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Thèse de doctorat/ PhD Thesis Citation APA:
Lambiotte, R. (2004). Inelastic gases: a paradigm for far-from-equilibrium systems (Unpublished doctoral dissertation). Université libre de Bruxelles, Faculté des Sciences – Physique, Bruxelles.
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Université Libre de Bruxelles Faculté des Sciences
Phénomènes non-linéaires et Mécanique statistique
Inelastic Gases: a paradigm for far-from-equilibrium Systems.
Dissertation originale présentée par Renaud Lambiotte
en vue de l’obtention du grade de Docteur en Sciences
Année académique 2003-2004
Université Libre de Bruxelles
003303313
AU rights reserved.
Title Page...
Table of Contents...
1 Introduction
1.1 What are granular media?...
1.2 Sketch of the thesis...
1.3 Theoretical and numerical methods...
1.3.1 A model for the grains...
1.3.2 Elastic and inelastic kinetic theory...
1.3.3 Numerical methods...
2 Granular hydrodynamics
2.1 Introduction...
2.1.1 Conservation équations ...
2.1.2 Non-equilibrium thermodynamics...
2.2 From kinetic theory to hydrodynamics...
2.2.1 Local equilibrium and normal solutions . . . 2.2.2 Chapman-Enskog method...
2.3 Simplified kinetic models...
2.3.1 Elastic BGK model...
2.3.2 Inelastic BGK models...
2.4 Granular hydrodynamics...
2.4.1 Local HCS ...
2.4.2 Chapman-Enskog scheme...
2.5 Température inversion...
2.6 Summary of results...
3 Anomalous velocity distribution
3.1 Introduction...
3.1.1 Isotropy and velocity corrélations...
3.1.2 Non-equilibrium stationary States...
3.1.3 Anomalous distribution in the IHS model . . . 3.2 Inelastic Maxwell Models ...
3.2.1 Introduction ...
3.2.2 Bobylev-Fourier method and moments hierarchy 3.2.3 Stationary Moments...
3.2.4 Multiscaling behaviour...
3.3 One-dimensional IMM...
3.3.1 Toy models...
3.3.2 Discrète time dynamics and random walks . . . iii
i iii 1 1 4 7 8 11 26 32 32 32 35 39 39 40 45 45 47 51 51 54 65 69 70
70
70
74
77
81
81
85
89
90
93
93
95
3.3.3 Central-Limit theorem... 97
3.3.4 Maxwell-Boltzmann scaling solution... 99
3.3.5 One dimensional Lorentz model... 102
3.4 Lévy based statistics... 106
3.4.1 Introduction ... 106
3.4.2 Physical relevance and simulations of Lévy distributions... 107
3.4.3 Equilibrium-like behaviour ... 109
3.5 Limitations of inelastic Maxwell models... 113
3.6 Summary of résulta... 118
4 Non-equipartition of energy 119 4.1 Introduction... 119
4.1.1 OverView... 119
4.1.2 Inelastic mixtures ... 121
4.2 Qualitative approach... 126
4.2.1 Inelastic Lorentz Models... 126
4.2.2 Inelastic exchange of energy... 127
4.2.3 Relaxation time compétition ... 131
4.3 Generalized Maxwell model... 136
4.3.1 In the Lorentz limit... 136
4.3.2 Arbitrary mixtures... 138
4.3.3 Multi-component mixtures ... 142
4.4 Non-Maxwellian velocity distribution... 150
4.5 Summary of results... 153
5 Prom the Maxwell démon to the granular dock 154 5.1 Maxwell Démon... 154
5.1.1 Historical bati^ground... 154
5.1.2 The granular Démon experiment... 157
5.2 Ehrenfest urn model... 163
5.2.1 Classical model... 163
5.2.2 Granular urn experiment ... 171
5.2.3 Thermodynamic-like approach ... 180
5.2.4 Asymmetric Démon and hystérésis... 185
5.2.5 Three compartments... 197
5.3 The granular dock... 204
5.3.1 Description of the experimental setting ... 204
5.3.2 Asymptotic régimes ... 206
5.3.3 Qualitative model ... 212
5.4 Summary of results... 219
6 Conclusion 220
A Granular hydrodynamics 223
Bibliography 226
Introduction
1.1 What are granular media?
Granular media are Systems composed of a large number of macroscopie 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 hâve expensive or dramatical conséquences. In the pharmaceutical industry, for instance, where Chemical components are usually made of granules or uowder, the properties of mixing and ségrégation 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 hâve striking conséquences, as for the storage of cereals in silos which may undergo dramatic breakages due to the huge internai 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 hâve then been considered as a part of applied physics and, except some notable attempts by Faraday and Coulomb, they hâve not reached the attention of the physicist community until recently.
There are several reasons for this new interest, beyond the économie impératives discussed above (ref. [34], [35], [147], [148], [158], [138]). A first reason cornes from the appearent simplicity of granular materials, which surprisingly leads to a very rich and often counterintuitive phenomenology. These features mainly originate from the macroscopie dimensions of the grains, which at the relevant scale of granular physics are considered as solid objects whose internai degrees of freedom are neglected.
This has two important conséquences: the ordinary température plays no rôle on the grain motion, and the interaction between the grains is dissipative. The non-relevance of température cornes from
1
Figure 1.1: Typical occurrences of granular media in nature and industrial processes: explosion of a silo due to the internai stress, formation of planetary rings, granular transport in a pharmaceutical industry and coal transport in a mine.
the fact that kgT ~ 0 at the scale of the giains. This implies that entropy considérations may be out-weighted by dynamical effects, and that the exploration of phase space is unusual as compared to that of an equilibrium System. The dissipativity of collisions also contributes to this anomalous exploration of phase space. Indeed, the transfer of kinetic energy to the internai degrees of freedom of the grains, for instance under the form of internai 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 irréversible behaviour may favour some régions 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 complété 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 répulsive 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
décades, their striking features hâve been put in parallel with phenomena occurring in more complex Systems, and hâve been used as a fruitful metaphor in order to describe dissipative dynamical Systems.
For instance, sand piles hâve been used by De Gennes (ref. [85]) as a macroscopie picture for the motion of flux Unes in superconductors, and by Bak (ref. [8]) as a paradigmatic expérience in order to lay the foundations of the Self-Organized Criticality concept. Similarly, other flelds of physics hâve 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 hâve been used in order to study the features of non-equilibrium Systems. Indeed, due to their internai 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 axe characterized by an energy flux from the external world toward the internai 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 paxt of their energy and momentum, and by grain-surface interactions during which the grai' s 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 température, 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 rôle in the description of granular fluids is nonetheless abusively called température. 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 température should exhibit the same features as a well-defined thermodynamic température, 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 régime, 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 équations for these Systems, by the use of Chapman-Enskog methods and to
dérivé a microscopie expression for the transport coefficients. This scheme, which haa led to the prédiction of new transport processes in granular fluids, has provided a microscopie 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 macroscopie 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 hâve been used as a paradigm in order to apply equilibrium methods to granular fluids. Amongst others, there are the identification of diffusion and self-difltusion (ref. [58], [82], [189]), Einstein relations between the diffusion coefficients and mobility (ref. [120]), the dérivation 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 ihis 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 spécifie 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 macroscopie conséquences
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 équation, in order to
dérivé hydrodynamic équations for inelastic dilute gases. First, we give a short introduction of Non-
Equilibrium Thermodynamics and show the reasons why the Fourier law q = —pd^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 socalled 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 dérivé a generalized Fourier law for the beat flux of the granular fluid q = —pLÔ^T —
kÔ
tU and isolate the mechanisms which make
k^Q. 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 équivalent 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 équation. This mean-fleld approach was studied during the flrst year of this thesis, with a model called wooled elastic hard spheres. In the last section, we illustrate the macroscopie effects of the new transport coefficient
kby discussing the phenomenon of température 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 spécifie tendency to hâve overpopulated high energy tails.
Indeed, contrary to the spécifie shape of the tail which dépends on the details of the model, overpop- ulation 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 ohtained 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 flrst section, we review the methods developed in the literature in order to characterize the tadl 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 simplifi
cations of the inelastic Boltzmann équation allowing an analytical treatment of the kinetic équation.
By studying a two-dimensional version of the model, we show that scaling solutions of IMM are char-
acterized 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 dépends on the number of collisions they hâve suffered during their history. Finally, we discuss the emergence of Lévy distributions in the one-dimensional model. These are infinité energy stationary solutions of the kinetic équation, and we show that they exhibit equilibrium-like features, such as the existence of a quantity analogous to a température. Furthermore, they satisfy a fractional Fokker-Planck équation. Their physical relevance is discussed and we show that, in more realistic situations, they correspond to truncated Lévy distri
butions which are quasi stationary in the course of time. This resuit 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 température, 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 m; :del for a mixture, namely the motion of an impurity in a bath. By focusing on suitable limits, we show the respective rôles played by the inelastic exchange of energy between the components and the intrinjic 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 équation 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 complété 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 température ratios for mixtures composed by an arbitrary number of species.
We also verify the theoretical prédictions 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 Démon to the granular clock
In this last chapter, we focus on the granular Démon 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 présents 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 présent 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 présent the Ehrenfest urn model, and the granular urn model which gen- eralize 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 clar
ifies the similarities of the granular Démon 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 Démon 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 Démon 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 dock. 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 présent 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 dérivation of the elastic and of the inelastic Boltzmann
équation, which are the basic kinetic équations 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 équation for energy dissipation in the inelastic case. Moreover, we insist on the
similarities and différences 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 hâve 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 régime, 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 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-sphcres will be called spheres or dises 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 .nomentum is transfered between the particles. This simplification permits to neglect this variable which has no incidence on the évolution of the System. Let us précisé that we do not prétend 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 délicate balance between reality and simplicity requirements.
For instance, in the présent 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 a, which we define by the relation:
(e.Vij)* = -a (e.Vij) (1.1)
Figure 1.2: Notations and variables describing a hard sphere collision.
where Vjj = v* — Vj, = rj — rj and e is a unitary vector along the axis joining the centers of the two colliding hard spheres i and j, e = |^. 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 dénoté vectorial variables. The smoothness condition and the momentum conservation imply that the other components of the velocities Vj and Vj are invariant during one collision. For instance, the mass center velocity is conserved u*^ .—' Uÿ. If the conservation of energy is further imposed, the inelasticity (or restitution) coefficient reduces to a = 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 a = 0, the whole perpendicular contribution to the relative kinetic energy ~ (e.Vÿ)^ is lost and the velocity transformation is no longer invertible, as a continuum of velocities is sent to zéro. For arbitrary values of a, the collision rule for the individual velocities reads directly:
(1 + a)
^ V,:--- --- e(e-Vy)
Vj = Vj +
(1+a) 2 e(e.vy) which may be inverted if a 0 into:
(1+a)
V,- =Vi---;---e(e.Vy)
V,- = V,- + 2a (1 +
q)
2a e(e.vy)
(1.2)
(1.3)
In (1.3), the prime and unprimed velocities dénoté the pre-collisional and post-collisional velocities
respectively. This convention will be maintained in the sequel. We also assume that the restitution coefficient a is constant in this work, i.e. we neglect the dépendance of a on the relative velocity of the particles. Obviously, this is an approximation since real world grains hâve a velocity dépendent a (ref. [230], [72], [73], [74]). One should stress that this simplification has a dramatic but unphysical conséquences 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 spécifie to inelastic fluids and consists in the occurrence of an infinité number of collisions in a finite time. One should note that this unphysical effect has awful conséquences in slowing down the computer simulations of the System, as they imply an infinité 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* =Vi-
V* = Vj +
2 ^ whose inverse transformation reads, when a ^ 0:
(1.4)
Vi = Vi
Vj = Vj
(1.5)
It is important to précisé that one-dimensional Systems tend toward trivial dynamics in the limit of elastic collisions. Indeed, in this case, their collision rules simplify into:
< = Vj = U-
Vj=Vi = v'j (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
équivalent 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 équation
In 1872, Boltzmann introduced the basic équation of transport theory for dilute gases. We shall see below that the latter is a closed équation for the évolution of the position and velocity distribution fonction 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 corrélations 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 dérivé the Boltzmann équation by closely following the steps exposed in (ref. [234]) and in (ref. [89]). We generalize this dérivation 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 répulsive core and a finite range cr, 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 ncr‘^ << 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 évolution by a sériés of almost discrète events, which are localized in a very small space région, 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 /(r,v;t) over the 2d-dimensional phase space (r, v), such that /(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 région 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 continuons process, and that f is continuons 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 /(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 At, the évolution is decomposed into:
Figure 1.3: Flux of particles through a small cubic cell during the time interval At.
A/(r,v;t) = {Af{r,v-t))fiow + (A/(r, v; t))co« (1.7) The time scale for which this last relation is valid also suffers restrictions. Indeed, At is much longer than the interaction time so that the interactions may be considered local in time, and At 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 différence between the number of particles entering and leaving the small région during At. 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-^Ax and x+^Ax:
{M{x)Yfiow = ViAtASAv[/(x - ^Ax) - /(x + ^Ax)] (1.8) where AS 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 sériés around x, and whose only flrst terms hâve to be retained. Neglecting A^
order terms and summing over ail dimensions leads to the well known expression for the free streaming term:
(A/(r, v; t))/;ow = -AvAtAr (v.clr)/(r, v; t) (1.9)
where dr is the spatial gradient. In the case of an interaction-less gas, the évolution reduces to this
contribution. Moreover, in the case of dissipative collisions, this term remains unchanged.
Figure 1.4: Geometry of a hard sphere collision.
The less trivial point is to détermine the change in velocity space due to collisions. This contribu
tion may be calculated for arbitrary répulsive 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 'oss term and count the number of collisions suffered by particles with velocity '.'i, in the région Al Av during the period At, noting that such collision results in a change of the parucle 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 Vi2 = vi — 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 At if it belongs to its collision cylinder whose surface is 2av\2At. Let us remark that this collision occurs if no other particle interfères, which we may assume thanks to the dilution hypothesis. These collision cylinders are called the (vi, 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||^ (1.10)
with:
r*v ni = V
The vector of impact e characterizes the angle of the collision, and is related to these variables by:
<re = b + 7—
V
(1.12)
where we define:
7 = \/(
t2 _ ^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 ail particles 2 with velocity v which hit the central one within a time At with impact parameters comprised between b and b+db, are contained in thin layers of thickness db, and of length vAt. The number of such particles is therefore:
/(r,v;t)Av db |v| At (1.14)
Hence, the total number of collisions involving the central particle during At is obtained by integrating this number over ail velocities and ail impact parameter:
1.3.2. Different approximation schemes may be devised in order to estimate these pre-collisional corrélations. The choice made by Boltzmann corresponds to the so-called Molecular Chaos assumption, or Stosszahlansatz. Namely, the numbers of pairs of molécules in the element Ar with respective velocities in the range (vi, vi + Avj) and (v2, V2 + AV2) which are able to participate in a collision is given by:
This assumption may be understood intuitively by noting that a binary collision between two particles which hâve already interacted together, directly or indirectly through a common set of other particles, is an improbable event in a very dilute régime. Indeed, in this limit, colliding particles corne from different région in space and hâve completely different collisional historiés. Therefore, these two particles may be considered uncorrelated, i.e. statistically independent when colliding, as assumed by (1.16). One should précisé that this assumption does not imply that no corrélations at ail are taken into account by the Boltzmann assumption. Indeed, the neglect of pre-collisional corrélations does not
(1.15) The total number of collisions suffered by particles 1 is then obtained by counting the number of particles in Ar whose velocity is vi, 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 définition, the knowledge of the one-particle velocity distribution is not suflicient for that purpose, as it does not take into account the possible corrélations between the particles before collisions.
This is a crucial step in the dérivation of the Boltzmann équation as discussed in detail in section
/(r,vi;t)ArAvi /(r,V2;t)ArAv2 (1.16)
remove the existence of post-collisional corrélations, 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 vi:
j dv2d6/(r,V2;t)/(r,vi;f)|vi2| At (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;
dv2 d‘^6(e.vi2)0(e.vi2)/(r,vi;t)/(r,v2;t) At (1.18) where the 9 function expresses the condition that the velocity vectors of the two particles are such that they are going to collide. The (e.vi2) factor is spécifie to the hard spheres model, but other microscopie 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 dépend on |vi2|, but rather on the average of that quantity.
In order to dérivé 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 vi. These collisions, which are at the heart of the gain process, are called the restituting collisions, while the previous collisions are called the direct collisions. Because of the microscopie 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 cre, the impact vector of the restituting collision is then —ae. This effect (fig. 1.6) is a direct conséquence of the collision rule (1.2) applied to elastic spheres, a = 1:
v* = vi - e(e.vi2)
V2 = V2 + e(e.vi2) (1-19)
One vérifiés that a collision between particles which hâve the velocities (v*,V2) and whose angle is characterized by —e, transform these velocities into their the initial values (vi,V2):
V2* = 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:
Original velocities
Reversed velocities
Figure 1.5: Typical collision between elastic hard spheres. On the right, we show that these Systems exhiba 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 vérifiés
that the new trajectories are identical, but they are followed in the opposite direction.
\v*2 /v*i \vi ^V2
■*—I—(
\ // e \ *
f V2
time
reversibility rotation
Figure 1.6: Construction of a restituting collision from a direct collision, by using time and space symmetry.
^<^‘^e(e.vi2)©(e.vi2)/(r,v'i;t)/(r,v2;t) dv'jAt (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 préserves the volume in phase space. Indeed, one vérifiés from (1.20) that the jacobian of the elastic collision transformation is unitary:
and therefore:
J = 9(vi,vj)
= 1( 1 . 22 )
d\*d\* = dvidvj = dv-dVj- (1.23)
The use of this relation into (1.21) allows to rewrite the gain term into the canonical form:
dv2 d‘^e(e.vi2)©(e.vi2)/(r,v'i;t)/(r,v2;t) dviAt (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 équation, or the kinetic équation for dilute gases:
df{r,vi;t)
+ V.
5/(r,vi;t)
= (
t‘^ dedv2(e.vi2)0(e.vi2)
dt dr
X [/(r,vi;t)/(r,v'i;t) -/(r,vi;t)/(r,vi;t)] (1.25) Remarkably, the Boltzmann approach reduces the évolution équations of —> oo particles to a closed équation for the one-particle probability distribution. Moreover, this réduction of dynamics leads to an irréversible kinetic équation, despite its réversible microscopie foundation. This fact is discussed further below.
Let us now review some basic properties of the Boltzmann équation. The only stationary solution of (1.25) is shown to be the well-known Maxwell-Boltzmann distribution:
ÎE
Tl _mv^
--- Te 2T (1.26)
(27rT)2
where d is the dimension of the System, n is the average density of the gas and T is the equilibrium température. This resuit is easily verified by inserting (1.26) into (1.25), and by using the conservation of energy during the elastic collision, -|- v^^ = -f Vj. In the remainder of this thesis, we write (1.25) into a more concise form:
(1.27)
dt dr
where K[f,f] dénotés the Boltzmann collision operator:
^[fJ] = J dedv
2{e.vi
2)
0{e.vi
2) [f{r,v[;t)f{T,y[;t)
- /(r,vi;t)/(r,vi;t)] (1.28)
Let us stress that (1.26) is not the only solution of the non-linear relation K[f,f] - 0. Indeed, straightforward calculations show that the following distribution is also solution:
9L{r) n(r)
m(v-u(r))^-■■ ■ 2r{r) (1.29)
(27rT(r))2
where n(r) is the local density, T(r) the local kinetic température, 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 équation is usually shown by examining the temporal évolution of the so-called H-functional:
/i(r;i) = J dv/(r,v)[ln(/(r,v;t) - 1]
Boltzmann (ref. [52]) showed that the équation of évolution for this quantity reads:
dth{r;t) -dr.{hu + Ü
h) + CTH
(1.30)
(1.31) where:
J H = JdvVf{lnf-l)
= j dvK[f,f]\nf (1.32) These expressions imply that the quantity h is not conserved locally, through the presence of a source term <
thin (1.31). This source term satisfies the so-called H-theorem:
aH < 0
an = 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:
/i(r; t) = -
p{
t- 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) = J dr h{r; t) (1.35)
This définition, together with relations (1.31) and (1.33) show clearly that:
dtH{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 équation for an isolated System approach irreversibly the equilibrium State, just like a real gas does.
The H-theorem is usually considered as a microscopie 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 résolve the apparent contradiction between the microscopie réversible dynamics and the irréversible Boltzmann équation, 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 interprétation 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 dérivation of the Boltzmann équation for inelastic hard spheres is now a well defined problem, whose details and limitations hâve been much discussed over the last decade (see for instance ref. [263]
and référencés therein). In the following, we dérivé it in an approach very similar to that of the
previous section, in order to point ont the main différences and physical features originating in the inelasticity. Let us first note that the inelastic Boltzmann équation has been derived in the literature by more fundamental methods (ref. [69], [239]), namely methods based upon the so-called pseudo- Liouville équation (ref. [234]) for inelastic hard spheres. The latter is an équation for the N particles phase space distribution, and is constructed by considering an ensemble of équivalent initial conditions which define an initial distribution over phase space:
/;v(R,V;0) (1.37)
where R and V dénoté respectively (ri, ...., r^) and (vi, ....,
vat). 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:
/iv(R,V;t) = /iv(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 équation may be rewritten into an integro-differentiaJ form, in the case of elastic or inelastic hard spheres, which is called pseudo-Liouville équation. The term pseudo cornes from the fact that dynamics of hard spheres are singular, as the individual trajectories of the particles are not différentiable. Hence, this forbids the existence of a Hamiltonian. The pseudo- Liouville équation gives rise to the so-called BBGKY hierarchy, from which the Boltzmann équation may be obtained from suitable approximations.
One should remark that the nature of the pseudo-Liouville équation is very different from that of the Boltzmann équation, for several reasons. Indeed, in the case of elastic interactions, the Liouville équation is as réversible as the microscopie dynamics, and does not yield to the observed macroscopie irreversibility. Moreover, this équation predicts the deterministic évolution of an ensemble of équivalent Systems that constitute the initial ensemble. The only probabilistic element in the Liouville équation is found in the statistical distribution of the initial conditions. In contrast, in the Boltzmann équation, the stochasticity appears at two levels: in the initial condition and at the level of the dynamics itself. Moreover, the Boltzmann équation is a macroscopie équation, since it describes the most probable évolution 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 fonction, whose time évolution almost surely behaves as the solution of the Boltzmann équation does. This feature will be discussed further in the following and in chapter 5.
From the dérivation of the elastic Boltzmann équation, 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 Ar, Av, At 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 cornes from the following observation: the developments used in order to dérivé the loss term (1.18) made no explicit mention on the elasticity of the collision. The main argument was that particles having velocity vi, and which perform a collision with a particle 2, hâve 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 équation:
J dx2 d‘^e(6.vi2)0(e.vi2)/(r,vi;t)/(r,v2;t) At (1.39) In contrast with this resuit, the time-reversible nature of collisions, as well as the microscopie conservation of energy were primordial in the dérivation 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. When the collisions were elastic, we noticed that the post-collisional velocities (v*, Vj) with impact angle e are equivalently pre-collisional velocities with the impact angle — e. 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 a ^ 1:
(e.vi2) = -(e.vi2) > (e.vi2)
a (1.40)
This relation is a direct conséquence 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 dises ~ (e.vi2). 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 conséquence of inelasticity. The next step
consists in counting these collisions (vj, v^) —* (vi, V2), where the prime velocities are given by the
collision rule (1.1). The above discussion directly leads to the following expression:
Original velocities
^ ‘ ^r/ ! <T^/
energy dissipation
Reversed velocities
\
\
il
il/
time y
asymmetry /
1 U''
t ii
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 (e.Vi2) during one collision. In the figures at
right, we ülustrate the irreversibility of the microscopie dynamics, by showing that the
inverted trajectories are no longer the same as the direct ones.
Figure 1.8: Construction of a restituting collision from a direct collision.
dv2 d‘^e(e.Vi2)0(e.vi2)/(r,Vi;t)/(r,V2;t) dv'iAi (1.41) where the ^ factor takes into account the increased collision frequency. In order to dérivé the inelastic Boltzmann équation as a standard gain-loss process, we express the volume element of the restituting velocities as a fonction of the volume element of the direct velocities. Contrary to the elastic case, this transformation does not préservés the phase space volume. Indeed, the jacobian of the transformation v —» v reads:
dvj dvj
9v' Sv'
dvj dvj
9v' 9v'
a (1.42)
This expression implies that the volume in phase space has an irréversible tendency to shrink, because of inelastic microscopie dynamics. This resuit is in agreement with the obvions existence of a lower dimensional attractor in phase space, namely the total rest State. The latter consists in a point in phase space for which the relative velocities of ail particles is zéro. Equivalently, this point is characterized by null velocities (0, ..., 0, ..., 0) in the center of mass reference frame. This effect, together with (1.41) lead to the standard expression for the inelastic gain term:
^cr‘^“^y'dv2 d‘^e(e.vi2)0(e.vi2)/(r,v'i;t)/(r,V2;t) dviAt (1.43)
and to the the Boltzmann équation for inelastic hard spheres:
df{r,vi;t) dt
X
^ ^ J dedv2(e.vi2)0(e.vi2)
[^/(r,vi;t)/(r,vî;t) - /(r, vi; t)/(r, vi; t)] (1.44) The inelastic Boltzmann équation is the fondamental équation for low density inelastic gases, and plays the same rôle as the elastic Boltzmann équation in classical kinetic theory. Let us insist on the fact that its structure is very similar to that of the elastic Boltzmann équation (1.25), except two main différences. First, the ^ factor which represents both the time asymmetry of the collision frequency and the contraction of phase space. Moreover, the primed velocities of the restituting collisions are those obtained from an inelastic collision obeying (1.3), with a ^ 1.
These différences lead to a qualitatively different behaviour, as compared to the elastic case, namely the fact that the mean energy in the System is not conserved. This is clearly seen from the équation for the average energy in the System:
= cr‘^ ^ y dedv2(e.vi2)0(e.vi2) U? [^/(r,v'i;t)/(r,vi;t) - /(r,vi;t)/(r,vi;t)]
0
