Inelastic Gases:

Physique Statistique et Plasmas

Inelastic Gases:

A paradigm for far-from-equilibrium systems.

Dissertation originale pr´ esent´ ee par Renaud Lambiotte

en vue de l’obtention du grade de Docteur en Sciences

Promoteur: L´ eon Brenig

Septembre 2004

Title Page . . . . i

Table of Contents . . . . iii

Acknowledgments . . . . v

1 Introduction 1

1.1 What are granular media? . . . . 1

1.2 Sketch of the thesis . . . . 4

1.3 Theoretical and numerical methods . . . . 7

1.3.1 A model for the grains . . . . 7

1.3.2 Elastic and inelastic kinetic theory . . . . 10

1.3.3 Numerical methods . . . . 26

2 Granular hydrodynamics 31

2.1 Introduction . . . . 31

2.1.1 Conservation equations . . . . 31

2.1.2 Non-equilibrium thermodynamics . . . . 34

2.2 From kinetic theory to hydrodynamics . . . . 37

2.2.1 Local equilibrium and normal solutions . . . . 37

2.2.2 Chapman-Enskog method . . . . 39

2.3 Simplified kinetic models . . . . 44

2.3.1 Elastic BGK model . . . . 44

2.3.2 Inelastic BGK models . . . . 45

2.4 Granular hydrodynamics . . . . 49

2.4.1 Local HCS . . . . 49

2.4.2 Chapman-Enskog scheme . . . . 53

2.5 Temperature inversion . . . . 64

2.6 Summary of results . . . . 67

3 Anomalous velocity distribution 69

3.1 Introduction . . . . 69

3.1.1 Isotropy and velocity correlations . . . . 69

3.1.2 Non-equilibrium stationary states . . . . 73

3.1.3 Anomalous distribution in the IHS model . . . . 76

3.2 Inelastic Maxwell Models . . . . 80

3.2.1 Introduction . . . . 80

3.2.2 Bobylev-Fourier method and moments hierarchy . . . . 83

3.2.3 Stationary Moments . . . . 87

3.2.4 Multiscaling behaviour . . . . 89

3.3 One-dimensional IMM . . . . 91

3.3.1 Toy models . . . . 91

3.3.2 Discrete time dynamics and random walks . . . . 94

iii

3.3.3 Central-Limit theorem . . . . 96

3.3.4 Maxwell-Boltzmann scaling solution . . . . 98

3.3.5 One dimensional Lorentz model . . . 100

3.4 L´ evy based statistics . . . 104

3.4.1 Introduction . . . 104

3.4.2 Physical relevance and simulations of L´ evy distributions . . . 106

3.4.3 Equilibrium-like behaviour . . . 108

3.5 Limitations of inelastic Maxwell models . . . 112

3.6 Summary of results . . . 116

4 Non-equipartition of energy 117

4.1 Introduction . . . 117

4.1.1 Overview . . . 117

4.1.2 Inelastic mixtures . . . 119

4.2 Qualitative approach . . . 124

4.2.1 Inelastic Lorentz Models . . . 124

4.2.2 Inelastic exchange of energy . . . 125

4.2.3 Relaxation time competition . . . 129

4.3 Generalized Maxwell model . . . 133

4.3.1 In the Lorentz limit . . . 133

4.3.2 Arbitrary mixtures . . . 136

4.3.3 Multi-component mixtures . . . 139

4.4 Non-Maxwellian velocity distribution . . . 147

4.5 Summary of results . . . 149

5 From the Maxwell demon to the granular clock 151

5.1 Maxwell Demon . . . 151

5.1.1 Historical background . . . 151

5.1.2 The granular Demon experiment . . . 153

5.2 Ehrenfest urn model . . . 159

5.2.1 Classical model . . . 159

5.2.2 Granular urn experiment . . . 167

5.2.3 Thermodynamic-like approach . . . 176

5.2.4 Asymmetric Demon and hysteresis . . . 181

5.2.5 Three compartments . . . 193

5.3 The granular clock . . . 200

5.3.1 Description of the experimental setting . . . 200

5.3.2 Asymptotic regimes . . . 202

5.3.3 Qualitative model . . . 208

5.4 Summary of results . . . 214

6 Conclusion 215

A Granular hydrodynamics 219

Bibliography 223

During these last five years, I have met several people who have helped me to finish this thesis, by giving me new ideas or insight about physical concepts, or by making my every day life at work as pleasant and productive as possible. I would like to thank them all as a preamble to this monograph.

First of all, of course, I would like to thank my supervisor L. Brenig, for these years of common work and the support he gave me at the right moments, J. Wallenborn for his patient interest in my work, as well as M. Malek Mansour for making me discover the piston and inviting me in a CECAM workshop. From the Universit´ e Libre de Bruxelles, I would also like to acknowledge D. Carati, M.

Nizette, C. Schomblond, F. De Neyn, O. Debliquy, D. Desmidts, R. Balescu, B. Knaepen, S. Viscardy, M. Henneaux, P. Gaspard, G. Nicolis and Z. Offer.

Over the last years, I have made several stays in CECAM (Lyon, France) thanks to the invitations of M. Mareschal. Therefore, and for many other reasons, I would like to thank him warmly and to pinpoint how important these invitations were in the writing of this work. From CECAM, I would also like to express my appreciation to L. Maragliano for his friendly invitation in Roma, T. Van Erp for discussing TPS around a pint, D. Tanguy for excursions in the Alps, F. Barmes for his exotic initiations to Alsacian food, R. Ramirez for making me discover the pleasures of C and E. Crespeau for her kindness and laugh.

I would also like to acknowledge M. Salazar for invitations in Dijon during August 2003 and for our collaboration about the granular clock, Y. Elskens for his early interest in my work, as well as people from the granular media community: J. Dufty for inviting me in a workshop in Paris, E. Trizac, I.

Goldhirsh, M.H. Ernst, V. Garzo, J. Brey and U.M.M. Marconi for fruitful discussions.

Finally, I would like to dedicate this work to my family, and especially to my grand parents, for

their constant support. This thesis has been done with the help of a FRIA fellowship.

Introduction

1.1 What are granular media?

Granular media are systems composed of a large number of macroscopic solid entities, which we call grains. The scale of these particles, which extends from sand to rocks, applies to a wide variety of physical systems. For example, the physics of planetary rings and of avalanches belongs to the realm of granular physics. More pragmatically, these kinds of systems are ubiquitous in a large number of industrial processes (fig. 1.1), and the lack of understanding of the static and dynamical properties of powders or cereals may have expensive or dramatical consequences. In the pharmaceutical industry, for instance, where chemical components are usually made of granules or powder, the properties of mixing and segregation of these systems is primordial in order to optimize the area of contact between the chemical components, and consequently the efficiency of the chemical reaction; moreover, in the agro-alimentary and food industry, these properties of homogenization go on a par with the standardization required by mass production. Transport and storage properties of these systems may also be problematic, and have striking consequences, as for the storage of cereals in silos which may undergo dramatic breakages due to the huge internal pressure forces inside. For these reasons, there has been a long-standing interest in describing and predicting the behaviour of granular materials in the engineering community. Granular systems have then been considered as a part of applied physics and, except some notable attempts by Faraday and Coulomb, they have not reached the attention of the physicist community until recently.

There are several reasons for this new interest, beyond the economic imperatives discussed above (ref. [34], [35], [147], [148], [158], [138]). A first reason comes from the appearent simplicity of granular materials, which surprisingly leads to a very rich and often counterintuitive phenomenology. These features mainly originate from the macroscopic dimensions of the grains, which at the relevant scale of granular physics are considered as solid objects whose internal degrees of freedom are neglected. This has two important consequences: the ordinary temperature plays no role on the grain motion, and the interaction between the grains is dissipative. The non-relevance of temperature comes from the fact that k

_B

T

∼

0 at the scale of the grains. This implies that entropy considerations may be out-weighted by dynamical effects, and that the exploration of phase space is unusual as compared to that of an

1

Figure 1.1:

Typical occurrences of granular media in nature and industrial processes: explosion of a silo due to the internal stress, formation of planetary rings, granular transport in a pharmaceutical industry and coal transport in a mine.

equilibrium system. The dissipativity of collisions also contributes to this anomalous exploration of phase space. Indeed, the transfer of kinetic energy to the internal degrees of freedom of the grains, for instance under the form of internal vibrations, causes a decrease of the total kinetic energy of the system, and leads to the irreversibility of the grains dynamics in the course of time. This irreversible behaviour may favour some regions in phase space, and generates some complex phenomena. Moreover, it also implies that if no external energy is applied to the system, the latter has a tendency to reach the total rest state, namely a state where the total kinetic energy is totally dissipated. Consequently, the exploration of phase space is slowed down, and before the complete rest the system will reach different asymptotic states depending on the initial condition, because of the lack of ergodicity which develops in the system. Let us stress that this expression of metastability does not rest on equilibrium- like mechanisms. However, we show in chapter 5 that equilibrium-like metastability may also occur in vibrated granular systems.

The simplicity of granular materials originates from the simple repulsive forces acting between their components, and has to be put in contrast with their very rich phenomenology, which may qualitatively differ from that observed in the familiar forms of matter, namely solid, liquid or gaseous. Over the last decades, their striking features have been put in parallel with phenomena occurring in more complex systems, and have been used as a fruitful metaphor in order to describe dissipative dynamical systems.

For instance, sand piles have been used by De Gennes (ref. [85]) as a macroscopic picture for the motion

of flux lines in superconductors, and by Bak (ref. [8]) as a paradigmatic experience in order to lay the foundations of the Self-Organized Criticality concept. Similarly, other fields of physics have been compared to granular features, such as the slow relaxation process which take place in vibrated sand piles as well as in glasses or flux lattices (ref. [149], [164]). More generally, granular materials have been used in order to study the features of non-equilibrium systems. Indeed, due to their internal dissipation of energy, granular materials may reach two kinds of asymptotic states. If they evolve freely, namely in the absence of external forces, their kinetic energy decreases until the moment the system has reached total rest state. Therefore, the system is non-stationary and out-of-equilibrium, except in the trivial state where no grains move. In contrast, when granular materials are in contact with an energy source, such as a vibrating wall for instance, their energy loss may be counterbalanced and the system may exhibit fluid-like properties and stationary features. However, these states are not at equilibrium, because they are characterized by an energy flux from the external world toward the internal degrees of freedom of the grains. These states are usually called non-equilibrium stationary states (NESS).

If the energy source is strong enough, the motion of the grain becomes very disordered and erratic:

the motion is made of free flights interrupted by grain-grain collision during which the grains exchange part of their energy and momentum, and by grain-surface interactions during which the grains enter into contact with the surroundings. Analogies with the motion of atoms in classical fluids has led many authors to try to adapt methods borrowed from liquid and gas theory into the context of granular media. For instance, a key concept introduced in the early eighties is the so-called granular temperature, which measures the degree of agitation of the grains. It is defined in a kinetic way as the variance of the velocity distribution of the grains. This quantity, which plays a central role in the description of granular fluids is nonetheless abusively called temperature. Indeed, this term may lead to confusions by suggesting thermodynamic grounds for it, which is obviously not the case given the non-equilibrium nature of granular materials. Consequently, there is a priori no reason that granular temperature should exhibit the same features as a well-defined thermodynamic temperature, like for instance the property of energy equipartition. This fact has to be kept in mind, and will be discussed in detail in chapter 4.

In this thesis, we mainly focus on very dilute systems, where the methods of kinetic theory may be generalized. This regime, which consists in a dilute assembly of inelastic grains, is usually called an inelastic gas. More particularly, we will focus on a dilute gas of inelastic hard spheres. This idealized model has been much studied over the last decade in order to highlight the influence of inelasticity on the properties of a fluid. Amongst others, it has been used in order to provide a kinetic foundation to hydrodynamic-like equations for these systems, by the use of Chapman-Enskog methods and to derive a microscopic expression for the transport coefficients. This scheme, which has led to the prediction of new transport processes in granular fluids, has provided a microscopic justification to the use of granular hydrodynamics and has been successfully verified by computer simulations.

Moreover, this approach has later been extended to the case of moderately dense fluids and to the case

of mixtures. The study of inelastic gases has now become a problem of his own, whose simple features

preserve most of the phenomena observed in denser granular fluids, such as the emergence of convection

rolls in vibrated granular media, the non-Maxwellian features of their velocity distributions or their characteristic tendency to clustering. The study of inelastic gases has also led to a clear identification of the main inelastic effects which alter the macroscopic dynamics, namely the coupling between the local energy and the local density in these systems, and the new time scale associated to the dissipative cooling. Moreover, they have been used as a paradigm in order to apply equilibrium methods to granular fluids. Amongst others, there are the identification of diffusion and self-diffusion (ref. [58], [82], [189]), Einstein relations between the diffusion coefficients and mobility (ref. [120]), the derivation of Green- Kubo expressions (ref. [126], [66], [93]), the emergence of normal solutions (ref. [93]), the generalization of Onsager symmetry relations (ref. [118]), fluctuation-dissipation theorems (ref [227], [14], [24]).

1.2 Sketch of the thesis

In this section, we briefly review, chapter by chapter, the results presented in this thesis. The latter is composed by four chapters which form the core of this work and focus on distinct features of granular media. The chapters are self-contained and may be read independently. The order between the chapters corresponds to a chronological sequence, from the beginning to the end of my PhD work. One should also stress that a short summary section has been added at the end of each chapter in order to clearly identify results derived in the literature from my results. Let us now briefly describe the content of the chapters.

Chapter 2:

Granular Hydrodynamics

Non-conservation of energy during collisions is responsible for the specific coupling between density

and energy in granular materials as well as for the emergence of a new time scale in these systems,

which is the cooling time of the system. In chapter 2, we discuss one of the macroscopic consequences

of these effects, namely the emergence of anomalous transport processes in inelastic gases. This is

done by applying a Chapman-Enskog procedure to the inelastic Boltzmann equation, in order to derive

hydrodynamic equations for inelastic dilute gases. First, we give a short introduction of Non-Equilibrium

Thermodynamics and show the reasons why the Fourier law

q

=

−µ∂_r

T applies in close-to-equilibrium

systems. There is, therefore, no reason for this law to apply for inelastic gases, given their strong non-

equilibrium nature. After introducing the so-called Local Equilibrium States in the elastic case, we show

that their inelastic counterparts are the Local Homogeneous Cooling States. In the Chapman-Enskog

expansion, the non-stationarity of this solution leads to technical and conceptual difficulties which are

discussed in detail. The objective of this chapter is to show qualitatively the different mechanisms

which contribute to the anomalous behaviour of the energy flux

q. Therefore, we introduce various

versions of the Bhatnagar-Gross-Krook (BGK) model, from which we derive a generalized Fourier law

for the heat flux of the granular fluid

q

=

−µ∂_r

T

−

κ∂

_r

n and isolate the mechanisms which make

κ

6= 0. These are the density-energy coupling discussed above, and the non-Maxwellian character of the

Homogeneous Cooling State (HCS) velocity distribution. Let us stress that one of the inelastic BGK models is physically equivalent to a system composed by elastic hard spheres where the inelasticity of the collisions is taken into account by an additional dissipative term in the Boltzmann equation. This mean-field approach was studied during the first year of this thesis, with a model called wooled elastic hard spheres. In the last section, we illustrate the macroscopic effects of the new transport coefficient κ by discussing the phenomenon of temperature inversion in vibrated granular systems.

Chapter 3:

Anomalous velocity distribution

In this chapter, we study in detail the non-Maxwellian features of velocity distributions arising in inelastic gases, and more precisely their specific tendency to have overpopulated high energy tails. In- deed, contrary to the specific shape of the tail which depends on the details of the model, overpopulation seems to be a generic feature of inelastic gases, which expresses their deep non-equilibrium nature. In inelastic gases, two kinds of asymptotic velocity distributions may be considered:

a) scaling solutions. These solutions occur when the grains evolve freely, without external forcing and correspond to the Homogeneous Cooling State described above.

b) heated stationary solutions. These solutions are obtained by injecting energy into the system in order to counterbalance the energy loss. The energy is usually introduced by stochastic forces, usually Langevin-type forces.

In the first section, we review the methods developed in the literature in order to characterize the tail of the velocity distributions, namely the use of Sonine polynomials, and the Krook-Wu methods.

Then, we introduce the so-called Inelastic Maxwell Models (IMM), which are mathematical simplifica-

tions of the inelastic Boltzmann equation allowing an analytical treatment of the kinetic equation. By

studying a two-dimensional version of the model, we show that scaling solutions of IMM are character-

ized by power laws, which are associated to multiscaling features of the velocity moments. In order to

highlight the mechanisms leading to overpopulated high energy tails, we restrict the scope to a simpler

one-dimensional model, to which methods borrowed from random walk theory are applied. By focusing

on a linear model, the Inelastic Lorentz Model (ILM), we show that the anomalous tails may originate

from the fact that the average energy of particles depends on the number of collisions they have suffered

during their history. Finally, we discuss the emergence of L´ evy distributions in the one-dimensional

model. These are infinite energy stationary solutions of the kinetic equation, and we show that they

exhibit equilibrium-like features, such as the existence of a quantity analogous to a temperature. Fur-

thermore, they satisfy a fractional Fokker-Planck equation. Their physical relevance is discussed and

we show that, in more realistic situations, they correspond to truncated L´ evy distributions which are

quasi stationary in the course of time. This result is verified by DSMC simulations. Unfortunately, we

also show in the last section that the existence of these states is limited to the Maxwell Models, and

that they do not appear for arbitrary grain interactions, as is the case for true inelastic hard spheres, for instance.

Chapter 4:

Non-equipartition of energy

We focus on non-equilibrium features of the granular temperature, by discussing in detail the problem of non-equipartition of energy in inelastic mixtures. The aim of this chapter is twofold. On the one hand, we study idealized systems in order to highlight the mechanisms responsible for the non-equipartition phenomenon. Therefore, we first consider the simplest model for a mixture, namely the motion of an impurity in a bath. By focusing on suitable limits, we show the respective roles played by the inelastic exchange of energy between the components and the intrinsic time dependence of the system. The theoretical results are obtained by using mean field methods, which are very similar to the Inelastic Maxwell Model applied to mixtures. Then, we compare these results with expressions obtained from the Boltzmann equation for inelastic hard spheres and from event-driven simulations of the mixture.

Simulations show discrepancies which occur because of the neglect of a relevant mechanism in the Maxwell model. This leads us to propose a more complete mean field model, the Two Rates Maxwell Model (TRMM), whose properties are in perfect agreement with inelastic hard spheres and with the simulation results in the small inelasticity limit. This model rests on a more detailed account for the collision frequencies of the different components. Moreover, we generalize the TRMM in order to predict the distribution of temperature ratios for mixtures composed by an arbitrary number of species. We also verify the theoretical predictions with DSMC simulations, and we give analytical expressions for systems composed by a continuum of species. In the last section, we discuss the influence of the form of the velocity distribution on energy non-equipartition and more precisely the reason why this dependence is not taken into account by mean field models.

Chapter 5:

From the Maxwell Demon to the granular clock

In this last chapter, we focus on the granular Demon experiment, which is a very simple experiment

conceived to visualize the energy-density coupling characteristic of granular media. It consists of a box

divided into two equal compartments by a vertical wall starting from the bottom of the box, in which a

hole allows the passage of the grains from one compartment to the other. The box is filled with inelastic

identical particles submitted to gravitation and the energy is supplied by a vibrating bottom wall. This

very simple and easily realizable system has been shown to exhibit an order-disorder transition. Indeed,

for a high energy input, the system presents a homogeneous steady state, while when the energy input

is decreased, a phase transition occurs and an asymmetric steady state prevails. In the first section

of chapter 5, we review the effusive model proposed by Eggers to explain this transition. Moreover,

we present the simulation and experimental results obtained for the original experiment, for a related

experiment in the absence of gravitation, and for a system composed by three compartments instead of two.

In the second section, we present the Ehrenfest urn model, and the granular urn model which generalize the Eggers effusive model and allows to take into account finite size fluctuations. This approach has the further advantage to define a phase space, and to introduce a canonical distribution which clarifies the similarities of the granular Demon experiment with a thermodynamic phase transition.

These similarities are strengthened by focusing on the three urn model, and on the asymmetric urn model. The latter leads to a clear parallel between the Demon experiment and the phase transition occurring in a ferromagnetic crystal. The results are derived both theoretically and by solving the urn models through Monte-Carlo methods. Some results are also studied by performing event-driven simulations of the true dynamical system. Finally, analytical solutions for a model composed by an arbitrary number of urns are also derived. In the last section, we generalize the granular Demon experiment to binary mixtures and show by event-driven simulations the spontaneous apparition of concentration oscillations in the system. This is what we call a granular clock. We discuss the different mechanisms responsible for this behaviour, and introduce a very simple dynamical model, based on the Eggers approach, that we solve numerically. This model, though very crude, reproduces qualitatively well the phenomenology observed in event-driven simulations.

1.3 Theoretical and numerical methods

Before closing this chapter, we review some necessary theoretical and numerical tools that we used throughout this work. This is done in the following sections. First, we present the inelastic hard sphere model, which has been shown over the last decade to reproduce qualitatively well the features of granular fluids. Then, we give a rapid derivation of the elastic and of the inelastic Boltzmann equation, which are the basic kinetic equations for elastic and inelastic gases. We discuss some of their properties, namely the H-theorem and the emergence of the Maxwell-Boltzmann distribution in the elastic case, and the equation for energy dissipation in the inelastic case. Moreover, we insist on the similarities and differences between the two cases. Finally, in section 1.3.3, we explain qualitatively the numerical methods (DSMC and MD) used in this thesis in order to perform computer simulations of granular fluids. Let us note that these sections do not provide a full description of these subjects, and have been included in this work for the sake of completeness. Therefore, we refer the reader to classical monographs (ref. [234], [226], [3]) for detailed discussions about these topics.

1.3.1 A model for the grains

In the fluid-like regime, the individual motion of the particles constituting a granular system may

be described as a free motion of a rigid particle punctuated by instantaneous collisions. Indeed, the

grains, which are considered here as electrically neutral, do not interact via long range interactions, such

as electromagnetic forces. Moreover, in many situations, the interstitial fluid between the grains has

v^j

e

=r^ij/|r^ij| vⁱ

v^ij u^ij

disc i vⁱ v^j disc j

r^j rⁱ

r^ij

Figure 1.2:

Notations and variables describing a hard sphere collision.

no influence on their dynamics and may be neglected (of course, this fact is no longer true in a dense viscous fluid, such as a liquid). Consequently, their motion reduces to a sequence of free motion and instantaneous collisions when two grains are in contact. Depending on the absence or the presence of gravity, the inter-collisional motion is rectilinear or ballistic respectively. The most idealized model for grains is the Smooth Inelastic Hard Sphere Model (IHS), where the grains are considered as hard spheres whose interactions are smooth and dissipative. Let us note that, in the following, we call sphere any d-spherical particle, whatever the dimension of the system (for instance, 2-spheres will be called spheres or discs equivalently). The smoothness of the interaction means that there are no forces tangentially to the contact surface of the spheres during the collisions, i.e. no angular momentum is transfered between the particles. This simplification permits to neglect this variable which has no incidence on the evolution of the system. Let us precise that we do not pretend to quantitatively reproduce experimental or industrial facts in this work. Our motivation is essentially the understanding at the fundamental level of the main mechanisms responsible for the peculiar phenomena observed in granular media. Therefore, we will simplify the physical constituents of the system in such a way that it becomes analytically or numerically tractable, while preserving its essential qualitative features. Of course, this simplification consists in a delicate balance between reality and simplicity requirements. For instance, in the present case, the neglect of angular momentum may hide interesting phenomena associated to it, such as the non-equipartition of energy between the translational and the rotational energy due to inelasticity.

However, most of the original features of inelastic fluids are well described by smooth spheres. We, thus, limit our scope to elucidate the properties of systems composed by smooth but inelastic spheres.

The energy dissipation is accounted through the inelasticity parameter α, which we define by the

relation:

(.v

ij

)

^∗

=

−α

(.v

ij

) (1.1) where

v_ij ≡ v_i−v_j

,

r_ij ≡r_i −r_j

and is a unitary vector along the axis joining the centers of the two colliding hard spheres i and j,

≡ _|r^rij

ij|

. The velocities with the * index are the velocities after the collision and the unleashed velocities are the pre-collisional velocities. Since now on, bold symbols always denote vectorial variables. The smoothness condition and the momentum conservation imply that the other components of the velocities

vi

and

vj

are invariant during one collision. For instance, the mass center velocity

uij

=

^vi+v₂ ^j

is conserved

u^∗_ij

=

uij

. If the conservation of energy is further imposed, the inelasticity (or restitution) coefficient reduces to α = 1. Hence, in this work, the inelastic grains consist in hard spheres whose inelasticity coefficient is a constant belonging to the interval [0, 1[.

In the case of a totally dissipative collision α = 0, the whole perpendicular contribution to the relative kinetic energy

∼

(.v

ij

)

²

is lost and the velocity transformation is no longer invertible, as a continuum of velocities is sent to zero. For arbitrary values of α, the collision rule for the individual velocities reads directly:

v_i^∗

=

v_i−

(1 + α)

2

(.v_ij

)

v^∗_j

=

v_j

+ (1 + α)

2

(.v_ij

) (1.2)

which may be inverted if α

6= 0 into:

v_i⁰

=

v_i−

(1 + α)

2α

(.v_ij

)

v_j⁰

=

v_j

+ (1 + α)

2α

(.v_ij

) (1.3)

In (1.3), the prime and unprimed velocities denote the pre-collisional and post-collisional velocities respectively. This convention will be maintained in the sequel. We also assume that the restitution coefficient α is constant in this work, i.e. we neglect the dependance of α on the relative velocity of the particles. Obviously, this is an approximation since real world grains have a velocity dependent α (ref. [230], [72], [73], [74]). One should stress that this simplification has a dramatic but unphysical consequences in MD simulations, the so-called collapse phenomenon (ref. [205]). This phenomenon, which has been studied in detail during the last decade, is an extreme expression of the clustering instability specific to inelastic fluids and consists in the occurrence of an infinite number of collisions in a finite time. One should note that this unphysical effect has awful consequences in slowing down the computer simulations of the system, as they imply an infinite number of operations. Fortunately, this effect may be avoided by using numerical tricks, such as the use of non-constant restitution coefficient tending to 1 for small velocities (ref. [127]).

In the following, we will also be concerned by one dimensional systems, i.e. systems composed by

a large number of inelastic hard rods. This case leads to the following collision rules:

v

_i^∗

= v

i−

(1 + α) 2 v

ij

v

_j^∗

= v

j

+ (1 + α)

2 v

ij

(1.4)

whose inverse transformation reads, when α

6= 0:

v

⁰_i

= v

_i−

(1 + α) 2α v

_ij

v

_j⁰

= v

_j

+ (1 + α)

2α v

_ij

(1.5)

It is important to precise that one-dimensional systems tend toward trivial dynamics in the limit of elastic collisions. Indeed, in this case, their collision rules simplify into:

v

^∗_i

= v

_j

= v

⁰_i

v

_j^∗

= v

_i

= v

_j⁰

(1.6)

Therefore, a collision reduces to an interchange of the velocities between the particles, or equivalently to an interchange of their index. Because of the indistinguishability of particles, this dynamics is fully equivalent to the collision-less motion of rods on a straight line. Let us stress that the dynamics is no longer trivial when the particles perform inelastic collisions and that these systems exhibit complex trajectories which justify the use of kinetic theory.

1.3.2 Elastic and inelastic kinetic theory

Boltzmann equation

In 1872, Boltzmann introduced the basic equation of transport theory for dilute gases. We shall see below that the latter is a closed equation for the evolution of the position and velocity distribution function in an atomic or molecular gas. This closure property is exceptional and relies on the dilution assumption, which allows to neglect any other collision than the binary ones in the system. Dilution also implies that the collisions may be assumed as instantaneous and local in space, and that the pre-collisional correlations between the particles are negligible. We discuss these requirements further below. In the following, we first focus on the case of conservative interaction forces, and we derive the Boltzmann equation by closely following the steps exposed in (ref. [234]) and in (ref. [89]). We generalize this derivation to the case of inelastic collisions in the following section.

Consider a vessel of volume V, containing N particles whose binary interaction is characterized by

a strong repulsive core and a finite range σ, such as in a system of elastic hard spheres. The system is

taken in the Boltzmann-Grad limit, which restricts the system to a dilution characterized by nσ

^d

<< 1,

where n is the density of the particles, and d is the dimension of the system. In this limit and neglecting

gravity, the particles move along their free, straight trajectory most of the time. Then, when two particles approach each other closely enough, they interact and their velocities are deflected during a short period, after which they continue their free motion, along new straight trajectories and with new velocities. One should remark that the Boltzmann-Grad limit defines two time scales in the system, the interaction time scale which is very short, and the free path time scale which takes place over longer periods. This distinction allows to idealize the gas evolution by a series of almost discrete events, which are localized in a very small space region, and which are almost instantaneous. The hard sphere model is therefore the limiting case of this process, namely a model for which collisions are truly instantaneous.

This feature justifies the use of hard spheres as a paradigm in order to study dilute gas dynamics.

We define the distribution function f (r,

v;

t) over the 2d-dimensional phase space (r,

v), such that

f (r,

v;

t)drdv gives the number of particles the system whose positions and velocities belong to the volume drdv around (r,

v) at time t. In the following, we assume that this region in phase space is large

enough to contain a lot of particles, but small as compared to the range of variation of f. These limits ensure respectively that the variation in time of f is a continuous process, and that f is continuous in the one particle phase space. Moreover, we consider a system isolated from the external world for the sake of simplicity. There are three mechanisms responsible for the variation of f (r,

v;

t), namely the free-streaming term which accounts for the flow into and out of the volume dr due to the free motion of the particles; a gain term accounting for the collisions which make particles enter the volume dv; and a loss term accounting for the collisions which make particles leave the volume dv. Therefore, during a small time interval ∆t, the evolution is decomposed into:

∆f (r,

v;

t) = (∆f (r,

v;

t))

f low

+ (∆f (r,

v;

t))

coll

(1.7) The time scale for which this last relation is valid also suffers restrictions. Indeed, ∆t is much longer than the interaction time so that the interactions may be considered local in time, and ∆t is much shorter than the relaxation time in the gas so that the variation of f is small during that time.

The free streaming term is derived as the difference between the number of particles entering and leaving the small region during ∆t. First, let us consider a cubic cell, and calculate the flux of particles across the faces perpendicular to the x axis, which are located at x-

¹₂

∆x and x+

¹₂

∆x:

(∆f (x))

^x_{f low}

= v

x

∆t∆S∆v[f (x

−

1

2 ∆x)

−

f (x + 1

2 ∆x)] (1.8)

where ∆S is the surface of the small volume, perpendicular to the x axis. The previous assumptions for the time scale, and the size of the cells in phase space imply that f is a smooth function, which can be expanded in Taylor series around x, and whose only first terms have to be retained. Neglecting ∆

²

order terms and summing over all dimensions leads to the well known expression for the free streaming term:

(∆f (r,

v;

t))

_{f low}

=

−∆v∆t∆r

(v.∂

r

)f (r,

v;

t) (1.9)

v^x Dt v^x Dt

D S

x-1/2 dx x+1/2 dx

small cell in position space

Figure 1.3:

Flux of particles through a small cubic cell during the time interval∆t.

se b g

v r

v*

vt

Figure 1.4:

Geometry of a hard sphere collision.

where ∂

r

is the spatial gradient. In the case of an interaction-less gas, the evolution reduces to this contribution. Moreover, in the case of dissipative collisions, this term remains unchanged.

The less trivial point is to determine the change in velocity space due to collisions. This contribution may be calculated for arbitrary repulsive short-range interactions. However, we restrict the scope to hard spheres because this is the model considered in the case of granular gases. We first focus on the loss term and count the number of collisions suffered by particles with velocity

v1

, in the region ∆r∆v during the period ∆t, noting that such collision results in a change of the particle velocity. We consider collisions with a particle whose velocity is

v2

, and we set up a coordinate system with origin at the center of the particle 1, and with x-axis along the relative velocity

v12

=

v1 −v2

. Moreover, for the sake of clarity, we focus on the two-dimensional case. By examining figure 1.4, one observes that a particle will collide the first particle within time ∆t if it belongs to its collision cylinder whose surface is 2σv

₁₂

∆t. Let us remark that this collision occurs if no other particle interferes, which we may assume thanks to the dilution hypothesis. These collision cylinders are called the (v

1

,

v2

) collision cylinders.

In order to characterize the collisions, it is useful to introduce the impact parameter

b, which we define

by:

b

=

r−

r

_||v

v (1.10)

with:

r

_||

=

r.v

v (1.11)

The vector of impact characterizes the angle of the collision, and is related to these variables by:

σ =

b

+ γ

v

v (1.12)

where we define:

γ =

p

σ

²−

b

²

(1.13)

In order to calculate the loss and the gain term, we consider a stream of incident particles which are scattered by the particle. One first note that all particles 2 with velocity

v

which hit the central one within a time ∆t with impact parameters comprised between b and b+db, are contained in thin layers of thickness db, and of length v∆t. The number of such particles is therefore:

f (r,

v;

t)∆v db

|v|

∆t (1.14)

Hence, the total number of collisions involving the central particle during ∆t is obtained by integrating this number over all velocities and all impact parameter:

Z

dvdb f (r,

v;

t)

|v|

∆t (1.15)

The total number of collisions suffered by particles 1 is then obtained by counting the number of particles in ∆r whose velocity is

v1

, and which suffer a collision with a particle 2. However, this step is non trivial, because we need to evaluate the number of pairs of colliding particles 1 and 2 and, by definition, the knowledge of the one-particle velocity distribution is not sufficient for that purpose, as it does not take into account the possible correlations between the particles before collisions. This is a crucial step in the derivation of the Boltzmann equation as discussed in detail in section 1.3.2. Different approximation schemes may be devised in order to estimate these pre-collisional correlations. The choice made by Boltzmann corresponds to the so-called Molecular Chaos assumption, or Stosszahlansatz. Namely, the numbers of pairs of molecules in the element ∆r with respective velocities in the range (v

1

,

v1

+ ∆v

1

) and (v

₂

,

v₂

+ ∆v

₂

) which are able to participate in a collision is given by:

f (r,

v₁

; t)∆r∆v

₁

f (r,

v₂

; t)∆r∆v

₂

(1.16) This assumption may be understood intuitively by noting that a binary collision between two particles which have already interacted together, directly or indirectly through a common set of other particles, is an improbable event in a very dilute regime. Indeed, in this limit, colliding particles come from different region in space and have completely different collisional histories. Therefore, these two particles may be considered uncorrelated, i.e. statistically independent when colliding, as assumed by (1.16). One should precise that this assumption does not imply that no correlations at all are taken into account by the Boltzmann assumption. Indeed, the neglect of pre-collisional correlations does not remove the existence of post-collisional correlations, which are very strong since two particles are not independent the one from the other just after a collision.

The use of the Stosszahlansatz allows to calculate explicitly the loss term. Equation (1.15) together with (1.16) lead to the following expression for the number of collisions involving a particle with velocity

v1

:

Z

dv

₂

dbf (r,

v₂

; t)f(r,

v₁

; t)|v

₁₂|

∆t (1.17) which we put in its standard form by straightforward calculations. In the case of hard spheres, it leads to the following expression, valid in arbitrary dimensions:

σ

^d−1 Z

dv

₂

d

^d(.v₁₂

)Θ(.v

₁₂

)f (r,

v₁

; t)f (r,

v₂

; t) ∆t (1.18) where the θ function expresses the condition that the velocity vectors of the two particles are such that they are going to collide. The (.v

12

) factor is specific to the hard spheres model, but other microscopic interactions lead to other velocity dependences of the scattering cross section. A very useful case, which we study in detail in this work is in the so-called Maxwell-molecules model, for which the scattering cross section does not depend on

|v₁₂|, but rather on the average of that quantity.

In order to derive the expression of the gain term, we need to know how two particles collide in such

a way that after the collision, one of them has a velocity

v1

. These collisions, which are at the heart of

Original velocities

Reversed velocities

Figure 1.5:

Typical collision between elastic hard spheres. On the right, we show that these systems exhibit time reversibility. Indeed, we first plot the trajectory with the initial velocities.

Then, we reverse the velocities after the collision and let the system evolve. One verifies that the new trajectories are identical, but they are followed in the opposite direction.

v

¹

v

²

v*

¹

v*

²

e

-v'

²

-v'

¹

-v

²

-v

¹

time reversibility

-e

v'

¹

v'

²

v

²

v

¹

rotation

e

Figure 1.6:

Construction of a restituting collision from a direct collision, by using time and space symmetry.

the gain process, are called the restituting collisions, while the previous collisions are called the direct collisions. Because of the microscopic time symmetry of the model (fig. 1.5), the restituting and the direct collisions are related the one to the other. Indeed, to each direct collision, there is a restituting collision which is obtained by inverting the velocities of the particles. Let us stress that if the initial collision was characterized by a impact vector σ, the impact vector of the restituting collision is then

−σ. This effect (fig. 1.6) is a direct consequence of the collision rule (1.2) applied to elastic spheres,

α = 1:

v^∗₁

=

v1−(.v12

)

v^∗₂

=

v2

+

(.v12

) (1.19)

One verifies that a collision between particles which have the velocities (v

^∗₁

,v

₂^∗

) and whose angle is characterized by

−, transform these velocities into their the initial values (v₁

,v

2

):

v^∗∗₁

=

v₁

v^∗∗₂

=

v₂

(1.20)

This symmetric feature implies that the direct and the restituting collisions are characterized by the same scattering cross section. Therefore, the gain term reads:

σ

^d−1 Z

dv

⁰₂

d

^d(.v12

)Θ(.v

12

)f (r,

v₁⁰

; t)f (r,

v⁰₂

; t) dv

⁰₁

∆t (1.21)

Another key feature of conservative dynamics stems from the fact that the pre-collisional and post- collisional velocities are related by a canonical transformation, which preserves the volume in phase space. Indeed, one verifies from (1.20) that the jacobian of the elastic collision transformation is unitary:

J =

|

∂(v

^∗_i

,

v^∗_j

)

∂(v

i

,

vj

)

|

= 1 (1.22)

and therefore:

dv

_i^∗

dv

_j^∗

= dv

_i

dv

_j

= dv

_i⁰

dv

⁰_j

(1.23) The use of this relation into (1.21) allows to rewrite the gain term into the canonical form:

σ

^d−1 Z

dv

₂

d

^d(.v₁₂

)Θ(.v

₁₂

)f (r,

v⁰₁

; t)f (r,

v⁰₂

; t) dv

₁

∆t (1.24) Together with the previous contributions, namely the free flux term and the collisional loss term, this expression leads, after a proper division by phase space volumes and time intervals, to the celebrated Boltzmann equation, or the kinetic equation for dilute gases:

∂f (r,

v1

; t)

∂t +

v.

∂f (r,

v1

; t)

∂r = σ

^d−1 Z

ddv

2

(.v

12

)Θ(.v

12

)

×

[f (r,

v⁰₁

; t)f (r,

v₂⁰

; t)

−

f (r,

v₁

; t)f(r,

v₂

; t)] (1.25) Remarkably, the Boltzmann approach reduces the evolution equations of N

→ ∞

particles to a closed equation for the one-particle probability distribution. Moreover, this reduction of dynamics leads to an irreversible kinetic equation, despite its reversible microscopic foundation. This fact is discussed further below.

Let us now review some basic properties of the Boltzmann equation. The only stationary solution of (1.25) is shown to be the well-known Maxwell-Boltzmann distribution:

f

E

= n

(2πT )

^d2

e

^−mv

2

2T

(1.26)

where d is the dimension of the system, n is the average density of the gas and T is the equilibrium temperature. This result is easily verified by inserting (1.26) into (1.25), and by using the conservation of energy during the elastic collision,

v^∗2₁

+

v^∗2₂

=

v²₁

+

v₂²

. In the remainder of this thesis, we write (1.25) into a more concise form:

∂f (r,

v₁

; t)

∂t +

v.

∂f(r,

v₁

; t)

∂r = K[f, f ] (1.27)

where K[f, f ] denotes the Boltzmann collision operator:

K[f, f ] = σ

^d−1 Z

ddv

₂

(.v

₁₂

)Θ(.v

₁₂

) [f (r,

v₁⁰

; t)f (r,

v⁰₂

; t)

−

f (r,

v1

; t)f (r,

v2

; t)] (1.28)

Let us stress that (1.26) is not the only solution of the non-linear relation K[f, f ] = 0. Indeed, straight- forward calculations show that the following distribution is also solution:

g

L

(r) = n(r) (2πT (r))

^d2

e

⁻

m(v−u(r))2

2T(r)

(1.29)

where n(r) is the local density, T(r) the local kinetic temperature, and

u(r) the local hydrodynamic

velocity. These solutions, which are usually referred to local equilibrium distributions, show clearly that elastic collisions do not affect the Maxwell-Boltzmann distribution locally and will be discussed in detail in chapter 2.

Irreversibility of the Boltzmann equation is usually shown by examining the temporal evolution of the so-called H-functional:

h(r; t) =

Z

dvf (r,

v)[ln(f

(r,

v;

t)

−

1] (1.30) Boltzmann (ref. [52]) showed that the equation of evolution for this quantity reads:

∂

t

h(r; t) =

−∂_r

.(hu +

JH

) + σ

H

(1.31)

where:

J_H

=

Z

dvVf (ln f

−

1) σ

H

=

Z

dvK[f, f] ln f (1.32)

These expressions imply that the quantity h is not conserved locally, through the presence of a source term σ

H

in (1.31). This source term satisfies the so-called H-theorem:

σ

_H ≤

0

σ

H

= 0

⇔

local equilibrium (1.33)

The stroke of genius of Boltzmann was to associate this quantity with the local entropy in the system through the relation:

h(r; t) =

−ρ(r;

t)s(r; t) (1.34) His intuition clearly was inspired by the analogy of this theorem with the mathematical expression of the second principle of thermodynamics in terms of entropy. The H-theorem implies, amongst other features, that there exists a time-dependent quantity, H(t), that never increases in the course of time.

This quantity is the extensive counterpart of h(r; t):

H(t) =

Z

dr h(r; t) (1.35)

This definition, together with relations (1.31) and (1.33) show clearly that:

∂

t

H(t)

≤

0 (1.36)

Let us stress that this relation is valid in an isolated system, where there is no influence of the external world on the gas. The quantity H is constant on time only when the system is in total equilibrium.

This behaviour implies that H decreases in the system, until is has reached its minimum value, i.e. until the velocity distribution has attained equilibrium (1.26). Consequently, solutions of the Boltzmann equation for an isolated system approach irreversibly the equilibrium state, just like a real gas does.

The H-theorem is usually considered as a microscopic foundation to the second law of thermodynamics, which states that entropy of an isolated system is a quantity that can only increase in the course of time. In order to resolve the apparent contradiction between the microscopic reversible dynamics and the irreversible Boltzmann equation, one has to interpret it as a description of the most probable behaviour of an ensemble of systems (ref. [172]), rather than a description of the exact behaviour of the gas dynamics, i.e. of one member of this ensemble. Let us stress that this interpretation will be illustrated and discussed further in section 5.2.1, which is dedicated to the Ehrenfest urn model.

Generalization to Inelastic Hard Spheres

The derivation of the Boltzmann equation for inelastic hard spheres is now a well defined problem, whose details and limitations have been much discussed over the last decade (see for instance ref. [264]

and references therein). In the following, we derive it in an approach very similar to that of the previous section, in order to point out the main differences and physical features originating in the inelasticity. Let us first note that the inelastic Boltzmann equation has been derived in the literature by more fundamental methods (ref. [69], [239]), namely methods based upon the so-called pseudo-Liouville equation (ref. [234]) for inelastic hard spheres. The latter is an equation for the N particles phase space distribution, and is constructed by considering an ensemble of equivalent initial conditions which define an initial distribution over phase space:

f

N

(R,

V; 0)

(1.37)

where

R

and

V

denote respectively (r

1

, ....,

rN

) and (v

1

, ....,

vN

). Each member of the ensemble evolves along its own trajectory starting from its initial position. This induces different trajectories in phase space for the different systems of the ensemble, and the N-particles phase space distribution of the ensemble at time t is obtained by:

f

N

(R,

V;

t) = f

N

(R

−t

,

V−t

; 0) (1.38)

where

Rt ≡ Rt

(R,

V) and Vt ≡Vt

(R,

V) define the coordinates in phase space at time t of a point

whose coordinates were (R,

V) at time 0. This equation may be rewritten into an integro-differential

form, in the case of elastic or inelastic hard spheres, which is called pseudo-Liouville equation. The term

pseudo comes from the fact that dynamics of hard spheres are singular, as the individual trajectories

of the particles are not differentiable. Hence, this forbids the existence of a Hamiltonian. The pseudo-

Liouville equation gives rise to the so-called BBGKY hierarchy, from which the Boltzmann equation

may be obtained from suitable approximations.

One should remark that the nature of the pseudo-Liouville equation is very different from that of the Boltzmann equation, for several reasons. Indeed, in the case of elastic interactions, the Liouville equation is as reversible as the microscopic dynamics, and does not yield to the observed macroscopic irreversibility. Moreover, this equation predicts the deterministic evolution of an ensemble of equivalent systems that constitute the initial ensemble. The only probabilistic element in the Liouville equation is found in the statistical distribution of the initial conditions. In contrast, in the Boltzmann equation, the stochasticity appears at two levels: in the initial condition and at the level of the dynamics itself.

Moreover, the Boltzmann equation is a macroscopic equation, since it describes the most probable evolution of one realization of the physical system, given the knowledge of its initial velocity distribution.

Indeed, it predicts the time-evolution of a quantity, the one particle distribution function, whose time evolution almost surely behaves as the solution of the Boltzmann equation does. This feature will be discussed further in the following and in chapter 5.

From the derivation of the elastic Boltzmann equation, most of the developments and assumptions may be transfered into the case of inelastic particles. This is due to the similarities between the elastic and the inelastic hard spheres model. For instance, IHS is also characterized by instantaneous collisions, which give rise to well defined time scales in the system. Consequently, we define the velocity distribution of the gas, which is based upon the mesoscopic scale ∆r, ∆v, ∆t as previously. The instantaneity of collisions ensures that relation (1.7) is valid. Of course, the expression for the free streaming term (1.9) is not modified by the inelasticity of collisions. Therefore, the whole difficulty consists in obtaining the collision term, i.e. in counting the direct and restituting collisions which affect the particle velocities.

The first step comes from the following observation: the developments used in order to derive the loss term (1.18) made no explicit mention on the elasticity of the collision. The main argument was that particles having velocity

v1

, and which perform a collision with a particle 2, have a different velocity after the collision. This property is also valid in the case of inelastic collisions. The remaining arguments were geometrical and probabilistic. By construction, the geometrical results are still applicable because both models deal with hard spheres. On the contrary, the molecular chaos assumption is not justified a priori, because inelasticity changes dramatically the nature of dynamics. We discuss further below the validity of this assumption, where we show that the Stosszahlansatz applies only in the combined limit of the low density and low inelasticity. Consequently, the loss term of the inelastic Boltzmann equation:

σ

^d−1 Z

dv

2

d

^d(.v12

)Θ(.v

12

)f (r,

v1

; t)f (r,

v2

; t) ∆t (1.39)

In contrast with this result, the time-reversible nature of collisions, as well as the microscopic

conservation of energy were primordial in the derivation of the gain term (1.24), especially in order to

find a relation between direct and restituting collisions. In the following, we generalize this relation into

the context of inelastic hard spheres. First, let us focus on direct and restituting collisions in that case.

Original velocities

Reversed velocities

timeasymmetry

energy dissipation

Figure 1.7:

Typical collision between inelastic hard spheres. Inelasticity makes the post-collisional velocities more parallel as compared to an elastic collision. This is a direct effect of inelasticity which lowers the quantity (.v₁₂) during one collision. In the figures at right, we illustrate the irreversibility of the microscopic dynamics, by showing that the inverted trajectories are no longer the same as the direct ones.

v

¹

v

²

v*

¹

v*

²

e -e

v'

¹

v'

²

v

²

v

¹

Figure 1.8:

Construction of a restituting collision from a direct collision.

When the collisions were elastic, we noticed that the post-collisional velocities (v

^∗₁

,

v^∗₂

) with impact angle are equivalently pre-collisional velocities with the impact angle

−. In the IHS, however, this

observation based upon time-reversibility no longer holds (fig. 1.7). This implies (fig. 1.8) that the relative velocity of the restituting collisions is higher than the relative velocity of the direct collisions, when α

6= 1:

(.v

₁₂

)

⁰

= 1

α (.v

₁₂

)

≥

(.v

₁₂

) (1.40)

This relation is a direct consequence of (1.1), and leads to an increase of the collision frequency of restituting collisions as compared to direct collisions. Indeed, expression (1.39) for instance, shows clearly that the collision frequency, or equivalently the scattering cross section is proportional to the relative velocity of the colliding discs

∼

(.v

12

). Therefore, the collision frequency of the restituting collisions is higher than that of the direct collisions and, equivalently, the relative velocity is higher before the collision than after the collision. This is a direct consequence of inelasticity. The next step consists in counting these collisions (v

⁰₁

,

v⁰₂

)

→

(v

₁

,

v₂

), where the prime velocities are given by the collision rule (1.1). The above discussion directly leads to the following expression:

1 α σ

^d−1

Z

dv

⁰₂

d

^d(.v12

)Θ(.v

12

)f (r,

v₁⁰

; t)f (r,

v⁰₂

; t) dv

⁰₁

∆t (1.41)

where the

_α¹

factor takes into account the increased collision frequency. In order to derive the inelastic

Boltzmann equation as a standard gain-loss process, we express the volume element of the restituting

velocities as a function of the volume element of the direct velocities. Contrary to the elastic case, this

transformation does not preserves the phase space volume. Indeed, the jacobian of the transformation

v⁰ →v

reads: