ukB > 0 . j

This yields the desired expressions for the new states uiK , uK and ukK at the node j for given states uiB , uB , ukB .

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3.4 Numerical Results In this section the derived coupling conditions are investigated numerically. The numerical solutions of the macroscopic equation (25) with F (u) = u2 are compared to those obtained for the kinetic model (24). As numerical scheme for the kinetic equations the Upwind method for the linear advective part is combined with an implicit Euler scheme for the source term. The solution of the macroscopic equation is approximated with a Godunov scheme. For all computations 1000 cells are used per edge, i.e. a gridsize of Δx = 10−3 , as spacial resolution. The time steps Δt are chosen according to the respective CFL conditions, Δt = Δx/v for the kinetic problem and Δt = Δx/(2|umax |) for the macroscopic equation. The simulations are computed up to time T = 0.5 and the relaxation parameter is chosen as  = 0.0005. The initial conditions are formulated in macroscopic states, the remaining values in the kinetic model are chosen according to the relaxed state, i.e. uˆ = F (u). The kinetic speeds are chosen in agreement with the subcharacteristic condition as v = 2. Larger values of v would lead to similar results, as the coupling conditions are independent of v. Here we consider a 1-2-junction. For a 2-1-junction we refer to [13]. The coupling conditions are tested with six different Riemann problems at the junction. The first edge is connected to the junction at x = 1, while the other two edges are connected to the node at x = 0. Thus in the following figures waves move to the left in the first edge but to the right in edge 2 and 3. In Fig. 10 the solutions to the initial conditions (u10 , u20 , u30 )(x) = (−1, 0.75, 0.5) and (u10 , u20 , u30 )(x) = (−1, 0.75, −0.5) are shown. These correspond to the cases RP1-1-1 and RP1-1-2 respectively. In RP1-1-1 the initial states only have Burgers

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characteristic speeds away from the junction and the coupling conditions enforce zero states in all edges. This leads to three rarefaction waves. On the right hand side RP1-1-2 is considered. Flow is entering from edge 3 but only exiting in edge 1. In edge 1 a rarefaction wave forms and moves to the left. On the slower end of the rarefaction wave a bump in the kinetic solution is present. As the initial states at t = 0 do not satisfy the coupling conditions, this small disturbance arises due to the transition in the new state at the junction. For smaller values of  and when refining the spacial and the temporal grid, this disturbance becomes narrower and more peaked. Such temporal layers due to the initial conditions will also occur in other test cases. On edge 2 a boundary layer connects the junction state and a rarefaction wave. The ingoing flow from edge 3 leads to a small boundary layer. In Fig. 11 on the left, the flow from the first edge is split to the outgoing edges. On the right hand side the flow from edge 2 and 3 is merged into edge 1. Two layers connect the backward going flows with the junction states. In the first edge a small rarefaction wave travels to the left, followed by a small temporal layer. This corresponds to the case1-2-2. Figure 12 shows the last two test-cases for this junction. On the left in the case RP2-1-2 the flow enters from the first and the third edge and exits into the second one, where a rarefaction wave moves to the right. The test on the right considers the case RP2-2-2. In all tests of the 1-2 junction the kinetic and the macroscopic solutions are very close. Especially the states at the junction are correctly represented by the derived coupling conditions. Since the value of  is small, also the boundary layers in the kinetic solution have a small spacial width. Burgers

kinetic 0

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4 Outlook Coupling conditions for macroscopic equations on networks have been derived from underlying kinetic models via a kinetic layer analysis at the nodes of the network. We have treated simple model equations like linear BGK-type kinetic equations and nonlinear relaxation approximations for scalar limit problems. The approach presented here will be extended to more general situations in future work. In particular, an investigation of linear models with more collision invariants leading, in the limit, for example, to the full linearized Euler model is under way. Moreover, discrete velocity relaxation systems leading in the limit to nonlinear macroscopic systems of equations similar to the isothermal Euler equations are currently investigated.

References 1. Andreianov, B., Sbihi, K.: Well-posedness of general boundary-value problems for scalar conservation laws. Trans. Am. Math. Soc. 367(6), 3763–3806 (2015) 2. Andreianov, B.P., Coclite, G.M., Donadello, C.: Well-posedness for vanishing viscosity solutions of scalar conservation laws on a network. Discrete Contin. Dyn. Syst. 37(11), 5913– 5942 (2017) 3. Aregba-Driollet, D., Milisic, V.: Kinetic approximation of a boundary value problem for conservation laws. Numer. Math. 97, 595–633 (2004) 4. Banda, M., Herty, M., Klar, A.: Coupling conditions for gas networks governed by the isothermal Euler equations. NHM 1(2), 295–314 (2006) 5. Banda, M., Herty, M., Klar, A. : Gas flow in pipeline networks. NHM 1(1), 41–56 (2006)

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6. Bardos, C., le Roux, A.Y., Nédélec, J.-C.: First order quasilinear equations with boundary conditions. Commun. Partial Differ. Equ. 4(9), 1017–1034 (1979) 7. Bardos, C., Santos, R., Sentis, R.: Diffusion approximation and computation of the critical size. Trans. Am. Math. Soc. 284(2), 617–649 (1984) 8. Bensoussan, A., Lions, J.L., Papanicolaou, G.C.: Boundary-layers and homogenization of transport processes. J. Publ. RIMS Kyoto Univ. 15, 53–157 (1979) 9. Borsche, R., Colombo, R., Garavello, M.: On the coupling of systems of hyperbolic conservation laws with ordinary differential equations. Nonlinearity 23, 112749 (2010) 10. Borsche, R., Göttlich, S., Klar, A., Schillen, P.: The scalar Keller-Segel model on networks. Math. Models Methods Appl. Sci. 24(2), 221–247 (2014) 11. Borsche, R., Kall, J., Klar, A., Pham, T.N.H.: Kinetic and related macroscopic models for chemotaxis on networks. Math. Models Methods Appl. Sci. 26(6), 1219–1242 (2016) 12. Borsche, R., Klar, A.: Kinetic layers and coupling conditions for macroscopic equations on networks. SIAM Sci. Comput. 40(3), 1784–1808 (2018) 13. Borsche, R., Klar, A.: Kinetic layers and coupling conditions for nonlinear scalar equations on networks. Nonlinearity 31, 351 (2018) 14. Bretti, G., Natalini, R., Ribot, M.: A hyperbolic model of chemotaxis on a network: a numerical study. ESAIM: M2AN 48(1), 231–258 (2014) 15. Bürger, R., Frid, H., Karlsen, K.H.: On the well-posedness of entropy solutions to conservation laws with a zero-flux boundary condition. J. Math. Anal. Appl. 326(1), 108–120 (2007) 16. Camilli, F., Corrias, L.: Parabolic models for chemotaxis on weighted networks. Math. Pures Appl. 108(4), 459–480 (2017) 17. Cercignani, C.: A variational principle for boundary value problems. J. Stat. Phys. 1(2), 297– 311 (1969) 18. Coclite, G.M., Garavello, M., Piccoli, B.: Traffic flow on a road network. SIAM J. Math. Anal. 36, 1862–1886 (2005) 19. Colombo, R., Herty, M., Sachers, V.: On 2 × 2 conservation laws at a junction. SIAM J. Math. Anal. 40(2), 605–622 (2008) 20. Colombo, R.M., Garavello, M.: On the Cauchy problem for the p-system at a junction. SIAM J. Math. Anal. 39, 1456–1471 (2008) 21. Colombo, R.M., Mauri, C.: Euler system for compressible fluids at a junction. J. Hyperbolic Differ. Equ. 5(3), 547–568 (2008) 22. Corli, A., di Ruvo, L., Malaguti, L., Rosini, M.D.: Traveling waves for degenerate diffusive equations on networks. NHM 12(3), 339–370 (2017) 23. Coron, F.: Computation of the asymptotic states for linear halfspace problems. TTSP 19(2), 89 (1990) 24. Coron, F., Golse, F., Sulem, C.: A classification of well-posed kinetic layer problems. CPAM 41, 409 (1988) 25. Dager, R., Zuazua, E.: Controllability of tree-shaped networks of vibrating strings. C. R. Acad. Sci. Paris 332, 1087–1092 (2001) 26. Egger, H. Kugler, T.: Damped wave systems on networks: exponential stability and uniform approximations (2016). https://arxiv.org/abs/1605.03066 27. Fermo, L., Tosin, A.: A fully-discrete-state kinetic theory approach to traffic flow on road networks. Math. Models Methods Appl. Sci. 25(3), 423–461 (2015) 28. Garavello, M.: A review of conservation laws on networks. NHM 5(3), 565–581 (2010) 29. Garavello, M., Piccoli, B.: Traffic Flow on Networks. AIMS (2006) 30. Golse, F.: Analysis of the boundary layer equation in the kinetic theory of gases. Bull. Inst. Math. Acad. Sin. 3(1), 211–242 (2008) 31. Golse, F., Klar, A.: Numerical method for computing asymptotic states and outgoing distributions for a kinetic linear half space problem. J. Stat. Phys. 80(5–6), 1033–1061 (1995) 32. Gugat, M., Herty, M., Klar, A., Leugering, G., Schleper, V.: Well–posedness of networked hyperbolic systems of balance laws. In: International Series of Numerical Mathematics, vol. 160, 175–198. Springer, Berlin (2011)

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33. Herty, M., Klar, A., Piccoli, B.: Existence of solutions for supply chain models based on partial differential equations. SIAM J. Math. Anal. 39(1), 160–173 (2007) 34. Herty, M., Moutari, S.: A macro-kinetic hybrid model for traffic flow on road networks. Comput. Methods Appl. Math. 9(3), 238–252 (2009) 35. Leugering, G., Schmidt, E.J.P.G.: On the modelling and stabilization of flows in networks of open canals. SIAM J. Control Optim. 41(1), 164–180 (2002) 36. Li, Q., Lu, J., Sun, W.: Half-space kinetic equations with general boundary conditions. Math. Comput. 86, 1269–1301 (2017) 37. Li, Q., Lu, J., Sun, W.: A convergent method for linear half-space kinetic equations. ESAIM: M2AN 51(5), 1583–1615 (2017) 38. Liu, T.P.: Hyperbolic conservation laws with relaxation. Commun. Math. Phys. 108, 153–175 (1987) 39. Liu, J.-G., Xin, Z.: Boundary-layer behavior in the fluid-dynamic limit for a nonlinear model Boltzmann equation. Arch. Ration. Mech. Anal. 135, 61–105 (1996) 40. Loyalka, S.K., Ferziger, J.H.: Model dependence of the slip coefficient. Phys. Fluids 108, 1833 (1967) 41. Loyalka, S.K.: Approximate method in the kinetic theory. Phys. Fluids 11(14), 2291 (1971) 42. Otto, F.: Initial-boundary value problem for a scalar conservation law. C. R. Acad. Sci. Paris Sér. I Math. 322(8), 729–734 (1996) 43. Siewert, C.E., Thomas, J.R.: Strong evaporation into a half space I. Z. Angew. Math. Phys. 32, 421 (1981) 44. Toro, E.F.: Riemann Solvers and Numerical Methods for Fluid Dynamics. Springer, Berlin (2009) 45. Ukai, S., Yang, T., Yu, S.-H.: Nonlinear boundary layers of the Boltzmann equation. I. Existence. Commun. Math. Phys. 236(3), 373–393 (2003) 46. Valein, J., Zuazua, E.: Stabilization of the wave equation on 1-d networks. SIAM J. Control Optim. 48(4), 2771–2797 (2009) 47. Wang, W.-C., Xin, Z.: Asymptotic limit of initial boundary value problems for conservation laws with relaxational extensions. Commun. Pure Appl. Math. 51(5), 505–535 (1998) 48. Xu, W.-Q.: Boundary conditions and boundary layers for a multi-dimensional relaxation model. J. Differ. Equ. 197(1), 85–117 (2004) 49. Yong, W.-A.: Boundary conditions for hyperbolic systems with stiff relaxation. Indiana Univ. Math. J. 48(1), 115–137 (1999)

Coagulation Equations for Aerosol Dynamics Marina A. Ferreira

Abstract Binary coagulation is an important process in aerosol dynamics by which two particles merge to form a larger one. The distribution of particle sizes over time may be described by the so-called Smoluchowski’s coagulation equation. This integrodifferential equation exhibits complex non-local behaviour that strongly depends on the coagulation rate considered. We first discuss well-posedness results for the Smoluchowski’s equation for a large class of coagulation kernels as well as the existence and nonexistence of stationary solutions in the presence of a source of small particles. The existence result uses Schauder fixed point theorem, and the nonexistence result relies on a flux formulation of the problem and on power law estimates for the decay of stationary solutions with a constant flux. We then consider a more general setting. We consider that particles may be constituted by different chemicals, which leads to multi-component equations describing the distribution of particle compositions. We obtain explicit solutions in the simplest case where the coagulation kernel is constant by using a generating function. Using an approximation of the solution we observe that the mass localizes along a straight line in the size space for large times and large sizes. Keywords Smoluchowski’s equation · Well-posedness · Stationary solutions · Non-equilibrium · Source term · Multi-component coagulation · Asymptotic localization

M. A. Ferreira () Department of Mathematics and Statistics, Gustaf Hällströmin katu 2b, Helsingin yliopisto, Finland e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 G. Albi et al. (eds.), Trails in Kinetic Theory, SEMA SIMAI Springer Series 25, https://doi.org/10.1007/978-3-030-67104-4_3

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1 Introduction We consider particle systems where moving particles undergo binary coagulation, forming larger particles. This simple system can be used to study the dynamics of aerosols in the atmosphere [19] as well as raindrop formation, smoke, sprays and galaxies [1, 9]. When the number of particles is very large it becomes more relevant to study the collective behaviour of the particles rather than individual particle behaviour. This motivates a statistical description of the system. In 1916 Smoluchowski [32] proposed an equation to describe the particle size distribution over time, assuming that the system is homogeneous in space. Let f (x, t) be the number density of particles of size x > 0 at time t ≥ 0. The Smoluchowski’s coagulation equation, or simply coagulation equation, is the following mean-field equation for the evolution of f ∂t f (x, t) =

1 2



x





K(x − y, y)f (x − y, t)f (y, t)dy −

0

K(x, y)f (x, t)f (y, t)dy 0

(1.1) where K(x, y) is the coagulation rate between particles of size x and y. The first term on the right hand-side is the gain term due to the coagulation between particles of size x − y and particles of size y to create a particle of size x. The second term is the loss term which describes the loss of particles of size x by merging with any other particle in the system. Equation (1.1) is an integrodifferential equation belonging to the class of kinetic equations. We also consider more general systems where a constant input of particles may be present. The number density in this case satisfies the coagulation equation with an extra source term η ≥ 0, 1 ∂t f (x, t) = 2



x

K(x − y, y)f (x − y, t)f (y, t)dy

0 ∞



K(x, y)f (x, t)f (y, t)dy + η(x).

(1.2)

0

Complementary research lines have expanded over the last decades on experimental, numerical and theoretical aspects of Eqs. (1.1) and (1.2). Algorithms to simulate these equations have been developed to test hypotheses drawn from atmospheric data [25, 34] (see [26] for a survey on numerical methods). On the other hand, theoretical results have clarified issues mainly related to Eq. (1.1), such as existence and uniqueness of solutions for general classes of kernels [18, 30] or the behaviour of solutions for explicitly solvable kernels [28] and general kernels [4, 5, 12, 17]. A particle may be characterized not only by its size but also by its composition, leading to multi-component coagulation equations where the size is described by

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a vector x ∈ Rd+ \{0} representing the size of each of the chemical components of a particle. An application of multi-component equations to aerosol dynamics is described in Sect. 2.2. In this paper we review analytic results related to the one-component equations (1.1) and (1.2) as well as to the corresponding discrete multi-component equations with x, y ∈ Nd \{0}. We start in Sect. 2 with a short overview on various topics related to properties of the solutions, applications and derivation from particle systems. We also introduce some notation that is used throughout the chapter. In Sect. 3 we study the one-component equations (1.1) and (1.2). Section 3.2 contains the main steps of the proof of one of the first well-posedness results for Eq. (1.1) with unbounded coagulation kernels obtained in 1999 by Norris [30]. Section 3.3 contains a review of the proofs of existence and non-existence of stationary solutions to coagulation equations to (1.2), obtained recently in [13]. In Sect. 4 we consider the discrete multi-component equation with constant kernel. Following the computations presented in [21], we compute in Sect. 4.1 explicit time-dependent solutions and in Sect. 4.2 we compute stationary solutions when an additional source at the monomers is present. We also obtain approximations of both solutions showing explicitly that mass localizes along a straight line in the multi-dimensional size space for large times and large sizes. Finally in Sect. 5 we mention some recent results in the literature and open questions.

2 Preliminaries 2.1 Conservation of Mass and Continuity Equation By multiplying (1.1) by x and integrating in x from 0 to ∞ one obtains formally an "∞ d equation for the mass M1 (t) = 0 xf (x, t)dx given by dt M1 (t) = 0. This shows that the mass is conserved, provided the integrals are well-defined. Associated to the mass-conservation, one may write a continuity equation that shows that mass is transported continuously along the size space: ∂t (xf (x, t)) = ∂x J (x, t)

(2.1)

where the flux of mass from small to large clusters is given by

x

J (x, t) =





yf (y, t)f (z, t)K(y, z)dzdy. 0

(2.2)

x−y

As we will see in Sect. 3.3.1 a (non-equilibrium) stationary solution has a constant flux of mass at large sizes, i.e., J (x) is constant for all x > L, for some positive L. Moreover this flux plays an important role in the proof of non-existence of stationary solutions in Sect. 3.3.2.

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Interestingly, if the coagulation rate behaves like a power law and if the power is sufficiently large, then mass-conservation is lost. Such phenomenon is called gelation and it corresponds to the formation of infinitely large clusters that are not seen any more by the equation. Therefore these big clusters leave the system and the total mass decreases. Gelation may be interpreted as a change in state from gas to gel. Mathematically, this phenomenon poses interesting challenges [11]. Since gelation has not been observed in atmospheric aerosols we do not discuss it further here. We note that, contrarily to the Boltzmann equation, the coagulation equation does not preserve number of particles, due to the sticky collisions.

2.2 Coagulation Kernels for Aerosols in the Atmosphere Atmospheric aerosols are suspensions of small particles in the air, whose diameter ranges approximately between 1 nanometre, in the case of molecular particles, to 100 micrometres, in the case of cloud droplets and dust particles [19]. Aerosols influence sunlight scattering by reflecting and absorbing radiation, and they constitute the seeds that originate the clouds. Therefore, they play an important role in weather and climate forecast [6]. Aerosols are subject to complex processes that influence their size distribution over time. One important process is the coagulation of particles to produce larger ones. Other processes include the formation of new small particles, or monomers, due to certain physical and chemical processes, the removal of particles due to gravity or diffusion, and the growth/shrinkage due to condensation/evaporation [25]. Atmospheric aerosols are typically constituted by different chemicals, leading to multi-component systems, which may alter the rate of the processes mentioned before and consequently, the particle size distribution [34]. We consider the regime in which the particles are uniformly distributed in space. Moreover, we assume that removal and growth of particles due to condensation is not important, which in practice may correspond roughly to sizes between 10 nanometers and 10 micrometers [19]. We are then led to the study of multicomponent systems where particles undergo binary coagulation in the presence of a source of small particles. Coagulation kernels K have been derived for atmospheric aerosols using kinetic theory under several assumptions on the shape and motion of particles [19]. Aerosol particles are commonly assumed to be spherical and to undergo elastic collisions with background air particles. The number of such collision events is assumed to be much larger than the number of collisions between two coalescing particles. This drives the system towards an equilibrium where the particle velocities follow a Maxwell-Boltzmann distribution. Moreover, any collision between coalescing particles yields a coalescing particle. Two different coagulation kernels have been derived under the previous conditions for two different regimes. Each regime is defined based on the relation between particle size and the average distance travelled by a particle between two collisions

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in air, called mean free path. Under normal pressure and temperature conditions, the mean free path in air, , is of the order of 10 nanometres. If the size (diameter) of a spherical particle, x, is much smaller than the mean free path x  , the particle is more likely to travel in straight lines before meeting another coalescing particle. In this case the rate of coagulation has been estimated by the free molecular coagulation kernel: K(x, y) = (x 1/3 + y 1/3)2 (x −1 + y −1 )1/2.

(2.3)

Otherwise, if the size of a particle is much larger than the mean free path, x  , the coalescing particle will meet many background air particles before meeting another coalescing particle. In this case, the air behaves like a fluid and the coalescing particle is more likely to diffuse. The coagulation rate has been estimated by the diffusive coagulation kernel: K(x, y) = (x −1/3 + y −1/3 )(x 1/3 + y 1/3).

(2.4)

This kernel was first derived in the original work by Smoluchowski [32]. Other kernels have been derived under different assumptions on the underlying background gas and particles, such as particles moving in a laminar shear or turbulent flow [19], and particles having electric charges [33, 34]. The behaviour and even the existence of solutions to Eq. (1.2) strongly depends on the coagulation kernel. In Sects. 3.3.1 and 3.3.2 we review the existence of stationary solutions for a large class of kernels which includes in particular the free molecular (2.3) and the diffusive kernels (2.4).

2.3 From Particle Models to Smoluchowski’s Coagulation Equation The Smoluchowski’s coagulation equation has been rigorously derived using different approaches that consider different types of particle systems. In one approach, a purely stochastic particle system is considered, where pairs of particles are randomly picked to originate a new particle. The associated stochastic process is usually called Marcus-Lushnikov process. A different approach considers deterministic particle systems, where particles move and when they collide they merge with a certain probability. The first approach is inspired in Kac-models for the derivation of the Boltzmann equation [16]. A common strategy is to start from an infinite stochastic particle system where particles of size x and y coalesce at a rate K(x, y) and to prove that the number density, after being conveniently rescaled, converges, as the unit volume tends to infinity, to a measure that solves the Smoluchowski’s coagulation equation with kernel K. This has been obtained for the additive kernel, product kernel as well as for a class of sub-multiplicative kernels using combinatorial techniques and random graphs. See [2] (Chapter 5.2) for an accessible exposition and [1] for a review on existing results and open problems.

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In the second approach, there are fewer rigorous results. The first result to the best of our knowledge is due to Lang and Xanh [22]. They consider Brownian particles moving in the three-dimensional Euclidean space according to Brownian motion with a diffusion coefficient D. The particles are assumed to move independently on each other provided they are at a distance greater than the sum of their radius 2R. Once they come closer than 2R they coalesce with probability 1/2, forming one Brownian particle with the same radius R and the same diffusion coefficient D. In the limit when the number of particles N goes to infinity and the radius R goes to zero, such that RN remains constant, the authors prove propagation of chaos and that the density function converges in probability to the solution to the Smoluchowski’s coagulation equation with constant coagulation kernel. The limit where RN remains constant is the so-called Boltzmann-Grad limit and is the limit of constant mean free time. A more general case of coalescing Brownian particles with diffusion coefficients changing after coalescence, but not the size R, has been treated in [20]. More recently, the change in size after coalescence has been considered in [31] (see also [29]) in the case of a tracer particle moving in a straight line and coalescing with randomly distributed fixed particles of different sizes. In this case, a linear coagulation equation with a simple shear kernel was derived in the kinetic limit where the volume fraction filled by the background of particles tends to zero.

2.4 Notation We use the notation R∗ := (0,∞), R+ := [0, ∞) and N+ = {0, 1, . . .}. We denote the 1 − norm in Rd by |α| = i=1,...,d |αi |, α ∈ Rd . We denote by M (I ) the space of signed Radon measures supported on I ⊂ R+ , i.e., the non-negative measures having finite total variation on any compact subset of I , and by · the total variation norm. We denote by M+ (I ) the space of measures on M (I ) that are nonnegative. The measures from M+ (I ) that are also bounded are denoted by M+,b (I ) := {μ ∈ M+ (I ) | μ(I ) < ∞}. The space M+,b (I ) equipped with the norm  ·  is a Banach space. The notation ft (x) will sometimes be used to denote f (t, x). We denote by Cc (I ) or Cb (I ) the spaces of continuous functions on I that are compactly supported or bounded, respectively. For simplicity, we use a generic constant C > 0 which may change from line to line.

3 One-Component Equation 3.1 Main Results We consider kernels K : (0, ∞)2 → [0, ∞) satisfying K(x, y) = K(y, x), K(x, y) ≥ 0,

(3.1)

K(x, y) ≥ c1 [x λ+γ y −λ + y λ+γ x −λ ],

(3.2)

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K(x, y) ≤ c2 [x λ+γ y −λ + y λ+γ x −λ ],

(3.3)

0 < c1 ≤ c2 < ∞,

(3.4)

and λ, γ ∈ R,

(3.5)

for some given constants c1 , c2 , λ and γ and for all (x, y) ∈ (0, ∞)2 . This class includes in particular the physical kernels (2.3) and (2.4). The parameter γ represents the homogeneity of the kernel, while λ represents the “off-diagonal” rate. The parameter γ yields the behaviour under the scaling of the particle size, while λ measures the importance of collisions between particles of different sizes. Note that the bounds in (3.2) and (3.3) are homogeneous, i.e., they satisfy for any k > 0, h(kx, ky) = k γ h(x, y), but the kernels are not necessarily homogeneous. We assume the following condition on the source η ∈ M+ (R∗ ) suppη ∈ [1, L], for some L > 1.

(3.6)

Note that then the source is bounded, i.e., η(R∗ ) < ∞. We consider the following definition of time-dependent solution to the Smoluchowski’s coagulation equation (1.1) [30]. Definition 3.1 Assume that K is a measurable function satisfying (3.1) and (3.3). We will say that the map t → ft ; [0, T ) → M+ (R∗ ), where T ∈ (0, ∞] is a local solution to (1.1) if it satisfies 1. for all compact sets B ⊂ R∗ , the map t → ft (B); [0, T ) → [0, ∞) is measurable 2. for all t < T and all compact sets B ⊂ R∗ t 0

B×R∗

K(x, y)fs (dx)fs (dy)ds < ∞,

3. for all bounded measurable functions ϕ of compact support and t < T it holds

t

ϕ, ft  = ϕ, f0  +

ϕ, L(fs )ds

(3.7)

0

where L(f ) is defined by ϕ, L(f ) =

1 K(x, y)[ϕ(x + y) − ϕ(x) − ϕ(y)]f (dx)f (dy), 2 R∗ R∗

" 4. R∗ x1x≤1f0 (dx) < ∞ and (3.7) holds with ϕ(x) = x1x≤1. If T = ∞ we call time-dependent solution to (1.1). One can easily check that condition 2 is the minimal one to have well-defined integrals. Condition 3 is the weak formulation commonly used in the literature and

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it is obtained by formally multiplying (1.1) by a test function and integrating in x. Condition 4 is a boundary condition imposing that no mass enters at 0. The existence and uniqueness of a time-dependent solution to (1.1) for sublinear kernels is established in the next Theorem [30]. Similar results have also been proven in [30] for kernels satisfying (3.2)–(3.3) with γ + λ = −λ and λ > −1/2. Theorem 3.2 Let K be a measurable function satisfying (3.1) and (3.3) with λ = 0 and γ < 1. If x 2 , f0  < ∞, then there exists a unique time-dependent solution (ft )t >0 to (1.1) in the sense of Definition 3.1. We consider now a source η = 0 of small particles entering into the system at a constant rate as described by Eq. (1.2). We study the existence of stationary injection solutions, i.e., solutions that satisfy f (t, x) = f (0, x) for all t > 0, as defined next [13]. Definition 3.3 Assume that K : R2∗ → R+ is a continuous function satisfying (3.1) and the upper bound (3.3). Assume further that η ∈ M+ (R∗ ) satisfies (3.6). We will say that f ∈ M+ (R∗ ) , satisfying f ((0, 1)) = 0 and R∗

x γ +λ f (dx) +

R∗

x −λ f (dx) < ∞ ,

(3.8)

is a stationary injection solution of (1.2) if the following identity holds for any test function ϕ ∈ Cc (R∗ ): 1 2

R∗

R∗

K (x, y) [ϕ (x + y) − ϕ (x) − ϕ (y)] f (dx) f (dy) +

R∗

ϕ (x) η (dx) = 0 .

(3.9) Condition (3.8) is the minimal one for the integrals in (3.9) to be well-defined. Stationary injection solutions have a constant in time flux of mass from small to large sizes, due to the source, therefore they are non-equilibrium solutions. Note that to be able to be stationary, the volume of particles entering the system has to balance the volume of particles leaving the system. Interestingly, there is an implicit removal of particles from the system at infinite sizes that allows the existence of these solutions. As we will see in the next two Theorems, for some class of coagulation rates, including the diffusive kernel (2.4), such balance exists, while for other class of kernels, including the free molecular kernel (2.3), such balance does not exist. Theorem 3.4 Assume that K satisfies (3.1)–(3.5) and |γ + 2λ| < 1. Let η = 0 satisfy (3.6). Then, there exists a stationary injection solution f ∈ M+ (R∗ ), f = 0, to (1.2) in the sense of Definition 3.3. Theorem 3.5 Suppose that K satisfies (3.1)–(3.5) as well as |γ + 2λ| ≥ 1. Let us assume also that η = 0 satisfies (3.6). Then, there is not any solution of (1.2) in the sense of Definition 3.3.

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Note that the diffusive kernel (2.4) satisfies the growth conditions (3.2)–(3.3) with γ = 0 and λ = 1/3, while the free molecular kernel (2.3) satisfies the growth conditions with γ = 1/6 and λ = 1/2. Therefore there exists a stationary solution for the diffusive but not for the free molecular kernel. The mass flux from small to large sizes associated to a stationary injection solutions is given in the next Lemma. Lemma 3.6 Suppose that the assumptions of Theorem 3.4 hold. Let f be a stationary injection solution in the sense of Definition 3.3. Then f satisfies for any R>0 J (R) = xη(dx) (3.10) (0,R]

where J (R) is the mass flux at size R and it is defined by



J (R) :=

K(x, y)xf (dx)f (dy) (0,R] (R−x,∞)

Remark 3.7 If R ≥ Lη , the right-hand side of (3.10) is constant equal to Jη = " xη(dx) > 0. Therefore, J (R) = Jη for R > Lη , i.e., the mass flux is constant [1,Lη ] in the regions that include large sizes. The main ideas to prove Lemma 3.6 are the following. For each ε > 0, we define the test function ϕ(x) = xχε (x) ∈ Cc (R∗ ) where χε ∈ Cc∞ (R∗ ) is such that 0 ≤ χε ≤ 1, χε (x) = 1, for 1 ≤ x ≤ R, and χε (x) = 0, for x ≥ R + ε. Using this test function in (3.9), the result can be obtained after letting ε → 0.

3.2 Well-Posedness for the Time-Dependent Problem We describe the main ideas of the proof of Theorem 3.2 obtained in [30] (Section 2). The first step is to prove well-posedness for a truncated problem. The second step is to obtain estimates that allow us to remove the truncation and to obtain wellposedness for the original problem. Let B ⊂ R∗ be a compact set. Note that all measures in M (B) are bounded. Note that from the hypotheses of Theorem 3.2 on the kernel we have that 0 ≤ K(x, y) ≤ w(x) + w(y),

with w(x) := x γ and γ < 1.

(3.11)

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M. A. Ferreira

The truncated operator LB : M (B) × R → M (B) × R is defined by (ϕ, a), LB (f, ξ ) := 1 {ϕ(x + y)1{x+y∈B} + aw(x + y)1{x+y ∈/ B} − ϕ(x) − ϕ(y)} 2 R∗ ×R∗ ×K(x, y)f (dx)f (dy) {aw(x) − ϕ(x)}w(x)f (dx) +ξ R∗

for all bounded measurable functions ϕ on R∗ and all a ∈ R, where (ϕ, a), (f, ξ ) denotes ϕ, f  + aξ . The truncated equation reads

t

(ϕ, a), (ft , ξt ) = (ϕ, a), (f0 , ξ0 ) +

(ϕ, a), LB (f, ξ )ds.

(3.12)

0

An interpretation of the dynamics associated with operator LB is the following (see [30] for more details). Particles of size x and y merge at a rate K(x, y) and they produce a new particle of size x + y. If the merging particle has size outside B, we add w(x + y). A solution to (3.12) is defined next. Definition 3.8 Let T ∈ (0, ∞). We will say that (ft , ξt )t ∈[0,T ] is a local solution to (3.12) if t → (ft , ξt ); [0, T ] → M (B) × R is a continuous map satisfying (3.12) for all t ∈ [0, T ]. Additionally, (ft )t ∈[0,T ] is called a solution to (3.12) when [0, T ] is replaced by [0, ∞). Proposition 3.9 Suppose that f0 ∈ M (B) with f0 ≥ 0 and ξ0 ∈ [0, ∞). Equation (3.12) has a unique solution (ft , ξt )t ≥0 starting from (f0 , ξ0 ). Moreover ft ≥ 0 and ξt ≥ 0 for all t ≥ 0. We will discuss the main ideas of the proof that is organized in three steps. The first step is to show that there is a constant T > 0 depending only on γ and B such that there exists a unique local solution (ft , ξt )t ∈[0,T ] to (3.12) starting from (f0 , ξ0 ). This is obtained by using an iterative scheme of continuous maps (ftn , ξtn ) : [0, ∞) → M (B) × R defined by (ft0 , ξt0 ) = (f0 , ξ0 ) (ftn , ξtn )



t

= (f0 , ξ0 ) + 0

LB (ftn−1 , ξtn−1 )

and proving that there exists a T > 0 such that (f n , ξ n ) converges in M (B) × R uniformly in t ≤ T to the desired local solution, which is also unique. The second step is to prove that ft ≥ 0, t ∈ [0, T ], which is obtained using again an iterative argument similar to the one used in the first step.

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Finally, the third step is to show that the solution exists for all times t ∈ [0, ∞). Choosing ϕ = w and a = 1 we obtain that d 1 (w, ft  + ξt ) = {w(x + y) − w(x) − w(y)}K(x, y)ft (dx)ft (dy) ≤ 0, dt 2 R∗ ×R∗ which implies that fT  + |ξT | ≤ w, fT  + ξT ≤ w, f0  + ξ0 . Using a scaling argument, we may assume without loss of generality that w, f0  + ξ0 ≤ 1, consequently fT  + |ξT | ≤ 1. We can start again from (fT , ξT ) at time T to extend the solution to [0, 2T ] and so on. Moreover, choosing ϕ = 0 and a = 1 in (3.12), we obtain 1 d ξt = {w(x + y)1B¯ (x + y)}K(x, y)f (dx)f (dy) + ξt w2 (x)f (dx), dt 2 R∗ ×R∗ R∗ which implies that ξt ≥ 0, for all t ≥ 0, due to ft ≥ 0, which ends the proof of the proposition. Proof of Theorem 3.2 Fix f0 ∈ M+ , such " that w, f0  < ∞. For each compact set B ⊂ R∗ define f0B = 1B f0 and ξ0B = B¯ w(x)f0 (dx). From Proposition 3.9 there is a unique solution (f0B , ξtB )t ≥0 to (3.12) starting from (f0B , ξ0B ). We now set ft = lim ftB and ξt = lim ξtB . Using (3.11), we obtain by dominated convergence,

B→R∗

B→R∗

1 d ϕ, ft  = {ϕ(x +y)−ϕ(x)−ϕ(y)}K(x, y)ft (dx)ft (dy)−ξt ϕw, ft , dt 2 R∗ ×R∗ for all bounded measurable functions ϕ. One can prove that for all t < T and for any local solution (gt )t 0, which allows to pass to the limit as B → R∗ in (3.12) and to deduce that ξt = 0, t > 0.

(3.14)

Then (3.13) and (3.14) imply that (ft )t ≥0 is a time-dependent solution to (1.1) and moreover, it is the only solution.  

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M. A. Ferreira

3.3 Stationary Solutions with Injection We now consider the Smoluchowski’s coagulation equation with source (1.2).

3.3.1 Existence of Stationary Solutions We present here the main ideas of the proof of the existence Theorem 3.4 obtained in [13]. The general strategy is similar to the strategy used in the proof of wellposedness presented in the previous section. First, we prove existence of a stationary solution for a truncated problem and second, we obtain estimates that allow to remove the truncation and hence the existence result for the original problem. Unfortunately the method used to prove existence does not give uniqueness, that problem needs a separate treatment (see [23] for a simple explanation of the available techniques). Let ε > 0 and R∗ ≥ Lη , where Lη is the upper bound of the support of the source η defined in (3.6). We will eventually make ε → 0 and R∗ → ∞. We consider kernels Kε,R∗ that are continuous, bounded and have compact support, such that Kε,R∗ (x, y) ≤ a2 (ε), Kε,R∗ (x, y) ∈ [a1 (ε), a2 (ε)], Kε,R∗ (x, y) = 0,

(x, y) ∈ R2+

(3.15)

(x, y) ∈ [1, 2R∗ ]

(3.16)

x ≥ 4R∗ or y ≥ 4R∗ ,

(3.17)

and lim Kε,R∗ (x, y) = Kε (x, y)

R∗ →∞

(3.18)

where Kε is continuous and satisfies Kε (x, y) ∈ [a1 (ε), a2 (ε)], for all (x, y) ∈ R2+ and lim Kε (x, y) = K(x, y).

ε→0

(3.19)

Additionally, in the evolution equation, we consider a cut-off of the gain term due to the coagulation that ensures that the measure solutions are supported in [1, 2R∗ ] and bounded at all times. To this end, we choose ζR∗ ∈ C (R∗ ) such that 0 ≤ ζR∗ ≤ 1, ζR∗ (x) = 1 for 0 ≤ x ≤ R∗ , and ζR∗ (x) = 0 for x ≥ 2R∗ . The regularized time evolution equation then reads as ∂t f (x, t) =

ζR∗(x) Kε,R∗ (x − y, y)f (x − y, t)f (y, t)dy 2 (0,x] Kε,R∗ (x, y)f (x, t)f (y, t)dy + η(x) . − R∗

(3.20)

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Definition 3.10 Let ε > 0 and R∗ ≥ Lη . Suppose that Kε,R∗ satisfies (3.15)– (3.17) and η ∈ M+ (R+ ) satisfies (3.6). Consider some initial data f0 ∈ M+ (R∗ ) for which f0 ((0, 1) ∪ (2R∗ , ∞)) = 0. Then f0 ∈ M+,b (R∗ ). We will say that f ∈ C 1 ([0, T ] , M+,b (R∗ )) satisfying f (·, 0) = f0 (·) is a time-dependent solution of (3.20) if the following identity holds for any test function ϕ ∈ C 1 ([0, T ] , Cc (R∗ )) and all 0 < t < T , d ϕ (x, t) f (dx, t) − ϕ˙ (x, t) f (dx, t) dt R∗ R∗ $ % 1 = Kε,R∗ (x, y) ϕ (x + y, t) ζR∗ (x + y) − ϕ (x, t) − ϕ (y, t) 2 R∗ R∗ ×f (dx, t) f (dy, t) + ϕ (x, t) η (dx) ,

(3.21)

R∗

where ϕ˙ denotes the Fréchet time-derivative of ϕ. Proposition 3.11 Let ε > 0 and R∗ ≥ Lη . Suppose that Kε,R∗ satisfies (3.15)– (3.17) and η ∈ M+ (R+ ) satisfies (3.6). Then, for any initial condition f0 satisfying f0 ∈ M+ (R∗ ), f0 ((0, 1) ∪ (2R∗ , ∞)) = 0 there exists a unique time-dependent solution f ∈ C 1 ([0, T ] , M+,b (R∗ )) to (3.20) which solves it in the classical sense. Moreover, f is a Weak solution of (3.20) in the sense of Definition 3.10 such that f ((0, 1) ∪ (2R∗ , ∞) , t) = 0 ,

for 0 ≤ t ≤ T ,

and the following estimate holds R∗

f (dx, t) ≤

R∗

f0 (dx) + Ct ,

t ≥ 0,

" for C = R∗ η(dx) ≥ 0 which is independent of f0 , t, and T . To prove Prop. 3.11 we observe that since the kernel is bounded, the result may be obtained using Banach fixed-point theorem. Definition 3.12 Let ε > 0 and R∗ ≥ Lη . Suppose that Kε,R∗ satisfies (3.15)– (3.17) and η ∈ M+ (R+ ) satisfies (3.6). We will say that f ∈ M+ (R∗ ), satisfying f ((0, 1) ∪ (2R∗ , ∞)) = 0 is a stationary injection solution of (3.20) if the following identity holds for any test function ϕ ∈ Cc (R∗ ): 0=

$ % 1 Kε,R∗ (x, y) ϕ (x + y) ζR∗ (x + y) − ϕ (x) − ϕ (y) f (dx) f (dy) 2 R∗ R∗ ϕ (x) η (dx) . + R∗

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M. A. Ferreira

We denote by S(t) the semigroup defined by the time-dependent solution f obtained in Proposition 3.11, S(t)f0 = f (·, t) that satisfies the semigroup property S(t + s)f = S(t)S(s)f, t, s ∈ R+ . The operators S(t) define mappings S(t) : XR∗ → XR∗ ,

for each t1 , t2 ∈ R+

with XR∗ = {f ∈ M+ (R∗ ) : f ((0, 1) ∪ (2R∗ , ∞)) = 0}. Proposition 3.13 Under the assumptions of Proposition 3.11, there exists a stationary injection solution fˆ ∈ M+ (R∗ ) to (3.20) as defined in Definition 3.12. Idea of the Proof The key point of the proof is to use Schauder fixed point theorem. The first step is to obtain the existence of an invariant region for the evolution problem (3.21). To that end, we choose a time independent test function ϕ(x) = 1 for x ∈ [1, 2R∗ ]. Using the lower bound for the kernel (3.16) and that f (·, t) has support in [1, 2R∗ ] we obtain the following estimate d dt



a1 f (dx, t) ≤ − 2 [1,2R∗ ]

2

 [1,2R∗ ]

f (dx, t)

+ c0

" where c0 = R∗ η (dx). This implies that for a large enough M > 0, the set   f (dx) ≤ M . UM = f ∈ XR∗ : [1,2R∗ ]

is invariant under the time evolution (3.20). Moreover, UM is compact in the ∗-weak topology due to Banach–Alaoglu’s Theorem (cf.[3]), since it is an intersection of a ∗-weak closed set XR∗ and the closed ball f  ≤ M. The second step is to prove that for each t > 0, both maps S(t) : UM → UM and t → S(t)f0 are continuous in the ∗-weak topology. Finally, the third step of the proof reads as follows. Since for each t, the operator S(t) is continuous and UM is compact and convex when endowed with the ∗−weak topology, we can apply Schauder fixed point theorem to conclude that for all δ > 0 there is a fixed point fδ of S(δ) in UM . Moreover, since UM is metrizable and hence sequentially compact, there is a convergent sequence {fδn }n∈N , i.e., there exists fˆ ∈ UM such that fδn → fˆ when δn → 0 in the ∗−weak topology. For each t we choose δn = t/n. Using the semigroup property we obtain that S(t)fδn = S(nδn )fδn = S(δn )fδn . Using the continuity of t → S(t)f0 and the fact that S(0)fˆ = fˆ, we

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83

obtain S(δn )fδn → fˆ. On the other hand using the continuity of S(t) we obtain that S(t)fδn → S(t)fˆ. Therefore S(t)fˆ = fˆ and thus fˆ is a stationary solution to (3.20), which concludes the proof.   The next Lemma provides uniform estimates for integrals. Lemma 3.14 Let a > 0, R ≥ a and b ∈ (0, 1) be such that bR > a. Suppose f ∈ M+ (R∗ ), ϕ ∈ C(R∗ ), g ∈ L1 (R∗ ), and g, ϕ ≥ 0. If 1 ϕ(x)f (dx) ≤ g(z) , for z ∈ [a, R] , z [bz,z] then

"

[a,R]

ϕ(x)f (dx) ≤

[a,∞) g(z)dz ln(b −1 )

+ Rg(R) .

We now extend the previous existence result to general unbounded kernels K supported in R2+ and satisfying the conditions of the theorem 3.4. Idea of the Proof of Theorem 3.4 Let fε,R∗ be a stationary injection solution to (3.20) as in Definition 3.12 provided by Proposition 3.13. The idea is to obtain estimates that are independent of both ε and R∗ that allow to pass to the limit as ε → 0 and R∗ → ∞ and to obtain the existence of a stationary injection solution to the original problem as defined in (3.3). First we obtain an estimate uniform in R∗ : fε,R∗ (dx) ≤ C¯ ε , R∗ > 0, [0,2R∗ /3]

where C¯ ε is a constant independent of R∗ . This estimate implies, that taking a subsequence if needed, there exists fε ∈ M+ (R+ ) such that fε ([0, 1)) = 0 and: fε,R∗n  fε as n → ∞ in the ∗ −weak topology with R∗n → ∞ as n → ∞. For any bounded continuous test function ϕ : [0, ∞) → R, one proves that fε satisfies 1 2



[0,∞)2

Kε (x, y) [ϕ(x+y)−ϕ(x)−ϕ(y)]fε (dx) fε (dy)+

[0,∞)

ϕ(x)η (dx) = 0.

where Kε is defined in (3.18). Second, we obtain estimates independent of ε: 1 z



⎛ ⎞1 2 ˜ 1 C ⎝ ⎠   f , ≤ ! ε (dx) 3 2z z 2 min zγ , 1ε 3 ,z

(3.22)

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M. A. Ferreira

and 1 z

2z 3 ,z

! fε

C˜ √ z ε

(dx) ≤

3 2

where C˜ is independent of ε. This estimate yields ∗−weak compactness of the family of measures {fε }ε>0 in M+ (R+ ) . Therefore, there exists f ∈ M+ (R+ ) such that: fεn  f as n → ∞ in the ∗ −weak topology for some subsequence {εn }n∈N with limn→∞ εn = 0. Using (3.19) and Lemma 3.14, one can prove that fε satisfies (3.9) for any ϕ ∈ Cc (R+ ). In particular, f = 0 due to η = 0. It only remains to prove (3.8). Taking the limit of (3.22) as ε → 0 we arrive at: 1 z

[2z/3,z]

f (dx) ≤

C z3/2+γ /2

for all z ∈ (0, ∞),

which implies 1 z

[2z/3,z]

x μ f (dx) ≤ C

zμ z3/2+γ /2

for all z ∈ (0, ∞),

for any μ ∈ R. From Lemma 3.14 we obtain the boundedness of the moment of order μ: x μ f (dx) < ∞. [0,∞)

for any μ satisfying μ < γ +1 2 . In particular, since |γ + 2λ| < 1, then the moments μ = −λ and μ = γ + λ are bounded, which proves (3.8).  

3.3.2 Nonexistence of Stationary Solutions We present the main ideas of the proof of Theorem 3.5 obtained in [13]. The proof is done by contradiction. Let the kernel K satisfy the power law bounds (3.2)–(3.3) with |γ + 2λ| ≥ 1. Suppose that f ∈ M+ (R∗ ) is a stationary injection solution of (1.1) in the sense of Definition 3.3. Then f satisfies the weak formulation (3.9) as well as the condition on the moments (3.8).

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The first step is to rewrite (3.9) using the flux formulation. Consider the function J : R∗ → R+ defined by J (R) =

K (x, y) xf (dx) f (dy)

(3.23)

R

where R = {x ≥ 1, y ≥ 1 : x + y > R, x ≤ R} . Let ε > 0, R ≥ 1 and χε ∈ C ∞ (R+ ) satisfy χε (x) = 1, x ≤ R and χε (x) = 0, x ≥ R + ε. Choosing a test function ϕ(x) = xχε (x) we obtain from (3.9) the flux formulation J (R) = xη(dx), R ≥ 1. [1,R]

We note that J describes the flux of particles " through R and that this flux

passing is constant for all R ≥ Lη and equal to J Lη = [1,∞) xη(dx) > 0, i.e., J (R) = J (Lη ), R ≥ Lη . The second step is to prove that the main contribution to the integral (3.23) as R → ∞ is due to collisions between particles of size close to R and particles of size of order 1. To that end, for a given δ > 0 small, we consider a partition of R = (1) (2) Dδ ∩ Dδ such that (1)

Dδ = {x ≥ 1, y ≥ 1 : y ≤ δx} , (2)

Dδ = {x ≥ 1, y ≥ 1 : y > δx} and we define Jk (R) =

R ∩Dδk

K (x, y) xf (dx) f (dy) , k = 1, 2.

Therefore J (R) = J1 (R) + J2 (R). Using the upper bound for the kernel (3.3), the moment condition (3.8) and the fact δR , ∞) one concludes after some computations that the that R ∩ Dδ2 ⊂ [1, R] × [ 1+δ contribution of J2 vanishes as R → ∞, i.e., lim J2 (R) = 0

R→∞

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M. A. Ferreira

which implies that lim J1 (R) = lim J (R) = J (Lη ).

R→∞

R→∞

In the remainder of the proof we will use the notation a := γ + λ and b := −λ if (γ + 2λ) ≥ 1, or a := γ + λ and b := −λ if (γ + 2λ) ≤ −1. Then, the assumption (3.8) may be rewritten as R∗

x a f (dx) < ∞.

(3.24)

The third step of the proof consists in obtaining a lower bound for the fluxes that implies a lower bound for the number of particles in some region of the size space. Using the upper bound for the kernel (3.3) we obtain after some computations  lim inf R





y f (dx) f (dy) ≥ (1)

a+1

R→∞

b

R ∩Dδ

J Lη

. c3 1 + δ |a−b|

(3.25)

For R sufficiently large we have that (1)

R ∩ Dδ ⊂ {(x, y) : 1 ≤ y ≤ δR, R < x + y, x ≤ R} whence, (3.25) implies the inequality

J Lη 1

y f (dy) f (dx) ≥ |a−b| R a+1 1 + δ 2c [1,δR] (R−y,R] 3

b

(3.26)

for R ≥ R0 with R0 large enough. We now consider two cases separately a ≥ 0 and a < 0. Let first a ≥ 0. Due to (3.24) we may define f (dx) , R ≥ 1 .

F (R) =

(3.27)

(R,∞)

Using (3.27) we can rewrite (3.26) as

J Lη 1

a+1 for R ≥ R0 . − [F (R − y) − F (R)] y f (dy) ≤ − |a−b| R 2c3 1 + δ [1,δR]

b

Then, using a comparison argument (see Lemma 4.1 in [13]), for some constant B > 0, it follows that F (R) ≥

B if R ≥ R0 , for a > 0, Ra

(3.28)

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87

and F (R) ≥ B log(R) if R ≥ R0 , for a = 0.

(3.29)

In the case a > 0, (3.28) implies B ≤ R a F (R) ≤

x a f (dx) (R,∞)

Taking the limit when R → ∞ and using (3.24) it follows that B ≤ 0, which leads to a contradiction. In the case a = 0, the contradiction follows from (3.29) in a similar way using (3.24). Let now a < 0. We define the function F by F (R) =

[1,R]

f (dx) , R ≥ 1.

(3.30)

Using (3.30) we can rewrite (3.26) as:

J Lη 1

a+1 for R ≥ R0 . − [F (R) − F (R − y)] y f (dy) ≤ − |a−b| R 2c3 1 + δ [1,δR]

b

As in the previous case, using a comparison argument (see Lemma 4.2 in [13]), it follows that there is B > 0 such that F (R) ≥

B if R ≥ R0 . Ra

For a small ε > 0 satisfying ε < B there exists M such that [M,∞)

Then for all R > M we have f (dx) ≤ R a B ≤ Ra [1,R]

x a f (dx) = ε .

[1,M]

f (dx)+

[M,R]

x a f (dx) ≤ R a

[1,M]

f (dx)+ε.

Since a < 0, taking the limit as R → ∞ we obtain B ≤ ε, which leads to a contradiction.

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4 Discrete Multi-Component Coagulation Equation with Constant Kernel In this section we consider discrete coagulation equations where the particle size is a discrete variable representing the number of monomers. In addition, we consider that particles may be composed of different types of monomers. A particle with d components is described by a vector α = (αi )i=1,...,d ∈ Nd+ \{0} where αi represents the size of the ith component. The number density nα (t) of particles with composition α at time t ≥ 0 satisfies the discrete multi-component coagulation equation ∂t nα (t) =

 1  Kα−β,β nα−β (t)nα (t) − Kα,β nα (t)nβ (t) 2 00

where D ⊂ Rd is the domain of convergence of the series and zα = z1α1 z2α2 . . . zdαd . Using ψα = zα in (4.4) we obtain an equation for the generating function F , ∂t F (z, t) = F (z, t)2 − 2F (z, t)N(t)

(4.6)

 where N(t) = F (0, t) = α>0 nα (t) is the total number of particles at time t. From (4.3) the initial number of particles is N(0) = 1. Using ψα = 1 in (4.4) we obtain an equation for N, ∂t N(t) = −N 2 (t), N(0) = 1

⇐⇒

N(t) =

1 . 1+t

(4.7)

If we subtract Eqs. (4.6) and (4.7) we obtain an equation for F − N. More precisely, we get ∂t (F − N) = (F − N)2 . Solving this equation and using (4.7) yields an expression for F F (z, t) =

F0 (z) (1 + t)(1 + t − tF0 (z))

(4.8)

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 where F0 (z) = F (z, 0) is given by F0 (z) = d1 di=1 zi after substituting (4.3) in (4.5). The expression for F will be used in the following to determine the solution to (4.2). We note that if {nα }α>0 is a solution to dthe multi-component coagulation equation (4.2), then {n } , where |α| = i=1 αi is the sum variable and n|α| |α| α>0 is defined by n|α| = β>0 nβ δ|α|,|β| , is a solution to the one-component equation with constant kernel K|α|,|β| = 2 and initial condition n|α| (0) = δ|α|,1 . This result may be obtained using the weak formulation (4.4) with a test function of the form ψα = ϕ|α| . We first solve the one-component equation to find an expression for {n|α| }α>0 . We consider the generating function f : D × R+ → R, D ⊂ R, associated to the one-component problem ∞ 

f (z, t) =

z|α| n|α| (t),

(4.9)

|α|=1

which may be expressed by (4.8) with f0 (z, t) = f (z, t) =

∞

|α|=1 z

|α| n (0) |α|

z (1 + t)(1 + t − tz)

= z, i.e. (4.10)

Using the Taylor series, we expand f around z = 0 and obtain f (z, t) =

∞  k=1

zk

t k−1 , (1 + t)k+1

z
0. t

(4.11)

Comparing each term of the two series (4.11) and (4.9) we conclude that the solution to the one-component equation is n|α| (t) =

t |α|−1 . (1 + t)|α|+1

(4.12)

The solution to the multi-component equation (4.2) can now be computed by expanding (4.8) and comparing with (4.5). Using the Taylor series in several variables we obtain the expansion of (4.8) around 0, f (z, t) =

∞  1 t k−1 |z|k , k d (1 + t)k+1 k=1

|z| < d

1+t , t > 0. t

(4.13)

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91

Comparing with (4.9) and using (4.12) and the fact that (z1 + . . . + zd )k =  α1 α2 αd k! α1 !α2 !...αd ! z1 z2 . . . zd we finally obtain the solution to the multi-component |α|=k

coagulation equation (4.2) expressed in terms of n|α| , nα (t) = n|α| (t)g(α)

with

g(α) =

|α|! 1 . d |α| α1 !α2 ! . . . αd !

(4.14)

t To study the long time behaviour, we use the fact that lim( 1+t t ) = e to obtain an approximation for n|α| (t) for large |α| and large time t

n|α| (t) ≈ t −2 exp(−

|α| ). t

(4.15)

Remark 4.2 In [28] it is shown that the solution to the continuous one-component

equation with constant kernel does approach the form f (x, t) = t −2 exp − xt for large times, provided the initial mass is either finite, which includes the case treated in this section, or its mass distribution function diverges sufficiently weakly. We also consider an approximation of the function g g(α) ≈ |α|−(d−1)/2 exp(− where |α|2− =

1 d

d  i,j =1

|α|2− ) 2|α|

(4.16)

(αi − αj )2 denotes the generalized mass difference variable.

Using (4.15) and (4.16) in (4.14) we obtain for large t and |α| the approximation nα (t) ≈ t −2 |α|−(d−1)/2 exp(−

|α|2 |α| ) exp(− − ). t 2|α|

(4.17)

We observe that, besides the mass scale |α| ∼ t imported from the solution√to the one-component equation, there is a second mass scale given by |α|− ∼ t. |α| √− we may then write the solution in a Introducing the variables ξ = |α| t and ρ = t scaling form nα (t) ≈ t −(d+3)/2φ(ξ, ρ)

(4.18)

where φ(ξ, ρ) = ξ −(d−1)/2 exp(−ξ ) exp(−

ρ2 ). 2ξ

(4.19)

Finally we note from (4.17) that for any fixed time, nα (t) reaches maximum values when |α|2− = 0. This condition defines a straight line in the size space given by

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{α ∈ Nd+ | α1 = α2 = . . . = αd }, indicating that mass concentrates along a line for large times and large sizes.

4.2 Mass Localization in Stationary Solutions Mass localization is also observed in stationary solutions to coagulation equations with source by applying a similar analysis as in the previous section. We consider the stationary multi-component coagulation equation with source and constant kernel Kα,β = 2, 0=



nα−β nβ − 2

β0

where sα is the source term. In analogy to the initial conditions in the timedependent case (4.3), the source term is given by sα =

h  δα,β , d

(4.21)

|β|=1

for some given h > 0. The constant kernel belongs to the class of kernels considered in Sect. 3. In particular, the constant kernel belongs to the subclass of kernels for which there is a stationary injection solution (see Theorem 3.4). Following the computations of [21] we compute in the following an explicit solution to the multi-component equation (4.20). Given a test function ψα , the weak formulation is now given by 0=

∞ 

[ψα+β − ψα − ψβ ]nα nβ +

h  ψα . d

(4.22)

|α|=1

α,β=1

The generating function F (z) =

∞ 

z|α| nα

(4.23)

|α|=1

satisfies F (z)2 − 2F (z)N + S(z) = 0

(4.24)

Coagulation Equations

93

√   where S(z) = dh |α|=1 zα = hd di=1 zi and N = h is obtained using an appropriate test function in (4.22). The solution to (4.24) reads & √ |z| (4.25) F (z) = h[1 − 1 − ]. d The solution to (4.20) is obtained by expanding F in powers of the variables zi and comparing with (4.23), yielding nα = n|α| g(α)

(4.26)

where g is defined in (4.14) and √ h(2|α|)! . (4.27) n|α| = (2|α| − 1)(2|α| |α|!)2 √ For large sizes we may approximate n|α| by h|α|−3/2 , therefore using also (4.16), we obtain nα ≈

√ |α|2 h|α|−(d+2)/2 exp(− − ). 2|α|

(4.28)

Like in the time-dependent problem, an additional size scale is observed |α|− ∼ √ |α|. Also here we can see from (4.28) that a stationary solution nα reaches maximum values at the straight line defined by {α ∈ Nd+ | α1 = α2 = . . . = αd }. A representation of (4.28) is shown in Fig. 1.

Fig. 1 Approximation of a stationary solution to the two-component coagulation equation with source and constant kernel (4.20). We observe a concentration of particles along a straight line

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5 Perspectives and Open Problems The existence and uniqueness of a time-dependent solution have also been established in [30] for coagulation kernels satisfying (3.2)–(3.3) with γ + λ = −λ and λ > −1/2 using a similar reasoning as the one we presented in Sect. 3.3.1 [30]. Moreover, in [10] existence is obtained using a functional framework, for a class of kernels satisfying (3.2)–(3.3) with c1 = c2 = 1, λ ∈ [−1, 1], γ ∈ [0, 2], γ ≤ −2λ, γ + λ ∈ [−1, 1] and (γ , λ) = (−λ, −1). In [18] uniqueness is proved globally in time for a class of kernels satisfying some regularity conditions as well as the bounds (3.2)–(3.3) with γ ≤ 1, λ = 0, and for a different class of kernels such that γ ∈ (1, 2], λ = 1 up to a gelation time T . For more general classes of kernels, both existence and uniqueness remain open problems. We refer to the survey [24] for further references. In the presence of a constant source of small particles, the existence and nonexistence of stationary solutions presented in Sect. 3 are the most recent existence results to the best of our knowledge. Previous results [8] were obtained for particular classes of kernels that are included in the more general setting presented here. In the case of multi-component equations with d components, source and kernel K satisfying c1 w(x, y) ≤ K(x, y) ≤ c2 w(x, y)

with

w(x, y) =

d 

x γi −λi y λi + y γi −λi x λi ,

i=1

(5.1) we expect the existence result (Theorem 3.4) to remain valid for a class of kernels satisfying |γi +2λi | < 1 for all i = 1, . . . , d [14]. In the same line, the nonexistence result (Theorem 3.5) should hold true if |γi + 2λi | ≥ 1 for some i. Moreover, stationary solutions are expected to exhibit mass localization along a straight line for a class of kernels satisfying growth bounds that are invariant under permutations of the components [15]. To the best of our knowledge, nothing is known about rigorous results for multi-component coagulation equations with general kernels. However, the wellposedness results for the one-component case, are expected to remain true in the multi-component case provided the kernel satisfies the bounds (5.1) with γi and λi satisfying the same conditions for well-posedness in dimension d = 1 for all i. Mass localization for large times is expected to hold for a class of kernels that satisfy the same bounds with the additional condition that w is invariant under any permutation of the components. Without this invariance condition, the mass may not localize, the mass may not localize in a straight line, due to the different rates of coagulation of each component. Multiscale behaviour is then expected to emerge that could break down the nice localization structure.

Coagulation Equations

95

Mass localization results are very important in the optimization of current algorithms as they allow to focus the computations on the region of the size space where the mass is localized. The computational complexity of the multi-component problem may in this way be reduced to the complexity of the one-component problem. There are also open problems on general coagulation equations with fragmentation, sink and growth terms, as well as on the derivation of these equations from particle systems. We refer to [7] for a brief overview on some of these topics. Acknowledgments The author is grateful to J. Lukkarinen, A. Nota and J. J. L. Velázquez for a very productive collaboration that led to the results presented in Sects. 3.3.1 and 3.3.2. The author acknowledges support of the Faculty of Science of University of Helsinki through the Atmospheric Mathematics (AtMath) collaboration as well as of the Hausdorff Research Institute for Mathematics (Bonn), through the Junior Trimester Program on Kinetic Theory, of the CRC 1060 The mathematics of emergent effects at the University of Bonn funded through the German Science Foundation (DFG).

References 1. Aldous, A.: Deterministic and stochastic models for coalescence (aggregation and coagulation): a review of the mean-field theory for probabilists. Bernoulli 5(1), 3–48 (1999) 2. Bertoin, J.: Random Fragmentation and Coagulation Processes, vol. 102. Cambridge University Press, Cambridge (2006) 3. Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, Berlin (2010) 4. Bonacini, M., Niethammer, B., Velázquez, J.J.L.: Self-similar solutions to coagulation equations with time-dependent tails: the case of homogeneity smaller than one. Commun. Part. Differ. Equ. 43(1), 82–117 (2018) 5. Bonacini, M., Niethammer, B., Velázquez, J.J.L.: Self-similar solutions to coagulation equations with time-dependent tails: the case of homogeneity one. Arch. Rat. Mech. Anal. 233(1), 1–43 (2019) 6. Carslaw, K.S., Lee, L.A., Reddington, C.L., Pringle, K.J., Rap, A., Forster, P.M., Mann, G.W., Spracklen, D.V., Woodhouse, M.T., Regayre, L.A., Pierce, J.R.: Large contribution of natural aerosols to uncertainty in indirect forcing. Nature 503(7474), 67–71 (2013) 7. da Costa, F.P.: Mathematical aspects of coagulation-fragmentation equations. In Bourguignon, J.P., Jeltsch, R., Pinto, A., Viana, M. (eds.) Mathematics of Energy and Climate Change; CIM Series in Mathematical Sciences, vol. 2, pp. 83–162. Springer, Cham (2015) 8. Dubovski, P.B.: Mathematical Theory of Coagulation. Lecture Notes Series, vol. 23. Seoul National University, Seoul (1994) 9. Elimelech, M., Gregory, J., Jia, X., Williams, R.A.: Particle Deposition and Aggregation— Measurement, Modelling and Simulation. Elsevier, Amsterdam (1995) 10. Escobedo, M., Mischler, S.: Dust and self-similarity for the Smoluchowski coagulation equation. Ann. I. H. Poincarè Anal. Non Linéaire 23(3), 331–362 (2006) 11. Escobedo, M., Mischler, S., Perthame, B.: Gelation in coagulation and fragmentation models. Commun. Math. Phys. 231(1), 157–188 (2002) 12. Escobedo, M., Mischler, S., Rodriguez Ricard, M.: On self-similarity and stationary problem for fragmentation and coagulation models. Ann. Inst. H. Poincaré Anal. Non Linéaire 22(1), 99–125 (2005)

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13. Ferreira, M.A., Lukkarinen, J., Nota, A., Velázquez, J.J.L.: Stationary non-equilibrium solutions for coagulation systems. Arch. Ration. Mech. Anal. 1–67 (2019) https://doi.org/10.1007/ s00205-021-01623-w 14. Ferreira, M.A., Lukkarinen, J., Nota, A., Velázquez, J.J.L.: Multicomponent coagulation systems: existence and non-existence of stationary non-equilibrium solutions (2021). arXiv preprint arXiv:2103.12763 15. Ferreira, M.A., Lukkarinen, J., Nota, A., Velázquez, J.J.L.: Localization in stationary non-equilibrium solutions for multicomponent coagulation systems (2020). arXiv preprint arXiv:2006.14840 16. Fournier, N., Giet, J.S.: Convergence of the Marcus–Lushnikov process. Methodol. Comput. Appl. Probab. 6, 219–231 (2004) 17. Fournier, N., Laurençot, P.: Existence of self-similar solutions to Smoluchowski’s coagulation equation. Commun. Math. Phys. 256(3), 589–609 (2005) 18. Fournier, N., Laurençot, P.: Well-posedness of Smoluchowski’s coagulation equation for a class of homogeneous kernels. J. Funct. Anal. 233, 351–379 (2006) 19. Friedlander, S.K.: Smoke, Dust, and Haze. Oxford University Press, Oxford (2000) 20. Hammond, A., Rezakhanlou, F.: The kinetic limit of a system of coagulating Brownian particles. Arch. Ration. Mech. Anal. 185(1), 1–67 (2007) 21. Krapivsky, P., Ben-Naim, E.: Aggregation with multiple conservation laws. Phys. Rev. E 53(10), 1103 (1995) 22. Lang, R., Xanh, N.X.: Smoluchowski’s theory of coagulation in colloids holds rigorously in the Boltzmann-Grad-limit. Z. Wahrscheinlichkeitstheor. verw. Geb. 54(3), 227–280 (1980) 23. Laurençot, P.: Weak compactness techniques and coagulation equations. In: Evolutionary Equations with Applications in Natural Sciences, pp. 199–253. Springer, Cham (2015) 24. Laurençot, P., Mischler, S.: On coalescence equations and related models. In: Degond, P., Pareschi, L., Russo, G. (eds.) Modeling and Computational Methods for Kinetic Equations. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, Boston (2004) 25. Lehtinen, K.E.J., Kulmala, M.: A model for particle formation and growth in the atmosphere with molecular resolution in size. Atmos. Chem. Phys. 3, 251–257 (2003) 26. Lee, M.H.: A survey of numerical solutions to the coagulation equation. J. Phys. A 34(47), 10219 (2001) 27. Lushnikov, A.A.: Evolution of coagulating systems. III. Coagulating mixtures. J. Colloid. Interf. Sci. 54(1), 94–101 (1976) 28. Menon, G., Pego, R.: Approach to self-similarity in Smoluchowski’s coagulation equation. Commun. Pure and Appl. Math. 57(9), 1197–1232 (2004) 29. Niethammer, B., Nota, A., Throm, S., Velázquez, J.J.L.: Self-similar asymptotic behavior for the solutions of a linear coagulation equation. J. Differ. Equ. 266(1), 653–715 (2019) 30. Norris, J.R.: Smoluchowski’s coagulation equation: uniqueness, nonuniqueness and a hydrodynamic limit for the stochastic coalescent. Ann. App. Probab. 9, 78–109 (1999) 31. Nota, A., Velázquez, J.J.L.: On the growth of a particle coalescing in a Poisson distribution of obstacles. Commun Math Phys 354(3), 957–1013 (2017) 32. Smoluchowski, M.: Drei Vorträge über diffusion, Brownsche Molekularbewegung und Koagulation von Kolloidteilchen. Phys. Z. 17, 557–599 (1916) 33. Su, T., Bowers, M.T.: Theory of ion-polar molecule collisions. Comparison with experimental charge transfer reactions of rare gas ions to geometric isomers of difluorobenzene and dichloroethylene. J. Chem. Phys. 58(7), 3027–3037 (1973) 34. Olenius, T., Kupiainen-Määttä, O., Ortega, I.K., Kurtén, T., Vehkamäki, H.: Free energy barrier in the growth of sulfuric acid–ammonia and sulfuric acid–dimethylamine clusters. J. Chem. Phys. 139(17), 084312 (2013)

Multibody and Macroscopic Impact Laws: A Convex Analysis Standpoint Félicien Bourdin and Bertrand Maury

Abstract These lecture notes address mathematical issues related to the modeling of impact laws for systems of rigid spheres and their macroscopic counterpart. We analyze the so-called Moreau’s approach to define multibody impact laws at the mircroscopic level, and we analyze the formal macroscopic extensions of these laws, where the non-overlapping constraint is replaced by a barrier-type constraint on the local density. We detail the formal analogies between the two settings, and also their deep discrepancies, detailing how the macroscopic impact laws, natural ingredient in the so-called Pressureless Euler Equations with a Maximal Density Constraint, are in some way irrelevant to describe the global motion of a collection of inertial hard spheres. We propose some preliminary steps in the direction of designing macroscopic impact models more respectful of the underlying microscopic structure, in particular we establish micro-macro convergence results under strong assumptions on the microscopic structure. Keywords Convex analysis · Granular media · Rigid bodies · Collisions · Homogenization · Maximal density constraint

1 Introduction The modeling of particle systems spreads over a wide range of approaches, which rely on various levels of description of the particles. At one end of this range, the microscopic/Lagrangian setting is based on an individual description of particles, which “simply” obey Newton’s Laws. At the other end, macroscopic models rely on

F. Bourdin Laboratoire de Mathématiques d’Orsay, Université Paris-Sud, Orsay, France e-mail: [email protected] B. Maury () Département de Mathématiques et Applications, Ecole Normale Supérieure, PSL University, Paris, France e-mail: [email protected]; [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 G. Albi et al. (eds.), Trails in Kinetic Theory, SEMA SIMAI Springer Series 25, https://doi.org/10.1007/978-3-030-67104-4_4

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a description of the collection of particles by a local density, and designing models amounts to elaborating equations verified by the velocity fields, under the implicit assumption that such a velocity is indeed well-defined. Between those extreme levels of descriptions, Boltzmann-like models are based on a kinetic description of the particle collection, namely a function f (x, v, t) which quantifies at time t the number of particles around x at velocity v. Note that this setting makes it possible to handle a diffuse limit (smooth f representing an infinite number of infinitely small particles), as well as finite collection (a single particle at a given velocity is represented as a Dirac mass in the (x, v) space). From this standpoint, the kinetic description can be considered as microscopic in a generalized sense. This setting is particularly relevant to describe the limit of a low-density gas with the underlying hypothesis of elastic binary collisions between particles, and it is a natural bridge between Lagrangian models, considered as untractable for many-body systems, and macroscopic models which can be used to investigate the global behavior of these systems, by means of theoretical analysis or numerical computations. Considerable energy has been, and is still, deployed to rigorously obtain macroscopic models from the Boltzmann equation, like Euler or Navier-Stokes equations (see e.g. [14, 34]). We are interested here in dense collections of finite size particles (more commonly called grains in this context), subject to possibly non-elastic collisions. The non-dilute character of the collections together with the non-elastic character of the collision is likely to rule out the hypothesis of sole binary collisions which prevails in the Boltzmann context: multiple (or quasi-simultaneous) collisions together with persistent contacts can be expected to be generic in this situation. As a consequence, Boltzmann-like equations can no longer be considered as a natural step between microscopic and macroscopic models, and most macroscopic models which have been proposed to describe the behavior of dense (up to jammed) granular media have indeed been built independently from any homogenization procedure. We propose here to investigate the possibility to identify some ingredients that might appear in relevant macroscopic models considered as limits of microscopic ones. Let us make it clear that we are far from proposing a full and rigorous construction of a macroscopic model from the microscopic one, which is clearly out of reach. We shall rather focus on a crucial part of microscopic models, namely collision laws, and investigate the possibility to infer collision laws at the macroscopic level which would be respectful of the microscopic structure. If one restricts to local interactions due to direct contact between entities, microscopic models based on finite size grains essentially rely on impact laws. Different strategies have been carried out to formalize this type of direct interactions. In the Molecular Dynamic approach (MD), see e.g. [1], one considers that grains are slightly deformable by implementing a short range force of the repulsive type. Note that this force is commonly taken as a computational trick to handle the nonoverlapping constraint. This makes it possible to circumvent the very question of collisions, for it leads to classical Ordinary Differential Equations which fit in a classical theoretical framework (Cauchy–Lipschitz theory), and which can be solved by standard numerical schemes to perform actual computations. Note that, in spite of the natively elastic character of such interactions, some ingredients can be included to account for inelastic collision, as well as shear forces (see [27]).

Multibody and Macroscopic Impact Laws: A Convex Analysis Standpoint

99

In order to directly address micro-macro issues, we shall restrict ourselves here to alternative approaches, called Contact Dynamics (CD), based on a hard-sphere setting. As detailed in a recent review [25], several strategies can still be carried out to formalize the behavior of the system whenever the non-overlapping constraint is up to be violated. A popular, sometimes called event-driven, strategy, consists in handling binary collisions only. In this setting multiple collisions are considered as so rare that they can be disregarded, which makes it possible to use explicit expressions of post-collisional velocities. Another strategy is based on extending the so-called Darboux–Keller shock dynamics to multibody collisions. It consists in changing the time scale in the neighborhood of a collision event, to set it at the impulse scale. The dynamics is then described as a sequence of compression and extension phases (see [16] for a detailed description of this method). We also refer to [13] for a very detailed account of thermodynamical aspects of collision problems. The developments we present here are based on an alternative approach, called Moreau’s approach in [25], which considers instantaneous impacts involving an arbitrary large number of grains, treated in a global way (see [3, 24, 28]). As detailed below, it relies on basic concepts of Convex Analysis, the principal of which being the cone of feasible direction associated to the set of admissible configurations (configurations with no overlapping), and the associated polar cone (set of vector which have a nonpositive scalar product with all feasible directions) which is the outward normal cone. Given a restitution coefficient e ∈ [0, 1], the post-collisional velocity is determined from the projection of the pre-collisional velocity on the outward normal cone. Since everything can be written as a simple expression of the projection on the cone of feasible directions, we shall actually focus on this very notion in the largest part of these notes. We shall end this introductory section by a few considerations on microscopic impact law following Moreau’s approach, and what appears to be the canonical extension of this approach to the macroscopic setting. Section 2 is then dedicated to a detailed analysis on these impact laws. Identifying similarities and discrepancies between these formally similar laws is the object of Sect. 3. We describe in particular Laplace-like operators which are canonically associated to the collision laws in both settings, We introduce a notion of Abstract Maximum Principle (detailed in the appendix), which is verified in the macroscopic setting but not in the microscopic one, which deeply differentiates both models, and enlighten in some way the poorness of the macroscopic law. In Sect. 4, we investigate the possibility to elaborate macroscopic impact models which are more respectful of the underlying microscopic structure. As detailed in Sect. 5, a rigorous homogenization procedure makes it possible to build such macroscopic models under very strong assumptions on the structure. Although the resulting evolution problems are out of the scope of our work, we dedicate Sect. 6 to some remarks on this aspect of the problem. In the microscopic setting, the question is delicate but well understood: the problem is well-posed for analytic data, but might admit multiple solution otherwise, even for infinitely smooth data. In the macroscopic setting, under oversimplifying assumptions, the

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expected model takes the form of the so called pressureless equations with maximal density constraint, which essentially fits into a sound framework in the onedimensional setting only [4, 6]. For higher dimension, little is known on this equation. Let us add that the system is commonly written without any collision law, the actual choice being usually made in an implicit way, depending on the approach which is followed. For instance, in [6], particular solutions are built by means of sticky blocks with a purely inelastic collision law, whereas in [9, 10], the approach is based on compressible Euler equation with a barrier-like pressure with respect to the density, natively leading to a purely elastic behavior. The largest part of this text is meant to be accessible to graduate students, so we tried to preserve self-consistency as far as possible, writing at some points full proofs of elementary results, in particular in the appendix. From Single Collision to Multibody Impact Laws We introduce here Moreau’s approach of impact laws, which fits in the general class of Contact Dynamics Methods (see [24, 29]). Let us start with a point particle subject to remain in the upper half plane R × R+ , with a purely inelastic collision law on the boundary. We denote by r = (x, y) its position, and by u its velocity. If this particle is not subject to any force, its motion follows  2 R if y > 0 u = PCr u , with Cr =  R × R+ if y = 0 +



(1)

where u− (resp. u+ ) is the pre- (resp. post-) collisional velocity, and PCr is the Euclidian projection on Cr . When the particle does not touch the wall, the velocity is constant. When a collision occurs, with pre-collisional velocity u− = (ux , uy ) (with uy < 0), the post-collisional velocity is u+ = (ux , 0). In the case of an elastic collision, we introduce a restitution coefficient e ∈ (0, 1]. The post-collisional velocity is now u+ = (ux , −euy ). This behavior can be written in a way which can generalized to the multi-collisional situation. We introduce the outward normal cone to K, defined as ( ' Nr = Cr◦ = v ∈ R2 , ß < v, w >≤ 0 ∀w ∈ Cr = {0} × R− . The collision law can be written (see Fig. 1) u+ = u− − (1 + e)PNr u− . In the multi-collisional situation, the Moreau’s approach consists in straightforwardly write the previous collision law, with the appropriate notion of cone of feasible velocities and outward normal cone. Consider a many-body system of hard spheres in Rd , centered at r1 , . . . , rn , with common radius R. The feasible set writes ' K = r ∈ Rdn , Dij = |rj − ri | − 2R ≥ 0

( ∀i = j .

Multibody and Macroscopic Impact Laws: A Convex Analysis Standpoint Fig. 1 Collision against a wall. Depending on the restitution coefficient e ∈ [0, 1], the post collisional velocity can take any value between (ux , 0) (purely non-elastic) and (ux , −uy ) (elastic)

Denoting eij =

101

y u+

u−

ux

−e u y uy

x

rj − ri , the set of admissible velocities is |rj − ri |

  ) * Cr = v , Dij (r) = rj − ri  − 2R = 0 ⇒ ß < eij , (vj − vi ) >≥ 0 .

(2)

(N.B.: we use the notation ß < a, b > to denote the dot product of vectors in the physical space Rd , while ·|· shall be used for generalized velocity vectors in Rnd , or elements in abstract Hilbert spaces.) Let r = (r1 , . . . , rn ) ∈ K be given. As previously, the outward normal cone to K at r is defined as the polar set to the cone of feasible velocities: ' Nr = Cr◦ = v ∈ Rdn , v|w ≤ 0

( ∀w ∈ Cr .

To alleviate notation, we shall now denote by U the pre-collisional velocity, and by u the post-collisional velocity. With these new notation, the collision model writes u = U − (1 + e)PNr U,

(3)

where e ∈ [0, 1] is the restitution coefficient. Since Nr and Cr are mutually polar, it holds that I = PNr + PCr , where I is the identity operator in Rdn (see [23], or the proof of Proposition 11 in the appendix). As a consequence, the post-collisional velocity can be expressed in terms of PCr U , for any e ∈ [0, 1], u = U − (1 + e)(U − PCr U ),

(4)

which simply reduces to u = PCr U for e = 0. For the sake of simplicity, we shall therefore focus on this purely inelastic situation, keeping in mind that the knowledge of PCr U makes it possible to recover the whole range of elastic collision laws through (4).

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Macroscopic Impact Laws We describe here informally how the Moreau’s approach described above can be developed at the macroscopic scale. More details will be given in the next section. We consider an infinite collection of inertial grains described by a macroscopic density ρ, which is subject to remain below a prescribed value, which we set at the corresponding set of densities of a given mass, which are 1. We denote by K assumed to be supported in some domain . We denote by U the velocity field at some instant, defined on the support of ρ, and we aim at defining a collision law which would give us the post-collisional velocity from this pre-collisional velocity U . In the purely inelastic setting, a natural candidate for this law amounts to define the post-collisional velocity u+ as the projection u of U on the set of all those vector fields which have a nonnegative divergence on the saturated zone, that is the macroscopic counterpart of Cr (defined by (2)). Indeed, having ∇ · u < 0 in the neighborhood of some point in the saturated zone would lead to an increase of ρ, thereby a violation of the constraint. As will be ρ can be described as the set of all those velocity fields detailed below, this cone C which have a nonnegative divergence (in a weak sense) over the saturated zone.

2 A Closer Look to Micro and Macro Impact Laws In this section, we give some details on the mathematical formulation of the impact laws presented above, in the microscopic and macroscopic settings, and we investigate their similarities and discrepancies.

2.1 Saddle Point Formulation of the Microscopic Impact Law We consider as previously a system of hard spheres in Rd , centered at r1 , . . . , rn , with common radius R. The feasible set writes ' ( K = r ∈ Rdn , Dij = |rj − ri | − 2R ≥ 0 ∀i = j . (5) The set of feasible velocities Cr is defined by (2). Let us denote by m ∈ N the number of contacts, i.e. the number of pairs {i, j } such that Dij = |rj −ri |−2R = 0. We introduce B ∈ Mm,n (R) the matrix which expresses the constraints, each row of which is

Gij = 0, . . . , 0, −eij , 0, . . . , 0, eij , 0, . . . , 0 ∈ Rdn ,

(6)

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where eij = (rj − ri )/|rj − ri |. The feasible set can be written Cr = {v , Bv ≤ 0} = B −1 o+ , + = Rm +,

(7)

−1 ◦ where ◦+ is the polar cone to + , that is Rm − , and B + its preimage by B. The problem which consists in projecting U ∈ Rnd on Cr fits into the abstract setting of Proposition 12 in the appendix, and it can be put in a saddle point form:

Proposition 1 Let Cr ∈ Rdn be defined by (2) (or equivalently by (7)). Denote by B the transpose of the matrix B. If u = PCr U then there exists p ∈ + = Rm + such that  u + B p = U   Bu (8) ≤0   Bu|p = 0. Conversely, if (u, p) ∈ Rdn × Rm + is a solution to (8), then u = PCr U . Proof Let us start by a preliminary remark: the fact that the image of B is closed (it is a finite dimensional linear space) is not sufficient to ensure that B (+ ) is closed (see Remark 11 in the appendix). This property is nevertheless true here, because B (+ ) is spanned by a finite number of vectors ⎧ ⎫ ⎨  ⎬ B (+ ) = − pij Gij , pij ≥ 0 ⎩ ⎭

(9)

ij

where Gij is defined by (6), which implies closedness by Lemma 3. Proposition 12 then ensures existence of p ∈ + = Rm + such that u + B p = U , with the complementarity condition Bu|p = 0.   If we furthermore assume that B is one-to-one, i.e. B is onto, then p is unique. The one-to-one character of B is lost as soon as the number of constraints is larger than the number of degrees of freedom (hyperstatic situation). For identical disks in 2d, it can appear as soon as n = 14 discs are involved (see Fig. 2). For many-body triangular lattices, the number of primal degrees of freedom is 2n, while the number Fig. 2 In the configuration represented here, the number of primal degrees of freedom is 2 × n = 28, whereas the number of contacts is m = 29

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of constraints is asymptotically 3n, which mean that the dimension of the kernel of B is asymptotically n. The problem nevertheless presents some sort of uniqueness property, restricted to the homogeneous problem (U = 0). The following proposition essentially states a very intuitive fact: if one considers any static configuration of a finite number of hard spheres in the open space (i.e. with no walls), under the assumption that interaction contact forces are only repulsive, then all forces are actually zero. This property will be used to show that the solution set for the pressure field (Proposition 1) is bounded. Lemma 1 We consider an admissible configuration r ∈ K, and the associated matrix B ∈ Mm,n (R) (the raws of which are given by (6)). The set * ) S = q ∈ Rm + , B q = 0 = ker B ∩ + is reduced to {0}. Proof Let us first establish the uniqueness for the homogeneous problem. We consider q = (qij ) ∈ Rm + such that B q=



qij Gij = 0.

i∼j

where i ∼ j means that the particles i and j are in contact. Let i0 denote the index of an extremal vertex of the convex hull conv(qi , 1 ≤ i ≤ n). By Hahn-Banach’s theorem, the compact {qi0 } and the set conv{qi , i = i0 }, which is closed and convex, can be separated in a strict sense by a plane in Rd . We denote by x an element of this plane, and by v a normal vector to it. One has ß < (qi0 − x), v >> 0 , ß < (qj − x), v >< 0

∀j = 1, . . . , n , j = i0 ,

so that ß < (qi0 −qj ), v >> 0 for j = i0 . Now the balance of contact forces exerted upon sphere i0 in the direction v reads 

qj i0 ß < ej i0 , v >

j =i0

where ß < ej i0 , v >> 0, and qj i0 ≥ 0 for all j . This quantity is positive unless qj i0 = 0 for all j = i0 . Therefore all multipliers associated to a contact with sphere i0 are equal to 0, and this approach can be iterated for the reduced family (qj , j = i0 ). By downward induction on the number of active spheres, we prove that S is reduced to {0}.   An important consequence of this expected property is the boundedness of the solution set for (8).

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Proposition 2 Under the assumptions of Proposition 1, the solution set for the dual component p ) *

S = q ∈ Rm + , B q = U − u = p + ker B ∩ + , where (u, p) is a solution to (8), is bounded. Proof This is a direct consequence of Proposition 15 (in the appendix) and Lemma 1.  

2.2 Saddle Point Formulation of the Macroscopic Impact Law In the macroscopic setting, we consider that the solid phase is represented by a density supported in a domain  ∈ Rd , subject to remain below the value 1. We the set of all those measures, which is the macroscopic counterpart of denote by K the set K of n-sphere configurations with no overlapping. We shall disregard here issues possibly related to wall conditions or mass at infinity: we assume that  is bounded, and that that the support of ρ is strongly included in  (i.e. the support of ρ is at a positive distance from ∂, which we denote by ω ⊂⊂ ). together with a preWe define a pre-collisional configuration as a density in K, collisional velocity field U∈

L2ρ ()d

  d 2 = v :  → R , ρ − measurable , |v| dρ < ∞ . 

We describe here a natural way to define a post collisional velocity u, natural in the sense that it directly follows the same principles as the microscopic law. This strategy to define a post-collisional velocity in the purely inelastic setting follows the framework proposed in [20, 21] for macroscopic crowd motion models. Feasible velocities are those which are non-concentrating in the saturated zone [ρ = 1]. For smooth velocities, it amounts to prescribe a nonnegative divergence in this zone. Such a set can be properly defined by duality as  ρ = v ∈ L2ρ ()d , C v · ∇q dρ ≤ 0

 ∀q ∈ ρ , q ≥ 0 a.e.

(10)



where the space ρ of pressure test functions is defined as ' ( ρ = p ∈ H 1 () , p(1 − ρ) = 0 a.e. .

(11)

Note that, since we assumed that the support of ρ is strongly included in , it holds that ρ ⊂ H01 ().

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It can be easily checked that, for a smooth velocity field v and a regular saturated ρ is equivalent to verifying ∇ · v ≥ 0 on [ρ = 1]. zone, belonging to C The non-elastic collision law writes u = PC ρ U, where the projection PC ρ is with respect to the L2ρ norm. Let us check that it fits into the abstract setting of Proposition 12. with V = L2ρ ()d , and  = ρ . We define + ⊂ ρ as the set of all those functions in ρ which are nonnegative almost everywhere: ) * + = q ∈ ρ , q(x) ≥ 0 a.e. in  .

(12)

We introduce ∈  , : v ∈ V = L2ρ (ω)d −→ Bv B is defined by where Bv .

/ p = Bv,

v · ∇p dρ.

(13)

maps V onto  , so that the Note that  and  are not identified here, and that B adjoint operator B is defined in L(, V ), the set of continuous linear mappings from  into V . The saddle-point formulation of the problem can be written   u + ∇p  “ − ∇ · u   p    u · ∇p dρ 

= U ρ-a.e. in , ≤0 in [ρ = 1]”, ≥ 0 ρ-a.e. in ,

(14)

= 0,



where the second equation (between quotation marks) is meant in a weak sense, i.e. u · ∇q dρ ≤ 0 

∀q ∈ + .

∈ − , where − is the This condition can also be written in an abstract way: Bu polar cone to + , i.e. the cone of all those linear forms in  which are nonpositive over + . We may now state the well-posedness result. be given as a density defined over a bounded domain Proposition 3 Let ρ ∈ K , with supp(ρ) strongly included in . Problem (14) admits a unique solution

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(u, p) ∈ V × + , and the primal component u of this solution is the projection of ρ . U on C Proof From Proposition (12) (more precisely Corollary 1), it is sufficient to prove (defined by (13)) is onto. Let us prove that there exists a constant β > 0 such that B that for every q ∈  0 0 0 B q 0L2 ≥ βqH 1 , ρ

which writes ∇qL2ρ ≥ βqH 1 in the present context. Due to Poincaré Inequality, which holds true because  ⊂ H01 (), it is sufficient to establish that the inequality ∇qL2ρ ≥ β∇qL2 holds for any q ∈ . For q ∈ , by Theorem 1.56 in [33] one has (1 − ρ) ∇q = (1 − ρ) 1q =0 ∇q = 0, has a closed range, and so does so that ∇qL2ρ = ∇qL2 . As a consequence, B by Banach-Steinhaus Theorem. The range of B is also dense thanks to the same B is onto. inequality, thus B  

3 Micro-Macro Issues We detailed in the previous section impact laws for a collection of rigid spheres, in the Moreau’s spirit, and we proposed a natural instantiation of the same principles at the macroscopic level. The macroscopic version may appear as a natural candidate to handle collision between clusters of infinitely many hard spheres represented by a diffuse density. We shall see here that some considerations may comfort this standpoint in the one-dimensional setting. Yet, for dimensions d ≥ 2, we shall prove that the macroscopic law presented in the previous section is not a relevant model for describing the impact between large collections of hard spheres. One Dimensional Setting In the one-dimensional setting (hard spheres move on a fixed line) the two approaches are mutually consistent, as we shall see here. First, the notion of maximal density is well defined at the microscopic level: a cluster of spheres (represented by segments in 1d) is saturated if the solid phase covers some zone of the real line, which corresponds to ρ = 1 in the macroscopic setting. Now consider such a cluster of n segments covering an interval I ∈ R, and the corresponding macroscopic density ρ = 1I (characteristic function of I ). We consider a pre-collisional velocity field U that pushes the configuration against the boundary of the feasible set, i.e. such that ∂x U ≤ 0. In this case the constraint will be

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saturated overall the cluster so that, at the macroscopic scale, −∇ · u = −∂x u = 0, and Problem (14) is a classical Darcy problem   u + ∂x p = U,   −∂x u =0 in I.

(15)

Eliminating the velocity yields a Poisson problem on the pressure − ∂xx p = −∂x U,

(16)

with Dirichlet boundary conditions on the boundary of I . At the microscopic level, we simply consider pre-collisional velocities U1 , . . . , Un , with Ui = U (qi ), and we make a slight abuse of notation by keeping U to denote the vector of velocities. Since the velocities U1 , . . . , Un , are non-increasing, the constraint will also be saturated, which leads to a Darcy-like problem  u + B p = U   Bu = 0.

(17)

Eliminating the velocity yields a Poisson-like problem BB p = BU,

(18)

with ⎛

1 ⎜0 ⎜ B=⎜ ⎜ ⎝0 0

−1 0 . . . 1 −1 . . . . . 0 .. .. 0 ... 1

⎞ 2 −1 0 · · 0 ⎟ ⎜ ... ⎜ −1 2 −1 0 · · ⎟ ⎟ ⎜ ⎟ ... ⎟ ⎜ ·⎟ ⎟ ∈ Mn−1 (R), ⎟ ∈ Mn−1,n (R) , BB = ⎜ 0 −1 · · ⎜ · ⎟ · · · ·⎟ ⎟ ⎜ ... ⎠ ⎟ ⎜ ⎝ · · 2 −1 ⎠ −1 0 · · 0 −1 2 ⎞



that is the discrete Laplacian matrix. The two formulations are mutually consistent in the sense that the linear system (18) is a standard finite difference discretization of the Poisson problem (16), which is covered by rigorous convergence results (see e.g. [2]).

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Fig. 3 Square (left) and triangular (right) lattices

Case d ≥ 2 In higher dimensions the situation is fully different. First, the notion of maximal density is not clearly defined at the microscopic level. Let us consider collections √ of identical discs. The maximal packing density ρmax = π/2 3 ≈ 0.9069 . . . , and corresponds to the triangular lattice (see Fig. 3, right). Yet the actual density of moving collections of rigid disks is generally strictly less than this maximal value, which does not mean that the flow is unconstrained (as the macroscopic setting would suggest). These considerations call for a clear identification of configurations which saturate the constraint. It is tempting to consider as maximal in some sense any density corresponding to such configurations, for which there are no free disks, so that constraints are activated everywhere. The triangular lattice is clearly jammed, but so is the Cartesian lattice (ρ = π/4 ≈ 0.79), and it is possible to build looser jammed configurations, for example by removing some non neighboring discs from the triangular lattice. We refer to [32] for a general review on the notion of maximal random packing. Beyond this difficulty to properly define the notion of maximal density, the microscopic and macroscopic projections exhibit deep discrepancies in dimensions higher than 1. We propose here to enlighten these discrepancies by considering the underlying Poisson problems for the pressure in both settings. Like in the one-dimensional setting, we first consider the macroscopic setting, which is in some manner simpler than the microscopic one, in spite of its infinite dimensional character. The pressure can be shown, under some assumptions, to verify a Poisson like problem in the saturated zone, The first step consists in proving that the problem verifies the abstract maximum principle (see Definition 3), that is ∈ −− = −◦+ $⇒ ∃p ∈ + s.t. B B p = BU. BU Proposition 4 We assume that supp(ρ) is strongly included in , and that  is + ) verifies the maximal principle (Definition 3). connected. The couple (B, ∈ −− , i.e. if U is is one to one, it amounts to check that, if BU Proof Since B such that U · ∇q ≥ 0 ∀q ∈  , q ≥ 0 a.e., (19) 

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then the (unique) solution p ∈  to

∇p · ∇q = 

U · ∇q

∀q ∈ 



takes nonnegative values almost everywhere, i.e. it lies in + . This property takes the form of a maximum principle for the Laplace operator, in an extended sense: the saturated zone [ρ = 1] may be not be the closure of an open domain, it may in particular have an empty interior, while having a positive measure (see Remark 1). This property is obtained by a standard procedure, which consists in taking a test function equal to the negative part of p, i.e. q = p− = − min(0, p). We have that ∇q = −∇p1p≤0 (see Theorem 1.56 in [33]), and q ≥ 0, so that

 − 2 ∇p  =

− 



U · ∇p− ≥ 0, 

which implies that ∇p− vanishes almost everywhere, i.e. p− is constant on . Since it is 0 in the neighborhood of the boundary, it vanishes on  i.e. p ≥ 0 a.e. in .   In other words, if the pre-collisional field is non-expansive, i.e. ∇ · U ≤ 0, then the pressure field p is a weak solution to the Poisson problem − p = −∇ · U,

(20)

on the saturated zone [ρ = 1], with homogeneous boundary conditions. Let us add that the PDE above can only we legitimately written under certain conditions on the saturated zone. If the latter presents some pathologies, for example if it has an empty interior (like the complement of a dense open set) and yet a positive measure, then p might be non-trivial ( is not reduced to {0}), whereas (20) is not even verified in the sense of distributions. Indeed, since  does not contain any smooth function (except 0), the formulation (19) is much weaker than (20) considered in the sense of distribution. A proper Poisson problem can be recovered under some additional assumptions, for instance if the saturated zone is “regular” in the sense that [ρ = 1] = ω where ω is a smooth domain. The condition that is actually needed is actually the following: ω is such that  (defined by (11)) is equal to the closure of Cc∞ (ω) in H 1 () (the functions of Cc∞ (ω) being extended by 0 outside ω). Under these conditions, (19) implies that p is a weak solution (in a standard sense) to the Poisson problem (20). Remark 1 We mentioned the fact that the space  might contain no smooth function, while being non-trivial. We prove in this remark that it is indeed the case. We propose to investigate the case where the saturated zone [ρ = 1], which contains the support of all those functions in , is the complement of a dense open set ω, which excludes any nontrivial smooth function. We describe below how to build a nontrivial function in . We assume d = 2, but the approach can be straightforwardly extended to higher dimensions. We consider a sequence

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N N (c n ) ∈2  that is dense in , and a sequence (Rn ) ∈ (0, +∞] such that πRn ≤ ||/2. For a given rn < Rn , we denote by γn the circle of radius rn , centered at cn , by !n the cocentric circle of radius Rn , and by n the ring domain between these circles. We denote by gn the solution to the following Dirichlet problem in n

  −g = 0 in n ,   g = 0 on γn ,   g = 1 on !n ,

(21)

extended by 0 inside the small disc, and by 1 outside the large one. Since the capacity of a point is 0 in R2 (see e.g. [22]), one can choose rn , with 0 < rn < Rn , sufficiently small to ensure that 1 |∇gn |2 ≤ 2n . 2  We denote by ω the union of the small discs (centered at cn , with radius rn ), which is open and dense by construction. Now consider the function Gn = g1 g2 . . . gn . It holds that ∇Gn =

n  k=0

∇gn



gj ,

j =k

so that (all the gj take values between 0 and 1 by construction), by the triangular inequality in L2 (), ∇Gn L2 () ≤

n  k=0

∇gk L2 () ≤

n  1 ≤ 2. 2k k=1

The sequence (Gn ) is therefore bounded in H 1 () (the gradient is bounded in L2 , and they all vanish in the first disc centered at c0 , with radius r0 ). One can extract a sub-sequence which weakly converges in H 1 () to some function G ∈ H 1 (). Since the convergence is strong in L2 by Rellich’s Theorem, the convergence (up to a subsequence) holds almost everywhere, so that G is by construction equal to 1 almost everywhere in the complement of the union of the large discs (centered at cn , with radius Rn ). By assumption on Rn , the measure of this set is positive (larger than ||/2), so that G is different from 0, while vanishing by construction in the dense union ω of the small discs. At the microscopic level the picture is different in general. In particular, the approach carried out in the proof on the previous proposition is no longer valid. The difficulty comes from the fact that BB , which is the straight analog of − in the one-dimensional setting, does not verify any maximum principle in general.

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Fig. 4 In the configuration represented here, considering that the distances are subject to remain 0 (constraint Bu = 0), the pre-collisional velocity tends to push any two grains in contact toward overlapping, and yet the pressure between the two grains in the center will be negative

Consider the simple situation represented in Fig. 4, with a pre-collisional velocity directed toward the center. If we consider (like in the proof of the previous proposition) the problem with an equality constraint (Bu = 0, which means that the hard grains are glued together), eliminating the velocity leads to a discrete Poisson problem BB p = BU. Since the horizontal velocities have a much larger magnitude, in spite of the fact that the pre-collisional velocity pushes the grains against each other (i.e. BU > 0), it is clear that the pressure associated with the contact between the two central grains will be negative, which rules out the maximum principle (in the sense of Definition 3). As a consequence, the solution to the impact problem, with a unilateral constraint, will not be the same: the grains at the center will be pushed apart during the collision, which implies (thanks to the complementarity constraint Bu|p = 0) that the corresponding pressure is 0. Note that some sort of Poisson Problem can be recovered for the pressure associated with the impact law, by removing the raws of B which correspond to non activated contacts, i.e. with −Gij · u < 0. If one denote by B the corresponding matrix, and by p the corresponding pressure, it holds that B B p = B U, with a reduced matrix B which may also not verify the maximum principle. This violation of the maximum principle for BB is generic in the hard-sphere, microscopic, setting, as soon a dense collections of grains are concerned. It can be checked for simple situation that the matrix BB , unlike in the one-dimensional setting, has positive off-diagonal entries. Square and Triangular Lattices As an illustration of the previous considerations, and as an introduction to the next section, let us make some remarks on very specific situations, where the overall

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behavior of a collection of rigid discs can be seen to significantly differ from the behavior given by the macroscopic impact law. Consider at first a jammed configuration structured according to a square lattice (see Fig. 3 (left)). On each row, the non-overlapping constraints impose horizontal velocities to be non-decreasing. Similarly, on each column, the vertical velocities must be non-decreasing also. Two fields of Lagrange multipliers can therefore be associated to the constraints in the main directions x and y, which act on the system independently from each other. As a consequence, two constraints must be verified, to be compared to the single scalar constraint of the macroscopic constraint ∇ · u ≥ 0. In the case of a triangular lattice (see Fig. 3 (right)), the monotonicity of the velocity is imposed in each of the 3 principal directions. In both cases, the microscopic constraints are much stronger than the macroscopic one, which is therefore obviously irrelevant to model at the macroscopic scale the collections of hard discs. The next section is dedicated to designing macroscopic models more respectful of the underlying microscopic structure, in the case of crystal-like configurations.

4 Anisotropic Macroscopic Collision Laws We develop here some macroscopic models intended to represent configuration of jammed grains introduced in Sect. 3, namely configurations that are structured in a periodic way. The approach is the following: starting from the constraints at the microscopic level, we extend them to the macroscopic level. Square Lattice We first propose a macroscopic model adapted from the microscopic configuration of spheres jammed on a Cartesian lattice, as depicted on Fig. 5. Let us study the constraints on the velocity of the central sphere denoted by 0 in the figure: there are

1 1

3

0

2

4

2 0

3 5

4 6

Fig. 5 The two structured jammed configuration. On both side, the spheres in contact with the sphere 0 are labelled from 1 to 4 or 6

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four, each one corresponding to a contact with an adjacent sphere. The microscopic constraint for spheres in contact described by (2) writes here   (u1 − u0 ) · ey   (u0 − u2 ) · ey   (u4 − u0 ) · ex   (u − u ) · e 0 3 x

≥0 ≥0 ≥0 ≥0

(22)

System (22) can be reformulated in a more concise way: the quantities ux = u · ex and uy = u · ey must be non decreasing along each axis. In a macroscopic setting, we want thus to translate this constraint by subjecting ∂ux /∂x and ∂uy /∂y to be nonnegative in some sense, considering that u has L2 regularity. To that purpose, we define the directional derivatives of the two components in a dual way, imposing −

ux 

∂p ≥0 ∂x

(23)

for every nonnegative test function p such that its weak partial derivative in x can be defined. In order to clarify this last condition, we will introduce anisotropic Sobolev spaces, naturally defined to formalize the notion of “weakly derivable along one direction”. The following description of these spaces is extracted from [15]. In what follows,  is a strictly convex bounded open set, with regular boundary. We refer to [15] for the study of more general domains. Definition 1 The anisotropic Sobolev space in the direction x on  is defined by: Hx1 () where “

  ∂f 2 2 weakly exists in L () = f ∈ L () , ∂x

∂f weakly exists in L2 () ” means that ∂x ∂f ∂g g=− f. ∀g ∈ C 1 () ,  ∂x  ∂x

This space is endowed with the norm

f 2H 1 x

=

f 22

(24)

(25)

0 02 0 ∂f 0 0 +0 0 ∂x 0 2

1 We then define H0,x () as the closure of Cc∞ () in Hx1 (), and Hx−1 () as 1 (). Since integrating along a single direction is sufficient to prove the dual of H0,x Poincaré inequality in the usual Sobolev space H01 () (see Proposition 9.18 and Corollary 9.19 in [8]), the anisotropic counterpart of Poincaré inequality holds :

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1 Proposition 5 There exists c > 0 such that for every f ∈ H0,x (), 0 0 0 ∂f 0 0 f 2 ≤ c 0 0 ∂x 0 2

(26)

We may now introduce the macroscopic model corresponding to a square microscopic structure. Let ρ ∈ L∞ (), such that ρ ≤ 1 a.e., we still denote by ρ the measure of density ρ with respect to the Lebesgue measure. In order to avoid boundary issues, we assume that all the measures we consider are supported on a set strongly included in  (that is to say, as previously, at positive distance to ∂). In order to alleviate notation (and to deal with realistic situations when the grains configuration is structured), we assume ρ to be saturated over all its support, so that there is no need in mentioning the dependence in ρ in the notation.

Problem 1 Given ρ ≤ 1 and U ∈ L2 ()2 , find u = ux , uy ∈ L2 ()2 that realizes the projection min x

u∈C ∩C y



u − U 22 dρ

(27)

where the constraints set C x and C y are defined by duality:  2 C = u ∈ L () , α



∂q uα ≤ 0 ∀q ∈ α , q ≥ 0 a.e ∂eα

' 1 () , q (1 − ρ) = 0 and α = q ∈ H0,e α

 (28)

( a.e. for α = x, y.

Since C x and C y are closed convex cones, the projection problem 1 admits a unique solution. We can write the saddle-point formulation of the problem, that is the instantiation of the abstract formulation (50) to the present situation. Proposition exists a unique pair of nonnegative Lagrange multipliers (or

6 There pressures) px , py ∈ x × y such that u+

∂py ∂px ex + ey = U ρ-a.e. ∂x ∂y

Proof The constraint set C = C x ∩ C y can be written ' ( /

.

C = u ∈ L2 ()2 , Bu| qx , qy ≤ 0 , ∀ qx , qy ∈ +

(29)

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where ⎧ 2 −1 −1 ⎨ L ()2 −→ H x () × Hy ()  ∂uy ∂ux B: ⎩ u ,− −→ − ∂x ∂y )

* and + = qx , qy ∈ x × y , qx ≥ 0 , qy ≥ 0 a.e. . By Poincaré inequality, there exists a constant β > 0 such that |B μ| ≥ β|μ|, with ⎧ 1 () × H 1 () −→ L2 ()2 ⎨ H0,x 0,y

∂qy B : ∂qx ⎩ ex + ey . qx , qy −→ ∂x ∂y Corollary 1 in the appendix guarantees the existence of a pair of Lagrange multipliers, the uniqueness comes from the one-to-one character of B given by the same inequality.   The saddle-point formulation (29) can be projected onto the two axes, leading to two independant systems. The problem then reduces to finding 1 (ux , px ) ∈ L2 () × H0,x () and



1 uy , py ∈ L2 () × H0,y ()

solutions to   ∂px  ρ-a.e.  ∂x + ux = Ux  ∂ux  ≤ 0 where ρ = 1,  − ∂x   ∂py  ρ-a.e.  ∂y + uy = Uy    − ∂uy ≤ 0 where ρ = 1.  ∂y

(30)

(31)

This is the macroscopic counterpart of what we had seen at the end of Sect. 3: two independant pressure fields appear, acting separately on each component of the velocity in order to correct the compressions in x and y. Remark 2 This model introduces anisotropy, so that the collision is no longer rotationally invariant: Fig. 6 shows the situation of two colliding blocks, under three angles of impact. In the case of an impact along one of the two principal directions, no perturbation occurs in the tangential direction whereas in the case of an impact involving both directions (second case in Fig. 6), a transverse velocity appear. Note that in the third case of two blocks colliding along a direction that is very close to one of the two axes, the post-impact velocity is mainly directed along the transverse direction.

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Fig. 6 Three impacts between opposing blocks, varying the angle of incidence. On the left, the velocity fields before the impact; on the right, the velocity fields after the impacts

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We shall now build a more pathological macroscopic model derived from the microscopic configuration of a triangular (or hexagonal) stack of particles (see Fig. 5, right). The well-posedness of the saddle-point formulation (i.e. existence and uniqueness of pressure fields) is more delicate than before: uniqueness is lost, as we shall see later on, and the existence is still an open problem. We introduce unit vectors along the principal directions:  √  √  3 3 1 1 , e2 = − , − . e0 = (1, 0) , e1 = − , 2 2 2 2 

In this case, any sphere has 6 neighbors, two along each axis directed by the ei . As for the previous configuration, one can write the microscopic constraints on the sphere 0   (u4 − u0 ) · e0 ≥ 0 , (u0 − u3 ) · e0 ≥ 0 , (u1 − u0 ) · e1 ≥ 0   (u0 − u6 ) · e1 ≥ 0 , (u5 − u0 ) · e2 ≥ 0 , (u0 − u2 ) · e2 ≥ 0

(32)

that can be reformulated by saying that ui = u · ei has to be increasing along each axis of constraint. We can thus write the macroscopic model reformulating this monotonicity exactly as above: the non-overlapping constraint becomes Bu ≤ 0, with  2 L ()2 −→ He−1 × He−1 () × He−1 () 1 2

0 B: (33) u −→ −∂e0 u0 , −∂e1 u1 , ∂e2 u2 where ui is the projection of u on the vector ei . The triangular macroscopic model then writes Problem 2 Given ρ ≤ 1 and U ∈ L2 ()2 , find u ∈ L2 ()2 that realizes the projection min

Bu≤0 

u − U 22 dρ.

(34)

The constraint Bu ≤ 0 is to be interpreted in a dual way: we then require Bu|q ≤ 0 for every nonnegative pressures q ∈ 0 × 1 × 2 , with ' 1 i = qi ∈ H0,e () , q (1 − ρ) = 0 i

( a.e. , i = 0, 1, 2.

Multibody and Macroscopic Impact Laws: A Convex Analysis Standpoint Fig. 7 Counterexample to the injectivity of B

119

−1

1

−1

1 −1

1

Since Problem 2 consists in projecting on a closed convex cone, it admits a unique solution. The saddle-point formulation reads:   Find (u, p0 , p1 , p2 ) ∈ L2 ()2 × 0 × 1 × 2 such that   2   u + ∂ei pi ei = U,   i=0  p ≥ 0,   2 i    ui ∂ei pi = 0.   i=0 

(35)

Remark 3 The operator B is not onto, as we show here by exhibiting a non-trivial element in ker (B ) . Consider an hexagon H included in , and f a piecewise constant function equal to 0 out of H , and alternatively 1 and −1 depending on the position in H , as represented on Fig. 7. Define now pi as the solution of   ∂i pi = f in H   pi = 0 in  \ H

(36)

Due to the symmetries of f , this equation is compatible with the limit condition p = 0 on ∂H : every line directed by any ei has an intersection of the same length 2 2   with zones labelled by 1 or −1. Moreover, we have ∂i pi ei = f ei = 0, so i=0

i=0

p lies in ker(B ). Remark 4 It is an interesting counterpart to the microscopic counterexample presented on Fig. 2. When the number of spheres increases, there is 3/2 times more constraints than degrees of freedom; thus the dimension of the kernel of B tends to infinity in the micro case. Accordingly, the macroscopic example above provides an infinite family of independent vectors in ker(B ).

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5 Homogenization Issues This section deals with issues pertaining to the convergence of microscopic models towards macroscopic ones. Let us make it clear that such convergence is out of reach in general. We shall rather describe a general framework to address these issues, and establish some convergence results in very particular situations, in the case when the microscopic situation is structured. The idea is the following: we start from a macroscopic velocity field, and we span the domain with a sequence of saturated configurations of spheres of radius tending to 0. At each scale, we project the field on the feasible set, which contains all those fields which comply with the non-overlapping constraint.

5.1 General Procedure We describe here a general procedure to formalize questions concerning micromacro convergence. First, we need to define a way to compare microscopic velocities to macroscopic fields. Given a field U ∈ L2 ()2 , and n non overlapping touching spheres in , denote D1 , . . . , Dn ⊂  the Voronoï cells associated to the spheres. Define U˜ i =

1 |Di |

U (x)dρ(x) ∀1 ≤ i ≤ n.

(37)

Di

Let u˜ be the solution of the microscopic problem associated to U˜ . Finally, let v be the piecewise constant function equal to u˜ on the cell Di . This mapping is depicted in Fig. 8. We have built an operator  φn :

L2 ()2 −→ L2 ()2 U −→ v

(38)

which maps a pre-collisional macroscopic velocity U to a post-collisional velocity, computed through projection at the microscopic level. We are now able to formulate the homogenization problem statement, in terms of two general questions:



? Homogenization of Impact Laws

Given a velocity field U ∈ L2 ()2 and a sequence (x n ) = (xin )1≤i≤n of collections of hard sphere configurations, with a common radius δn (with δn → 0), such that n  i=1

1B(xin,δn )

(39)

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weakly converges to some limit density when n goes to +∞, what are the possible limits of φn (U ) ? Is it possible to prescribe constraints on the microscopic structure so that φn converges to some projection operator at the macroscopic level, which would encode the characteristics of the microscopic structure ?

These questions should be seen as a wide research program which is way beyond the scope of these notes. We shall restrict ourselves to some short comments, and to providing a detailed answer in very specific situations (see Sect. 5.2). First, various sorts of constraints can be expected: isotropic ones like in Sect. 2, anisotropic ones according to some principal directions like in Sect. 4, or possibly not linked to any underlying regular structure in the grain configuration. The notion of local maximal value, already discussed in Sect. 2, is also an issue: consider e.g. a configuration where a part of the saturated domain is spanned by a square lattice, and another part is spanned by a triangular mesh. As we shall see below, the projection operators will actually converge √ towards an operator activated respectively when ρ = (1 − π/4) and ρ = π/2 3, depending on the local microscopic structure, so that the maximal density is not defined uniformly over the saturated zone. In all generality, when there is no reason to assume any regularity/periodicity in the microscopic structure, one may expect some sort of averaging in the direction of contacts, with a local constraint on the density based on the so-called Random Maximal Packing, that is around 0.64 for three-dimensional collections of identical hard spheres [31]. This may legitimate an isotropic approach like the one presented in Sect. 2.2, based on a uniform maximal density, and an isotropic constraint on the velocities. Yet, as extensively described in the literature on granular media (see

Fig. 8 On the left, the construction of a microscopic vector field U˜ from the macroscopic one U , in the square configuration. The green lines delimit the Voronoï cells associated to the spheres, the red arrows are the mean of the vector field on each cell (expanded for a sake of clarity). On the right, construction of a macroscopic vector field v (black) from a microscopic field u˜ (red)

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e.g. [30]), complex force networks are observed within collections of grains, over scales that go way beyond the microscopic size of the grains. Such observations advocate for the need to develop macroscopic models which would reflect some anisotropy at the mesoscopic scale, in the spirit of what is done in the next section for highly structured configurations. Remark 5 One could question the choice of using Voronoï Cells instead of defining the field v to be constant on every sphere, and null elsewhere. The reason is that we aim at showing strong L2 convergence results, which will not hold for velocities supported on spheres. For instance, consider the constant field U = ex for the squared configuration with radius tending to 0. As no constraint is activated, v is automatically equal to U everywhere, and similarly, u˜ ni = ex for every 1 ≤ i ≤ kn (kn being the number of spheres needed to span the domain). If we define wn piecewise constant on every sphere equal to 0 elsewhere, there subsists an irreducible gap 0 0 0U − wn 02 = (1 − π/4) λ() + o(1) 2 π/4 being the proportion of  spanned by the spheres for a square lattice.

5.2 Homogenization for Structured Configurations We detail here micro-macro convergence results in very specific situations, namely when the microscopic spheres on which we interpolate in the previous subsection are organized on square or triangular lattices. More precisely, under the framework presented at the beginning of Sect. 5.1, we establish that, given a pre-collisional velocity field U , the velocities obtained by projection at the microscopic level (operators φn defined by (38)) converge to a velocity obtained by projection at the macroscopic level, according to the projection operators detailed in Sect. 4. Here, for the sake of simplicity we consider the whole set ω = [0, 1] ×[0, 1] to be spanned by spheres disposed on a square or triangular lattice, as in Fig. 5. We can thus disregard the issues of maximal density raised in the previous section, as we know that the microscopic density measure will weakly converge to a constant, that we can set to 1. First, we need to fix some common notation for the two structures considered. Let V = L2 (ω)2 be the set of macroscopic velocities, and for n ∈ N, kn the number of spheres of radius 1/n needed to n ω, either for the square or the triangular

span configuration. Denote then V˜n = R2 the set of microscopic velocities, Vn ⊂ V the set of functions constant on each Voronoï cell Di ⊂ ω associated to the sphere configuration. We are going to use a classical theorem in Numerical Analysis, designed to estimate errors for approximated problems of optimisation. Here Vn is seen as an approximation space for V . The main idea here is to approximate not only the space

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of functions V , but also the space of constraints. We denote by Cρ ⊂ V the set of velocities satisfying the macroscopic anisotropic constraint (defined in Sect. 4), and Cn ⊂ Vn the set of velocities satisfying the microscopic constraint once the velocity of a cell is attributed to the sphere in the cell: Cn is seen as an approximation of the constraint space Cρ . We then have the following error estimate between the exact solution u and the approximate solution v: Theorem 1 (Adapted from Falk, 74’ [12]) There exists a, b > 0 such that for every f ∈ Cn and g ∈ Cρ , 1/2  u − v2 ≤ au − f 22 + bu − U 2 (u − f 2 + g − v2 )

(40)

Given U ∈ V , denote by U˜ n the approximation defined by (37) and vn = φn (U ). Denote as before u and u˜ n the solutions of the macroscopic and the microscopic problems associated to U and U˜ n , with respect to the constraint sets Cρ and Cn . Figure 8 illustrates this construction. Proposition 7 The sequence vn converges towards u in L2 (ω)2 .

1 u(x)dx on |Di | Di every Di . Since u is in Cρ , one can verify that uint n ∈ Cn . On the other hand, since u˜ n is the solution to the microscopic problem, vn is in Cρ . Therefore, we can use Theorem 1, that guarantees the existence of a, b some positive constants such that for every n ∈ N

Proof Let uint n be the piecewise constant function equal to

0 02 0 0 0 0 0 int 0 u − vn 22 ≤ a 0u − uint u  − u + b − U 0 0u 2 n n 0 2

2

(41)

It is then sufficient to show that the piecewise constant approximation uint n tends to u in L2 (ω)2 . Let  > 0 and f ∈ C 0 (ω)2 be such that u − f 2 ≤ ; denote fnint the piecewise constant approximation of f . We have 0 0 0 0 0 0 0 0 int 0 0 int 0 int 0 u  − f ≤ − f + + − u 0f 0f 0 0 0u − uint 2 n n n n 0 2

2

2

(42)

Using that f is uniformly continuous on ω, define n0 ∈ N such that for every 2 x, y ∈ ω satisfying |x − y| ≤ , |f (x) − f (y)| ≤ . Thus for n ≥ n0 n0 kn 0 02  0 0 0f − fnint 0 ≤ 2

i=1



 i∈I

Di

2    f (x) − fnint (x) dx 2 = 2.

Di

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On the other hand, using Jensen inequality kn 0 02  0 int 0 0un − fnint 0 = 2

i=1

=

kn 

Di

2    int un (x) − fnint (x) dx

  λ(Di ) 

i=1



kn  i=1

1 λ(Di )

Di

2  (u(y) − f (y)) dy 

|u(y) − f (y)|2 dy Di

= u − f 22 2 2 Thus uint n converges in L (ω) towards u, and so does vn .

 

Remark 6 In the previous proof, two ingredients can be identified as essential in the process of elaborating general homogenization results: • uint ∈ Cn : a field that respects the macroscopic constraint must check the n microscopic constraints once integrated on the Voronoï cells; and reciprocally the piecewise constant approximation vn of the corrected microscopic field must satisfy the macroscopic constraint. Thus the macro/micro constraints must be compatible under the mapping that we defined above. 2 • uint n needs to converge for the L norm toward u: this is in particular true if the spheres span the whole saturation area, in the sense that the diameter of the Voronoï cells tends to 0.

6 Evolution Models We describe here the evolution problems which are associated to the impact laws that have been described in the first sections of these notes. Let us first make it clear that writing an evolution problem associated to the impact laws studied in Sects. 4 and 5 is irrelevant a priori. Indeed, the assumptions which can be made on the microscopic structure of a granular medium are instantaneously ruled out as soon as the medium undergoes any deformation. A macroscopic model respectful of the current state of the medium in terms of microscopic structure should rely on some parameters to reflect the local organization of grains, which strongly conditions the impact law as we detailed in the previous sections. We shall rather present evolution problems for the microscopic setting, which takes the form of a second-order in time differential inclusion, and for the macroscopic scale we shall consider the isotropic setting only (the divergence is nonnegative on the saturated zone).

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Microscopic Evolution Problem Like in Sect. 2.1, we consider n moving rigid spheres centered at r1 , . . . , rn , with common radius R, subject to forces f = (f1 , . . . , fn ). We denote by m1 , . . . , mn , the masses of the grains, and by M ∈ Mnd the associated mass matrix. We denote by K the feasible set (defined by (5)), by Cr the cone of feasible direction (defined by (7)), and by Nr = Cr◦ the outward normal cone. Note that Nr is {0} as soon as r lies in the interior of K, i.e. when there is no contact. We shall consider1 that Nr = ∅ for r ∈ / K. The most concise way to write a class of evolution problems for this system, considering that impacts are frictionless, is the following (see e.g.[28]): M

d 2r + Nr ' f. dt 2

(43)

When there is no contact, Nr = {0}, and we recover n independent ODE’s in Rd . The contraint r ∈ K is implicitly prescribed because Nr = ∅ as soon as r ∈ / K. Note also that we have (see the proof of Proposition 1) ⎧ ⎫ ⎨  ⎬ Nr = − pij Gij , pij ≥ 0, ⎩ ⎭

(44)

ij

where Gij is defined by (6). It guarantees that contact forces verify the Law of Action-Reaction, and that only repulsive forces are exerted (grains do not glue to each other). Yet, Inclusion (43) is essentially compatible with all impact laws which do not violate the Law of Action-Reaction, including some laws which would lead to an increase of kinetic energy. An impact law of the type (3) has to be prescribed. We shall now write the full evolution system, in the purely inelastic setting, and with an explicit involvement of interaction forces. In the dynamic setting, these forces are generically singular in order to instantaneously change the velocities of the grains, and we shall represent them by positive measures in time, denoted by M+ (0, T ) . In the purely inelastic setting, the system writes   d 2r M  dt 2    pij   supp(pij )   u+

1 This

=f +



pij Gij

ij

∈ M+ (0, T ) ) * ⊂ t , Dij (r(t)) = 0 = PCr u− .

(45)

convention is consistent with the definition of Nr as the Fréchet subdifferential of the indicatrix function IK of K, which is indeed ∅ outside of K.

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More general impact laws can be considered, by setting u+ = u− − (1 + e)PNr u− for e ∈ [0, 1]. As detailed in [3, 17], the relevance of the impact law is ensured by the fact that the velocity has bounded variations in time, so that it admits at each time left and right limits. The system is formally well-posed in the sense that it fits in classical Cauchy– Lipschitzss theory when there is no contact, and whenever a contact occurs the impact law univocally expresses the post-collisional velocity with respect to the pre-collisional one. It can be checked that kinetic energy is preserved for e = 1, and part of it is lost during each collision for e < 1. There is indeed a well-posedness results for this system, under the condition that the forcing term f is analytic (see e.g. [3]). Counter-examples to uniqueness exist for the case of a single grain and a wall, in the elastic setting (e = 1), with a forcing term which is infinitely differentiable (see [28]). A similar counter-example can be built in the purely non-elastic case (e = 0), we again refer to [3, 28] for the analytic expression of the forcing term. In order to illustrate the principle of these counter-examples, we plot in Fig. 9 a numerical computation of two distinct solutions associated to the same forcing term, for a single particle forced toward a wall. As detailed in [17], the plotted numerical solutions correspond to two different sequences of time steps. The macroscopic counterpart of (45) is the so-called Pressureless Euler equations with maximal density constraint, which describes the motion of a granular fluid made of particles which do not interact unless saturation (set at ρ = 1) is reached. In the purely inelastic setting, the system writes ⎧ ∂t ρ + ∇ · (ρu) = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂t (ρu) + ∇ · (ρu ⊗ u) + ∇p = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ρ ≤ 1, ⎪ (1 − ρ)p = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ p ≥ 0, ⎪ ⎪ ⎪ ⎪ ⎩ + u = PC ρ (u− ).

(46)

ρ is the cone of feasible velocities defined by (10). These equation must where C be understood in a weak sense. In particular the pressure p is likely to be very singular in time, like in the microscopic setting, and the momentum equation is meant in a distributional sense. Little is known concerning this system, which is usually written without the impact law (last equation of the system). Note that this law can be replaced, at least formally, by any law of the type   u+ = u− − (1 + e) u− − PC ρ u− , with e = 1 for the elastic case. This equation is well-understood in the onedimensional setting, see e.g.[6] where particular “sticky-blocks” solutions are built,

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1 0.5 0 -0.5 -1 -1.5 -2 0.8

1/4

1/2

1

2

4

1/4

1/2

1

2

4

1/4

1/2

1

2

4

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

Fig. 9 Two distinct solutions associated to the same forcing term

and can be used to build solutions of the system. This class of solutions corresponds to the situation where the initial density is the sum of characteristic functions of segments, each one initially moving at a uniform velocity. Since no forcing term is involved, segments remains segments, possibly merging to form larger segments, and the model can be treated exactly according to the microscopic model (45). Note that this approach, presented in the purely inelastic setting, could be extended to various impact laws (e ∈ (0, 1]). Note also that, since sticky blocks reproduce the microscopic setting, the non-uniqueness result which we mentioned obviously extends to the macroscopic problem, if one accounts for a time-dependent forcing term.

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This constructive approach does not straightforwardly extend to higher dimensions for obvious reasons: the saturated zone is likely to deform in a complex way, which makes the macroscopic model fully different from the microscopic one. An extension has nevertheless been proposed recently in [5] to build solutions to (46) (without the impact law), again in a purely non-elastic spirit. A numerical approach is proposed in [9] to approximate candidate solutions to (46). It is based on barotropic Euler equations, i.e. compressible Euler equations where the pressure is assumed to be a function of the local density of barrier type: it is taken in the form p = εp(ρ), where p is smooth on [0, 1) and blows up to +∞ at 1− . When ε goes to 0, the action of the pressure disappears in non saturated zones, whereas the blow-up at 1 prevents the density to pass the maximal value. Again, the impact law is not integrated in the global limit system, but this approach natively recovers the elastic setting (e = 1). In [10], a similar approach is carried out in the case of a variable congestion (the constraint ρ ≤ 1 is replaced by ρ ≤ ρ , where ρ is a given, nonuniform, barrier density). We also refer to [26] for an analysis of a similar system with additional memory effects induced by the presence of an underlying viscous fluid. An alternative approach, also of the constructive type, is proposed in [19], it is based on a time discretization scheme of the splitting type: at each time step the density is transported according to the pre-collisional velocity (the congestion constraint is disregarded), possibly leading to a violation of the constraint. The according to the Wasserstein distance (like in the density is then projected on K crowd motion model presented in [20, 21]), and the post-collisional velocity is then a posteriori built from the projected density. This approach natively restricts to the purely non-elastic setting. Let us add that these exploratory approaches do not provide a full theoretical framework to the full system (46) (including the impact law). Let us add a few comments on the difficulty to handle the collision law, in the process of building solutions to the full system. The impact law (last equation of (46)) implicitly assumes that left and right limits exist for the velocity field, which is far from being obvious. In the microscopic setting, it is linked to the BV regularity in time of the velocity field, which makes clear sense in this purely Lagrangian setting. In the purely non-elastic setting the velocity of a given particle may undergo jumps, but each of these jumps also corresponds to a decrease of the kinetic energy. If the forcing term is controlled, the total variation due to these jumps can be shown to be bounded. In the macroscopic setting, the velocity field is defined in a Eulerian way, i.e. u(x, t) corresponds to the current velocity of the medium at x, and BV character of velocities for Lagrangian particles has no clear counterpart in this Eulerian description. Stability Issues As suggested by the non-uniqueness result for the evolution problems, the problem is unstable with respect to data, and in particular to grain positions. A striking illustration of this instability is given by the so-called Newton’s craddle, which can be described as follows: a straight raw of touching identical hard spheres is hit on one of its end by a hard sphere. Actual experiments on this setting show that the

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u−

PC r u −

PN r u−

u+ Fig. 10 Newton’s craddle: computation of u+ with Moreau’s approach, with initially touching discs (left), and slightly pulled apart discs (right)

apparent post-collisional velocity affects the sphere on the opposite side only, while the other spheres (including the hitting one) stay still. A straight application of the approach we presented (Moreau’s approach) in the elastic setting leads to a fully different picture, presented in Fig. 10 (left): the hitting sphere is pushed backward (i.e. rightward), almost as if it had hit a wall (the speed is slightly reduced), while the rest of the spheres are pushed leftward at a small velocity, in such a way that total momentum and kinetic energy are conserved. Yet, by considering an initial situation where grains are slightly pulled apart (initial distances set at an arbitrary small value), the experimentally observed behavior is recovered, after a series of quasi-simultaneous binary collisions as illustrated again in Fig. 10 (right). Similar examples of the high sensitivity of the impact law to the configuration, possibly inducing significant changes in the future behavior of the system, can straightforwardly be built for the macroscopic one-dimensional problem, in the elastic setting.

Appendix We gather here some well-known theoretical results, and some less classical ones, on the saddle-point formulation of cone-constrained minimization problems. Let V be a Hilbert space, and J : V −→ R a continuously differentiable functional. We denote by DJ (u) ∈ V  its differential at u, and by ∇J (u) its gradient: J (u + h) = J (u) + DJ (u), h + o(h) = J (u) + ∇J (u)|h + o(h).

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Linear Constraints Proposition 8 Let K be a linear subspace of V , and u a local minimizer of J over K. Then ∇J (u) is orthogonal to K, which we can write ∇J (u) + ξ = 0 , ξ ∈ K ⊥ . Proof Fix any h ∈ K. For t ∈ R in a neighborhood of 0, J (u + th) = J (u) + t ∇J (u)|h + o(t) ≥ J (u), which yields ∇J (u)|h = 0.

 

We now assume that K = ker B, where B ∈ L(V , ), and  is a Hilbert space, identified to its dual space. We furthermore restrict ourselves to the case of a quadratic functional v −→ J (v) =

1 |v − U |2 , 2

(47)

for a given U ∈ V . Proposition 9 Let K = ker B be a linear subspace of V , and u a local minimizer of J (defined by (47)) over K, the linear functional ξ defined in the previous proposition lies in B (). If we assume that B has a closed range, then ξ ∈ B (). If we identify V with its dual space, considering accordingly that B maps  to V , it means that there exists p ∈  such that 

u+B p = U Bu =0

(48)

Conversely, without any assumption on B, if (u, p) ∈ V ×  verifies (48), then u is the projection of U on K = ker B. Proof We have ξ ∈ K ⊥ = B () (see e.g. [8]). Now if B has a closed range, B has also a closed range, so that B () = B (), which yields (48). Conversely, if (48) is verified, then U − u ∈ K ⊥ , which implies that u is the projection of U on K = ker B.   Proposition 10 Under the assumption of Proposition 9, if we furthermore assume that B is onto, then λ is unique. Proof This is a straightforward consequence of ker(B ) = B(V )⊥ = {0}.

 

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Remark 7 Problem (48) is commonly called saddle-point formulation of the constrained minimization problem. Indeed, if we define the Lagrangian of the problem as L : (v, q) ∈ V ×  −→ L(v, q) = J (v) + Bv|q , then (u, p) verifies (48) if and only if it is a saddle point for L in V × , i.e. L(u, q) ≤ L(u, p) ≤ L(v, p)

∀v ∈ V , q ∈ .

Unilateral Constraints We now consider the projection of an element on a closed convex cone C. This cone, like all the cones we shall consider in this section, admits the origin as a pole, i.e. R+ C ⊂ C. More precisely, U ∈ V being given, we aim at minimizing v −→ J (v) =

1 |v − U |2 2

over C. We denote by N the polar cone to C: N = C ◦ = {v ∈ V , v|w ≤ 0 ∀w ∈ C} . Proposition 11 (Moreau [23]) Let C be a closed convex cone in V and N = C ◦ its polar cone. Then the identity in V decomposes as the orthogonal sum of the projections on C and N. In other words, for any U ∈ V , it holds that u + ξ = U , u = PC U , ξ = PN U , u|ξ  = 0. Conversely, if U = u + ξ with u ∈ C, ξ ∈ N, and u|ξ  = 0, then u = PC U and ξ = PN U . Proof For the sake of completeness, we give here a short proof of this standard result established in [23]. Let us first recall that, for any closed convex set, u is the projection of U on K if and only if u ∈ K and U − u|w − u ≤ 0

∀w ∈ K.

Applying this to the closed convex set C, and using the fact that C is a cone, we have that U − u|tw − u ≤ 0

∀w ∈ C , t ∈]0, +∞[.

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By dividing by t, and having t go to +∞, we obtain U − u|w ≤ 0, i.e. U − u ∈ N = C ◦ . Let us now prove that ξ = U − u is the projection of U on N. For any w∈N U − (U − u)|w − (U − u) = u|w + U − u|0 − u . The first term is nonnegative by polarity, and so is the second one because u is the projection of U on C. Finally, u|ξ  ≤ 0 and, since 0 ∈ N, 0 ≥ U − ξ |0 − ξ  = − u|ξ  , so that u|ξ  = 0. Conversely, if U = u + ξ with u ∈ C, ξ ∈ N, u|ξ  = 0, then for any w ∈ C U − u|w − u = ξ |w − u = ξ |w ≤ 0, so that u = PC U . The proof is similar for ξ = PN U .

 

We assume now that C is defined by duality as C = {v ∈ V , Bv|μ ≤ 0 , ∀μ ∈ + } ,

(49)

where B ∈ L(V , ),  is a Hilbert space identified to its dual, and + is a closed convex cone in  (see Fig. 11). Remark 8 One may interrogate the motivation for defining a convex cone by means of another convex cone. This approach will be proven fruitful in many situations where K is natively described in an implicit way, i.e. as the collection of elements which verify certain unilateral constraints, whereas + is defined in a explicit way, like Rd+ in the finite dimensional setting, or as a subset of real functions taking nonnegative values, so that projecting on + can be computed straightforwardly.

B

u

U

Λ− = Λ°+

C

N

h =B p B

Fig. 11 Mutually polar cones

Λ+

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Lemma 2 Let C be a closed convex cone in V , defined by (49). It holds that N = C ◦ = {w ∈ V , w|v ≤ 0 , ∀v ∈ C} = B + . Proof For any μ ∈ + , any v ∈ C, it holds B μ|v = Bv|μ ≤ 0, so that B (+ ) ⊂ N, so that B (+ ) ⊂ N. Now assume that the inclusion is strict: there exists w ∈ N, w ∈ / B (+ ). By Hahn-Banach separation Theorem, there exists h ∈ V , α ∈ R such that . / h|B μ ≤ α < h|w

∀μ ∈ + .

Since μ goes over a cone, the left hand side inequality implies that h|B μ ≤ 0 for all μ ∈ + , so that α ≥ 0 and h ∈ C by definition of C. We then have h|w > 0, which contradicts the fact that h ∈ C, w ∈ N = C o .   Let us now introduce the so-called saddle-point formulation of the projection problem  u + B p   Bu  p   Bu|p

=U ∈ − ∈ + = 0.

(50)

Remark 9 The condition p ∈ + will correspond in actual applications (impact laws) to p ≥ 0. It can be written the same way in the abstract setting, if one considers the partial order associated to the closed convex cone + (see e.g. [11]). Remark 10 The term saddle-point formulation comes from the fact that there is an equivalence between (50) and (u, p) ∈ V × + being a saddle point for the Lagrangian L(v, q) =

1 |v − U |2 + Bv|q , 2

i.e. L(u, q) ≤ L(u, p) ≤ L(v, p)

∀v ∈ V , q ∈ + .

Proposition 12 Let C be a closed convex cone in V , defined by (49), and U ∈ V . Let u be the projection of U on C. If the cone B (+ ) is closed, then there exists p ∈ + such that (u, p) is a solution to System (50), Conversely, if there exists (u, p) solution to System (50), then u is the projection of U on C.

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Proof If B (+ ) is closed, it identifies with N = C ◦ by Lemma 2, so that there exists p ∈ + such that U = u + B p by Proposition 11. Since U = u + B p is the decomposition of U over two mutually convex cones (see again Proposition 11), the two terms are orthogonal, i.e. B p|u = 0 Conversely, if (u, p) solution to System (50), then u = PC U (and B p = PN U ), thanks to Proposition 11.   Corollary 1 Under the assumptions of the previous proposition, if B is onto, then Problem (50) admits a solution (u, p), and it is unique. Proof Uniqueness is straightforward: if B is onto, then ker B = B(V )⊥ = {0}, i.e. B is one-to-one, and there exists at most one p ∈ + such that U = u + B p. Since B has a closed range, so does B by the Banach-Steinhaus theorem. As a consequence, there exists β > 0 such that |B μ| ≥ β|μ|. Now if a sequence (B μn ), with μn ∈ + , converges to w ∈ V , then (μn ) is a Cauchy sequence by the previous inequality, thus it converges to μ ∈ + , so that w = B μ ∈ B (+ ).   Remark 11 In the case of a linear space, assuming B has a closed range is enough to ensure that B () is closed (see Proposition 9 ). In the case of unilateral constraints, a stronger assumption is needed: B has to be assumed onto for ensuring the closed character of B (+ ). Indeed, the image of a closed convex cone by a linear mapping with closed range is not necessarily closed, even in the finite dimensional setting. Consider e.g.  = R3 , and the parabola ( ' P = (x, y, z) , z = 1 , y = x 2 . Now consider the closed convex cone spanned by this parabola, i.e.   3 + = conv R+ P R + ey , where ey is the unit vector in the direction y. The projection of + on the (x, y) plane is R×]0, +∞[∪ {(0, 0)}, which is not closed. Yet, an important family of cones enjoys the property of being linearly mapped onto a closed set, those are the cones spanned by a finite number of vectors. Lemma 3 Let V be a Hilbert space, and N a convex cone spanned by a finite number of vectors: N=

 n 

4 αi Gi , (α1 , . . . , αn ) ∈

Rn+

i=1

Then N is closed, as is its image by any linear mapping.

.

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Proof We give a full proof of this classical result to enlighten the importance of the fact that N is spanned by a finite number of vectors. We prove the result by induction on the number of vectors. For n = 1, the result is obvious. Assume that the property is true for n ≥ 1, and consider the cone N associated to n + 1 vectors. If the Gi s are independent, we call W the space spanned by these vectors, and we introduce G : α ∈ Rn+1 −→

n+1 

αi Gi ∈ W.

i=1

This map is invertible, and its reciprocal G−1 is linear and continuous from W to n+1 k R . Now consider v = α k Gi converging to v ∈ W . then G−1 v k converges to G−1 v, i.e. α k converges toward α in Rn+1 by continuity (G−1 is a linear mapping + between finite dimensional spaces). Now if the family is linearly dependent, there exists μ1 , . . . , μn+1 , not all equal to 0, such that n+1 

μi Gi = 0.

(51)

i=1

We consider a sequence (α k ) in Rn+1 + such that n+1 

α k Gi −→ v.

i=1

We assume (without loss of generality) that one of the coefficient of (51) is negative. We now consider, for any k, the largest β k ≥ 0 such that α k + β k μi ≥ 0 for 1 ≤ i ≤ n + 1. Equality holds for at least one of the indices. Since at least one index i0 realizes equality an infinite number of times, we extract the corresponding subsequence (without changing the notation). The limit v writes v = lim



(αik + β k μi )Gi

i =i0

which lies in the cone spanned by the n vectors (Gi )i =i0 (by the induction hypothesis), so it is in N.   We now address some theoretical issues related to the description of solution sets for the pressure p ∈  for equations of the type (50). Like in the case of equality constraints (Proposition 10), the solution p is unique a soon as B is onto, and uniqueness is lost whenever the range of B is not dense in . Yet, in the finite dimensional setting, the solution set can be proven to be bounded under some conditions which are typically met for impact laws in granular media. The approach is based on the notion of asymptotic cone (see e.g. [7]):

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Definition 2 Let V be a Hibert space, K ⊂ V a closed convex subset, and u ∈ K. The set → 5 − K = t (K − u), t >0

which does not depend on the choice of u ∈ K, is called the asymptotic cone of K (see e.g. [18]). Proposition 13 Let V be a Hibert space, and K ⊂ V a closed convex subset. For → − any u ∈ K, the asymptotic cone K is the set of directions h such that the half line u + R+ h is contained in K. → − → − Proof If u + R+ h ⊂ K, then h is in K by definition. Conversely, if h ∈ K , h writes t (v − u) for some t > 0, with v = u + h/t ∈ K, so that u + R+ h ⊂ K.   This notion provides a criterium to identify bounded convex sets (in the finite dimensional setting). Proposition 14 Let V be a finite dimensional Hibert space, and K ⊂ V a closed convex subset which contains 0. Then → − K is bounded ⇐⇒ K = {0} . → − Proof If K is bounded by M, then tK is bounded by tM, so K contains only 0. Conversely, if K is not bounded, there exists a sequence (un ) in K, with |un | → +∞. Let u be any element of K. Since V is finite-dimensional the unit sphere is compact, and one can extract a subsequence from (un −u)/|un − u|, which converges to some v ∈ H , with |v| = 1. Now consider t > 0, and θn = t/|un − u|. By convexity of K, it holds that (1 − μn )u + μn un = u + μn (un − u) = u + t

un − u ∈ K. |un − u|

→ − Since K is closed, having n go to infinity yields u + tv ∈ K. As a consequence K contains the nonzero vector v.   Note that the finite dimension is crucial in the previous proposition. Consider for example the case where V = 2 and K is the hypercube {x = (xn ) ∈ V , 0 ≤ xn ≤ 1}. The closed convex set K does not contain any half-line, while being not bounded. We may now establish the main property Proposition 15 Let V be a finite dimensional Hilbert space, C ⊂ V a closed convex cone defined by (49), U ∈ V , and u the projection of U on C. We assume that B (+ ) is closed, so that (by Proposition 12) there exists p ∈ + such that

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u + B p = U . If ker B ∩ + = {0}, then the solution set * ) S = q ∈ + , B q = B p = U − u is bounded. Proof The solution set can be written S = (p + ker B ) ∩ + , → − it is a closed convex set. Consider h ∈ S . By Proposition 13, the half line p + R+ h is contained in S ⊂ p + ker B , which implies h ∈ ker B . Since S is also contained in the cone + is a cone, it also implies that p/t + h ∈ + , for any t > 0, which yields, by having t go to 0, h ∈ + . To sum up, h ∈ ker B ∩ + = {0}. We proved → − that S = {0}, therefore (by Proposition 14), S is bounded.   We end this appendix by defining a notion which is relevant to classify problems according to some sort of abstract maximum principle. In the context of collisions, the issue can be formulated as follows: if the pre-collisional velocity fields tends to violate all the constraints, it can be expected that all contacts will be active, i.e. that all interaction forces will be positive, and the unilateral constraints turn out to be equalities. It is an essential tool to exhibit a Poisson like problem for the pressure in impact laws (see the end of Sect. 3). We shall see that this intuitive fact is sometimes ruled out, when a general property is not verified. Definition 3 (Abstract Maximum Principle) Let C be a closed convex cone in V , associated to B ∈ L(V , ) through Eq. (49). Like in proposition 12, we assume that B () and B (+ ) are closed, so that, for any U ∈ V , the system (50) admits at least a solution (u, p) ∈ V × + , where u is the projection of U on C. We say that the couple (B, + ) (which encodes the structure of the projection problem) verifies the maximum principle if BU ∈ −− = −◦+ $⇒ ∃p ∈ + s.t. BB p = BU. Proposition 16 If B verifies the abstract maximum principle defined above then, for any U such that BU ∈ −− , there exists a solution (u, p) to (50) such that BB p = BU. Proof Let us consider the problem with an equality constraint, i.e. u ∈ ker B. We denote by u the projection of U on C. From the maximum principle there exists p ∈ λ+ such that BB p = BU , which implies that u = U − B p is in K = ker B, so that u = PK U by Proposition 9. Since Bu = 0 ∈ − and p ∈ + , the couple (u, p) is also a solution to the problem with unilateral constraints (50), which ends the proof.  

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References 1. Alder, B.J., Wainwright, T.E.: Studies in molecular dynamics. I. General method. J. Chem. Phys. 31(2), 459 (1959) 2. Ames, W.F.: Numerical Methods for Partial Differential Equations. Academic Press, New York (2014) 3. Ballard, P.: The dynamics of discrete mechanical systems with perfect unilateral constraints. Arch. Ration. Mech. Anal. 154, 199–274 (2000) 4. Berthelin, F.: Existence and weak stability for a pressureless model with unilateral constraint. Math. Models Methods Appl. Sci. 12(2), 249–272 (2002) 5. Berthelin, F.: Theoretical study of a multi-dimensional pressureless model with unilateral constraint. SIAM J. Math. Anal. 49(3), 2287–2320 (2017) 6. Bouchut, F., Brenier, Y., Cortes, J., Ripoll, J.-F.: A hierarchy of models for two-phase flows. J. Nonlinear Sci. 10(6), 639–660 (2000) 7. Bourbaki, N.: Espaces Vectoriels Topologiques. Masson, Paris (1981) 8. Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, Berlin (2011) 9. Degond, P., Hua, J., Navoret, L.: Numerical simulations of the Euler system with congestion constraint. J. Comput. Phys. 230(22), 8057–8088 (2011) 10. Degond, P., Minakowski, P., Navoret, L., Zatorska, E.: Finite volume approximations of the Euler system with variable congestion. Comput. Fluids 169, 23–39 (2017). https://doi.org/10. 1016/j.compfluid.2017.09.007 11. Ekeland, I., Temam, R.: Analyse convexe et problèmes variationnels. Dunod 12. Falk, R.S.: Error estimates for the approximation of a class of variational inequalities. Math. Comput. 28(128), 963–971 1974 13. Frémond, M.: Non-Smooth Thermomechanics. Springer, Berlin (2002) 14. Golse, F., Saint-Raymond, L.: The Navier–Stokes limit of the Boltzmann equation for bounded collision kernels. Invent. Math. 155, 81–161 (2004) 15. Joly, P.: Some trace theorems in anisotripic Sobolev spaces. SIAM J. Math Anal 23(3), 799– 819 (1994) 16. Liu, C., Zhao, Z., Brogliato, B.: Frictionless multiple impacts in multibody systems: part I. Theoretical framework. Proc. R. Soc. A Math. Phys. Eng. Sci. 464(2100), 3193–3211 (2008) 17. Maury, B., A time-stepping scheme for inelastic collisions. Numer. Math. 102, 649–679 (2006) 18. Maury, B., Analyse fonctionnelle, exercices et problèmes corrigés. Ellipses, Paris (2004) 19. Maury, B., Preux, A.: Pressureless Euler equations with maximal density constraint: a timesplitting scheme. Topol. Optim. Optimal Transp. Appl. Sci. 17, 333 (2017) 20. Maury, B., Roudneff-Chupin, A., Santambrogio, F.: A macroscopic Crowd motion model of the gradient-flow type. Math. Models Methods Appl. Sci. 20(10), 1787–1821 (2010) 21. Maury, B., Roudneff-Chupin, A., Santambrogio, F., Venel, J.: Handling congestion in Crowd motion modeling. Netw. Heterog. Media 6(3), 485–519 2011 22. Maz’ya, V.: Sobolev Spaces: With Applications to Elliptic Partial Differential Equations, Grundlehren der mathematischen Wissenschaften, vol. 342. Springer, Berlin (2011) 23. Moreau, J.-J.: Décomposition orthogonale d’un espace hilbertien selon deux cônes mutuellement polaires. C. R. Acad. Sci. Paris, 255, 238–240 (1962) 24. Moreau, J.J.: Some numerical methods in multibody dynamics: application to granular materials. Eur. J. Mech. A Solids 13(4), 93–114 (1994) 25. Nguyen, N.S., Brogliato, B.: Comparisons of multiple-impact laws for multibody systems: Moreau’s law, binary impacts, and the LZB approach. In: Leine R., Acary V., Brüls O. (eds.) Advanced Topics in Nonsmooth Dynamics. Springer, Cham (2018) 26. Perrin, C., Westdickenberg, M.: One-dimensional granular system with memory effects. SIAM J. Math. Anal. 50(6), 5921–5946 (2018) 27. Ristow, G.: Simulating granular flow with molecular dynamics. J. Phys. I EDP Sci. 2(5), 649– 662 (1992)

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28. Schatzmann, M.: A class of nonlinear differential equations of second order in time. Nonlinear Anal. Theory Methods Appl. 2, 355–373 (1978) 29. Radjaï, F., Dubois, F., (eds.): Discrete-element Modeling of Granular Materials. Wiley, London (2011) 30. Radjai, F., Roux, S., Moreau, J.-J.: Contact forces in a granular packing. Chaos: Interdiscip. J. Nonlinear Sci. 9, 544–544 (1999) 31. Torquato, S., Stillinger, F.H.: Jammed Hard-particle packings: from Kepler to Bernal and beyond. Rev. Mod. Phys. 82, 2633 (2010) 32. Torquato, S., Truskett, T.M., Debenedetti, P.G.: Is random close packing of spheres well defined? Phys. Rev. Lett. 84, 2064–2067 (2000) 33. Troianiello, G.M.: Elliptic Differential Equations and Obstacle Problems, University Series in Mathematics. Springer, Berlin (1987) 34. Villani, C.: Limites hydrodynamiques de l’équation de Boltzmann [d’après C. Bardos, F. Golse, D. Levermore, P.-L. Lions, N. Masmoudi, N., L. Saint-Raymond]. Séminaire Bourbaki, vol. 2000–2001, Exp. 893

An Introduction to Uncertainty Quantification for Kinetic Equations and Related Problems Lorenzo Pareschi

Abstract We overview some recent results in the field of uncertainty quantification for kinetic equations and related problems with random inputs. Uncertainties may be due to various reasons, such as lack of knowledge on the microscopic interaction details or incomplete information at the boundaries or on the initial data. These uncertainties contribute to the curse of dimensionality and the development of efficient numerical methods is a challenge. After a brief introduction on the main numerical techniques for uncertainty quantification in partial differential equations, we focus our survey on some of the recent progress on multi-fidelity methods and stochastic Galerkin methods for kinetic equations. Keywords Uncertainty quantification · Kinetic models · Boltzmann equation · Euler equations · Multi-fidelity methods · Stochastic Galerkin methods

1 Introduction Many physical, biological, social, economic, financial systems involve uncertainties that must be taken into account in the mathematical models, for example partial differential equations (PDEs), describing these systems [4, 24, 32, 46, 50, 55, 65]. These may be due to incomplete knowledge of the system (epistemic uncertainties) or they may be intrinsic to the system and cannot be reduced through improvements in measurements, etc. (aleatoric uncertainties). Examples include uncertainty in the initial data and in the boundary conditions, or in the modeling parameters, like microscopic interactions, source terms and external forces. From the point of view of numerical methods, there are no relevant differences between the types and sources of uncertainty, so we will not be concerned about the nature of the uncertainties in the description of the system.

L. Pareschi () Department of Mathematics and Computer Science, University of Ferrara, Ferrara, Italy e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 G. Albi et al. (eds.), Trails in Kinetic Theory, SEMA SIMAI Springer Series 25, https://doi.org/10.1007/978-3-030-67104-4_5

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In this context, a particularly challenging case is represented by kinetic equations with random inputs. The construction of numerical methods for uncertainty quantification (UQ) in kinetic equations is a problem of considerable interest that has recently attracted the attention of many researchers (see [2, 8, 9, 11, 12, 15, 16, 18, 19, 21, 26, 27, 35, 36, 38, 40, 41, 57, 58, 61] and the collection [32]). Some of the main difficulties that characterize the development of efficient methods for these equations concern the high dimensionality of the problems that, besides the variables that characterize the phase space, contain stochastic parameters, and the constraints imposed by physical properties such as the positivity of the solution and the equilibrium states. The latter problem is closely related to the construction of stochastic asymptotic preserving methods [3, 31, 33, 34, 39, 66]. In this survey, we will address some recent developments in this direction based on the use of Monte Carlo-type (MC) techniques [15, 16, 30] and on the use of Stochastic Galerkin-type (SG) approaches [8, 18, 19, 52]. This short survey and the selected bibliography are obviously biased by the personal contributions and knowledge of the author, and are not intended to provide a complete overview of all the different techniques that have been developed for the quantification of uncertainty in kinetic equations. Uncertainty quantification is such a broad and active field of research that it is impossible to give credit to all relevant contributions. The rest of the manuscript is organized as follows. After a general introduction, in Sect. 2, on uncertainty quantification for PDEs and related numerical techniques, we will focus our survey on the case of kinetic equations. In the first part, we will introduce multi-fidelity techniques to accelerate the convergence of MC methods. These techniques are particularly effective in the context of kinetic equations thanks to the presence in the literature of several surrogate models constructed with the aim of reducing the computational cost of the full model, typically represented by a Boltzmann-type collision equation. Sections 3, 4, and 5 are dedicated to these aspects. In the second part, we will address the problem of the loss of structural properties of numerical schemes in the case of intrusive SG approaches. In this context, we will first illustrate a technique based on micro-macro decomposition that allows us to efficiently and accurately approximate the equilibrium states. Subsequently, we will introduce a novel hybrid approach based on a random space SG method combined with a particle approximation of the kinetic equation in the physical space. This latter technique, allows to build efficient solvers that retain all the main physical properties, including the non-negativity of the solution.

2 Uncertainty Quantification for PDEs The recent growth of interest in UQ for PDEs can be traced back mainly to three factors: widespread availability of data resulting from advances in technology, the increased development of high-performance computing and the construction and

Uncertainty Quantification for Kinetic Equation and Related Problems

Statistics about uncertain inputs

PDE

Uncertain solution of the PDE and postprocessing

143

Statistics about uncertain outputs of interest

Fig. 1 The uncertainty quantification process for PDEs

analysis of new algorithms for solving differential equations with random inputs. In presence of uncertainties it becomes necessary to quantify these effects on the solution of the PDE, or on any quantity of interest (a quantity that depends on the solution of the PDE for which we want to know some statistical information), derived from the solution. The complete UQ task then consists of determining information about the uncertainty in an output of interest that depends on the solution of a PDE, given information about the uncertainty in the inputs of the PDE (see Fig. 1).

2.1 PDEs with Random Inputs Assume a set of random parameters (a finite set of random numbers) z1 , . . . , zdz which may be collected in a vector z = (z1 , . . . , zdz )T ∈  ⊆ Rdz . The solution of the PDE is not only function of the physical variables in the phase space but also of the random vector. For example, the scalar conservation law with random inputs ∂t u(z, x, t) + ∇x · F(z, u(z, x, t)) = 0,

(1)

or the Fokker–Planck equation with uncertainty ∂t u(z, x, t) + ∇x · [B(z, x, t)u(z, x, t) − ∇x (D(z, x, t)u(z, x, t))] = 0

(2)

given the (eventually uncertain) initial data u(z, x, 0) = u0 (z, x), x ∈ Rdx , and where the terms F, B and D depend on the random parameters. A realization of a solution of the PDE is a solution obtained for a specific choice of the random parameters. One instead wants to obtain statistical information on a quantity of interest e.g., expected values, variances, standard deviations, covariances, higher statistical moments, etc. Therefore, multiple solutions of the PDE are necessary in order to achieve such information.

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The statistical quantities of interest are usually determined as follows: • From the solution u(z, x, t) of the PDE, define an output of interest F (u). • Choose what statistical information about F (u) is desired and define "(w) such that the quantity of interest is given by E["(F (u))] :=

(3)

"(F (u))p(z) dz, 

where p(z) is the probability density function (PDF) of the input parameters. Below we give some examples of quantities of interests. Example 1 (i)  F (u) = uLp (Rdx ) =

1/p Rdx

|u(z, x, t)|p dx

,

"(w) = w

will give as quantity of interest E[uLp (Rdx ) ] the expected value of the Lp norm. (ii) F (u) = u,

"(w) = (w − E[w])2

yields Var[u] the variance of the solution. (iii) "(w, v) = (w − E[w])(v − E(v)) applied to F (u1 ) and F (u2 ), where u1 and u2 are two solutions of the PDE gives Cov(F (u1 ), F (u2 )) the covariance. One of the main challenges for numerical methods, is that the computational cost associated with UQ increases with the number of parameters used to model the uncertainty (curse of dimensionality). This is a general problem but it is particularly relevant for kinetic equations where the dimension of the phase-space is very high. We can follow two main strategies to alleviate this problem: • one can try to use relatively few solutions of the PDE and replace the PDE with a surrogate, low-fidelity, model which is much cheaper to solve. Correlation between the two models may then be used in a control variate setting. • for smooth solutions one can design methods which permit an accurate evaluation of E["(F (u))] using few quadrature points obtained from stochastic orthogonal polynomials with respect to the PDF.

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We have tacitly assumed that we know p(z), the PDF of the input parameters. In practice, one usually does not know much about the statistics of the input variables and need to deal with the corresponding stochastic inverse problem [59].

2.2 Overview of Techniques Many methods have been devised in the literature for approximating statistics of quantities of interest. We summarize shortly some of the main methods below (see [4, 22–24, 32, 37, 55, 65] for recent monographs and surveys) • Monte Carlo sampling: one generates independent realizations of random inputs based on their PDF (which may be known or not and not necessarily smooth). For each realization the problem is deterministic and can be solved by standard methods in a non intrusive way. The advantage is its simplicity but on the other hand it implies a slow convergence and fluctuations in the solution statistics [6, 25]. • Multi-fidelity, multi-level methods: Accelerate Monte Carlo sampling methods by using multiple surrogate models with different levels of fidelity in a control variate setting [15, 16, 41, 53, 54]. Low-fidelity models may also be obtained from a multi-level hierarchy of numerical discretizations in the phase space [20, 22, 30, 44]. • Stochastic-Galerkin: solutions are expressed as orthogonal polynomials of the random inputs accordingly to their PDF. Spectral convergence for smooth solutions in the random space [40, 65]. They require smoothness and knowledge of the PDF. The intrusive nature may lead to the loss of physical properties and suffers of the curse of dimensionality [27, 61]. • Other methods: moment methods where the unknowns are the moments of the solution, stochastic collocation methods based on orthogonal polynomials but selecting the quadrature points [47, 67].

2.2.1 Monte Carlo (MC) Sampling Methods Let us quickly describe the simple Monte Carlo sampling method. Assume u(z, x, t), x ∈ R, solution of a PDE with uncertainty only in the initial data u0 (z, x), z ∈  ⊂ Rdz . The method does not depend on the particular solver used for the PDE or the dimension dz , and consists of three main steps. Algorithm (Simple Monte Carlo Method) 1. Sampling: Sample M independent identically distributed (i.i.d.) initial data uk,0 ,

k = 1, . . . , M

from the random initial data u0 and approximate on a grid x to get uk,0 x .

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2. Solving: For each realization uk,0 x the PDE is solved by a deterministic numerical method to obtain at time t n = nt, n > 0 uk,n x ,

k = 1, . . . , M.

3. Estimating: Estimate the desired statistical information of the quantity of interest by its statistical mean E["(F (u(·, t n )))] ≈ EM ["(F (unx ))] :=

M 1  "(F (uk,n x )). M k=1

In the sequel we will consider F (u) = u and "(w) = w, namely the quantity of interest is E[u]. Let su recall that, from the central limit theorem, the root mean square error satisfies [6, 43] E (E[u] − EM [u])2

!1/2

= Var(u)1/2 M −1/2 .

(4)

Assume that the deterministic solver satisfies an estimate of the type u(·, t n ) − unx L1 (R) ≤ Cx p ,

(5)

where p ≥ 1 and to keep notations simple we ignored the time discretization error. Let us define the following norms !1/2 02 0 (i) E[u(·, t n )] − EM [unx ]Lp (R;L2 ()) := E 0E[u(·, t n )] − EM [unx ]0Lp (R) 0 ! 0 0

0 n )] − E [un ] 2 1/2 0 E E[u(·, t (ii) E[u(·, t n )] − EM [unx ]L2 (;Lp (R)) := 0 M x 0 0

Lp (R)

Note that, by the Jensen inequality for any convex function φ we have φ (E[ · ]) ≤ E [φ(·)]

$⇒

(ii) ≤ (i).

(6)

Considering norm (ii) and p = 1, we have the error estimate E[u(·, t n )] − EM [unx ]L2 (;L1 (R)) ≤ E[u(·, t n )] − EM [u](·, t n )]L2 (;L1(R)) 6 78 9 I1 =statistical error

+ EM [u(·, t n )] − EM [unx ]L2 (;L1(R)) 6 78 9 I2 =spatial error

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These errors are bounded by I1 =

R

E (E[u(·, t n )] − EM [u(·, t n )])2

M 1  I2 ≤ |uk (·, t n ) − uk,n x |dx ≤ M R k=1



!1/2

dx ≤ C1 σu M −1/2

 M 1  Ck x p = C2 x p M k=1

with σu = Var(u(·, t n ))1/2 L1 (R) . We get the final estimate   E[u(·, t n )] − EM [unx ]L2 (;L1 (R)) ≤ C σu M −1/2 + x p .

(7)

To equilibrate the discretization and the sampling errors we should take M = O(x −2p ). Therefore, for a method of order p changing the grid from x to x/2 requires to multiply the number of samples by a factor 22p .

2.2.2 Stochastic Galerkin (SG) Methods To describe the method, let us assume that the solution of the PDE, u(z, x, t), x ∈ R, has an uncertain initial data which depends on a one-dimensional random variable z ∈  ⊂ R. The method is based on the construction of a set of orthogonal polynomials {m (z)}M m=0 , of degree less or equal to M, orthonormal with respect to the probability density function p(z) [55, 65] n (z)m (z)p(z) dz = E[m (·)n (·)] = δmn ,

m, n = 0, . . . , M.



Note that, {m (z)}M m=0 are hierarchical, in the sense that m (z) has degree m. The solution of the PDE is then represented as uM (z, x, t) =

M 

uˆ m (x, t)m (z),

(8)

m=0

where uˆ m is the projection of the solution with respect to m uˆ m (x, t) =

u(z, x, t)m (z)p(z) dz = E[u(·, x, t)m (·)]. 

(9)

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For the quantity of interest we have E["(F (uM ))] =

"(F (uM ))p(z) dz, 

which can be evaluated by the same quadrature (Gaussian) used to compute uˆ m . In case the quantity of interest is the expectation of the solution we have E[uM ] =

uM (z, x, t)p(z) dz = 

M 

uˆ m (x, t)E[m (·)] = uˆ 0 ,

m=0

whereas for the variance we get Var(uM ) =

E[u2M ] − E[uM ]2

=

M 

uˆ m uˆ n E[m n ] − uˆ 20

m,n=0

=

M 

uˆ 2m − uˆ 20 .

m=0

Stochastic Galerkin approximation in the field of random PDEs are better known under the name of generalized polynomial chaos (gPC). The solution of the PDE, is obtained by standard Galerkin approach, first replacing u with uM and then projecting the PDE to the space generated by {m (z)}M m=0 . Let us consider a general PDE in the form ∂t u(z, x, t) = J(u(z, x, t)), the stochastic Galerkin method corresponds to take E [∂t uM (·, x, t) h ] = ∂t uˆ h (x, t) = E [J(uM (·, x, t)) h ] ,

h = 0, . . . , M.

For a linear PDE J(u(z, x, t)) = L(u(z, x, t)) we get ∂t uˆ h (x, t) = L(uˆ h (x, t)),

h = 0, . . . , M.

However, for nonlinear problems, for example a bilinear PDE J(u(z, x, t)) = Q(u(z, x, t), u(z, x, t)) we get an additional quadratic cost O(M 2 ) ∂t uˆ h (x, t) =

M 

uˆ m uˆ n E[Q(m , n ) h ],

h = 0, . . . , M.

m,n=0

A general problem, is the loss of physical properties (like positivity of the solution or other invariants) due to the approximation in the orthogonal polynomial space.

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The main interest in SG methods is due to their convergence properties, known as spectral convengence, for smooth solutions in the random space. If the solution u(z, x, t) ∈ H r (), r > 0, the weighted Sobolev space, we have[65] E[(uM (·, x, t) − u(·, x, t))2 ]1/2 = uM (·, x, t) − u(·, x, t)L2 () ≤C

u(·, x, t)H r () . Mr

(10)

For analytic functions, spectral convergence becomes exponential convergence. Therefore, we must equilibrate an error relation of the type M = O(x −p/r ),

rp

and then very small values of M are sufficient to balance the errors in the method. For multi-dimensional random spaces, assuming the same degree M in each dimension, the number of degrees of freedom of the polynomial space is K=

(dz + M)! . dz !M!

For example, M = 5, dz = 3 gives K = 320 (curse of dimensionality) and typically sparse grid approximations are necessary to avoid explosive growth in the number of parameters [23, 61].

3 Uncertainty in Kinetic Equations Let us focus our attention on the specific case of kinetic equations of Boltzmann and mean-field type. More precisely, we consider kinetic equations of the general form [10, 14, 32, 64] ∂t f (z, x, v, t) + v · ∇x f (z, x, v, t) =

1 Q(f, f )(z, x, v, t), ε

(x, v) ∈ Rdx × Rdv (11)

where ε > 0 is the Knudsen number and z ∈  ⊆ Rdz is a random vector. The particular structure of the interaction term Q(f, f ) depends on the kinetic model considered. Well know examples are given by the Boltzmann equation Q(f, f )(z, x, v, t) =

B(z, v, v∗ , ω)(f (v  )f (v∗ )∗ − f (v)f (v∗ )) dv∗ dω

Sdv −1 ×Rdv

(12)

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L. Pareschi

where B(z, v, v∗ , ω) ≥ 0 is the collision kernel and v =

|v − v∗ | v + v∗ + ω, 2 2

v∗ =

|v − v∗ | v + v∗ − ω, 2 2

(13)

or by mean-field Vlasov-Fokker–Planck type models Q(f, f ) = ∇v · (P[f ]f + ∇v (D f ))

(14)

where P[·] is a non–local operator, for example of the form

P[f ](z, x, v, t) =

Rdx

Rdv

P (z, x, x∗ ; v, v∗ )(v − v∗ )f (z, x∗ , v∗ , t)dv∗ dx∗ , (15)

and D(z, v, t) ≥ 0 describes the local relevance of the diffusion.

3.1 The Boltzmann Equation with Random Inputs In the classical case of rarefied gas dynamic, we have the collision invariants R3

Q(f, f )φ(v) dv = 0,

φ(v) = 1, v, |v|2 ,

(16)

and in addition the entropy inequality R3

Q(f, f ) ln(f (v))dv ≤ 0.

(17)

The equality holds only if f is a local Maxwellian equilibrium f (z, v) = M(ρ, u, T )(z, v) =

  |u(z) − v|2 ρ(z) , exp − (2πT (z))dv /2 2T (z)

(18)

where the dependence from x and t has been omitted, and ρ=

Rdv

f dv,

u=

1 ρ

Rdv

vf dv,

T =

1 dv ρ

Rdv

(v − u)2 f dv,

(19)

are the density, mean velocity and temperature of the gas depending on (z, x, t). Integrating the Boltzmann equation against the collision invariants φ(v) yields 

∂t

R3

f (z, x, v, t)φ(v) dv + ∇x ·

R3

 vf (z, x, v, t)φ(v) dv = 0,

φ(v) = 1, v, |v|2 .

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151

These equations descrive the balance of mass, momentum and energy. However, the system is not closed since it involves higher order moments of f . The simplest way to find an approximate closure is to assume f ≈ M to obtain the compressible Euler equations with random inputs ∂t ρ(z, x, t) + ∇x · (ρu)(z, x, t) = 0 ∂t (ρu)(z, x, t) + ∇x · (ρu ⊗ u + p)(z, x, t) = 0 ∂t E(z, x, t) + ∇x · (Eu + pu)(z, x, t) = 0,

p = ρT =

(20) 1 (2E − ρu2 ). dv

Other closure strategies, like the Navier–Stokes approach, lead to more accurate macroscopic approximations of the moment system.

3.2 Numerical Methods for UQ in Kinetic Equations Two peculiar aspects of kinetic equations are the high dimensionality and the structural properties (nonnegativity of the solution, conservation of physical quantities, . . .) which represent a challenge for numerical methods. These difficulties are even more striking in the context of UQ. We summarize below the main advantages and drawbacks of MC and SG methods. MC Methods for UQ 1. easy non intrusive application as they rely on existing numerical solvers. Efficiency and structural properties are inherited from the existing solvers. 2. lower impact on the curse of dimensionality. Easy to parallelize and convergence is independent of the dimension of the random space. 3. can be applied even if the PDF of the random vector is not known or lacks of regularity. 4. convergence behavior is slow. SG Methods for UQ 1. application is intrusive and problem dependent. Hard to combine with stochastic methods (phase space) and structural properties often are lost. 2. suffer the curse of dimensionality, in particular for nonlinear problems, and special techniques are required to reduce the computational cost. 3. require knowledge and smoothness of the PDF. 4. can achieve high accuracy, spectral accuracy for smooth solutions, in the random space. In the next sections we will focus on some of the recent progress on MC methods based on multi-fidelity techniques and on stochastic Galerkin methods using micromacro decomposition and hybrid approaches.

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4 Single Control Variate (bi-Fidelity) Methods In this section we describe the construction of multiscale control variate (MSCV) methods based on a single low fidelity model [15]. To simplify the presentation, we restrict to kinetic equations with random initial data f (z, x, v, 0) = f0 (z, v, x) and focus on E[f ] as quantity of interest. We introduce some preliminary notations. For a random variable X taking values in a Banach space B(R) we define XB(R;L2 ()) = E X2

!1/2

B(R) ,

XL2 (;B(R)) = E X2B(R)

!1/2 .

We assume that the equation has been discretized by a deterministic solver on a grid v and x, which satisfies [17, 60]

n f (·, t n ) − fx,v B(R) ≤ C x p + v q ,

(21)

where, for example, B(R) = L12 (D × Rdv ) with p f (z, ·, t)Lp (D×Rdv ) s

=

D×Rdv

|f (z, x, v, t)|p (1 + |v|s ) dv dx.

(22)

For the Monte Carlo method therefore we have the error estimate

n E[f ](·, t n )−EM [fx,v ]B(R;L2 ()) ≤ C1 σf M −1/2 +C2 x p + v q

(23)

with σf = Var(f )1/2 B(R) .

4.1 Space Homogeneous Case First, we describe the method for the space homogeneous problem ∂f = Q(f, f ), ∂t

(24)

where f = f (z, v, t) with initial data f (z, v, 0) = f0 (z, v). Under suitable assumptions [62, 63], f (z, v, t) → f ∞ (z, v) exponentially as t → ∞, where f ∞ (z, v) is the Maxwellian equilibrium state s.t. Q(f ∞ , f ∞ ) = 0. We denote the moments as mφ (f )(z, t) := φ(v)f (z, v, t)dv, φ(v) = 1, v, |v|2 /2. Rdv

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153

MSCV methods aim at improving the MC estimate by considering the solution of a low-fidelity model f˜(z, v, t), whose evaluation is significantly cheaper than f (z, v, t), s.t. mφ (f˜) = mφ (f ) and that f˜ → f ∞ as t → ∞. The hope is that the cheaper low-fidelity model can be used to speed up, without compromising accuracy, the approximation of the quantities of interest corresponding to the highfidelity model.

4.1.1 Local Equilibrium Control Variate Let us recall that a Monte Carlo estimator for E[f ] based on M samples gives E[f ](v, t) − EM [f ](v, t)B(R;L2 ()) ≤ Cσf M −1/2 .

(25)

Now introduce the micro-macro decomposition [18, 19, 42] g(z, v, t) = f (z, v, t) − f ∞ (z, v),

(26)

we have g(z, v, t) → 0 as t → ∞ and so Var(g) → 0 as t → ∞. We can decompose the expected value of the solution as E[f ](v, t) =

E[f ∞ ](v) 6 78 9 Accurate evaluation

+

E[g](v, t). 6 78 9

(27)

Monte Carlo estimate

Since f ∞ (z, v) is known, we can assume E[f ∞ ](v) is evaluated with a negligible error and use the Monte Carlo estimator only on E[g] to get

E[f ](v, t) − E[f ∞ ](v) + EM [g](v, t) B(R;L2 ()) = E[g](v, t) − EM [g](v, t)B(R;L2 ()) ≤ Cσg M −1/2 .

(28)

The resulting estimate now depends on σg = Var(g)1/2  instead of σf , where now σg → 0 as t → ∞. Therefore, the statistical error vanish asymptotically in time.

4.1.2 Time Dependent Control Variate For a time dependent low-fidelity model f˜(z, v, t) given M samples f k (v, t), k = 1, . . . , M we define   λ [f ](v, t) := EM [f ](v, t) − λ EM [f˜](v, t) − E[f˜](v, t) E[f ](v, t) ≈ EM   M M 1  ˜k 1  k ˜ f (v, t) − λ = f (v, t) − f(v, t) , M M k=1

k=1

(29)

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L. Pareschi

λ [f ] with ˜f = E[f˜] or an accurate approximation. It is immediate to verify that EM is an unbiased estimator for any choice of λ ∈ R. In particular, the above estimator includes 0 [f ] = E [f ] is the simple MC estimator. • λ = 0 $⇒ EM M 1 [f ] is the local equilibrium control variate estimator. • λ = 1, f˜ = f ∞ $⇒ EM

Let us consider the random variable f λ (z, v, t) = f (z, v, t) − λ(f˜(z, v, t) − ˜f(v)).

(30)

λ [f ](v, t) = E [f λ ](v, t) and its variance is We have EM M

Var(f λ ) = Var(f ) + λ2 Var(f˜) − 2λCov(f, f˜). We can minimize the variance by direct differentiation to get ∂Var(f λ ) = 2λVar(f˜) − 2Cov(f, f˜) = 0. ∂λ As a consequence we have the following result. Proposition 1 The quantity λ∗ =

Cov(f, f˜) minimizes Var(f λ ) at (v, t) and gives Var(f˜)

 Cov(f, f˜)2 2 Var(f ) = (1 − ρf, )= 1− )Var(f ), f˜ Var(f˜)Var(f ) 

Var(f

λ∗

(31)

where ρf,f˜ ∈ [−1, 1] is the correlation coefficient of f and f˜. We have lim λ∗ (v, t) = 1,

t →∞



lim Var(f λ )(v, t) = 0

t →∞

∀ v ∈ Rdv .

(32)

The resulting MSCV method can be implemented as follows. Algorithm (Space Homogeneous MSCV Method) 0 on 1. Initialize the control variate: From the random initial data f0 compute f˜v 0 ]. the mesh v and denote by ˜f0v an accurate estimate of E[f˜v k 2. Sampling: Sample M i.i.d. initial data f0 , k = 1, . . . , M from the random initial data f0 . n at time t n by a 3. Solving the control variate: Compute the control variate f˜v n n suitable scheme and denote by ˜fv an accurate estimate of E[f˜v ].

Uncertainty Quantification for Kinetic Equation and Related Problems

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4. Solving: For each realization f0k , k = 1, . . . , M the kinetic equation and the control variate are solved by the deterministic schemes. Denote the solutions at k,n k,n time t n by fv , and f˜v , k = 1, . . . , M. n n CovM (fv , f˜v ) 5. Estimating: Estimate λ∗ using the M samples as λ∗,n . M = n ˜ VarM (fv )) Compute the expected value of the random solution as M 1  k,n λ∗ n [fv ] = fv − λ∗,n E˜ M M M k=1



 M 1  ˜k,n ˜n fv − fv . M k=1

Compared to standard MC no additional cost is required until the low-fidelity models can be evaluated off line, for example if we take f˜(z, v, t) = e−t f0 (z, v) + (1 − e−t )f ∞ (z, v) . 78 9 6

f˜(z, v) = f ∞ (z, v), 6 78 9 equilibrium state

BGK approximation

Using such an approach one obtains an error estimate of the type ∗

λ n [fv ]B(R;L2 ()) E[f ](·, t n ) − EM ∗

λ ≤ E[f ](·, t n ) − EM [f ](·, t n )B(R;L2 ()) ∗



λ λ n +EM [f ](·, t n ) − EM [fv ]B(R;L2 ())   ≤ C σf λ∗ M −1/2 + v q ,

(33)

where σf λ∗ = (1 − ρ 2 ˜ )1/2Var(f )1/2 B(R) . The statistical error depends on the f,f

correlation between f and f˜. Since ρf,f˜ → 1 as t → ∞ the statistical error will vanish for large times. In Fig. 2 we report the results for the space homogeneous Boltzmann equation with uncertain initial data using various control variates and values of λ. In Fig. 3 we report the time evolution of the optimal value function λ in the case of a control variate approach based on the BGK approximation. The initial condition is a two bumps problem with uncertainty ρ0 f0 (z, v) = 2π

     |v − (2 + sz)|2 |v + (1 + sz)|2 exp − + exp − σ σ

(34)

with s = 0.2, ρ0 = 0.125, σ = 0.5 and z uniform in [0, 1]. The deterministic solver adopted for the Boltzmann equation is the fast spectral method [17, 45] and the discretization parameters are such that the stochastic error dominates the computation (see [15] for more details). We can see from the computations that

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Fig. 2 Homogeneous relaxation. Error in E[f ] using various control variate strategies. Left M = 10, right M = 100 1 1.1

1 0.9999 0.9

0.9998

0.8

0.7 0.9997 0.6

0.5 0.9996 0.4

0.9995

0.3

Fig. 3 Optimal λ∗ (v, t) for the MSCV method based on a BGK control variate at t = 10 (left) and t = 50 (right)

with the optimal method based on the BGK model we can gain almost two digits of precision for the same computational cost. Note that, to divide the MC error by a factor 100 we need to multiply the number of samples by 10,000!

4.2 Non Homogeneous Case Consider now, a general space non homogeneous kinetic equation with random inputs ∂t f (z, x, v, t) + v · ∇x f (z, x, v, t) =

1 Q(f, f )(z, x, v, t). ε

(35)

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157

For an approximated (low fidelity) solution f˜(z, x, v, t) the estimator reads   λ [f ](x, v, t) = EM [f ](x, v, t) − λ EM [f˜](x, v, t)] − ˜f(x, v, t) , EM where ˜f(x, v, t) is an accurate approximation of E[f˜](x, v, t). The simplest control variate choice, which naturally generalizes the equilibrium control variate in the space homogeneous case, is to consider the solution of the compressible Euler system as control variate. Namely f˜ = fF∞ the equilibrium state corresponding to UF = (ρF , uF , TF ) solution of the fluid model (20). Improved control variates are obtained using more accurate fluid-models, like the Navier– Stokes system, or a simplified kinetic model, like a relaxation model of BGK type with QBGK (f˜, f˜) = ν(f˜∞ − f˜),

ν > 0.

The fundamental difference between the space homogeneous and the space non homogeneous case, is that now the variance of f λ (z, x, v, t) = f (z, x, v, t) − λ(f˜(z, x, v, t) − ˜f(x, v, t))

(36)

will not vanish asymptotically in time, unless the kinetic equation is close to the surrogate model (fluid regime), namely for small values of the Knudsen number. Proposition 2 The quantity λ∗ =

Cov(f, f˜) minimizes Var(f λ ) at (x, v, t) and Var(f˜)

gives ∗

2 )Var(f ), Var(f λ ) = (1 − ρf, f˜

(37)

where ρf,f˜ ∈ [−1, 1] is the correlation coefficient between f and f˜. In addition, we have ∗

lim λ∗ (x, v, t) = 1,

lim Var(f λ )(x, v, t) = 0

ε→0

∀ (x, v) ∈ Rdx × Rdv .

ε→0

(38) Contrary to the space homogeneous case, one cannot ignore the computational cost of solving the macroscopic fluid equations or the BGK model, although considerably smaller than that of the Boltzmann collision operator. Using ME  M samples for the control variate, we get the error estimate ∗

λ n [fx,v ]B(R;L2 ()) E[f ](·, t n ) − EM,M E

'

−1/2

≤ C σf λ∗ M −1/2 + τf λ∗ ME

+ x p + v q

where σf λ∗ = (1 − ρ 2 ˜ )1/2 Var(f )1/2 B(R) , τf λ∗ = ρf,f˜ Var(f )1/2 B(R) . f,f

(39) (

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L. Pareschi

Again the statistical error depends on the correlation between f and f˜. In this case, ρf,f˜ → 1 as ε → 0, therefore the statistical error will depend only on the fine scale sampling in the fluid limit. We point out that, the optimal value of λ depends on the quantity of interest and in practice does not depend on (x, v, t) unless one is interested in the details of the distribution function. For a general moment mφ (f ), the optimal value depends on (x, t) and is given by λ∗ =

Cov(mφ (f ), mφ (f˜)) Var(mφ (f˜))

Remark 1 We remark that, by the central limit theorem we have Var(EM [f ]) = M −1 Var(f ). Therefore, taking into account the number of effective samples in the minimization process and using the independence of the estimators EM [·] and EME [·] we get λ Var(EM,M [f ]) = M −1 Var(f − λf˜) + ME−1 Var(λf˜) E   = M −1 Var(f ) − 2λCov(f, f˜) + (M −1 + ME−1 )λ2 Var(f˜).

Minimizing with respect to λ yields the effective optimal value λ˜ ∗ which reads λ˜ ∗ =

ME λ∗ , M + ME

λ∗ =

Cov(f, f˜) . Var(f˜)

(40)

Let Cost(·) denote computational cost to compute the solution of a given model for a fixed value of the random parameter. The total cost is MCost(f ) + ME Cost(f˜). Fixing a given cost for both models MCost(f ) = ME Cost(f˜), we obtain λ˜ ∗ =

Cost(f ) Cost(f ) + Cost(f˜)

λ∗ .

In our setting since Cost(f )  Cost(f˜), or equivalently ME  M, we have λ˜ ∗ ≈ λ∗ . As a numerical example of the performance of the method let us consider the Boltzmann equation with dx = 1, dv = 2 for the following Sod test with uncertain initial data ρ0 (x) = 1, T0 (z, x) = 1 + sz

if 0 < x < L/2

ρ0 (x) = 0.125, T0 (z, x) = 0.8 + sz

if L/2 < x < 1

(41)

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159

Fig. 4 Sod test with uncertain initial data. E[T ] and confidence bands at t = 0.875 (top). Left ε = 10−2 , right ε = 10−3 . Error in E[T ] with M = 10 and ε = 10−2 (bottom). Left ME = 103 , right ME = 104 . Here Nx = 100, Nv = 32

with s = 0.25, z uniform in [0, 1] and equilibrium initial distribution   |v|2 ρ0 (x) exp − f0 (z, x, v) = . 2π 2T0 (z, x) Since we are interested only in the accuracy in the random variable, the numerical parameters of the deterministic discretization Nv , Nx and t have been selected such that the deterministic error is smaller than the stochastic one (see [15] for further details). In Fig. 4 (top), we report the expectation of the solution at the final time together with the confidence bands. In the same figure (bottom) we also report the various errors using different control variates for the expected value of the temperature as a function of time. The optimal values of λ∗ (x, t) have been computed with respect to the temperature. The improvements obtained by the various control variates are evident and, as expected, becomes particularly striking close to fluid regimes. Next we consider in the same setting the sudden heating problem with uncertain boundary condition. Initial condition is a local equilibrium with ρ0 = 1, u0 = 0,

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L. Pareschi

Fig. 5 Sudden heating with uncertain boundary condition. E[T ] and confidence bands at t = 0.9 (top). Left ε = 10−2 , right ε = 10−3 . Error in E[T ] with M = 10 and ε = 10−3 (bottom). Left ME = 103 , right ME = 104 . Here Nx = 100, Nv = 32

T0 = 1 for x ∈ [0, 1] with diffusive equilibrium boundary condition with uncertain wall temperature Tw (z, 0) = 2(T0 + sz)

(42)

with s = 0.2, z uniform in [0, 1]. The results are summarized in Fig. 5, where we report the expectation of the temperature at the final time and the various errors using different control variates. In this case, due to the source of uncertainty at the boundary there is no relevant difference between the Euler and BGK control variates and the results is less sensitive to the choice of the Knudsen number.

5 Multiple Control Variate (Multi-Fidelity) Methods The bi-fidelity approach developed in the previous sections is fully general and accordingly to the particular kinetic model studied one can select a suitable approximated solution as control variate which acts at a given scale. In this section we extend the methodology to the use of several approximated solutions as control

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161

Fig. 6 High fidelity model and low fidelity surrogate models vs. cost

variates with the aim to further improve the variance reduction properties of MSCV methods (see Fig. 6). Given f˜1 , . . . , f˜L approximations of f (z, v, t) we can consider the random variable f λ1 ,...,λL (z, v, t) = f (z, v, t) −

L 

λh (f˜h (z, v, t) − ˜fh (v, t)),

(43)

h=1

where ˜fh (v, t) = E[f˜h ](v, t). The variance is given by Var(f λ1 ,...,λL ) = Var(f ) +

L 

λ2h Var(f˜h )

h=1



+2

L  h=1

⎞ L  ⎜ ⎟ λh ⎝ λk Cov(f˜h , f˜k ) − Cov(f, f˜h )⎠ , k=1 k =h

or in vector form Var(f  ) = Var(f ) + T C − 2T b

(44)

where  = (λ1 , . . . , λL )T , b = (Cov(f, f1 ), . . . , Cov(f, fL ))T and C = (cij ), cij = Cov(fi , fj ) is the symmetric L × L covariance matrix. Proposition 3 Assuming the covariance matrix C is not singular, the vector ∗ = C −1 b, minimizes the variance of f  at the point (v, t) and gives   bT (C −1 )T b ∗ Var(f ). Var(f  ) = 1 − Var(f )

(45)

(46)

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In fact, the optimal values λ∗h , h = 1, . . . , L are found by equating to zero the partial derivatives with respect to λh . This corresponds to the linear system Cov(f, f˜h ) =

L 

λk Cov(f˜h , f˜k ),

h = 1, . . . , L,

(47)

k=1

or equivalently C = b. Example 2 Let us consider the case L = 2, where f˜1 = f0 and f˜2 = f ∞ . The optimal values λ∗1 and λ∗2 are readily found and are given by λ∗1 =

Var(f ∞ )Cov(f, f0 ) − Cov(f0 , f ∞ )Cov(f, f ∞ ) , 

Var(f0 )Cov(f, f ∞ ) − Cov(f0 , f ∞ )Cov(f, f0 ) , λ∗2 = 

(48)

where  = Var(f0 )Var(f ∞ ) − Cov(f0 , f ∞ )2 . Using M samples the optimal estimator reads λ∗ ,λ∗2

EM1

(v, t) =EM [f ](v, t) − λ∗1 (EM [f0 ](v) − f0 (v))

− λ∗2 EM [f ∞ ](v) − f∞ (v) .

(49)

Since lim f (v, t) = f ∞ (v) we get t →∞

lim λ∗ t →∞ 1

= 0,

lim λ∗ t →∞ 2

= 1,

and thus, the variance of the estimator vanishes asymptotically in time λ∗ ,λ∗2

lim EM1

t →∞

(v, t) = f∞ (v).

In Fig. 7 we report the results obtained for the homogeneous relaxation problem with uncertain initial data. Compared to the optimal BGK control variate, at the same computational cost, using the estimator based on two control variates described above we can gain one additional digit of accuracy.

5.1 Hierarchical Methods Now, let us assume f1 , . . . , fL control variates with an increasing level of fidelity. The idea is to apply recursively a bi-fidelity approach where the level fh−1 is used as control variate for the level fh .

Uncertainty Quantification for Kinetic Equation and Related Problems

163

10-2 10-3

L2 error E[f]

10-4 10-5 10-6 10-7 MC MSCV MSCV2

10-8 10-9 0

1

2

3

4

5

6

7

8

9

10

time

Fig. 7 Homogeneous relaxation. Error for E[f ] over time with M = 10 for the MSCV method based on BGK and the MSCV2 method based on the two control variates f0 and f ∞

To start with, we estimate E[f ] with ML samples using fL as control variate

E[f ] ≈ EML [f ] − λˆ L EML [fL ] − E[fL ] . Next, to estimate E[fL ] we use ML−1  ML samples with fL−1 as control variate

E[fL ] ≈ EML−1 [fL ] − λˆ L−1 EML−1 [fL−1 ] − E[fL−1 ] . Similarly, in a recursive way, we can construct estimators for the remaining expectations of the control variates E[fL−1 ], E[fL−2 ], . . . , E[f2 ] using respectively ML−2  ML−3  . . .  M1 samples until

E[f2 ] ≈ EM1 [f2 ] − λˆ 1 EM1 [f1 ] − E[f1 ] , and we stop with the final estimate using M0  M1 E[f1 ] ≈ EM0 [f1 ].

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By combining the estimators of each stage we define the hierarchical estimator

ˆ EL [f ] := EML [f ] − λˆ L EML [fL ] − EML−1 [fL ]

+ λˆ L−1 EML−1 [fL−1 ] − EML−2 [fL−1 ]

(50)

...



+ λˆ 1 EM1 [f1 ] − EM0 [f1 ] . . . .

The estimator can be recast in the form ˆ

EL [f ] = EML [fL+1 ] −

L 

λh (EMh [fh ] − EMh−1 [fh ])

h=1

(51) L  (λh+1 EMh [fh+1 ] − λh EMh [fh ]), = λ1 EM0 [f1 ] + h=1

ˆ = (λˆ 1 , . . . , λˆ L )T and where we defined  λh =

L :

λˆ j ,

h = 1, . . . , L,

λL+1 = 1,

fL+1 = f.

(52)

j =h

The total variance of the resulting estimator is ˆ

Var(EL [f ]) = λ21 M0−1 Var(f1 ) +

L 

(53) ( ' Mh−1 λ2h+1 Var(fh+1 ) + λ2h Var(fh ) − 2λh+1 λh Cov(fh+1 , fh ) .

h=1

By direct differentiation we get the tridiagonal system for h = 1, . . . , L −1 {λh Var(fh ) − λh−1 Cov(fh , fh−1 )} Mh−1

+ Mh−1 {λh Var(fh ) − λh+1 Cov(fh+1 , fh )} = 0,

(54)

which under the assumption Mh  Mh−1 leads to the quasi-optimal solutions λ∗h =

L : j =h

λˆ ∗j ,

λˆ ∗j =

Cov(fj +1 , fj ) . Var(fj )

(55)

Uncertainty Quantification for Kinetic Equation and Related Problems

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In the case of a space homogeneous kinetic equation the hierarchical MSCV estimator (51) satisfies the error bound ˆ∗

n ]B(R;L2 ()) E[f ](·, t n ) − EL [fv

 ≤C

L 

 (56) −1/2

ξh σh Mh

−1/2

+ ξ0 M0

+ v q

h=1

0 0 1/2 0 0 2 1/2 0 1 − ρ where σh = 0 Var(f ) , τh = ρfh ,fh−1 Var(fh )1/2 B(R) h fh ,fh−1 0 0 B(R) ; and ξh = L j =h+1 τj . If the control variates share the same behavior as t → ∞, namely fh → f ∞ for h = 1, . . . , L, we get ρf2h ,fh−1 → 1 as t → ∞ the statistical error depends only on the finest level of samples M0 . Similar considerations hold in the space non homogeneous case as ε → 0. In Fig. 8 the results obtained in the case of the sudden heating problem (42) using a three models hierarchy based on the Euler system, the BGK model and the full Boltzmann equation are reported.

5.2 Multi-Level Monte Carlo Methods There is a close link between multi-fidelity methods and multi-level Monte Carlo methods. Let us consider as control variates a hierarchy of discretizations of the

Fig. 8 Sudden heating with uncertain boundary condition. Error for E[T ] over time for ε = 10−2 (left) and ε = 10−3 (right). MSCV method based on BGK and MSCVH2 based on BGK and Euler. M2 = 10 for Boltzmann, M1 = 102 for BGK and M0 = 104 for Euler

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kinetic equation. For example, in the homogeneous case, with a cartesian grid we take vh = 21−h (v1 ),

h = 1, . . . , L

where v1 is the mesh width for the coarsest resolution, which corresponds to the solution with the lowest level of fidelity. Our full model is, therefore, represented by the fine scale solution obtained for vL . The hierarchy of numerical solutions fh (z, v, t), h = 1, . . . , L, at time t with mesh vh represents the setting for the multi-level control variate estimators. In particular, fixing all λh = 1, h = 1, . . . , L, we get the classical Multi-level MC estimator [22] EL1 [f ](v, t) = EM0 [f1 ] +

L  (EMh [fh+1 − fh ]),

(57)

h=1

where we used the notation 1 = (1, . . . , 1)T . Using the quasi-optimal values (or the optimal values) for λh with the hierarchical grid constructed above we obtain a quasi-optimal (optimal) MLMC [16, 30]. The main difference, compared to multi-fidelity models is the possibility to compute accuracy estimates between the various levels. On the other hand, the approach depends on additional parameters (the various grid sizes) which make its practical realization more involved to achieve optimal performances. In Fig. 9 we report the results in the case of a space non homogeneous BGK model close to the fluid limit ε = 10−6 for the Sod shock tube problem with uncertainty on the interface location (see [30] for more details). Improvements in the error curves obtained using the quasi-optimal and optimal MLMC over standard MLMC can be observed. 10-3

7

1.1 MLMC quasi-MLMC optimal-MLMC Ref

1 0.9

MLMC quasi-MLMC optimal-MLMC

6 5

0.8 0.7

4

0.6 3

0.5 0.4

2

0.3

1

0.2 0.1

0

0.2

0.4

0.6

0.8

1

0

0

0.2

0.4

0.6

0.8

1

Fig. 9 Multilevel MC for BGK. E[ρ] (left) and error in space (right) for the various MLMC using L = 3, M0 = 320, M1 = 80, M2 = 20 with xh = 21−h (x1 ), h = 1, 2, 3, x1 = 0.1

Uncertainty Quantification for Kinetic Equation and Related Problems

167

6 Structure Preserving Stochastic-Galerkin (SG) Methods For notation simplicity, let us assume a one-dimensional random space z ∈  ⊂ R, with z distributed as p(z), for a space homogeneous kinetic equation. We approximate f (z, v, t) by its generalized Polynomial Chaos (gPC) expansion [65] fM (z, v, t) =

M 

fˆm (v, t)m (z),

(58)

m=0

where {m (z)}M m=0 are a set of orthogonal polynomials, of degree less or equal to M, orthonormal with respect to p(z) n (z)m (z)p(z) dz = E[m (·)n (·)] = δmn ,

m, n = 0, . . . , M.



In (58) the coefficients fˆm are the projection of the solution with respect to m fˆm (v, t) =

f (z, v, t)m (z)p(z) dz = E[f (·, v, t)m (·)].

(59)



Stochastic Galerkin (SG) methods for kinetic equations based on the use of deterministic methods in the phase space have demonstrated numerical and theoretical evidence of spectral accuracy[11, 27–29, 32, 40, 61]. However, their practical application presents some drawbacks. • SG methods lead to the loss of the physical properties, like positivity and " conservation of moments. For example, if we denote by mφ (f ) = f φ(v) dv the moments of f , we have mφ (f ) = mφ (fM ) =

M 

mφ (fˆm )m .

m=0

These properties are essential to characterize the long time behavior of the system and the Maxwellian equilibrium states [17, 51]. • One possibility, is to modify the coefficients fˆm in the gPC expansion or the polynomial basis in such a way that the macroscopic moments of f (or positivity) are preserved. This approach, however, is rather difficult in general and typically leads to the loss of spectral accuracy [7, 49]. • Additionally, for nonlinear hyperbolic conservation laws, like the Euler system in the fluid-dynamic limit, the generalized polynomial chaos expansion may lead to the loss of hyperbolicity of the resulting approximated system (see [12, 13, 28, 35, 56]).

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6.1 Equilibrium Preserving SG Methods for the Boltzmann Equation In this section we describe a general approach based on SG methods that permits to recover the correct long time behavior. To this aim, let us consider the space homogeneous Boltzmann equation ∂t f (z, v, t) = Q(f, f )(z, v, t). The standard SG method reads ∂t fˆh = Qˆ h (fˆ, fˆ),

h = 1, . . . , M

where fˆ = (fˆ0 , . . . , fˆM )T and ˆ h (fˆ, fˆ) = Q

M 

fˆm fˆn E [Q(m , n )h ] .

m,n=1

Using the decomposition f (z, v, t) = f ∞ (z, v) + g(z, v, t) from the bilinearity of Q and the fact that Q(f ∞ , f ∞ ) = 0 we get Q(f, f )(z, v, t) = Q(g, g)(z, v, t) + L(f ∞ , g)(z, v, t), where L(·, ·) is a linear operator defined as L(f ∞ , g)(z, v, t) = Q(g, f ∞ )(z, v, t) + Q(f ∞ , g)(z, v, t). Thus we can apply the SG projection to the transformed problem ∂t g(z, v, t) = Q(g, g)(z, v, t) + L(f ∞ , g)(z, v, t), which admits g ∞ (z, v) ≡ 0 as unique equilibrium state. We can write the equilibrium preserving SG method as ˆ h (fˆ∞ , g), ˆ g) ˆ +L ˆ ∂t gˆh = Qˆ h (g, fˆh = fˆh∞ + gˆh

h = 0, . . . , M

Uncertainty Quantification for Kinetic Equation and Related Problems

169

∞ )T and where gˆ = (gˆ0 , . . . , gˆM )T , fˆ∞ = (fˆ0∞ , . . . , fˆM

ˆ h (fˆ∞ , g) ˆ = L

M 

fˆm∞ gˆn E [L(m , n )h ] .

m,n=1 ∞ = 0) are a local equilibrium of the SG The values gˆh = 0 (or equivalently gM ∞ ∞ ) are a local equilibrium scheme and thus fˆh = fˆh (or equivalently fM = fM state. By substituting to gˆ h = fˆh − fˆh∞ the SG scheme can be rewritten as

ˆ fˆ, fˆ) − Q( ˆ fˆ∞ , fˆ∞ ), ∂t fˆh = Q(

h = 1, . . . , M,

or equivalently ∞ ∞ ∂t fM (z, v, t) = QM (fM , fM )(z, v, t) − QM (fM , fM ),

QM (fM , fM )(z, v, t) =

M 

Qˆ m (fˆ, fˆ)m (z).

m=0

If we have a spectral estimatefor Q(f, f ), namely for f ∈ H r () Q(f, f ) − QM (fM , fM )L2 () ≤

C

f H r () + Q(fM , fM )H r () Mr

and the equilibrium state f ∞ ∈ H r (), since Q(f ∞ , f ∞ ) = 0, we have ∞ ∞ QM (fM , fM )L2 () ≤

C ∞ ∞ ∞ f H r () + Q(fM , fM )H r () Mr

which provide a spectral estimate for the equilibrium preserving SG method.

6.2 Generalizations for Nonlinear Fokker–Planck Problems The approach just described applies to a large variety of kinetic equations where the equilibrium state is know. In the case of Fokker–Planck equations, the method can be generalized to the situation where the steady state is not known in advance [19]. The idea is based on the notion of quasi-equilibrium state. To this aim given a one-dimensional Fokker–Planck equation characterized by Q(f, f ) = ∂v (P[f ]f (z, v, t) + ∂v (D(z, v)f (z, v, t))) ,

(60)

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L. Pareschi

we can consider solutions of the following problem P[f ]f (z, v, t) + ∂v (D(z, v)f (z, v, t)) = 0, which gives

D(z, v)∂v f (z, v, t) = P[f ](z, , v, t) + D  (z, v) f (z, v, t). The above problem can be solved analytically for f = f ∞ (v) only in some special cases. More in general we can represent a quasi-stationary solution in the form  f q (z, v, t) = C exp −

v

−∞

P[f ](z, v∗ , t) + D  (z, v∗ ) dv∗ D(z, v∗ )

 (61)

being C > 0 a normalization constant. Therefore, f q is not the global in time equilibrium of the problem but have the property to annihilate the flux for each time t ≥ 0 and that f q (z, v, t) → f ∞ (v) as t → ∞. Using the decomposition f (z, v, t) = f q (z, v) + g(z, v, t) it is clear that the formulation presented in Sect. 6.1 applies and we obtain a steady state preserving method for large times. We refer to [19] for more details. As an example, let us consider the swarming model with self-propulsion defined by P[f ](z, v, t) = α(z)(|v|2 − 1)v + (v − u(z, t)),

(62)

" where the mean velocity u(z, t) = Rdv vf (z, v,"t) dv is not conserved in time. In the above definition we assumed for simplicity Rdv f (z, v, t) dv = 1. The quasistationary state is computed as    |v|4 |v|2 1 α(z) + (1 − α(z)) − uf (z, t)v . f q (z, v, t) = C exp − D(z) 4 2 (63) In Fig. 10 we report a comparison of the results obtained using a standard SG scheme and the micro-macro SG approach. We considered a diffusion coefficient D(z) = 1/5 + z/10 with z ∼ U([−1, 1]) and deterministic self-propulsion α = 2. In both cases the velocity space has been discretized by simple central differences using N points in the domain v ∈ [−2, 2]. It is evident that the error in the standard SG method saturates at the O(v 2 ) order of the solver in the velocity space, while

Uncertainty Quantification for Kinetic Equation and Related Problems 10 1

10 0

10 0

10-2

10 -1

10 -4

10 -2

10 -6

10

-3

10

10 -4 0

2

4

6

8

10

171

-8

10 -10

0

5

10

15

20

Fig. 10 Evolution of the L2 error for the swarming model (62) with standard SG scheme (left) and with the micro-macro SG scheme (right). The error has been computed with respect to a reference solution obtained with M = 40, N = 321, t = 10−1 and final time T = 20

the micro-macro SG approach, thanks to its equilibrium preserving property, is able to achieve spectral accuracy asymptotically.

7 Hybrid Particle Monte Carlo SG Methods The idea is to combine SG methods in the random space with particle Monte Carlo methods for the approximation of f in the phase space. This novel hybrid formulation makes it possible to construct efficient methods that preserve the main physical properties of the solution along with spectral accuracy in the random space [8, 9, 52].

7.1 Particle SG Methods for Fokker–Planck Equations We concentrate on a Vlasov Fokker–Planck (VFP) for the evolution of f = f (z, x, v, t) characterized by Q(f, f ) = ∇v · (P[f ]f + ∇v (Df )) where P[f ](z, x, v, t) = and diffusion D(z).

Rdv ×Rdx

P (z, x, x∗ )(v − v∗ )f (z, x∗ , v∗ , t)dv∗ dx∗

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L. Pareschi

The VFP equation can be derived from the following system of stochastic differential equations for (Xi (z, t), Vi (z, t)) ∈ Rdv × Rdx , i = 1, . . . , N with random inputs ⎧ ⎪ ⎪ ⎨dXi (z, t) = Vi (z, t)dt N  1  ⎪ dV (z, t) = P (z, Xi , Xj )(Vj − Vi )dt + 2D(z)dWi , ⎪ ⎩ i N j =1

being {Wi }N i=1 independent Brownian motions. We consider the empirical measure associated to the particle system f (N) (z, x, v, t) =

N 1  δ(x − Xi (z, t)) ⊗ δ(v − Vi (z, t)) N i=1

Under suitable assumptions it can be shown that as N → ∞, the empirical measure f (N) → f solution of the VFP problem [8]. We consider the SG approximation of the particle system, given by XiM (z, t) =

M 

Xˆ i,m (t)m (z),

ViM (z, t) =

m=0

M 

Vˆi,m (t)m (z),

i = 1, . . . , N

m=0

where Xˆ i,m , Vˆi,m are the projections of the solution with respect to m Xˆ i,m (t) = E[Xi (·, t)m (·)],

Vˆi,m (t) = E[Vi (·, t)m (·)].

The particle SG method is then obtained as ⎧ ⎪ d Xˆ i,h ⎪ ⎪ ⎨

= Vˆi,h dt

⎪ ⎪ d Vˆ ⎪ ⎩ i,h

=

N M 1   ij ˆ Phk (Vj,k − Vˆi,k )dt + Dh dWi N j =1 k=0

√ ij and Phk = E[P (·, XiM , XjM )h (·)k (·)], Dh = E[ 2D(·)h (·)]. Moments are recovered from the empirical measure as (N)

fM (z, x, v, t) =

N 1  δ(x − XiM (z, t)) ⊗ δ(v − ViM (z, t)) N i=1

(N) mφ (fM )=

N 1  δ(x − XiM (z, t))φ(ViM (z, t)) N i=1

Uncertainty Quantification for Kinetic Equation and Related Problems

173

The method just described has the usual quadratic cost O(N 2 ) of a mean field problem, where each particle at each time step modifies its velocity interacting with all other particles. In addition, this cost has to be multiplied by the quadratic cost O(M 2 ) of the SG method. Therefore the overall computational cost is O(M 2 N 2 ). A reduction of the cost is obtained using a suitable Monte Carlo evaluation of the interaction dynamics to mitigate the curse of dimensionality [1] ⎧ ⎪ d Xˆ i,h ⎪ ⎪ ⎨

= Vˆi,h dt

⎪ ⎪ d Vˆ ⎪ ⎩ i,h

=

M 1   ij ˆ Phk (Vj,k − Vˆi,k )dt + Dh dWi S j ∈Si k=0

where Si is a random subset of size S ≤ N of the particles indexes {1, 2, . . . , N}. Using the Euler-Maruyama method to update the particles we have the following algorithm. Algorithm (Particle SG Algorithm) 1. Consider N samples (Xi , Vi ) from f0 (x, v) and  fix S ≤N. 2. Perform gPC representation on the particles : Xˆ i,h , Vˆi,h , for h = 0, . . . , M 3. For n = 0, . . . , T − 1 • Generate N random variables {ηi }N i=1 ∼ N(0, 1) • For i = 1, . . . , N – Sample S particles {j1 , . . . , jS } := Si uniformly without repetition – Compute the space and velocity change n+1 n n = Xˆ i,h + Vˆi,h t Xˆ i,h M √ t   ij ˆ n n+1 n n ˆ ˆ Vi,h = Vi,h + Pkh (Vj,k − Vˆi,k ) + t Dh ηi S j ∈Si k=0

4. Reconstruct the quantity of interest E["(F (f ))]

 

The last step can be performed directly using the empirical distribution f (N) or some suitable reconstruction of f using standard techniques. Thanks to the random subset evaluation of the interaction sum the overall cost is reduced to O(M 2 SN), with S  N. Remark 2 • The advantage of considering a SG scheme for the particle system lies in the preservation of the typical spectral convergence in the random space together with the physical properties of the original system.

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3

10-1

2

10-2

1

10-3

0 -1

-0.5

0

0.5

1

10-4

0

1

2

3

4

5

Fig. 11 Left: expected density at time T = 1 obtained through a standard SG method (gPC) and the particle gPC scheme (MCgPC) with S = 5 at each time step. The gPC expansion has been performed up to order M = 5. Right: convergence for the expected temperature of the MCgPC method with fixed S = 50 and an increasing number of particles. The reference temperature has been computed with a standard SG method for the mean-field problem

• In the case S = N we obtain the typical convergence rate O(N −1/2 ) due to Monte Carlo sampling in the phase

√ space. The fast evaluation of the interactions induces an additional error O 1/S − 1/N with S < N. We report in Fig. 11 the result of a simulation concerning the simple space homogeneous one-dimensional alignment process corresponding to P (z, x, x∗ ) = 1 + sz, z ∼ U (0, 1), s = 0.5 and D = 0. The initial data is given by a bimodal density  (v−μ)2  (v+μ)2 f0 (v) = β e 2σ 2 + e 2σ 2 " with σ 2 = 0.1, μ = 0.25 and β such that R f0 (v) dv = 1. It is clear that a very small value M suffices to match the accuracy in the random and in the phase space.

7.2 Direct Simulation Monte Carlo SG Methods The extension of the particle SG approach just discussed to Boltzmann type equations is non trivial. We recall here the basic methodology in the simple case of Maxwell molecules and refer to [52] for details on its extension to the variable hard sphere cases.

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175

To this aim, we will focus on the space homogeneous Boltzmann equation, and observe that, in the case of Maxwell molecules B ≡ 1, the collision operator can be rewritten as Q(f, f )(z, v, t) = Q+ (f, f )(z, v, t) − μf (z, v, t),

(64)

" where μ > 0 is a constant and we assumed Rdv f (z, v∗ , t) dv∗ = 1, ∀ z ∈ . We consider a set of N samples vi (z, t), i = 1, . . . , N from the kinetic solution at time t and approximate vi (z, t) by its generalized polynomial chaos (gPC) expansion viM (z, t) =

M 

vˆi,m (t)m (z).

m=0

where vˆi,m (t) = E[vi (·, t)m (·)]. To define the DSMC-SG algorithm we consider the projection on the above space of the collision process in the DSMC method (see [48]). In the case of the uncertain Boltzmann collision term (64) we have 1 (vi (z, t) + vj (z, t)) + 2 1 vj (z, t) = (vi (z, t) + vj (z, t)) − 2 vi (z, t) =

1 |vi (z, t) − vj (z, t)|ω, 2 1 |vi (z, t) − vj (z, t)|ω. 2

Let us observe that |vi (z, t) − vj (z, t)| = |vi (z, t) − vj (z, t)|,

(65)

so that the modulus of the relative velocity is unchanged during collisions. We first substitute the velocities by their gPC expansion 1 M (v (z, t) + vjM (z, t)) + 2 i 1  viM (z, t) = (viM (z, t) + vjM (z, t)) − 2 

viM (z, t) =

1 M |v (z, t) − vjM (z, t)| ω, 2 i 1 M |v (z, t) − vjM (z, t)| ω 2 i

and then project by integrating against m (z) p(z) on  to get for m = 0, . . . , M 1 (vˆi,m (t) + vˆj,m (t)) + 2 1  vˆj,m (t) = (vˆi,m (t) + vˆj,m (t)) − 2  vˆi,m (t) =

1 ˆm V ω, 2 ij 1 ˆm V ω 2 ij

(66) (67)

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L. Pareschi

where Vˆijm =



|viM (z, t) − vjM (z, t)|m (z) p(z) dz,

(68)

is a time independent matrix thanks to (65) which consists of a total of (M + 1)N 2 elements that can be computed accurately and stored once for all at the beginning of the simulation. Thus, each Monte Carlo collision can be performed at a computational cost of O(M) which is the minimum cost to update the M modes of each velocity. The SG extensions of the DSMC algorithms by Nanbu for Maxwell molecules is reported below. Algorithm (DSMC-SG for Maxwell Molecules) 1. Compute the initial gPC expansions {viM,0 , i = 1, . . . , N}, from the initial density f0 (v) 2. Compute the collision matrix Vˆijm , i, j = 1, . . . , N, m = 0, . . . , M, using (68). 3. for n = 0 to nTOT − 1 given {vˆim,n , i = 1, . . . , N, m = 0, . . . , M} ◦ set Nc = Iround(μNt/2) ◦ select Nc pairs (i, j ) uniformly among all possible pairs, – perform the collision between i and j , and compute   vˆi,m and vˆj,m according to (66)–(67) n+1 n+1   – set vˆi,m = vˆi,m , vˆj,m = vˆj,m n+1 n ◦ set vˆi,m = vˆi,m for all the particles that have not been selected end for

 

As a numerical example we consider the 2D case with uncertain initial data corresponding to the exact solution[5, 52]    v2 1 − α(z)s(z, t) v2 1 1− 1− e− 2s(z,t) , f (z, v, t) = 2πs(z, t) α(z)s(z, t) 2s(z, t)

(69)

2 − e−t /8 . We will consider α(z) = 2 + κz, with z ∼ U (−1, 1). 2α(z) To emphasize the good agreement of the computed approximation for all times, we depict in Fig. 12 the evolution at times t = 0, 1, 5 of the marginal of E[f ] and Var(f ). Next, in Fig. 13 we present spectral convergence of the scheme computed through the fourth order moment of the 2D model with α(z) = 2 + κz, κ = 0.25 and κ = 0.75 with z ∼ U (−1, 1). As reference solution we considered the fourth order moment at time T = 5 obtained with N = 106 particles and M = 25 Galerkin projections and the evolution is computed with t = 10−1 . In the right plot we present the decay of the L2 () error for increasing M = 0, . . . , 14 in semilogarithmic scale. In the left plot we represent also the whole evolution of M4 where s(z, t) =

(e)

(d)

(f)

(c)

Fig. 12 Evolution at times t = 0, 1, 5 of the marginal E[f ] and Var(f ) from exact solution (69) and DSMC-SG approximation of the 2D Boltzmann model for Maxwell molecules with uncertain temperature. We considered N = 106 particles with M = 5 Galerkin projections and t = 10−1 . The reconstruction step has been performed in [−5, 5]2 through 1002 gridpoints. (a) t = 0. (b) t = 1. (c) t = 5. (d) t = 0. (e) t = 1. (f) t = 5

(b)

(a)

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10

0

10 -5

10 -10

10

-15

0

2

4

6

8

10

12

14

Fig. 13 Left: Convergence of the L2 () error with respect to the fourth order moment obtained from a reference solution computed with N = 106 particles and M = 25 from the DSMC-SG methods. Right: evolution of the fourth order moment in the interval [0, 5] for exact and DSMCSG approximation with N = 106 and M = 5

computed through exact solution and through its DSMC-SG approximation. We obtain numerical evidence of spectral convergence. Remark 3 In a space non-homogeneous setting, the relative velocity changes at each time step due to the transport process. Therefore, the largest part of the computational cost is due to the computation of the matrix (68) at each time step. Note, however, that since i and j are selected at random, we may not need all elements in the matrix in the collision process. Thus, for fixed values of i and j we approximate the vector Vˆijm by Gauss quadrature Vˆijm (t) ≈

H 

wh |viM (zh , t) − vjM (zh , t)|m (zh ).

(70)

h=0

The resulting scheme requires O(MH ) operations to compute viM (zh , t) and vjM (zh , t) for all h’s and O(MH ) operations to evaluate Vˆijm (t) for all m’s. Taking H = M the total cost of a Monte Carlo collision at each time step is therefore O(M 2 ). Acknowledgments This work has been supported by the Italian Ministry of Instruction, University and Research (MIUR) under the PRIN Project 2017, No. 2017KKJP4X, “Innovative numerical methods for evolutionary partial differential equations and applications”.

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A Brief Introduction to the Scaling Limits and Effective Equations in Kinetic Theory Mario Pulvirenti and Sergio Simonella

Abstract The content of these notes is based on a series of lectures given by the first author at HIM, Bonn, in May 2019. They provide the material for a short introductory course on effective equations for classical particle systems. They concern the basic equations in kinetic theory, written by Boltzmann and Landau, describing rarefied gases and weakly interacting plasmas respectively. These equations can be derived formally, under suitable scaling limits, taking classical particle systems as a starting point. A rigorous proof of this limiting procedure is difficult and still largely open. We discuss some mathematical problems arising in this context. Keywords Kinetic theory · Propagation of chaos · Scaling limits · Boltzmann and Landau equations

1 The Foundations of Kinetic Theory Many interesting systems in physics and applied sciences consist of a large number of identical components so that they are difficult to analyze from a mathematical point of view. On the other hand, quite often, we are not interested in a detailed description of the system but rather in its collective behaviour. Therefore, it is necessary to look for all procedures leading to simplified models, retaining the interesting features of the original system, cutting away redundant information. This is the methodology of statistical mechanics and of kinetic theory. Here we want to

M. Pulvirenti () Dipartimento di Matematica, Università di Roma La Sapienza, Roma, Italy International Research Center M&MOCS, Università dell’Aquila, Cisterna di Latina (LT), Italy e-mail: [email protected] S. Simonella ENS de Lyon, UMPA UMR 5669 CNRS, Lyon Cedex 07, France e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 G. Albi et al. (eds.), Trails in Kinetic Theory, SEMA SIMAI Springer Series 25, https://doi.org/10.1007/978-3-030-67104-4_6

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outline the limiting procedure leading from the microscopic description of a large particle system (based on the fundamental laws like the Newton or Schrödinger equations) to the more practical picture dictated by kinetic theory. Although the methods of kinetic theory are frequently applied to a large variety of complex systems (consisting of a huge number of individuals), we will discuss only models arising in physics and more precisely in classical mechanics. The starting point is a system of N identical particles in the physical space. A microscopic state of the system is a sequence z1 , · · · , zN where zi = (xi , vi ) denotes position and velocity of the i-th particle. The equations of motion are given by Newton’s laws of dynamics. We are interested in a situation where N is very large (for instance, a cubic centimeter of a rarefied gas contains approximately 1019 molecules). The knowledge of the microscopic states becomes useless, and we turn to a statistical description. We introduce a probability measure W0N (ZN )dZN (absolutely continuous with respect to the Lebesgue measure), defined on the phase space R3N × R3N , where ZN = (z1 , · · · , zN ) = (xi , vi , · · · , xN , vN ) .  differing W0N assigns the same statistical weight to two different vectors ZN and ZN only for the order of particles, i.e., identifying the same physical configuration. The time-evolved measure is defined by

W N (ZN , t) = W0N (−t (ZN )) .

(1)

Here t (ZN ) denotes the dynamical, measure-preserving flow constructed by solving the equations of motion. We can establish a partial differential equation, called the Liouville equation, describing the evolution of the measure (1). However, this equation is also not tractable in practice. To have an efficient reduced description, one can focus on the time evolution for the probability distribution of a given particle (say particle 1), all the particles being identical. To this end, we define the j -particle marginals fjN (Zj , t) :=

R3N ×R3N

dzj +1 · · · dzN W N (Zj , zj +1 , · · · , zN , t) ,

j = 1, · · · , N ,

(2) and we look for an equation describing the evolution of f1N . We deduce, in most of the physically relevant situations, an evolution equation of the form ∂t f1N = −v · ∇f1N + Q .

(3)

The first term in the right-hand side is due to the free transport of particles, while the term Q should describe the interaction of particle 1 with the rest of the system.

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We face a big difficulty. Since the interaction is binary, Q will depend on f2N , namely the two-particle marginal. In other words, (3) is still useless: to know f1N we need to know f2N , and to know f2N we need to know f3N , and so on. We handle a hierarchy of equations, called BBGKY hierarchy (from the names of the physicists Bogolyubov, Born, Green, Kirkwood, Yvon). Here enters the property called propagation of chaos, that is, f2N (x1 , v1 , x2 , v2 , t) * f1N (x1 , v1 , t)f1N (x2 , v2 , t).

(4)

Accepting (4), Q becomes an operator acting on f1N and (3) is a closed equation. We have thus replaced a huge ordinary differential system by a single PDE. The price we pay is that (3) is nonlinear. The equality in Eq. (4) is certainly false, since it expresses the statistical independence of particle 1 and particle 2 which, even if assumed at time 0, cannot hold at later times. Indeed, the dynamics creates correlations. Nevertheless, one can hope to recover this property in some asymptotic situation described by a suitable scaling limit. This is what happens in two different physical contexts: the low-density and the weak-coupling limits, yielding two different kinetic equations, namely the Boltzmann and the Landau equations, respectively. The passage from hamiltonian mechanics to this kinetic description is actually very delicate. As we shall see later on, we go from a deterministic time-reversible system to an irreversible equation. A different scaling procedure is the so-called mean-field limit. This leads to the Vlasov equation, which has still a time-reversible, hamiltonian nature. It is a sort of continuum limit and hence much simpler than the previous two. Some challenging and interesting problems concerning the mean-field limit are anyway still open, but we shall not discuss them in this note.

2 Low-Density Limit and Boltzmann Equation Ludwig Boltzmann established an evolution equation to describe the behaviour of a rarefied gas in 1872, starting from the mathematical model of elastic balls and using mechanical and statistical considerations [3]. The importance of this equation is twofold. On one side, it provides (as well as the hydrodynamical equations) a reduced description of the microscopic world. On the other, it is also an important tool for applications, especially for dilute fluids when the hydrodynamical equations fail to hold. According to the general paradigm of kinetic theory, the starting point of Boltzmann’s analysis is to renounce to study the gas in terms of the detailed motion of the molecules of the full system. It is preferable to investigate a function f (x, v), the probability density of a given particle, where x and v denote its position and velocity. Or, following the original approach proposed by Boltzmann, f (x, v)dxdv

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is to rather be interpreted as the fraction of molecules happening to be in the cell of the phase space of size dxdv around (x, v). The two quantities are not exactly the same, but they are asymptotically equivalent (when the number of particles diverges) if a law of large numbers holds. Boltzmann considered a gas as microscopically described by a system of elastic (hard) balls, colliding according to the laws of classical mechanics. In this case, the Boltzmann equation for the one-particle distribution function reads (∂t + v · ∇x )f = QB (f, f )

(5)

where QB , the collision operator, is defined by

QB (f, f )(x, v) :=

R3

dv1

2 S+

dn (v − v1 ) · n [f (x, v  )f (x, v1 ) − f (x, v)f (x, v1 )] ,

(6) with v  = v − n[n · (v − v1 )] v1 = v1 + n[n · (v − v1 )]

(7)

2 = {n ∈ S 2 | n · (v − v ) ≥ 0}. and n a unit vector (impact vector) varying in S+ 1   Note that v and v1 are the outgoing velocities after a collision of two elastic balls with incoming velocities v and v1 and centers x and x+εn, with ε the diameter of the spheres. The collision takes place if n · (v − v1 ) > 0. Formulas (7) are consequences of the conservation of energy and momenta. Note that ε does not enter (5) as a parameter.

v1

v n

v1 v

As a fundamental feature of (5), one has the formal conservation (in time) of the five quantities

dx

dvf (x, v, t)v α

(8)

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with α = 0, 1, 2, expressing conservation of probability, " "momentum and energy, respectively. From now on, we shall often abbreviate = R3 . Moreover, Boltzmann introduced the (kinetic) entropy defined by H (f ) =

dx

dvf log f (x, v)

(9)

and proved the famous H theorem asserting the decrease of H (f (t)) along the solutions of (5). Finally, in the case of bounded domains or homogeneous solutions (f = f (v, t) independent of x), the distribution defined for some β > 0, ρ > 0 and u ∈ R3 by M(v) =

ρ 2 e−β/2|v−u| , (2π/β)3/2

(10)

called Maxwellian distribution, is stationary for the evolution given by (5). In addition, M minimizes H among all distributions with given total mass ρ, mean velocity u and mean energy. The parameter β is interpreted as the inverse temperature. In conclusion, Boltzmann was able to introduce an evolution equation with the remarkable properties of expressing mass, momentum and energy conservation and also the tendency to thermal equilibrium. In this way, he tried to conciliate Newton’s laws with the second principle of thermodynamics. The H Theorem is apparently in contrast with the laws of mechanics, which are time-reversible. This fact caused skepticism among the scientific community, and the work of Boltzmann was attacked repeatedly. We refer the reader to the monograph by C. Cercignani [5], which is a beautiful compromise between historical account and scientific divulgation, to have a faithful idea of the debate at the time. To formally derive (5), let us consider a system of N identical hard spheres of diameter ε and unitary mass, interacting by means of the collision law (7). We denote by ε the diameter of the particles which, for the moment, is fixed and not necessarily small. The phase space !N of the system is the subset of R3N × R3N fulfilling the hardcore condition, namely |xi − xj | ≥ ε for i = j . The dynamical flow ZN → t (ZN ) is defined as the free flow, i.e., ZN → t (ZN ) = (x1 + v1 t, v1 , · · · , xN + vN t, vN ) up to the first impact time (when |xi − xj | = ε); then an instantaneous collision takes place according to the law (7), and the flow goes on up to the next collision instant. The well-posedness of the hard-sphere dynamics is not obvious, due to the occurrence of multiple collisions or to the a priori possibility that collision times accumulate at a finite limiting time. However, such pathologies cannot occur outside a set of initial conditions ZN of vanishing measure. Indeed following [1] (see also [6]), the flow ZN → t (ZN ) can be defined for all t ∈ R almost everywhere with respect to the Lebesgue measure, which is enough for what will follow (even the proof of this result is not relevant in the following, so that we omit further details).

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Given a probability measure with density W0N on !N , thanks to the invariance of the Lebesgue measure under the above evolution, we define the time-evolved measure as the measure with density given by (1). Notice that this density is now restricted to !N , however we can, equivalently and at any time, extend W N to zero outside !N and work with densities “with holes” in R3N × R3N . We recall that we consider probability distributions W N which are initially (hence at any positive time) symmetric in the exchange of the particles. The probability density of j particles is then given by the j -particle marginal (2). Note also that here !N , t , W N , fjN · · · should exhibit a double dependence on N and ε. We shall soon fix a precise ε = ε(N) so that the notation becomes unambiguous. Cercignani [4] derived a hierarchy of equations for the marginals (in exactly the same spirit of the BBKGY hierarchy for smooth potentials), and the first of such equations (j = 1) is (∂t + v · ∇x )f1N = Coll ,

(11)

where Coll denotes the variation of f1N due to the collisions, which takes the form Coll = (N − 1) ε

2

dv2

S2

dn f2N (x, v, x + nε, v2 ) (v2 − v) · n .

(12)

In the next section, we will comment on the justification of this equation. Here, let us accept it and argue on its consequences. Two given particles should be (almost) uncorrelated if the gas is rarefied enough. This leads to the propagation of chaos f2N (x, v, x2 , v2 ) * f (x, v)f (x2 , v2 ) ,

(13)

which might seem contradictory at first sight. In fact, if two particles collide, correlations are created. Even assuming (13) at some given time, if particle 1 collides with particle 2, such an equation cannot be satisfied at any time after the collision. Before discussing the propagation of chaos further, we notice that, in practical situations, for a rarefied gas, Nε3 (total volume occupied by the particles) is very small, while Nε2 = O(1). This implies that the collision operator given by (12) is O(1). Therefore, since we are dealing with a huge number of particles, we are tempted to perform the limit N → ∞ and ε → 0 in such a way that ε2 = O(N −1 ). As a consequence, the probability that two tagged particles collide (which is of the order of the surface of a ball, that is O(ε2 )), is negligible. Instead, the probability that a given particle collides with any of the remaining N − 1 particles (which is O(Nε2 ) = O(1)) is not negligible. On the other hand, condition (13) refers to two preselected particles (say 1 and 2) and it is not unreasonable to conceive that it holds in the limiting situation in which we work.

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Nevertheless, we cannot simply insert (13) into (12), as the integral operator refers to times both before and after the collision. Let us assume (13) only when the pair of velocities (v, v2 ) are incoming ((v − v2 ) · n > 0). If the two particles are initially uncorrelated, it is unlikely that they have collided before a given time t, hence we assume their statistical independence. This is a standard argument in textbooks of kinetic theory, but some extra care is needed. If particles 1 and 2 have not collided directly before a given time t, this does not imply that they are uncorrelated. Indeed there may exist a chain of collisions involving a group i1 , i2 , · · · of particles 1 → i1 → i2 → · · · → 2 , correlating particles 1 and 2. As we shall see later, this is excluded (at least for a short time) by a more rigorous analysis. The two clusters of particles influencing the dynamics of particles 1 and 2 are disjoint with large probability. Coming back to (12), for the outgoing pair velocities (v, v2 ) (satisfying (v2 − v) · n > 0), we shall make use of the continuity property f2N (x, v, x + nε, v2 ) = f2N (x, v  , x + nε, v2 ) ,

(14)

where the pair (v  , v2 ) is precollisional. On the two-particle distribution expressed in terms of precollisional variables, we apply now condition (13), obtaining



Coll = (N − 1)ε2

dv2

2 S+

dn (v − v2 ) · n

× [f (x, v  )f (x − nε, v2 ) − f (x, v)f (x + nε, v2 )]

(15)

2 for the after a change n → −n in the positive part of Coll (remind the notation S+ 2 hemisphere {n ∈ S | (v − v2 ) · n > 0}). Finally, in the limit N → ∞ and ε → 0 with Nε2 = λ−1 > 0, we find:

(∂t + v · ∇x )f = λ−1



dv2 S+

dn (v − v2 ) · n [f (x, v  )f (x, v2 ) − f (x, v)f (x, v2 )].

(16) The parameter λ represents, roughly, the typical length a particle can cover without undergoing any collision (mean free path). (In (6), we just chose λ = 1.) It may be worth remarking that, after having taken the limit N → ∞ and ε → 0, there is no way to distinguish between incoming and outgoing pair velocities. This is because no trace of the parameter ε is left in (16) and n plays the role of a random variable. However, keeping in mind the way the Boltzmann equation was derived, one shall conventionally maintain the name incoming for velocities satisfying the condition (v − v2 ) · n > 0 (and consequently the pair (v  , v2 ) would be outgoing in (16)).

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Equation (16) (or equivalently (5)–(6)) is the Boltzmann equation for hard spheres. Such an equation has a statistical nature, and it is not equivalent to the hamiltonian dynamics from which it has been derived. Indeed the H theorem shows that it is not reversible in time in contrast with the laws of mechanics. By the analysis on the order of magnitude of the quantities in the game, we deduced that the Boltzmann equation works in special situations only. The condition Nε2 = O(1) means that we consider a rarefied gas, with almost vanishing volume density. After Boltzmann established the equation, Harold Grad [8, 9] postulated its validity in the limit N → ∞ and ε → 0 with Nε2 → O(1) as discussed above (this is often called, indeed, the Boltzmann-Grad limit). There is no contradiction in the irreversibility or in the trend to equilibrium obtained after the limit, when they are strictly speaking false for mechanical systems. However, the arguments above are delicate and require a rigorous, deeper analysis. If the Boltzmann equation is not a purely phenomenological model derived by assumptions ad hoc and justified by its practical relevance, but rather a consequence of a mechanical model, we must derive it rigorously. In particular, the propagation of chaos should not be a hypothesis but the statement of a theorem. After the formulation of the mathematical validity problem by Grad, Cercignani [4] obtained the evolution equation (hierarchy) for the marginals of a hard-sphere system, and this was the starting point to rigorously derive the Boltzmann equation, as accomplished by Lanford in his famous paper [14], even though only for a short time interval. Lanford’s theorem is probably the most relevant result regarding the mathematical foundations of kinetic theory. In fact, it dispelled the many previous doubts on the validity of the Boltzmann equation (although some authors refuse a priori the problem of deriving the equation starting from mechanical systems [22]). Unfortunately, the short-time limitation is a serious one. Only for special systems, as is the case of a very rarefied gas expanding in a vacuum, can we obtain a global validity [11, 12]. The possibility of deriving the Boltzmann equation globally in time, at least in cases when we have a global existence of good solutions, is still an open, challenging problem. We conclude this section with a few historical remarks. Before Boltzmann, Maxwell proposed a kinetic equation that is just the Boltzmann equation integrated against test functions [16, 17]. He considered also more general potentials, in particular, inverse-power-law potentials, motivated essentially by the special properties of their cross-section. After Lanford’s result, the case of smooth short-range potentials has been studied by other authors [7, 13, 19]. It is a nontrivial extension, in particular when the interacting potential is not “close enough” to a hard-sphere potential. The validity (or nonvalidity) of the Boltzmann equation in the case of genuine longrange interactions is open, in absence of techniques suited to deal with collisional and mean-field terms simultaneously.

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2.1 Hard-Sphere Hierarchies In this and in the following section we give more details on the derivation of (5) from N hard spheres of diameter ε, discussed above heuristically. We remind the reader that we are interested in the behaviour of the system in the limit N → ∞, ε → 0 fixing ε2 N = 1 (1 chosen for simplicity), according to the Boltzmann-Grad limit. Namely we have a single scaling parameter ε (or N), and we study the asymptotics ε → 0 (N → ∞). We start with the justification of (1). Let A be a measurable set in R3N × R3N . Then the probability of finding the system in A at time t > 0 is given by Pt (A) = P0 (−t (A)) where −t (A) = {ZN | t (ZN ) ∈ A} (dropping the dependence on N = ε−2 ). If χA is the characteristic function of A, we have that N N W (ZN , t)χA (ZN ) = W0 (ZN )χ−t (A) (ZN ) = W0N (ZN )χA (t (ZN )) , which implies that

W N (ZN , t) u(ZN ) =

W0N (ZN ) u(t (ZN ))

(17)

for any bounded Borel function u. Here the integral is extended over all the phase space !N . By using the Liouville theorem on the transformation ZN → t (ZN ), it follows that W N (t (ZN ), t) = W0N (ZN ) , or (1) by the invertibility of the same transformation. This probability distribution is not expected to converge. Thus, we focus immediately on the collection of marginal distributions (fjN )j ≥1 , given by (2), for which the evolution equation has the form (∂t + Lεj )fjN = (N − j ) ε2 Cjε+1 fjN+1 ,

j = 1, · · · , N − 1 .

(18)

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Here Lεj is the generator of the dynamics of j hard spheres of diameter ε (Liouville operator of a j −particle system), while Cjε+1 = ε N Ck,j +1 fj +1 (Zj ) =

ε Ck,j +1 ,

(19)

k=1

dvj +1

j 

S2

dn (vj +1 − vk ) · n fjN+1 (Zj , xk + εn, vj +1 ) (20)

is the j -particle collision operator (generalizing (12) to higher orders). For j = N, we are left with the Liouville equation in a differential form, namely fNN = W N and (∂t + LεN )W N = 0 .

(21)

To derive Eq. (18) formally, we would like to give some description of Lεj as j differential operator. This poses a difficulty, in fact Lεj = i=1 vi · ∇xi on functions vanishing on ∂!j and the interacting dynamics is completely coded on the boundary. In [4, 6], boundary conditions are imposed using (14), and its higher order versions, and Eq. (18) is derived integrating by parts over !N . However if one is not afraid of working with delta functions, it is more convenient to use the following compact description: Lεj =

j 

vi · ∇xi − Tεj

(22)

i=1

where Tεj =



i,k Tε; j

(23)

i 0 (depending only on z0 , β) such that, for t < t0 we have, for all j ≥ 1, lim fjN (t) = f (t)⊗j

ε→0

(30)

where f (t) is the unique solution to the Boltzmann equation. The convergence holds almost everywhere. Following Lanford, the proof can be organized in two steps. We first give an a priori bound on the series expansions (27) (uniform in ε) and (28), using that the time t is small enough. To give a rough idea of this step, let us cutoff large velocities. In particular, we ignore the factors |vj +1 − vk | in (20). Then, the string of operators can be estimated brutally by |S ε (t − t1 ) Cjε+1 S ε (t1 − t2 ) · · · S ε (tn ) fjN+n (0)| ≤ C j +n j (j + 1) · · · (j + n − 1) for some C > 0, where the factorial growth comes from the sum in (19). On the other hand, the ordered time-integration yields t n /n!, so that the series expansion is  j bounded by a geometric series n C1 (C2 t)n , for positive C1 , C2 . In the second step, one shows the term by term convergence of (27)–(28). Here the short time restriction does not enter anymore. For more details on Lanford’s proof, we refer to [6, 14, 21]. We conclude with some remarks. 1. The time t0 is explicitly computable. It turns out to be a fraction of the mean free time between collisions. This time limitation is purely technical. 2. Lanford’s original proof was qualitative: it does not make explicit the rate of convergence. This can be obtained with some extra care, along the same arguments [7, 19].

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3. Initial conditions fulfilling Hypotheses 1 and 2 can be easily constructed. The most natural initial state is maximally factorized, meaning that the only source of correlation is due to the hard-core exclusion. In this case, the N-particle measure is W0N (ZN ) :=

1 ⊗N f (ZN ) ZN 0

:

1{|xi −xk |>ε} (ZN ) ,

1≤iε} (ZN )

1≤i 1. The collision operator QB is given by (6), with (7) replaced by v  = v − ω[ω · (v − v1 )] v1 = v1 + ω[ω · (v − v1 )]

(31)

where ω is the unit vector in the direction of the transferred momentum, while n is the impact parameter.1 The potential φ enters in the determination of ω. v  − v1

ω

θ

n

v − v1

According to the weak-coupling-limit prescription discussed above, we rescale √ the potential as φε = εφ( xε ), and simultaneously increase the density. The new collision operator reads QεB (f, f )(x, v) =

1 ε



dv1

2 S+

dn (v − v1 ) · n

× {f (x, v + p)f (x, v1 − p) − f (x, v)f (x, v1 )}

(32)

√ where p = −ω · (v − v1 ) ω is the transferred momentum, which is typically O( ε). It follows that, for any smooth test function u = u(v), setting U = v − v1 (omitting the spatial dependence), dv u(v) QεB (f, f )(v) =

1 2ε



dvdv1

2 S+

' ( dn U · n u(v + p) + u(v1 − p) − u(v) − u(v1 ) f (v)f (v1 )

1 Note that this is not the conventional form for the Boltzmann equation and usually the factor (v − v1 ) · n is rewritten in terms of ω, which amounts to introduce the differential cross-section.

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1 2ε

dvdv1

 1

dn U · n

2 S+

' p · (∇v u(v) − ∇v1 u(v1 )) +

199

(33)

( 1 2 2 ∂α,α ∂α,α u(v1 )pα pα f (v)f (v1 ) ,  u(v)pα pα  + 2 2

α,α

where we Taylor-expanded up to second order in p in order to compensate the divergence 1ε , and the α’s run over the three vector components. We first analyze the second order. Let x(s) be the trajectory of one particle scattering in the central potential φε with incoming velocity U and initial time fixed by x(0) = εn. To evaluate 1 2ε

Tα,α  (U ) :=

2 S+

dn U · n pα pα  ,

we write pα = −

+∞

1 ds √ ∇xα φ ε −∞



x(s) ε



 3 x(s) 1 1 ˆ . ds √ dk i kα ei k· ε φ(k) =− √ 3 ε 2π R

Then  Tα,α  = −

ds1

But x(s) ε ≈ n+ have that

1 2π

ds2 Us ε .

3

1 2ε2 dk1



2 S+

dn U · n

x(s1 ) x(s2 ) ˆ 1 )φ(k ˆ 2 ). dk2 (k1 )α (k2 )α  eik1 · ε eik2 · ε φ(k

Therefore, setting y(s) = n + U s (after rescaling times) we



 1 31 T =− dn U · n 2π 2 S+2 ˆ 1 )φ(k ˆ 2 ). ds1 ds2 dk1 dk2 (k1 )α (k2 )α  eik1 ·y(s1) eik2 ·y(s2) φ(k α,α 

Next we write eik1 ·y(s1) eik2 ·y(s2) = ei(k1 +k2 )·y(s1) eik2 ·U (s2 −s1 )

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and change variables in the following way. Setting τ = s2 − s1 , parametrize the points ξ of the cylinder with axis −U/|U | and basis the unit circle through the origin, by (n, s1 ) → n + U s1 . Then dξ = dn ds1 (U · n)+ and 

Tα,α 

1 ≈− 2π

3

1 2



ˆ 2) , ˆ 1 )φ(k dξ dτ dk1 dk2 (k1 )α (k2 )α  ei(k1 +k2 )·ξ eik2 ·U τ φ(k (34)

hence we arrive to Tα,α  ≈ −

(2π) 2



dk φˆ 2 (k) δ(k · U ) kα kα  =: aα,α  (U ) .

(35)

ˆ This matrix can be handled conveniently by means of polar coordinates k = kρ, ˆk = k : |k| aα,α  (U ) = −

(2π)4 1 2 |U |



dρ φˆ 2 (ρ) ρ 3



ˆ , d kˆ δ(Uˆ · k)

(36)

where Uˆ is the versor of U . Here we are using that, due to the spherical symmetry, φˆ depends on k through |k| only. Setting



B=π

dρ φˆ 2 (ρ) ρ 3

(37)

0

and computing

"

ˆ we conclude that d kˆ δ(Uˆ · k), aα,α  (U ) =

 B  δα,α  − Uˆ α Uˆ α  . |U |

(38)

B is the kinetic constant coding all the information on the microscopic potential. We turn now to the evaluation of the first order terms in (33), i.e. 1 1 ⊥ dn U · n pU , T⊥ (U ) := dn U · n pU T (U ) := 2ε S+2 2ε S+2 ⊥ ) and p > 0 is the projection of p over −U/|U |. Note that T where p = (pU , pU U ⊥ is vanishing by symmetry. On the other hand, pU = (ω · U )2 /|U | = p2 /|U | so that

T (U ) =

1  1  2B Tα,α (U ) ≈ aα,α (U ) = . |U | α |U | α |U |2

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In conclusion, ε dvdv1 Lu(v, v1 ) f (v)f (v1 ) dv u(v) QB (f, f )(v) ≈

(39)

where Lu(v, v1 ) := −2

B (v − v1 ) · (∇v u(v) − ∇v1 u(v1 )) + T r(a ⊗ D 2 u)(v, v1 ) |v − v1 |3

 where T r(a ⊗ D 2 u)(v, v1 ) = α,α  aα,α  (v − v1 )∂v2α ,vα u(v) and a = (aα,α  )α,α  is given by (38). This leads to introduce the Landau operator, defined by QL (f, f )(x, v) :=

dv1 ∇v a(v − v1 ) (∇v − ∇v1 )f (x, v)f (x, v1 ) .

(40)

By a straightforward integration by parts, we get that

dv u(v) QεB (f, f )(v)



dv u(v) QL (f, f )(v) .

(41)

The collision operator QL has been introduced by Landau in 1936 for the study of a weakly interacting dense plasma [15] and (∂t + v · ∇x )f = QL (f, f ) is called the Landau equation (sometimes, Fokker–Planck–Landau equation).2 The qualitative properties of the solutions to the Landau equation are the same as for the Boltzmann equation regarding the basic conservation laws and the H theorem. The procedure described above is a grazing collision limit. To the best of our knowledge, there is no rigorous version of the formal statement (41). The available rigorous results on grazing collision limits concern a suitable rescaling of the differential cross-section (rather than the potential): see [10] and references therein. Even a rigorous proof of (41) would be not completely satisfactory. Indeed the Landau equation is expected to be a fundamental equation, derivable from particle systems in the weak-coupling limit. A rigorous proof of this fact seems to be hard, even for short times. We will present a formal derivation, outlining the difficulties, in Sect. 3.2.

2 The Landau equation was obtained from the Boltzmann equation for cutoffed Coulomb potential (truncated both at short and large distances). Actually the word “Coulomb” is frequently used for the Landau equation with kernel singularity |U1 | (see (38)), which is somehow misleading. In fact as we have seen, this singularity is always present.

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3.1 Remarks on the Scaling Limits Let us give a unified picture of the different regimes discussed so far, leading to the Boltzmann and the Landau equation. The starting point is always a classical system of N identical particles of unit mass. Microscopic positions and velocities are denoted by q1 , · · · , qN and v1 , · · · , vN . Let τ be the microscopic time. The Newton’s equations read: d qi = vi , dτ

d vi = dτ



F (qi − qj )

(42)

j =1,··· ,N j =i

where F = −∇φ denotes the interparticle (conservative) force, φ the two-body, spherically symmetric potential. There is a unique scaling parameter ε, which can be interpreted as the ratio between typical macroscopic and microscopic units. In practice we introduce macroscopic variables x = ε q,

t = ετ ,

and ε has to be sent to zero to extract the essential macroscopic features. Note that the velocity remains unscaled. In these new variables, the system reads: d xi = vi , dt

d 1 vi = dt ε



 F

j =1,··· ,N j =i

xi − xj ε

 .

(43)

In order to have a finite density we should postulate N ∼ ε−3 in three dimensions. Instead, in the low-density limit we chose N ∼ ε−2 so that, for a test particle, the change of momentum (or velocity) for each collision is δv ∼ 1ε δt = O(1) , where the typical interaction time δt is O(ε) (if F has short range); on the other hand the collision frequency scales as the number of particles in the tube of radius ε (which has volume ε2 ), therefore it is finite in the Boltzmann-Grad limit. We have been dealing with this scaling in the most favourable situation, the system of hard spheres. In this case, the collision is instantaneous with transferred momentum of O(1). We are now interested in a situation where the interaction is very weak for which we rescale the potential as φ → εα φ , α ∈ (0, 1) and the equations of motion become    xi − xj d d xi = vi , vi = εα−1 F . (44) dt dt ε j =1,··· ,N j =i

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We should scale the number density as N −β with suitable β, to get a kinetic equation. The heuristic argument for the weak-coupling limit discussed in the previous section implies that, setting β = 2(1 + α), one should get diffusion in velocity, preserving mass, momentum and energy. Thus we expect that this regime is ruled out by the Landau equation, with the only exception α = 0, for which we recover the low-density scaling and the Boltzmann equation. Frequently, “weakcoupling limit” refers to the special case α = 1/2, which is also the case considered in the next section.

3.2 Weak-Coupling Limit for Classical Systems We start from the weak-coupling dynamics in macroscopic variables d 1 vi = − √ dt ε

d xi = vi , dt

 j =1,··· ,N j =i

 ∇φ

xi − xj ε

 ,

(45)

where we pose N = ε−3 . Once again, W N = W N (ZN ) is a symmetric probability density on the phase space R3N × R3N , obeying the Liouville equation  ∂t +

N  i=1



1 vi · ∇xi W N = √ TNε W N ε

(46)

where TNε W N =



ε Tk, WN ,

(47)

· (∇vk − ∇v )W N .

(48)

1≤k 0 because of the dynamics. Since the interaction between two given particles is vanishing in the limit ε → 0, we can hope for propagation of chaos. The physical mechanism producing chaos is however quite different from the one discussed in Sect. 2. Here, two given particles can interact, the force is strong but the net effect of the collision is small (because the interaction time is small), while in the low-density regime collisions are always strong and unlikely. Let us investigate the convergence of f1N to the Landau equation, in the limit ε → 0, using the hierarchy (49). Expanding fjN (t) as a perturbation of the free flow S(t) (as in (28)) we find that fjN (t)

⊗j =S(t)f0

1 √ ε



t 0

N −j + √ ε3 ε

0

t

S(t − t1 )Cjε+1 fjN+1 (t1 )dt1 +

S(t − t1 )Tjε fjN (t1 )dt1 .

It is now reasonable to assume that dX F (X) = 0 ,

(52)

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205

which implies Cjε+1 fjN+1 = O(ε) , provided that the second derivatives Dv2 fjN+1 (t)

√ = O(ε − 2 ) , we see that the second term in are bounded uniformly in ε. Since N−j ε the right-hand side of (52) does not give any contribution in the limit. In the same assumptions, 7



t 0

S(t − t1 )Tjε fjN (t1 )dt1 =





t

F 0

i =k

(xi − xk ) − (vi − vk )(t − t1 ) ε

 · g(Zj , t1 )dt1

where g is a smooth j -particle function, which is again O(ε) so that the last term in the right-hand side of (52) is also vanishing in the limit. We are therefore facing the alternative: either the limit is trivial, or the time evolved marginals are not smooth. This is indeed bad news: a rigorous derivation of the (expected) Landau equation seems problematic. The above difficulty suggests to split fjN (t) into two parts, namely we conjecture that: fjN = gjN + γjN , where gjN is the main part of fjN and is smooth, while γjN is small, but strongly oscillating (hence with large derivatives). The two parts satisfy, by definition, ⎛ ⎝∂t +

j 

⎞ vk · ∇xk ⎠ gjN =

k=1

⎛ ⎝∂t +

j  k=1

N −j 3 ε N −j √ ε Cj +1 gjN+1 + √ ε3 Cjε+1 γjN+1 , ε ε ⎞

1 1 vk · ∇xk ⎠ γjN = √ Tjε γjN + √ Tjε gjN , ε ε

with initial data ⊗j

gjN = f0 ,

γjN = 0 .

The remarkable feature of this decomposition is that the singular part can be eliminated. In fact we have that t   N −1 f1N (t) =S(t)f0 + √ ε3 S(t − t1 ) C2ε g2N (t1 ) + γ2N (t1 ) dt1 , ε 0

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M. Pulvirenti and S. Simonella

where γ2N (t)

1 =√ ε

0

t

ds U2ε (s) T2ε g2N (t − s)

and U2ε is just the two-particle interacting flow. Indicating by Z2ε (−s) s∈(0,t ) this flow with final condition Z2ε (0) = Z2 , we have that 1 γ2N (Z2 , t) = √ ε



t

 ds ∇φ

0

x1ε (−s) − x2ε (−s) ε





· ∇v1 − ∇v2 g2N (Z2ε (−s), t − s).

Based on the conjecture, we present now a formal derivation of the Landau equation (assuming g2N smooth). We have that

N −1 N −1 3 ε ε C2 ∂t + v1 · ∇x1 f1N (t) = √ ε3 C2ε g2N (t) + ε ε



t 0

ds U2ε (s) T2ε g2N (t − s) .

Let u ∈ D be a test function. As already mentioned the first term on the right-hand side is negligible: 

√ N −1 3 √ ε u, C2ε g2N (t) = O ε . ε The last term gives t N −1 ds ∇v1 u(z1 ) dz1 dz2 ε 0     x1 − x2 x1 (−s) − x2 (−s) F F · (∇v1 − ∇v2 )g2N (Z2ε (−s), t − s) ε ε ∞ ds ∇v1 u(z1 ) ≈ − dz1 dr dv2



 F (r) F

0

x1ε (−εs) − x2ε (−εs) ε

after having changed to variables r =

 · (∇v1 − ∇v2 )g2N (x1 , v1 , x1 , v2 , t) , x1 −x2 ε

and s → εs . Here, setting U = v1 − v2 ,

x1ε (−εs) − x2ε (−εs) ≈ r + Us . ε

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This term is then approximately equal to







drdv2

dz1

0

ds∇v1 u(z1 )F (r)F (r + U s) · (∇v1 − ∇v2 )g2N

×(x1 , v1 , x1 , v2 , t)   ≈ u, QL (g1N , g1N ) , where in the last step we invoked propagation of chaos (g2N ≈ (g1N )⊗2 ) and used definition (40). Indeed it is not hard to show that





dr

dsF (r)F (r + U s) =

0

1 2



dr

∞ −∞

dsF (r)F (r + U s) = a(U )

where a(U ) is the matrix given by (38). Indeed expressing the above identity in terms of the Fourier transforms we readily arrive to the right-hand side of (34). Unfortunately, very little is known about the mathematical derivation. We mention here the only result we are aware of. Consider the first order (in time) approximation g˜jN of gjN given by N −j 3 t = + √ ε S(t − τ )Cjε+1 S(τ )gjN+1 dτ ε 0 τ N −j 3 t ⊗(j +1) ε + dτ dσ S(t − τ )Cjε+1 Ujε+1 (τ − σ )Tjε+1 S(σ )f0 . ε 0 0

g˜jN (t)

⊗j S(t)f0

(53) Then we can prove: Theorem ([2]) Suppose that f0 ∈ C03 (R3 × R3 ) is the initial probability density satisfying: |D r f0 (x, v)| ≤ Ce−b|v|

2

for

r = 0, 1, 2

(54)

where D r is any derivative of order r and b > 0. Assume φ ∈ C 2 (R3 ) and φ(x) = 0 if |x| > 1. Assume that the marginals factorize exactly at time zero. Then lim

ε→0

g˜1N (t)

t

= S(t)f0 +

dτ S(t − τ )QL (S(τ )f0 , S(τ )f0 )

(55)

0

where Nε3 = 1 and the above limit is considered in D . Since the right-hand side of Eq. (55) is the first order approximation of the Landau equation, we can consider the theorem as a consistency result.

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References 1. Alexander, R.K.: The infinite hard sphere system. Ph.D.Thesis, Department of Mathematics, University of California at Berkeley (1975) 2. Bobylev, A.V., Pulvirenti, M., Saffirio, C.: From particle systems to the landau equation: a consistency result. Commun. Math. Phys. 319(3), 683–721 (2013) 3. Boltzmann, L.: Lectures on Gas Theory (English edition annotated by S. Brush). University of California Press, Berkeley (1964, reprint) 4. Cercignani, C.: On the Boltzmann equation for rigid spheres. Transport Theory Stat. Phys. 2(3), 211–225 (1972) 5. Cercignani, C.: Ludwig Boltzmann: The Man Who Trusted Atoms. Oxford University, Oxford (1998) 6. Cercignani, C., Illner, R., Pulvirenti, M.: The Mathematical Theory of Dilute Gases. Applied Mathematical Sciences, vol. 106. Springer, New York (1994) 7. Gallagher, I., Saint Raymond, L., Texier, B.: From Newton to Boltzmann: hard spheres and short–range potentials. Zurich Advanced Lectures in Mathematics Series, vol. 18. EMS, Lewes (2014) 8. Grad, H.: On the kinetic theory of rarefied gases. Commun. Pure Appl. Math. 2(4), 331–407 (1949) 9. Grad, H.: Principles of the kinetic theory of gases. In: Flügge, S. (ed.) Handbuch der Physik, vol. 12, pp. 205–294. Springer, Berlin (1958) 10. He, L.: Asymptotic analysis of the spatially homogeneous Boltzmann equation: grazing collisions limit. J. Stat. Phys. 155, 151–210 (2014) 11. Illner, R., Pulvirenti, M.: Global validity of the Boltzmann equation for a two–dimensional rare gas in the vacuum. Commun. Math. Phys. 105, 189–203 (1986) 12. Illner, R., Pulvirenti, M.: Global validity of the Boltzmann equation for a two– and three– dimensional rare gas in vacuum: erratum and improved result. Commun. Math. Phys. 121, 143–146 (1989) 13. King, F.: BBGKY hierarchy for positive potentials. Ph.D. Thesis, Department of Mathematics, University of California, Berkeley (1975) 14. Lanford, O.E.: Time evolution of large classical systems. In: Moser, J. (ed.) Dynamical Systems, Theory and Applications (Seattle, 1974). Lecture Notes in Physics, vol. 38, pp. 205– 294. Springer, Berlin (1975) 15. Lifshitz, E.M., Pitaevskii, L.P.: Course of Theoretical Physics “Landau-Lifshits”, vol. 10. Pergamon Press, Oxford (1981) 16. Maxwell, J.C.: On the dynamical theory of gases. Philos. Trans. Roy. Soc. London Ser. A 157 49–88 (1867) 17. Maxwell, J.C.: The Scientific Letters and Papers of James Clerk Maxwell, vol. 2, pp. 1862– 1873. Cambridge University Press, Cambridge (1995) 18. Pulvirenti, M., Simonella, S.: On the evolution of the empirical measure for the Hard-Sphere dynamics. Bull. Inst. Math. Acad. Sin. 10(2), 171–204 (2015) 19. Pulvirenti, M., Saffirio, C., Simonella, S.: On the validity of the Boltzmann equation for short– range potentials. Rev. Math. Phys. 26(2), 1450001 (2014) 20. Simonella, S.: Evolution of correlation functions in the hard sphere dynamics. J. Stat. Phys. 155(6), 1191–1221 (2014) 21. Spohn, H.: Large Scale Dynamics of Interacting Particles. Texts and Monographs in Physics. Springer, Heidelberg (1991) 22. Truesdell, C., Muncaster, R.G.: Fundamentals of Maxwell’s Kinetic Theory of a Simple Monatomic Gas: Treated as a Branch of Rational Mechanics. Pure and Applied Mathematics, vol. 83. Academic, New York (1980)

Statistical Description of Human Addiction Phenomena Giuseppe Toscani

Abstract We study the evolution in time of the statistical distribution of some addiction phenomena in a system of individuals. The kinetic approach leads to build up a novel class of Fokker–Planck equations describing relaxation of the probability density solution towards a generalized Gamma density. A qualitative analysis reveals that the relaxation process is very stable, and does not depend on the parameters that measure the main microscopic features of the addiction phenomenon. Keywords Social phenomena · Generalized Gamma distributions · Netic models · Fokker–Planck equations

1 Introduction A powerful way to describe the collective behavior of a multi-agent system of individuals with respect to a selected social phenomenon, whose intensity can be characterized by the parameter x, is to resort to kinetic theory, and to study the evolution of the number density f = f (x, t) of individuals which are characterized by the value x ∈ R+ at time t ≥ 0, in terms of its changes in time by microscopic interactions (Boltzmann-type description [49, 51]). These interactions are built taking into account the main features of the phenomenon under study, which are quantified resorting to a precise mathematical description. This approach has been originally applied to the statistical description of wealth distribution in a multi-agent systems of individuals to better understand the reasons behind the formation of Pareto curves [15–17, 19, 20, 22, 23, 57]. More recently, this modeling activity moved to social sciences, where, together with the investigation of opinion formation [5–8, 10–12, 18, 24, 28–31, 54, 55], other aspects of modern

G. Toscani () Department of Mathematics, IMATI Institute of CNR, Pavia, Italy e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 G. Albi et al. (eds.), Trails in Kinetic Theory, SEMA SIMAI Springer Series 25, https://doi.org/10.1007/978-3-030-67104-4_7

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societies, like conflicts, criminality, and city size formation have been considered [2–4, 32, 35]. Since social phenomena are deeply based on behavioral aspects of agents, the microscopic kinetic interactions have been often modeled to reproduce these features. To our knowledge, the first kinetic model in which psychological and behavioral components of the agents have been explicitly considered has been proposed in [47] to model the price formation of a good in a multi-agent market, consisting of two different trader populations. The kinetic description in [47] was inspired by the microscopic Lux–Marchesi model [45, 46] (cf. also [41, 42]). The microscopic trading rules of agents were assumed to depend both on the opinion of traders like in [55], and on the way they interact with each other and perceive risks. This last aspect has been done by resorting, in agreement with the pioneering prospect theory by Kahneman and Twersky [36, 37], to interactions involving a suitable value function. Analogous microscopic mechanism has been considered in [33], where the choice of a particularly adapted value function justified the statistical shape of the service time distribution in a call center, and, in more generality, the formation of a number of social phenomena which can be described by a lognormal distribution [34]. Starting from [19], where the formation of Pareto tails in the wealth distribution in a western society was studied at different scales, the analytical description of the stationary distribution of the social phenomenon under investigation was obtained resorting to a particular asymptotic limit of the kinetic equation of Boltzmann type, which results in a Fokker–Planck type equation [51, 59], still reminiscent of the microscopic interaction mechanism. This is particularly evident in the kinetic description of [33, 34], where the shape of the value function to insert in the microscopic interaction is deeply connected to the steady state of the Fokker–Planck asymptotic equation. The leading idea in [33] was recently applied to the study of the phenomenon of alcohol consumption in [21]. There, the choice of a new class of value functions, suitable to model the possibility of addiction in the microscopic interaction, led to a class of Fokker–Planck equations with a steady state given by a generalized Gamma distribution [43, 53]. The findings of [21] are in agreement with the exhaustive fitting analysis presented in [38, 52]. In these papers the analysis of the fitting of real data about alcohol consumption in a huge number of countries, pushed the authors to conclude that, among various probability distributions often used in this context, Gamma and Weibull distributions (particular cases of the generalized Gamma [53]), appeared to furnish a better fitting with respect to the Log-normal distribution, first proposed by Ledermann [40] as a reasonable model for the consumption problem.

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Let f = f (x, t) denote the probability density of individuals which are characterized by an alcohol consumption value equal to x ∈ R+ at time t ≥ 0. Then, the time evolution of the density f is shown in [21] to obey to a linear Fokker– Planck equation, that reads     ∂2  2 ∂ ∂f (x, t) δ δ = x f (x, t) + x − (κ + 1) xf (x, t) . ∂t ∂x 2 ∂x θδ

(1)

In Eq. (1) θ, κ and δ are positive constants related to the relevant characteristics of the phenomenon under study. Moreover, in all cases considered in [21], in agreement with the fitting analysis in [38, 52], the constant δ belongs to the interval (0, 1]. The equilibrium state of the Fokker–Planck equation is given the generalized Gamma density [43, 53] f∞ (x; θ, κ, δ) =

* ) 1 δ x κ−1 exp − (x/θ )δ . κ θ ! (κ/δ)

(2)

In this paper, we aim at improving the analysis of [21], by generalizing it to different addiction phenomena, and by taking into account a new modeling assumption, in the spirit of the recent paper [27]. This leads to describe the collective behavior of a multi-agent system of individuals subject to some addiction phenomena, whose intensity can be measured in terms of a positive parameter x, in terms of a new class of Fokker–Planck equations. Given the probability density f = f (x, t) of individuals which are characterized by an addiction value equal to x ∈ R+ at time t ≥ 0, the time evolution of the density f is shown to obey to a Fokker–Planck equation similar to (1), that now reads     ∂ 2  2−δ δ ∂f (x, t) ∂ 1−δ = f (x, t) . x f (x, t) + x − (κ + 1 − δ)x ∂t ∂x 2 ∂x θδ (3) Similarly to Eq. (1), θ , κ and δ are positive constants related to the relevant characteristics of the phenomenon under study, and δ ≤ 1. The main difference between the Fokker–Planck equation (1) and the present one, is that both the coefficient of diffusion and the drift term are scaled by a factor x δ . This scaling has no effect on the steady state, so that the new equation (3) has the same steady state (2) of Eq. (1). However, this new Fokker–Planck equation seems to be better adapted to describe the addiction phenomena, since it allows to obtain an explicit rate of relaxation of the solution towards the equilibrium. Given an initial probability density f0 (x) with a bounded variance, it can be proven by classical entropy methods that the (unique) solution to the initial-boundary value problem for the Fokker–Planck equation (3) converges towards the equilibrium density (2) exponentially fast in time with explicit rate [56], a result that seems not available for the solution to (1).

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The analysis of the present paper underlines the importance of the generalized Gamma density in the statistical description of social phenomena. Previous findings in this direction were concerned with event history and survival analysis [13]. Addiction phenomena which can be described by the Fokker–Planck equation (3) include alcohol consumption [21], on line gambling [58], as well as the abuse of the insights of social networking sites [39]. This new form of addiction is very recent, since online social networking sites reached a very high popularity only in the last decade, involving more and more individuals of the society to connect with others who share similar interests. The perceived need to be online was noticed to often result in compulsive use of these sites, which in extreme cases may produce symptoms and consequences traditionally associated with substancerelated addictions [39]. In more details, in Sect. 2 we will briefly describe the modeling assumptions of [21] and [58], relative to the addiction phenomena of alcohol consumption and, respectively, to web gambling activity. In particular, we will outline the importance to resort to a variable collision kernel in the underlying linear Boltzmann equation. A grazing collision limit [59] procedure finally allows to recover the Fokker–Planck equation (3). This will be the argument of Sect. 3. A short review of the qualitative analysis of the Fokker–Planck equation (3), recently obtained in [56], will be done in Sect. 4.

2 Kinetic Description of Addiction Phenomena The goal of kinetic modeling is to describe the collective behavior of a multiagent system of individuals with respect to a certain hallmark by resorting to the typical elementary (microscopic) variations of the hallmark itself. In the case under investigation, the hallmark to be studied is the degree of addiction of the population of individuals relative for example to gambling, alcohol consumption or abuse of insights of social networking sites, measured by a variable x which varies continuously from 0 to +∞. Following the well-consolidated approach furnished by the kinetic theory [26, 49, 51], the statistical description of the addiction variable will be described by resorting to a linear Boltzmann-type equation in which the unknown is the probability distribution f = f (x, t) of the agents with a degree of addiction equal to x at time t ≥ 0. The kinetic model is built up by taking into account some basic hypotheses we enumerate below [21, 58]. To fix ideas, and to fully understand the main steps of the kinetic construction, we will refer to the description of the possible abuse of the insights of social networking sites in a society of individuals. In this case, we assume that the meaning of the variable x is the daily time (in seconds) spent to visit web sites. A further key assumption is to consider the population homogeneous with respect to the phenomenon, assumption that requires to restrict it with respect to some characteristics, like age, sex and social class [39].

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Once the homogeneity assumption is satisfied, individuals in the system can be considered indistinguishable [51], so that the state of any individual at any instant of time t ≥ 0 is completely characterized by the daily of time x ≥ 0 spent in web activities. It is important to remark that both the variables x and t in this model measure time, and that the variable x satisfies the upper bound x ≤ t. However, since we are concerned with the correct identification of the statistical distribution of the daily time at equilibrium (corresponding to let t → ∞), the assumption x ≤ t can be dropped, and consequently the variables x and t can be considered independent each other. Note moreover that the variable x is subject to the physical upper bound 86,400, namely the number of seconds in a day, that could never be crossed. To avoid inessential difficulties linked to the presence of this upper bound, we will assume instead that x can vary on R+ . Consequently, the unknown is the density (or distribution function) f = f (x, t), where x ∈ R+ and the time t ≥ 0, and the target is to study its time evolution towards a certain equilibrium. In general, the density function is normalized to one R+

f (x, t) dx = 1.

The density changes in time since individuals connect (and disconnect) many times in the given period of a day, thus continuously upgrading the time x spent in web activities. In agreement with the classical kinetic theory of rarefied gases, we will always denote a single upgrade of the quantity x as a microscopic interaction. In the phenomena under study, we will focus on two aspects, which appear to be common and essential in the eventual formation of addiction. • Assumption A: There is an entry level (represented by values of the variable x below a certain value x) ¯ that is accepted by most societies. The assumption of a moderate quantity of alcohol, an occasional gambling activity or a limited use of the mobile phone are indeed seen as completely normal. • Assumption B: There is an objective pleasure in spending time in these activities. Consequently, it is normally easier to increase the value of the quantity x than to decrease it. To prevent addiction, it is usual to fix an alarm level (represented by a suitable value x¯L of the variable x, with x¯L > x), ¯ that individuals should not exceed, and to continuously advertise about the dangers associated with addiction values x > x¯L . Assumption B, strongly related to human behavior, has been fully considered in the kinetic modeling, at the level of individual microscopic interactions, in various papers [21, 33, 34], taking inspiration from the pioneering prospect theory by Kahneman and Twersky [36, 37] and their representation of value functions. On the contrary, in [21, 33, 34] Assumption A, mostly related to the collective behavior of the system of individuals, was not taken into account. The mathematical translation of the entry level corresponds to assign a different value (frequency) to the elementary interactions in terms of the value x. A reasonable hypothesis is

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to assume that the frequency of interactions relative to a value x of the addiction variable is inversely proportional to x. This relationship takes into account both the highly probable access of individuals to the entry level, and the rare possibility to reach very high values of the x variable. The choice of a variable interaction frequency has been fruitfully applied in a different context [27], to better describe the evolution in time of the wealth distribution in a western society. There, the frequency of the economic transactions has been proportionally related to the values of the wealth involved, to take into account the low interest of trading agents in transactions with small values of the traded wealth. As discussed in [27], the introduction of a variable kernel into the kinetic equation does not modify the shape of the equilibrium density, but it allows a better physical description of the phenomenon under study, including an exponential rate of relaxation to equilibrium for the underlying Fokker–Planck type equation. Following [21, 33, 34], we will now illustrate the mathematical formulation of Assumption B. The microscopic updates of time spent on social networks by individuals will be taken in the form x∗ = x − (x/x¯L )x + ηx.

(4)

In a single update (interaction) the value x of time can be modified for two reasons, expressed by two terms, both proportional to the value x. In the first one the coefficient (·), which can assume both positive and negative values, characterizes the predictable behavior of agents. The second term takes into account a certain amount of human unpredictability. The usual choice is to assume that the random variable η is of zero mean and bounded variance, expressed by η = 0, η2  = λ, with λ > 0. Small random variations of the interaction (4) will be expressed simply √ by multiplying η by a small positive constant , with   1, which produces the new (small) variance λ. The function  plays the role of the value function in the prospect theory of Kahneman and Twersky [36, 37], and contains the mathematical details of the expected human behavior in the phenomenon under consideration, namely the fact that it is normally easier to increase the value of x than to decrease it, in relationship with the alarm value x¯L . In terms of the variable s = x/x¯L the value functions considered in [21] to describe alcohol consumption are given by δ (s) = μ

δ −1)/δ

−1

δ e(s −1)/δ

+1

e(s

,

s ≥ 0,

(5)

where 0 < δ ≤ 1 and 0 < μ < 1 are suitable constants characterizing the intensity of the individual behavior, while the constant  > 0 is related to the intensity of the interaction. Hence, the choice   1 corresponds to small variations of the mean difference x∗ − x. In (9), the value μ denotes the maximal amount of variation of

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x that agents will be able to obtain in a single interaction. Note indeed that the value function δ (s) is such that − μ ≤ δ (s) ≤ μ.

(6)

Clearly, the choice μ < 1 implies that, in absence of randomness, the value of x ∗ remains positive if x is positive. As proven in [21], the value function satisfies − δ (1 − s) > δ (1 + s) ,

(7)

d  d   (1 + s) <  (1 − s) . ds δ ds δ

(8)

and

These properties are in agreement with the expected behavior of agents, since deviations from the reference point (s = 1 in our case), are bigger below it than above. Letting δ → 0 in (5) allows to recover the value function 0 (s) = μ

s − 1 , s + 1

s ≥ 0.

(9)

introduced in [33, 34] to describe phenomena characterized by the lognormal distribution [1, 44]. Hence, this choice is in full agreement with the data fitting of alcohol consumption proposed by Ledermann in 1956 [40], choice which is still used in present times (cf. also the recent paper [48] and the references therein). Given the interaction (4), for any choice of the value function  the study of the time-evolution of the distribution of the length x of periods spent on web follows by resorting to kinetic collision-like models [14, 51]. The variation of the density f (x, t) obeys to a linear Boltzmann-like equation, fruitfully written in weak form. The weak form corresponds to say that the solution f (x, t) satisfies, for all smooth functions ϕ(x) (the observable quantities) d dt

R+

ϕ(x) f (x, t) dx =

< R+

=

χ(x) ϕ(x∗ ) − ϕ(x) f (x, t) dx .

(10)

Here expectation · takes into account the presence of the random parameter η in the microscopic interaction (4). The function χ(x) measures the interaction frequency. The right-hand side of Eq. (10) measures the variation in density between individuals that modify their value from x to x∗ (loss term with negative sign) and agents that change their value from x∗ to x (gain term with positive sign). In [21], the simplification of the Maxwell molecules, leading to a constant interaction kernel χ, has been assumed. This simplification, maybe not so well justified from a modeling point of view, is the common assumption in the Boltzmann-type description of socio-economic phenomena [26, 51].

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In [27], the Maxwellian assumption has been analyzed in its critical aspects. There, starting from a careful analysis of the microscopic economic transactions of the kinetic model, allowed to conclude that the choice of a constant collision kernel included as possible also interactions which human agents would exclude a priori. This was evident for example in the case of interactions in which an agent that trades with a certain amount of wealth, does not receive (excluding the risk) some wealth back from the market. In strong analogy with the rarefied gas dynamics [9], where the analysis of the Boltzmann equation for Maxwell pseudo-molecules leads to the possibility to make use of the Fourier transformed version, this makes clear that, in the socio-economic modeling, the main advantages of the Maxwellian assumption are linked to the possibility to obtain analytical results. Following the approach in [27], we express the mathematical form of the kernel χ(x) by taking into account Assumption A, which implies that the frequency of changes which leads to increase the amount of time x is inversely proportional to x. Hence, in the addiction setting, it seems natural to consider collision kernels in the form χ(x) = αx −β ,

(11)

for some constants α > 0 and β > 0. This kernel assigns a low probability to happen to interactions in which individuals are subject to a high degree of addiction, and assigns a high probability to happen to interactions in which the value of the addiction variable x is close to zero. The values of the constants α and β can be suitably chosen by resorting to the following argument. For small values of the x variable, the rate of growth of the value function (9) is given by d   dx δ



x x¯L

 ≈

μ δ−1 x . x¯Lδ

(12)

This shows that, for small values of x, the mean individual growth predicted by the value function is proportional to x δ−1 . Then, the choice β = δ would correspond to a collective growth independent of the parameter δ characterizing the value function. A second important fact is that the individual rate of growth (12) depends linearly on the positive constant , and it is such that the intensity of the variation decreases as  decreases. Then, the choice α=

ν 

is such that the collective growth remains bounded even in presence of very small values of the constant . With these assumptions, the weak form of the Boltzmanntype equation (10), suitable to describe addiction phenomena, is given by d dt

R+

ϕ(x) f (x, t) dx =

=

ν < x −δ ϕ(x∗ ) − ϕ(x) f (x, t) dx .  R+

(13)

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Note that, in consequence of the choice made on the interaction kernel χ, the evolution of the density f (x, t) is tuned by the parameter , which characterizes both the intensity of interactions and the interaction frequency.

3 Fokker–Planck Description and Equilibria For any choice of the value function (9), the linear kinetic equation (13) describes the evolution of the density consequent to interactions of type (4). The parameter  is closely related to the intensity of interactions. In particular, values   1 describe the situation in which a single interaction determines only an extremely small change of the value x. This situation is well-known in kinetic theory of rarefied gases, where interactions of this type are called grazing collisions [51, 59]. At the same time [26], the balance of this smallness with the random part is achieved by setting η→



η.

(14)

In this way the scaling assumptions allow to retain the effect of all parameters appearing in (4) in the limit procedure. An exhaustive discussion on these scaling assumptions can be found in [26] (cf. also [33] for analogous computations in the case of the Log-normal case). For these reasons, we address the interested reader to these review papers for details. Letting  → 0, the weak solution f (x, t) to the kinetic model (13) converges towards f (x, t), solution of a Fokker–Planck type equation [26]. Indeed, the limit density f (x, t) is such that the time variation of the (smooth) observable quantity ϕ(x) satisfies d ϕ(x) f (x, t) dx = dt R+     (15)  δ μ λ x  1−δ  2−δ ν − 1 + ϕ (x)x −ϕ (x) x f (x, t) dx 2δ x¯L 2 R+ Hence, provided the boundary terms produced by the integration by parts vanish, Eq. (15) coincides with the weak form of the Fokker–Planck equation       μ ∂ 1  x δ ∂f (x, t) λ ∂ 2  2−δ 1−δ x f (x, t) + =ν −1 x f (x, t) . ∂t 2 ∂x 2 2 ∂x δ x¯L (16) Equation (16) describes the evolution of the distribution density f (x, t) of the weekly time x ∈ R+ spent on social networks related activities in the limit of the

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grazing interactions. The steady state density can be explicitly evaluated [21], and, by setting γ = μ/λ, it results to be the function  f∞ (x) =

f∞ (x¯L )x¯L2−δ x γ /δ+δ−2 exp

γ − 2 δ



x x¯L

4

δ −1

.

(17)

By fixing the mass of the steady state (17) equal to one, the consequent probability density is the generalized Gamma f∞ (x; θ, κ, δ) defined by (2), characterized in terms of the shape κ > 0, the scale parameter θ > 0, and the exponent δ > 0 that in the present situation are given by κ=

γ + δ − 1, δ

 θ = x¯ L

δ2 γ

1/δ .

(18)

It has to be remarked that the shape κ is positive, only if the constant γ = μ/λ satisfies the bound γ > δ(1 − δ).

(19)

Note that condition (19) holds, independently of δ, when μ ≥ 4λ, namely when the variance of the random variation in (4) is small with respect to the maximal variation of the value function. Note that the smallness assumption (19) is typical of addiction phenomena, where individuals live their addiction without large unpredictable variations. The limit δ → 0 in the Fokker–Planck equation (16) corresponds to the drift term induced by the value function (9). In this case, the equilibrium distribution (17) takes the form of a lognormal density [33]. Note that for all values δ > 0 the moments are expressed in terms of the parameters denoting respectively the alarm level x¯L , the variance λ of the random effects and the values δ and μ characterizing the value function φδ defined in (5). Going back to the fitting analysis presented in [38, 52], that lead to identify as correct statistical distributions for alcohol consumption the Gamma and Weibull ones, we recall that these cases are obtained by choosing δ = 1 and δ = κ, respectively. In particular, the case of Gamma distribution leads to a mean value of the addiction equal to the alarm level x¯L , while for the Weibull distribution, where γ = δ, the mean value of the addiction is given by R+

x f∞ (x) dx = x¯L δ

1/δ−1

  1 . ! δ

(20)

Note that, since δ < 1, in this case the mean value is strictly less than the alarm level x¯L . If for example δ = 1/2, the mean value is equal to x¯L /2. Hence, the Weibull case corresponds to the situation in which the addiction phenomenon is sensible to

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the advertisements about possible dangers. In the general case, the mean takes the value !  2 1/δ ! 1 γ + δ δ δ δ (21) x f∞ (x) dx = x¯L ! .

γ 1 γ R+ ! +δ−1 δ

δ

Exact computation of the mean can be done by choosing, for 1/2 < δ < 1, the value γ = δ 2 . This choice is such that condition (19) is satisfied. In this case R+

x f∞ (x) dx = x¯L



1

! 2−

1 δ

 > x¯L .

This shows that the alarm level can be exceeded in the presence of a small variance of the random variation (with respect to the maximal variation of the value function), which corresponds to a strong addiction phenomenon.

4 Relaxation to Equilibrium Scaling time in the Fokker–Planck equation (16) allows to write it in the clean form (3), which outlines the dependence of both the diffusion and drift terms on the shape κ > 0, the scale parameter θ > 0, and the exponent δ > 0. Also, Eq. (3) allows to directly recover the generalized Gamma density (2) in terms of the same parameters. It has to be remarked once more that the limit procedure described in Sect. 3 leads to Eq. (15), namely to a weak version of the Fokker–Planck equation (16). Then, suitable boundary conditions have to be considered, to guarantee the equivalence of the two equations, and consequently the correct evolution of the main macroscopic quantities, and among them the mass conservation. The most used are the so-called no–flux boundary conditions, expressed by    δ  ∂  2−δ 1−δ  f (x, t) x − (κ + 1 − δ)x = 0, x f (x, t) +  ∂x θδ x=0

t > 0. (22)

In presence of the no-flux boundary conditions (22) one can study, without loss of generality, the initial-boundary value problem for Eq. (3) with a probability density function, say f0 (x), as initial datum. Then, mass conservation implies that the solution f (x, t) remains a probability density for all subsequent times t > 0. The qualitative analysis of the Fokker–Planck equations (3) has been done in the recent paper [56]. There, the analysis was extended to values of δ in the interval 0 < δ ≤ 2, thus covering generalized Gamma densities that range from the Lognormal density, corresponding to δ → 0, to Chi-densities, obtained for δ = 2.

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In the following, for the sake of completeness, we will briefly present the main results obtained in [56], as well as the main properties of this class of Fokker– Planck equations. As extensively discussed in [26], Fokker–Planck type equations of type (3) can be rewritten in different equivalent forms, each one useful for various purposes. For given t > 0, let

x

F (x, t) =

(23)

f (y, t) dy 0

denote the probability distribution induced by the probability density f (x, t), solution of the Fokker–Planck equation (3). In [56], the writing the Fokker–Planck equations (3) in terms of the distribution F (x, t), highlighted an interesting feature of their solutions. Integrating both sides of Eq. (3) on the interval (0, x), and applying condition (22) on the boundary x = 0, it is immediate to verify that F (x, t) satisfies the equation ∂F (x, t) ∂2 = x 2−δ 2 F (x, t) + ∂t ∂x



δ x − (κ − 1)x 1−δ θδ



∂ F (x, t), ∂x

(24)

The no-flux boundary conditions (22) then guarantee that, for any t ≥ 0 F (0, t) = 0;

lim F (x, t) = 1.

x→+∞

(25)

The second condition in (25) corresponds to mass conservation. Given a positive constant m > 0, let us consider the transformation F (x, t) = G(y, τ ),

y = y(x) = x m ,

τ = τ (t) = m2 t.

Then it holds ∂ ∂ F (x, t) = m2 G(y, τ ), ∂t ∂τ while ∂ ∂ F (x, t) = mx m−1 G(y, τ ), ∂x ∂y and 2 ∂2 ∂ 2 2m−2 ∂ G(y, τ ). F (x, t) = m x G(y, τ ) + m(m − 1)x m−2 ∂x 2 ∂y 2 ∂y

(26)

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221

Hence, substituting into (24) the above identities and using the inverse relation x = y 1/m , it follows that G(y, τ ) satisfies the Fokker–Planck equation ∂2 ∂G(y, τ ) = y 2−δ/m 2 G(y, τ ) + ∂τ ∂y



  κ δ/m ∂ 1−δ/m −1 y G(y, τ ). y − (θ m )δ/m m ∂y (27)

Moreover, if F (x, t) satisfies conditions (25) for any t ≥ 0, G(y, τ ) still satisfies the same conditions for any τ ≥ 0. Note that Eq. (27) has the same structure of Eq. (24), with the constants κ, θ , and δ substituted by θ m , κ/m and δ/m. Consequently, its equilibrium distribution is given by the generalized Gamma density   κ δ . f∞ y; θ m , , m m

(28)

It is interesting to remark that, if X(t) is the random process with probability distribution given by F (x, t), by construction G(x, t) is the probability distribution of the process Xm (t/m2 ). In [53], Stacy noticed that the generalized Gamma densities satisfy a similar property. Given a constant m > 0, when a random variable is distributed according to (2), Xm is distributed according to (28). Using this property, in [56] two special cases, corresponding to the choices m = δ and m = δ/2, were considered. These cases correspond to simplify the drift term and the diffusion coefficient, respectively. Indeed, the choice m = δ leads to the Fokker–Planck equation with linear drift ! ∂2 κ ∂f (x, t) ∂ x = f (x, t) . − (x, t)) + (xf ∂t ∂x 2 ∂x θ δ δ

(29)

The steady state of Eq. (29) is the standard Gamma distribution of shape κ¯ = κ/δ and scale θ¯ = θ δ ¯ κ, f∞ (x; θ δ , κ/δ, 1) = f∞ (x; θ, ¯ 1)

) * 1 1 ¯ x κ−1 = exp −x/θ¯ . κ ¯ ¯ ! (κ) ¯ θ

(30)

Likewise, the choice m = δ/2 leads to the Fokker–Planck equation with constant coefficient of diffusion      ∂2 2 ∂ 2κ ∂f (x, t) −1 = − 1 x f (x, t) . (31) f (x, t) + x − ∂t ∂x 2 ∂x θδ δ

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In this second case, the steady state of Eq. (31) is the Chi-distribution of shape κ˜ = 2κ/δ and scale θ˜ = θ δ/2 f∞ (x; θ δ , κ/δ, 2) = f∞ (x; θ˜ 2 , κ/2, ˜ 2) =

( ' 1 ˜ x κ−1 exp −x 2/θ˜ 2 . ˜ ! (κ/2) ˜ θ˜ κ/2 (32) 2

Equation (29) has been exhaustively studied in a pioneering paper by Feller [25], who studied the initial boundary value problem with no-flux boundary conditions (22), and proved existence and uniqueness of solutions, positivity and mass conservation. It is interesting to remark that, when the shape κ¯ = κ/δ > 1, there exists a positive and norm preserving solution of the initial-boundary value problem such that both it and its flux vanish at x = 0 [25]. This means that when κ/δ > 1 the boundary x = 0 acts both as absorbing and reflecting barrier and that no homogeneous boundary conditions need to be imposed. Mass conservation holds even without no flux boundary conditions. In view of the aforementioned connections among the Fokker–Planck equations (3), the existence and uniqueness results relative to the exponent δ = 1 still hold for the initial-boundary value problem for Eq. (3) characterized by a parameter δ = 1. For a given initial probability density f0 (x), there exists a unique positive and mass preserving solution in presence of boundary conditions (22). Moreover, if κ > δ, there exists a unique positive and norm preserving solution of the initial value problem such that both it and its flux vanish at x = 0. Mass conservation holds even without no flux boundary conditions. The Fokker–Planck equation (31), with constant diffusion coefficient, allows to prove, using the strategy of Otto and Villani [50], that, provided κ ≥ δ/2, the generalized Gamma densities (2) satisfy the weighted logarithmic Sobolev inequality H (f |f∞ (θ, κ, δ)) ≤

θδ I2−δ (f |f∞ (θ, κ, δ)), δ2

(33)

where, given two probability densities f (x) and g(x), with x ∈ R+ , H (f, g) denotes the Shannon entropy of f relative to g H (f |g) =

R+

f (x) log

f (x) dx, g(x)

and, for a given constant β ≥ 0, Iβ (f, g) denotes the weighted Fisher information of f relative to g 



d f (x) x f (x) Iβ (f |g) = log dx g(x) R+ β

2 dx.

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Inequality (33) then implies exponential convergence in relative entropy of the solution to the Fokker–Planck equation (3) at the explicit rate θ δ /δ 2 . Note that, provided κ ≥ δ/2, the convergence rate does not depend on the value of κ. When the parameter θ is given by (18), the constant in the weighted logarithmic Sobolev inequality (33) takes the value x¯Lδ λ x¯Lδ θδ = . = δ2 γ μ Note that the rate of exponential convergence towards the equilibrium density increases with the alarm level x¯L and with the variance λ of the stochastic part of the microscopic interaction, while it decreases with respect to the maximal amount of variation μ of the value function (5). Also, the behaviour with respect to the parameter δ that characterizes the value function (5) is different depending of the value of x¯L . The rate of convergence decreases with δ if x¯L < 1, while it increases in the opposite situation. It is remarkable that there is no dependence on δ when the alarm level x¯L = 1.

5 Conclusions Recent results on fitting of the statistical distribution of addiction phenomena in a multi-agent system [38, 52] lead to conjecture that these phenomena are well represented by a generalized Gamma distribution. In this paper we show that this type of probability distributions can be obtained as steady states of Fokker– Planck equations modeling addiction phenomena in terms of suitable microscopic interactions. A qualitative analysis of these equations then verifies that equilibrium is reached exponentially in time, thus justifying the fitting analysis. Acknowledgments This work has been written within the activities of GNFM (Gruppo Nazionale per la Fisica Matematica) of INdAM (Istituto Nazionale di Alta Matematica), Italy. The research was partially supported by the Italian Ministry of Education, University and Research (MIUR) through the “Dipartimenti di Eccellenza” Programme (2018–2022)—Department of Mathematics “F. Casorati”, University of Pavia and through the MIUR project PRIN 2017TEXA3H “Gradient flows, Optimal Transport and Metric Measure Structures”.

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Boltzmann-Type Description with Cutoff of Follow-the-Leader Traffic Models Andrea Tosin and Mattia Zanella

Abstract In this paper we consider a Boltzmann-type kinetic description of Followthe-Leader traffic dynamics and we study the resulting asymptotic distributions, namely the counterpart of the Maxwellian distribution of the classical kinetic theory. In the Boltzmann-type equation we include a non-constant collision kernel, in the form of a cutoff, in order to exclude from the statistical model possibly unphysical interactions. In spite of the increased analytical difficulty caused by this further non-linearity, we show that a careful application of the quasi-invariant limit (an asymptotic procedure reminiscent of the grazing collision limit) successfully leads to a Fokker–Planck approximation of the original Boltzmann-type equation, whence stationary distributions can be explicitly computed. Our analytical results justify, from a genuinely model-based point of view, some empirical results found in the literature by interpolation of experimental data. Keywords Microscopic traffic models · Boltzmann-type kinetic description · Fokker-Planck asymptotics · Headway distribution · Log-normal distribution · Gamma distribution

1 Introduction Follow-the-Leader (FTL) traffic models are a class of microscopic models of vehicular traffic introduced in the fifties to describe the flow of vehicles along a one-directional road with no passing. Their basic assumption is that each vehicle adjusts its speed depending only on the speed of the vehicle ahead.

A. Tosin () Department of Mathematical Sciences “G. L. Lagrange”, Politecnico di Torino, Torino, Italy e-mail: [email protected] M. Zanella Department of Mathematics “F. Casorati”, University of Pavia, Pavia, Italy e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 G. Albi et al. (eds.), Trails in Kinetic Theory, SEMA SIMAI Springer Series 25, https://doi.org/10.1007/978-3-030-67104-4_8

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If the road is identified with the real axis and the position of the ith vehicle at time t ≥ 0 is denoted by xi = xi (t) ∈ R, a general FTL model is expressed by the following system of ordinary differential equations, cf. [8]: ⎧ ⎨x˙i = vi ⎩v˙i =

avim (vi+1 − vi ) , (xi+1 − xi )n

i = 1, 2, . . . ,

(1)

where vi = vi (t) ∈ R+ stands for the speed of the ith vehicle whereas a ∈ R+ and m, n ∈ N are parameters characterising the interaction of the ith vehicle with the (i + 1)th vehicle ahead. In essence, (1) prescribes that the acceleration v˙i is proportional to the relative speed of the two interacting vehicles through the nonconstant factor avim , (xi+1 − xi )n called the sensitivity of the driver. In this paper, we will derive from (1) binary interaction rules on which we will ground a “collisional”, viz. Boltzmann-type, kinetic description of traffic. Our ultimate goal is to deduce from the kinetic model the asymptotic distributions, i.e. the analogous of the Maxwellian distribution in classical gas dynamics, which depict several traffic features emerging at equilibrium. The latter include, for instance, the headway (sometimes also called clearance) and the time headway (sometimes simply referred to as the headway) statistical distributions, which in the transportation engineering literature are often estimated empirically and then interpolated by means of some known classes of probability density functions [1, 12, 29]. By exploiting the renowned potential of classical methods of kinetic theory to deal with multi-agent systems [18], we will show that those statistical distributions can actually be obtained from a genuinely model-based approach inspired by (1). In our opinion, this constitutes both a further interesting validation of the microscopic model (1) and a contribution to a deeper understanding and interpretation of the empirical data beyond their interpolation. As far as the advancement of kinetic methods for vehicular traffic is concerned, the contribution of this paper is twofold. On one hand, we introduce kinetic traffic models based on binary interaction rules which are non-standard with respect to the mainstream in the reference literature and built on well consolidated microscopic traffic models. Virtually all kinetic models of traffic flow, from the pioneering ones [19, 21] to the most contemporary ones, see e.g. [6, 11, 14, 22, 24], describe the microscopic state of the vehicles by means of their speed. Nevertheless, we show that if, rather than reinventing some ad hoc though reasonable interaction rules, one wants to rely on the microscopic dynamics (1), a more natural microscopic descriptor is the headway si := xi+1 − xi ,

(2)

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229

i.e. the space gap between a vehicle and the vehicle ahead. The advantage is that from the kinetic model one can then readily recover a statistical description of the traffic distributions mentioned before, which would instead be much less straightforward from a speed-based model. On the other hand, we consider “collisional” models with cutoff, which is a form of non-constant collision kernel quite rare in the kinetic literature of vehicular traffic and also, more in general, of multi-agent systems, see [4, 7, 21, 25]. In particular, we prove that it is still possible to obtain a precise analytical characterisation of the asymptotic distributions in spite of the increased non-linearity of the Boltzmanntype equation caused by the non-constant kernel. It is worth anticipating that the introduction of a kinetic model with cutoff is not just a theoretical speculation. As it will be clear in the sequel, it is fundamental in order to ensure the physical consistency of the interaction schemes derived from (1). In more detail, the paper is organised as follows. In Sect. 2 we focus on the binary interaction schemes that may be derived from (1) for m = n and we consider, in particular, those obtained for n = 1, 2, which will be relevant for the subsequent development of the theory. In Sect. 3 we introduce a Boltzmann-type kinetic model of the FTL dynamics based on the previous interaction rules and we show explicitly that a cutoff interaction kernel is needed, in general, to guarantee the physical consistency of the statistical description of the system. We anticipate that the role of such a kernel will be to exclude possible interactions leading to unphysical negative values of the post-interaction headway. In Sect. 4 we discuss the application of the asymptotic procedure called the quasi-invariant interaction limit to our Boltzmann-type setting with cutoff. In particular we show that, in a suitable regime of the parameters of the binary interactions, it permits to recover a Fokker– Planck approximation of the original “collisional” equation, whence we compute explicitly the stationary distributions of the kinetic model. In Sect. 5 we present some numerical tests which show that, consistently with the theoretical predictions in the appropriate regime of the microscopic parameters, the numerical solution of the Boltzmann-type equation approaches for large times the analytically computed stationary solution of the Fokker–Planck equation. Finally, in Sect. 6 we summarise the contents of the paper and we propose some concluding remarks.

2 FTL-Inspired Binary Interactions We observe that, using the headway (2), we may rewrite model (1) in the form v˙i s˙i m = a n, vi si

i = 1, 2, . . . ,

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which allows for a direct integration of the ith equation depending on the values of the exponents m, n. Throughout the paper, we will focus in particular on the case m = n, which for n = 1 gives vi = Csia

(C > 0),

(3)

while for n > 1 gives si vi =   1 n−1 a + Csin−1

(C ∈ R).

(4)

In both cases, C is an arbitrary integration constant. Since si ∈ [0, +∞) and a > 0, we observe that in (3) vi grows unboundedly for every C > 0. Conversely, in (4) vi 1 increases from 0 to 1/C n−1 , which suggests to fix in this case C = 1 so as to obtain a unitary maximum dimensionless speed of the vehicles.

2.1 The Case n = 1 Writing (1) with m = n = 1 for the ith and the (i + 1)th vehicle, subtracting the corresponding equations and using (3), we determine the following equation for the headway si : %

a d $ s˙i − C si+1 − sia = 0, dt which implies

a − sia + c s˙i = C si+1

(5)

for an arbitrary integration constant c ∈ R. We may fix c by imposing, for instance, that the jammed traffic state, namely the one with si (t) = 0 for all i = 1, 2, . . . and all t ≥ 0, be a particular solution to this equation. Then c = 0. Having obtained a first order model, we are now in a position to apply the idea illustrated in [2] to get a binary interaction rule: we approximate (5) in a short time interval of length t > 0 (understood e.g., as the reaction time of the drivers) with the forward Euler formula, denoting s := si (t), s∗ := si+1 (t) and s  := si (t + t):

s  = s + Ct s∗a − s a . Since the (i + 1)th vehicle does not modify instead its headway when interacting with the ith vehicle behind, the analogous binary rule for it reads simply s∗ = s∗ .

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In order to deal more realistically with partly random binary interactions, which model the non-deterministic aspects of driver behaviour, we further add to s  a zero-mean stochastic fluctuation, which does not modify on average the main FTL dynamics. To this purpose, we introduce a random variable η ∈ R such that Var(η) = η2  > 0,

η = 0,

(6)

where · denotes the expectation with respect to the law of η, and we finally write

s  = s + γ s∗a − s a + s δ η,

s∗ = s∗

(7)

with γ := Ct > 0 for brevity. The coefficient s δ with δ > 0 gives the intensity of the stochastic fluctuation. We assume that it increases with s, so that when a vehicle is close to the leading vehicle it mostly follows the deterministic FTL model. Conversely, when it is far from the leading vehicle it is mostly prone to the randomness of the driver behaviour.

2.2 The Case n = 2 For n = 2, which here we regard as the prototype of the cases n > 1, from (4) we have vi =

si . a + si

(8)

Proceeding like in Sect. 2.1, we determine now the following equation for the headway si :    1 1 d s˙i − a = 0, − dt a + si a + si+1 namely  s˙i = a

1 1 − a + si a + si+1

 +c

(9)

for an arbitrary integration constant c ∈ R. In particular, we fix again c = 0 in order for the jammed traffic state to be a solution also in this case. A forward-in-time discretisation of (9) produces 

s = s + at



1 1 − a+s a + s∗

 .

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Without loss of generality, here we may conveniently choose t = γa for γ > 0, as we anticipate that in this case we will be mainly interested in the regime of large a (cf. Sect. 4.2). Finally, adding a stochastic contribution to the interaction dynamics, we obtain the form of the binary interaction rules that we will consider in the sequel: s = s + γ



1 1 − a+s a + s∗

 + s δ η,

s∗ = s∗ ,

(10)

where η ∈ R satisfies (6) and δ > 0.

3 Boltzmann-Type Kinetic Description with Cutoff Both interaction rules (7), (10) can be recast in the form s  = s + I (s, s∗ ) + s δ η,

(11)

s∗ = s∗ ,

where the interaction function I has the property that I (s, s∗ ) = −I (s∗ , s). In order to be physically admissible, these rules have to be such that s  , s∗ ≥ 0 for all s, s∗ ≥ 0, which is clearly obvious for s∗ but not for s  . In general, the possibility to guarantee s  ≥ 0 depends strongly on I and on the exponent δ of the coefficient of the stochastic fluctuation η. For instance, in the case (10) with δ = 1 it can be proved that the conditions η≥

γ − 1, a2

γ < a2

are sufficient to ensure a priori s  ≥ 0 for all possible choices of s, s∗ ≥ 0, see [20] for the details. They amount to saying that the support of η is bounded from the left, however in such a way that η can take also negative values, which are essential in order to meet the requirements (6). The same is instead not true if, for the same interaction rule (10), we consider e.g., δ = 12 . Indeed, assume that we bound the support of η from the left as η ≥ −η0 for some 0 < η0 < +∞. Then, no matter how small η0 is, if η takes any negative value η = η¯ ∈ [−η0 , 0) and furthermore s = η¯ 2 we have 

s =γ



1 1 − 2 a + η¯ a + s∗

 ,

thus every s∗ ∈ [0, η¯ 2 ) produces s  < 0. A totally analogous situation occurs also for the interaction rule (7) with δ = 12 . These examples demonstrate that, in general, not all the interactions modelled by (7) and (10) are physically admissible. Those which are not have to be discarded

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from the statistical description of the system dynamics, in order to get the correct aggregate trends based only on the admissible interactions. This may be achieved by considering a Boltzmann-type description with cutoff: d dt

R+

ϕ(s)f (s, t) ds

1 = 2λ



R+

(12) 

R+



χ(s ≥ 0)(ϕ(s ) − ϕ(s))f (s, t)f (s∗ , t) ds ds∗ ,

where the kinetic distribution function f = f (s, t) : R+ × R+ → R+ is such that f (s, t)ds is the proportion of vehicles whose headway at time t > 0 is comprised between s and s + ds. Moreover, ϕ : R+ → R is an arbitrary observable quantity (test function) and, like before, · denotes the expectation with respect to the law of η contained in s  . The term χ(s  ≥ 0) :=

 1 0

if s  ≥ 0 otherwise

plays the role of the cutoff (in particular, non-constant) collision kernel. Specifically, it discards the interactions producing s  < 0, which in this way do not contribute to 1 the evolution of f . Finally, the coefficient 2λ on the right-hand side comes from the general form of Boltzmann-type equations with non-symmetric interactions, cf. [18], the parameter λ > 0 representing a relaxation time (in other words, λ1 is the interaction frequency). The presence of the non-constant collision kernel χ(s  ≥ 0) makes it more difficult to extract from (12) information on the aggregate trends of the system, such as e.g., the evolution of the statistical moments of the distribution function f : Mk (t) :=

R+

s k f (s, t) ds

(k ∈ N).

Choosing ϕ(s) = 1 in (12) we obtain however d dt

R+

f (s, t) ds = 0,

namely the conservation of the mass of the vehicles. This condition also implies that it is possible to understand f as a probability density, up to possibly normalising it with respect to the constant total mass. Choosing instead ϕ(s) = s in (12) we discover 1 dM1 = dt 2λ



R+

R+

χ(s  ≥ 0)(I (s, s∗ ) + s δ η)f (s, t)f (s∗ , t) ds ds∗ .

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We notice that if the binary interactions are such that the condition s  ≥ 0 may be guaranteed a priori, like in the case (10) with δ = 1, then χ(s  ≥ 0) ≡ 1 and 1 dM1 = dt 2λ



R+

R+

I (s, s∗ )f (s, t)f (s∗ , t) ds ds∗ = 0,

because I is antisymmetric with respect to the line s∗ = s. In this case, also the first moment of f , namely the mean headway of the vehicles, is conserved. However, this is in general not the case of the models that we are considering. The difficulty to deal with the strongly non-linear Boltzmann-type equation (12) may be bypassed in suitable asymptotic regimes, which allow one to transform (12) in a kinetic model more amenable to analytical investigations. This does not only include the determination of the statistical moments Mk but also the explicit computation of the stationary distribution, say f ∞ = f ∞ (s), which in this context plays the role of the Maxwellian distribution of the classical kinetic theory in that it depicts the emerging trend when interactions are close to equilibrium.

4 Fokker–Planck Asymptotics An asymptotic regime in which a detailed study of a collisional kinetic model is often possible is that of the quasi-invariant interactions, which has been introduced in [4, 23] and is inspired by the grazing collision regime of the classical kinetic theory [26, 27]. The idea is to consider a regime of the parameters of the model in which each interaction produces a small variation of the microscopic state of the particles, so that a suitable approximation of the collision operator (righthand side of (12)) is possible. At the same time, in order to balance the little effect of the interactions and observe aggregate trends, it is necessary to increase correspondingly the interaction frequency, viz. to make the relaxation time λ small. We now illustrate in detail this procedure, which is very much inspired by Cordier et al. [4], with reference to the interaction models introduced in Sect. 2.

4.1 The Case n = 1 Let us consider model (7) with δ =

1 2

and let us set1

a = Var(η) = ,

λ=

 2

choose λ = 2 rather than λ =  so as to absorb in the scaling the coefficient front of the collision operator in (12).

1 We

(13)

1 2

appearing in

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where 0 <   1 is a parameter. Then the interactions are quasi-invariant, i.e. s  ≈ s, because s  , s∗ ≈ 1 and the distribution√ of η is nearly the Dirac delta centred in zero. In particular, we can represent η = Y , where Y is a random variable with zero mean and unitary variance. On the whole, the scaled interactions that we consider are √

s  = s + γ s∗ − s  + sY, s∗ = s∗ . The idea is now to manipulate the Boltzmann-type equation (12) by taking advantage of the assumed smallness of  and finally to approximate it, in the limit  → 0+ , with a Fokker–Planck equation. In the following, we will obtain such a limit equation in a formal fashion. Next, we will justify numerically our derivation by comparing the stationary solution of the obtained Fokker–Planck equation with the numerical solution to (12) with  small and t large. For technical reasons, we will assume that: Assumption 4.1 (i) s, log s ∈ Lp (R+ ; f (·, t)ds) for some p > 0 and all t ≥ 0, i.e.:

R+

s p f (s, t) ds < +∞,

R+

|log s|p f (s, t) ds < +∞

∀ t ≥ 0;

(ii) Y is symmetric about 0, i.e. Y and −Y have the same law; (iii) Y has bounded moments up to the order 3 + ν with ν > 0, i.e. |Y |α  < +∞

for 0 ≤ α ≤ 3 + ν.

Remark 1 (i) Assumption 4.14.1 implies, in particular, that f has a minimum number of  moments bounded. Moreover, it implies that log s ∈ Lp (R+ ; f (·, t) ds) for every p ∈ [0, p]. Indeed, since |log s| ≥ 1 for s ∈ (0, e−1 ) ∪ (e, +∞), we have:





R+

1 e

|log s|p f (s, t) ds ≤ +

|log s|p f (s, t) ds

0 e 1 e



+∞

f (s, t) ds +

≤1+

|log s|p f (s, t) ds

e

R+

|log s|p f (s, t) ds < +∞.

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(ii) For every a ≥ 0, Assumption 4.14.1 implies that P(Y < −a) = P(Y > a), hence in particular that P(Y < −a) = 12 P(|Y | > a). To begin with, we observe that χ(s  ≥ 0) = 1 − χ(s  < 0), therefore we may rewrite (12) as d 1 ϕ(s)f (s, t) ds = ϕ(s  ) − ϕ(s)f (s, t)f (s∗ , t) ds ds∗ dt R+  R+ R+ 1 − χ(s  < 0)(ϕ(s  ) − ϕ(s))f (s, t)f (s∗ , t) ds ds∗  R+ R+ =: A (f, f )[ϕ](t) + R (f, f )[ϕ](t).

(14)

Let now ϕ ∈ Cc∞ (R+ ). Since s∗ − s  =  log

 1  s∗ +  2 s∗¯ log2 s∗ − s ¯ log2 s s 2

( → 0+ )

(15)

with ¯ ∈ (0, ) and since s  < 0 is equivalent to Y    s∗  s    |Y | + o(1) χ(Y < b (s, s∗ )) ϕ (s) γ log  + ≤ s  R+ R+   √  √  1 s∗  √ + ϕ  (s) 2 γ log  s |Y | + sY 2 + o( ) 2 s ?  √  1    √ 3/2 3 ϕ (¯s ) s |Y | + o( ) (17) + f (s, t)f (s∗ , t) ds ds∗ , 6 where √ s¯ ∈ (min{s, s∗ }, max{s, s∗ }). Using (15), we see that the remainders o(1), o( ) denote terms which are bounded in s, s∗ because: (i) s is bounded away from 0 and +∞ thanks to the compactness of the support of ϕ and all of its derivatives; (ii) Assumption 4.14.1 and Remark 11 ensure the f -integrability of the powers of s∗ and |log s∗ |, hence also of their products owing to Hölder’s inequality, on R+ for p sufficiently large.

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The goal is now to take  → 0+ in (17). Passing formally to the limit under the integrals, we have to handle expressions of the form |Y |k χ(Y < b (s, s∗ )) for k = 0, . . . , 3. From Hölder’s inequality we get 1

1

1

1

|Y |k χ (Y < b (s, s∗ )) ≤ |Y |kq  q χ (Y < b (s, s∗ ))r  r = |Y |kq  q P(Y < b (s, s∗ )) r ,

where q, r ≥ 1 are such that

1 q

+

1 r

= 1. Choosing q ≤

3+ν k ,

in view of

|kq

Assumption 4.14.1 we obtain |Y  < +∞ for every k = 0, . . . , 3. On the other hand, from the definition (16) of b (s, s∗ ) together with the expansion (15) we see that, for all fixed s ∈ suppϕ and s∗ > 0, we can choose  > 0 so small that b (s, s∗ ) < 0. Consequently, owing to Assumption 4.14.1, cf. also Remark 11, and to Chebyshev’s inequality,2 we have 1

P(Y < b (s, s∗ )) r = ≤

1 21/r

1

P(|Y | > |b (s, s∗ )|) r

1 (s)1/r =



2/r . 21/r b (s, s∗ )2/r 21/r s + γ s∗ − s 

This shows that all the terms under the # integrals in√(17) tend pointwise to zero when  → 0+ , including the one with s because  at the denominator can be compensated by the factor  1/r in the estimate above provided r < 2. Consequently, we obtain →0+

R (f, f )[ϕ] −−−→ 0. Concerning the term A (f, f )[ϕ], analogous calculations yield



  s∗ ϕ  (s) γ log + o(1) f (s, t)f (s∗ , t) ds ds∗ s R+ R+   s∗ 1 + ϕ  (s) s + γ 2  log2 + o() f (s, t)f (s∗ , t) ds ds∗ 2 R+ R+ s √ √  1 + ϕ  (¯s ) s 3/2 Y 3  + o( ) f (s, t)f (s∗ , t) ds ds∗ , 6 R+ R+

A (f, f )[ϕ](t) =

recall that Chebyshev’s inequality states that P(|X − μ| ≥ kσ ) ≤ k12 , where X is a realvalued random variable with finite expectation μ and finite non-zero variance σ 2 and k > 0. Here we apply it for X = Y , with μ = 0 and σ 2 = 1, and k = |b (s, s∗ )|.

2 We

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where we have taken into account that Y  = 0, Y 2  = 1. Using the compactness of suppϕ and Assumption 4.1, we get then →0+

A (f, f )[ϕ](t) −−−→







R+

R+

 s∗ 1  γ ϕ (s) log + ϕ (s)s f (s, t)f (s∗ , t) ds ds∗ . s 2 

On the whole, in the limit  → 0+ we obtain from (14) d dt



R+

ϕ(s)f (s, t) ds = γ

R+

+





1 2

ϕ (s)



R+

 R+

log s∗ f (s∗ , t) ds∗ − log s f (s, t) ds

ϕ  (s)sf (s, t) ds.

(18)

If we denote L(t) :=

R+

log s∗ f (s∗ , t) ds∗ ,

(19)

which is well defined in view of Assumption 4.14.1, integrating back by parts in (18) and using the arbitrariness of ϕ ∈ Cc∞ (R+ ) we recognise that f satisfies the following Fokker–Planck equation in strong form with non-constant coefficients: ∂t f =

$ % 1 2 ∂s (sf ) − γ ∂s (L(t) − log s)f . 2

(20)

In summary, (18) and (20) represent the weak and the strong form of the asymptotic model which approximates (12) in the quasi-invariant regime (13) of the interactions (7). Notice that, because of the compactness of suppϕ, the Fokker–Planck equation (20) comes without conditions at s = 0 and s → +∞. Boundary conditions may be set by imposing, for instance, the fulfilment of some conservation properties. In particular, as it will be clear in a moment, in this context it is useful to guarantee that model (20) conserves in time the first moment of f , i.e. the mean headway of the vehicles. To study the evolution of M1 , we multiply (20) by s and we integrate on R+ . Recalling the definition (19), we discover: dM1 = dt



+∞  1 2 s ∂s f (s, t) − γ L(t)sf (s, t) + γ s log sf (s, t) , 2 0

therefore M1 is conserved if, for all t > 0, the terms sf (s, t), s 2 ∂s f (s, t) and s log sf (s, t) vanish when s → 0+ and s → +∞. Sufficient conditions for this are that, for all t > 0, f (s, t) and ∂s f (s, t) are bounded in s = 0 and are infinitesimal of order greater than 2 for s → +∞.

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Next, we may use (20) to obtain the stationary distribution f ∞ , which satisfies 1 ∂s (sf ∞ ) − γ (L∞ − log s)f ∞ = 0, 2 where L∞ := limt →+∞ L(t) is so far unknown. This differential equation can be easily solved by separation of variables. Its unique solution with unitary mass is the function √ γ ∞ 2 ∞ f (s) = √ e−γ (log s−L ) , s π namely a log-normal probability density function with parameters L∞ ∈ R and √1 > 0. Notice that such an f ∞ satisfies the boundary conditions stated above. 2γ From the known formulas of the moments of a log-normally distributed random variable we deduce, in particular, that the mean of f ∞ is M1∞

:=

R+

sf ∞ (s) ds = e

1 L∞ + 4γ

,

which, owing to the conservation in time of M1 , has to coincide with the constant 1 mean headway of the system, say h > 0. Therefore we can express L∞ = log h− 4γ and finally write √ γ −γ f ∞ (s) = √ e s π

 !2 1 log s− log h− 4γ

,

(21)

see Fig. 1. In the transportation engineering literature, the log-normal distribution has often been reported to fit well the empirical data of vehicle interspacings, see e.g., [12, 16].

Fig. 1 The log-normal distribution (21) predicted by model (7) in the quasi-invariant regime (13) for: h = 1 and various γ > 0 (left); γ = 1 and various h > 0 (right)

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This motivated some attempts to justify, either analytically or computationally, the emergence of the log-normal distribution using particle models of traffic, which however rely often on case-specific assumptions [9, 12]. Recently, a much more limpid theoretical explanation of the emergence of the log-normal distribution from microscopic agent dynamics has been provided in [10] using kinetic theory methods which also inspire the present work. Nevertheless, in [10] the authors do not consider actual interactions among the agents; rather, they assume that the agents change independently their state, trying to approach a recommended optimal state. On the basis of the prospect theory by Kahneman and Tversky [13], such a change is assumed to require an asymmetric effort, depending on whether the current state is above or below the optimal one. It is then such an asymmetry which generates the log-normal distribution. In [10] the authors recast vehicular traffic in this conceptual scheme by assuming that each driver adjusts the distance s from the leading vehicle aiming at an optimal headway s¯ . The asymmetric effort depends on the fact that it should be easier to approach the optimal headway from above, i.e. for s > s¯ , because this corresponds to accelerating to get closer to the leading vehicle; while it should be harder to approach it from below, i.e. for s < s¯, because this corresponds to braking to get farther from the leading vehicle. While certainly reasonable and embraceable, unlike (7) such a behavioural model is not grounded on existing particle descriptions of traffic acknowledged in the literature. Our contribution has instead the merit to show that the log-normal distribution (21) can be obtained organically from true binary interactions motivated by well consolidated microscopic traffic models. Recalling (3), we also deduce the following relationship between the time headway τ and the headway s: τ :=

s s 1−a = . v C

(22)

Without loss of generality, let us fix C = 1. If, consistently with the quasiinvariant regime (13), we assume that a is small, in particular a < 1, we can use the distribution (21) together with the transformation (22) to obtain the stationary distribution g ∞ = g ∞ (τ ) of the time headway: a 1 τ 1−a f ∞ (τ 1/(1−a)) 1−a  !2 √ 1 γ log τ −(1−a) log h− 4γ − γ , √ e (1−a)2 = τ (1 − a) π

g ∞ (τ ) =

namely in turn a log-normal probability density function. The experimental literature widely acknowledges that the measured time headways distribute, with good approximation, according to a log-normal profile, see e.g., [3, 28] and references therein. Also in this case, ad hoc particle models have already been proposed [3] to justify the emergence of such a distribution. Nevertheless, we believe that the kinetic

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approach presented here offers a more general and organic explanation grounded on simpler and sounder first principles. Finally, from (21) and the transformation (3) with C = 1 we derive the stationary distribution k ∞ = k ∞ (v) of the speed v in the quasi-invariant limit (13), i.e. in particular for a small: 1 1−a ∞ 1/a v a f (v ) a  !2 √ γ − γ2 log v−a log h− 4γ1 a = √ e . va π

k ∞ (v) =

We observe that this is again a log-normal probability density function, hence it has in particular a slim tail for v → +∞. This partially mitigates the drawback of the unbounded speed allowed by the relationship (3) because it implies that, at least in the quasi-invariant regime (13), very high speed values are quite rarely produced by   the microscopic interaction model. In particular, the mean speed is a

a− 1

2 ha e 2γ . Interestingly, in [15] the authors suggest that a log-normal profile may provide an acceptable fitting of the experimental speed distribution, at least as far as the empirical data used in their study are concerned.

4.2 The Case n = 2 We now consider model (10) with δ = the parameters: 1 a= √ , 

1 2

and we focus on the following regime of

Var(η) = ,

λ=

 , 2

(23)

with 0 <   1 as usual. The scaled interaction rules take then the form s = s + γ 

(1 +

√ s∗ − s + sY, √ √ s)(1 + s∗ )

s∗ = s∗ , whence we see that they are quasi-invariant because s  ≈ s for  small. To obtain from (12) the Fokker–Planck equation in the quasi-invariant limit we proceed along the lines of Sect. 4.1, requiring in particular the validity of Assumption 4.1 except for the integrability of log s claimed at point 4.1.

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After rewriting (12) in the form (14), we observe that s  < 0 implies  s + γ

1 Y < −√ s

s∗ − s √ √ (1 + s)(1 + s∗ )

 ≤

γ  − 1√ √ s =: b (s), 

(24)

    s∗ −s √ whence χ(s  < 0) ≤ χ(Y < b (s)). Moreover,  (1+√s)(1+  ≤ |s∗ − s|. Thus, s∗ ) ∞ for ϕ ∈ Cc (R+ ) we estimate:

|R (f, f )[ϕ](t)| ≤

R+

&    >   s |Y | χ (Y < b (s)) ϕ  (s) γ |s∗ − s| +  R+     √ 1 + ϕ  (s) γ 2 (s∗ − s)2 + 2γ s |s∗ − s| |Y | + sY 2 2  √ 1 + ϕ  (¯s ) γ 3  2 |s∗ − s|3 + 3γ  s(s∗ − s)2 |Y | 6 ? √ 3/2 3  2 +3γ s |s∗ − s| Y + s |Y | f (s, t)f (s∗ , t) ds ds∗ .

To manipulate the terms |Y |k χ(Y < b (s)), k = 0, . . . , 3, we resort again to Hölder’s inequality: 1

1

1

1

|Y |k χ(Y < b ) ≤ |Y |kq  q χ(Y < b (s))r  r = |Y |kq  q P(Y < b (s)) r , where q, r ≥ 1 are chosen like in Sect. 4.1. In view of Assumption 4.14.1, it results |Y |kq  < +∞ for k = 0, . . . , 3. Furthermore, from (24) we see that we can take  so small, in particular  < γ1 , that b (s) < 0 for all s > 0. Consequently, invoking Assumption 4.14.1 and Remark 11 together with Chebyshev’s inequality, we obtain 1

1

P(Y < b (s)) r =

2

1

P(|Y | > |b (s)|) r ≤ 1/r

1  1/r = . 21/r b (s)2/r (2s)1/r (γ  − 1)2/r

Plugging this into the estimate of |R (f, f )[ϕ](t)|, and recalling that s ∈ suppϕ is bounded away from 0, +∞ while the powers of s∗ are f -integrable thanks to Assumption 4.14.1 with p sufficiently large, we conclude →0+

R (f, f )[ϕ](t) −−−→ 0. # In particular, we stress that the term containing s vanishes in the limit because √  at the denominator is compensated by the factor  1/r with r < 2.

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Concerning the term A (f, f )[ϕ](t), by means of analogous calculations and taking into account that Y  = 0, Y 2  = 1 and that |Y |3  < +∞, cf. Assumption 4.14.1, we find:



s∗ − s f (s, t )f (s∗ , t ) ds ds∗ √ √ (1 + s)(1 + s∗ ) R+ R+   1 γ 2 (s∗ − s)2 + ϕ  (s) + s f (s, t )f (s∗ , t ) ds ds∗ √ 2 √ 2 R+ R+ (1 + s) (1 + s∗ )2  1 3γ (s∗ − s)s γ 3  2 (s∗ − s)3 √ √ √ √ + ϕ  (¯s ) + 6 R+ R+ (1 + s)3 (1 + s∗ )3 (1 + s)(1 + s∗ )  √ + s 3/2 Y 3  f (s, t )f (s∗ , t ) ds ds∗ ϕ  (s)

A (f, f )[ϕ](t ) = γ

→0+

−−−→

R+



R+

 1 γ ϕ  (s)(s∗ − s) + ϕ  (s)s f (s, t )f (s∗ , t ) ds ds∗ , 2

hence for  → 0+ we finally get from (14) d dt





1 ϕ(s)f (s, t) ds = γ ϕ (s) (M1 (t) − s) f (s, t) ds + 2 R+ R+ 

R+

ϕ  (s)sf (s, t) ds.

Integrating back by parts and invoking the arbitrariness of ϕ ∈ Cc∞ (R+ ), we deduce that f satisfies the Fokker–Planck equation ∂t f =

1 2 ∂ (sf ) − γ ∂s ((M1 (t) − s)f ), 2 s

(25)

which comes again without conditions at s = 0 and for s → +∞ because of the compactness of suppϕ. Like in Sect. 4.1, it is convenient to fix these conditions in such a way that M1 is conserved in time. To this purpose, we multiply (25) by s and we integrate on R+ to discover: dM1 = dt



+∞  1 2 2 s ∂s f (s, t) − γ M1 (t)sf (s, t) + γ s f (s, t) . 2 0

1 From here we see that, analogously to Sect. 4.1, sufficient conditions for dM dt = 0 are the fact that, for all t > 0, f (s, t) and ∂s f (s, t) are bounded at s = 0 and be infinitesimal of order greater than 2 for s → +∞. Under such conditions we can set M1 (t) = h for all t ≥ 0, so that from (25) we obtain in particular the following unique stationary distribution with unitary mass:

f ∞ (s) =

(2γ )2γ h 2γ h−1 −2γ s s e , !(2γ h)

(26)

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Fig. 2 The gamma distribution (26) predicted by model (10) in the quasi-invariant regime (23) for: h = 1 and various γ > 0 (left); γ = 1 and various h > 0 (right)

namely a gamma probability density function with shape parameter 2γ h > 0 and rate parameter 2γ > 0, see Fig. 2. In the transportation engineering literature, also the gamma distribution is sometimes used to fit the experimental measurements of the vehicle interspacings, see e.g., [5]. Our derivation demonstrates that it may be justified out of Followthe-Leader microscopic dynamics (1) with an appropriate choice of the exponents m, n. Recalling (8), we see that the time headway is simply τ=

s = a + s, v

hence its asymptotic distribution g ∞ , which is supported in the interval [a, +∞) because s ≥ 0 implies now τ ≥ a, is obtained by translating f ∞ rightward: g ∞ (τ ) = f ∞ (τ − a)χ(τ ≥ a). Instead, the asymptotic distribution k ∞ of v resulting from the transformation (8) reads   a av ∞ ∞ k (v) = f (1 − v)2 1−v =

v v 2γ h−1 (2γ a)2γ h · e−2γ a 1−v 2γ h+1 !(2γ h) (1 − v)

(27)

and is naturally supported in [0, 1], see Fig. 3. Notice that for v → 1− we have k ∞ (v) → 0. Conversely, for v → 0+ we may have k ∞ (v) → 0 if 2γ h > 1; k ∞ (v) → (2γ a)2γ h/ !(2γ h) if 2γ h = 1; or k ∞ (v) → +∞ if 2γ h < 1. In the latter case, the singularity of k ∞ at v = 0 is however integrable.

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Fig. 3 The speed distribution (27) with a = 10 for: h = 5 and various γ > 0 (left); γ = 1 and various h > 0 (right)

We stress that, consistently with the quasi-invariant regime (23) motivating the form (26) of f ∞ , in both expressions of g ∞ and k ∞ the parameter a has to be understood as sufficiently large. 4.2.1 The Case δ = 1 If we consider model (10) with δ = 1 then, owing to the discussion at the beginning of Sect. 3, we can guarantee a priori the fulfilment of the condition s  ≥ 0 for all s, s∗ ≥ 0 with an appropriate choice of the parameters a, γ and of the random variable η. This implies that χ(s  ≥ 0) ≡ 1 in (12), hence, under the same scaling (23), the quasi-invariant limit simplifies considerably (it basically requires to deal only with the term A (f, f )[ϕ]) and yields finally the Fokker–Planck equation ∂t f =

1 2 2 ∂ (s f ) − γ ∂s ((h − s)f ), 2 s

which differs from (25) only in the coefficient of f in the second order derivative. The unique stationary solution with unitary mass is now 2γ h

f ∞ (s) =

(2γ h)1+2γ e− s · , !(1 + 2γ ) s 2(1+γ )

namely an inverse gamma probability density function with shape parameter 1 + 2γ > 0 and scale parameter 2γ h > 0. Unlike the stationary distributions (21) and (26), this f ∞ features a fat tail, indeed it behaves like s −2(1+γ ) for s → +∞. Interestingly, fat tailed headway distributions are also reported in the experimental literature [1] and justified with the presence of high occupancy vehicles in the traffic stream.

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Algorithm 1 Nanbu-Babovsky Monte Carlo scheme with rejection for (12) 1: fix N > 1 (number of particles, even) and t ∈ (0, ] (time step) 2: sample N particles from the initial distribution f 0 ; let {si0 }N i=1 be their microscopic states 3: for  = 0, 1, 2, . . . do 4: set N˜ := t  N ˜

5: sample uniformly N2 pairs of indexes (i, j ) with i, j ∈ {1, . . . , N}, i = j and no repetition 6: for every sampled pair (i, j ) do δ 7: let si := si + I (si , sj ) + (si ) η, cf. (11), with -scaled I , η (quasi-invariant regime)  8: if si ≥ 0 then 9: set si+1 := si 10: else 11: set si+1 := si 12: end if 13: set sj+1 := sj 14: end for 15: set si+1 := si for all indexes i which were not sampled in step 5 16: end for

5 Numerical Tests We present now several numerical tests, which illustrate the theoretical results obtained in Sect. 4. In particular, they show that the large time numerical solution to the Boltzmann-type equation with cutoff (12) is consistently approximated, for  > 0 small, by either stationary distribution (21) and (26) depending on the assumed model of binary interactions. For the numerical solution of the Boltzmann-type equation with cutoff (12), we adopt a direct simulation Monte Carlo (MC) method. We refer the interested reader to [17, 18] for an introduction. Here, we simply report an essential algorithm which implements an MC scheme suited to our equation, see Algorithm 1. In particular, unlike standard MC algorithms, we take into account that some binary interactions may need to be rejected, if they produce negative post-interaction headways (see lines 8 to 12 in Algorithm 1). It is worth remarking that, besides updating the microscopic states of the particles with the MC scheme, we also need to reconstruct their probability density function at every time step. For this, we recall that several approaches are possible, such as e.g., standard histograms (which we use in this paper), the weighted area rule or kernel density estimation-type strategies. In the following tests, we invariably use a sample of N = 105 particles. Moreover, for density reconstruction purposes, we take s in a bounded interval [0, S] ⊂ R+ and we discretise the latter by means of a certain number NS of grid points. In particular, for the model with n = 1 we use S = 20 and NS = 200, while for the model with n = 2 we use S = 10 and NS = 100.

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5.1 Log-Normal Equilibrium n = 1 We consider first the binary interaction scheme (7) with δ = 12 and the quasiinvariant scaling (13). In particular, we take for η a centred uniform law, so as to meet Assumption 4.14.1. Moreover, we prescribe the following initial condition:  f (s, 0) =

1 5

if 0 ≤ s ≤ 5

0

otherwise,

(28)

whence the mean headway is initially h = 52 . In Fig. 4, we show the numerical solution of (12) in the scaled regimes  = 0.5, 10−1 , 10−2 obtained with Algorithm 1 after T = 20 time steps. A direct comparison with the log-normal equilibrium distribution (21), also plotted in Fig. 4, confirms that if  is sufficiently small ( = O(10−2) in this case) the Fokker–Planck asymptotics provides a consistent approximation of the large time Boltzmann-type solution. Conversely, if  is not small enough, the large time Boltzmann-type solution may differ consistently from the Fokker–Planck equilibrium (cf. e.g., the case  = 0.5). One of the main reasons is that when  is large many interactions produce s  < 0 and are therefore discarded by the collision kernel χ(s  ≥ 0). Consequently, the statistical description provided by (12) is considerably different from that provided by (20). To further investigate the latter aspect, we track the cumulative number of rejections performed by the MC Algorithm 1. In Fig. 5, we show the evolution of such a number in time, starting from the initial condition (28). We observe that, when  is small enough, this number remains constant in time, which indicates that the binary interactions tend to produce only physically acceptable microscopic states. The non-zero cumulative number of rejections is simply due to the arbitrarily chosen initial condition, as the jump at t = 0 in the curve for  = 10−2 clearly shows.

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Fig. 5 Follow-the-leader model with n = 1. Cumulative number of particles rejected by the MC algorithm 1 in time (semi-logarithmic scale)

5.2 Gamma Equilibrium (n = 2) We repeat the same tests as in Sect. 5.1 for the binary interaction scheme (10) with δ = 12 under the quasi-invariant scaling (23). Hence, we compare the large time numerical solution of the Boltzmann-type equation (12) with the gamma equilibrium distribution (26) of the Fokker–Planck equation (25) obtained in the quasi-invariant limit. Figure 6 confirms that, for  sufficiently small ( = O(10−3 ) in this case), the large time Boltzmann solution approaches consistently the analytical Fokker–Planck equilibrium. Moreover, Fig. 7 shows that, for decreasing , the cumulative number of rejections performed by the MC algorithm 1 diminishes and remains constant in time.

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Fig. 6 Follow-the-leader model with n = 2. Comparison of the large time numerical solution of (12) with the Fokker–Planck equilibrium distribution (26) for a decreasing scaling parameter  and two different values of the parameter γ in (10)

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Fig. 7 Follow-the-leader model with n = 2. Cumulative number of particles rejected by the MC algorithm 1 in time (semi-logarithmic scale)

6 Conclusions In this paper we have shown that a Boltzmann-type kinetic approach may be successfully applied to Follow-the-Leader (FTL) traffic models to explain the emergence of various statistical distributions used to interpolate empirical traffic data. Specifically, we have recovered the log-normal and the gamma profiles of the headway and time headway distributions from FTL models of the form ⎧ ⎪ ⎨x˙i = vi  ⎪ ⎩v˙i = a

vi xi+1 − xi

n (vi+1 − vi )

with a > 0 and n = 1, 2, respectively. The further inclusion of stochastic fluctuations at the level of microscopic vehicle interactions, modelling the random behaviour of the drivers superimposed to the purely deterministic FTL dynamics, has turned out to be a crucial point. Indeed, the type of stationary distribution resulting from the kinetic model depends on the rate at which energy is introduced in the system by the interactions. We have described the stochastic fluctuations by means of a term of the form s δ η, where s ≥ 0 is the headway, δ > 0 is a parameter and η ∈ R is a centred random variable with nonzero variance. In this setting, the input rate of the energy is s δ , which increases with s to model the fact that for close vehicles the deterministic FTL dynamics dominate over the stochastic fluctuations while for far apart vehicles the converse holds. The log-normal and gamma distributions have been obtained for δ = 12 . Conversely, still in the case n = 2, we have shown that for δ = 1 an inverse gamma distribution is obtained, which belongs to the class of fat tailed distributions sometimes also cited in the experimental literature.

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From the technical point of view, treating the cases with δ = 12 has required to deal with “collisional” models with cutoff. This means that in the Boltzmanntype equation we have considered a non-constant collision kernel of the form χ(s  ≥ 0), where χ denotes the characteristic function and s  is the post-interaction headway. Such a kernel discards from the statistical description of the system possible interactions leading to unphysical negative headways and turns out to be necessary because for δ = 12 it is impossible to rule out a priori such interactions. On the other hand, for δ = 1 a more standard Maxwellian description may be adopted, because a priori bounds on η and the parameters of the interactions can be established which guarantee the non-negativity of the post-interaction headway. The analytical determination of the stationary distributions mentioned above has been possible in a particular regime of the microscopic parameters, called the quasi-invariant regime. Essentially, it corresponds to the case in which each vehicle interaction produces a very small variation of the headway but the interaction frequency is very high. In this sense, it is reminiscent of the grazing collision regime of the classical kinetic theory. In such a regime, the Boltzmann-type equation can be consistently approximated by a Fokker-Planck equation, which is more amenable to analytical investigations including the possible explicit computation of the large time distributions. Nevertheless, the application of this theory to kinetic models with cutoff is non-standard and has represented the main difficulty to overcome in this paper from both the analytical and the numerical points of view. We believe that the techniques discussed in this paper may further foster the application of kinetic theory methods to new problems in the wide realm of multiagent systems, which for various reasons may require non-constant interaction kernels, see e.g., [7, 25], and whose investigation might have been partly discouraged so far by the lack of proper analytical and numerical tools. Acknowledgments This research was partially supported by the Italian Ministry for Education, University and Research (MIUR) through the “Dipartimenti di Eccellenza” Programme (2018– 2022), Department of Mathematical Sciences “G. L. Lagrange”, Politecnico di Torino (CUP: E11G18000350001) and Department of Mathematics “F. Casorati”, University of Pavia; and through the PRIN 2017 project (No. 2017KKJP4X) “Innovative numerical methods for evolutionary partial differential equations and applications”. This work is also part of the activities of the Starting Grant “Attracting Excellent Professors” funded by “Compagnia di San Paolo” (Torino) and promoted by Politecnico di Torino. Both authors are members of GNFM (Gruppo Nazionale per la Fisica Matematica) of INdAM (Istituto Nazionale di Alta Matematica), Italy.

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