Open access peer-reviewed chapter

# Kinetic Equations of Granular Media

Written By

Viktor Gerasimenko

Submitted: May 18th, 2019 Reviewed: October 4th, 2019 Published: August 26th, 2020

DOI: 10.5772/intechopen.90027

From the Edited Volume

## Progress in Fine Particle Plasmas

Edited by Tetsu Mieno, Yasuaki Hayashi and Kun Xue

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## Abstract

Approaches to the rigorous derivation of a priori kinetic equations, namely, the Enskog-type and Boltzmann-type kinetic equations, describing granular media from the dynamics of inelastically colliding particles are reviewed. We also consider the problem of potential possibilities inherent in describing the evolution of the states of a system of many hard spheres with inelastic collisions by means of a one-particle distribution function.

### Keywords

• granular media
• inelastic collision
• Boltzmann equation
• Enskog equation

## 1. Introduction

It is well known that the properties of granular media (sand, powders, cements, seeds, etc.) have been extensively studied, in the last decades, by means of experiments, computer simulations, and analytical methods, and a huge amount of physical literature on this topic has been published (for pointers to physical literature, see in [1, 2, 3, 4, 5, 6]).

Granular media are systems of many particles that attract considerable interest not only because of their numerous applications but also as systems whose collective behavior differs from the statistical behavior of ordinary media, i.e., typical macroscopic properties of media, for example, gases. In particular, the most spectacular effects include with the phenomena of collapse or cooling effect at the kinetic scale or clustering at the hydrodynamical scale, spontaneous loss of homogeneity, modification of Fourier’s law and non-Maxwellian equilibrium kinetic distributions [1, 2, 3].

In modern works [4, 5, 6], it is assumed that the microscopic dynamics of granular media is dissipative, and it is described by a system of many hard spheres with inelastic collisions. The purpose of this chapter is to review some advances in the mathematical understanding of kinetic equations of systems with inelastic collisions.

As is known [7], the collective behavior of many-particle systems can be effectively described by means of a one-particle distribution function governed by the kinetic equation derived from underlying dynamics in a suitable scaling limit. At present the considerable advance is observed in a problem of the rigorous derivation of the Boltzmann kinetic equation for a system of hard spheres in the Boltzmann–Grad scaling limit [7, 8, 9, 10]. At the same time, many recent papers [5, 11] (and see references therein) consider the Boltzmann-type and the Enskog-type kinetic equations for inelastically interacting hard spheres, modelling the behavior of granular gases, as the original evolution equations and the rigorous derivation of such kinetic equations remain still an open problem [12, 13].

Hereinafter, an approach will be formulated, which makes it possible to rigorously justify the kinetic equations previously introduced a priori for the description of granular media, namely, the Enskog-type and Boltzmann-type kinetic equations. In addition, we will consider the problem of potential possibilities inherent in describing the evolution of the states of a system of many hard spheres with inelastic collisions by means of a one-particle distribution function.

## 2. Dynamics of hard spheres with inelastic collisions

As mentioned above, the microscopic dynamics of granular media is described by a system of many hard spheres with inelastic collisions. We consider a system of a non-fixed, i.e., arbitrary, but finite average number of identical particles of a unit mass with the diameter σ>0, interacting as hard spheres with inelastic collisions. Every particle is characterized by the phase coordinates: qipixiR3×R3,i1.

Let Cγ be the space of sequences b=b0b1bn of bounded continuous functions bnCn defined on the phase space of n hard spheres that are symmetric with respect to the permutations of the arguments x1,,xn, equal to zero on the set of forbidden configurations Wn{q1qnR3nqiqj<σ for at least one pair ij:ij1n} and equipped with the norm: bCγ=maxn0γnn!bnCn=maxn0γnn!supx1,,xnbnx1xn. We denote the set of continuously differentiable functions with compact supports by Cn,0Cn.

We introduce the semigroup of operators Snt,t0, that describes dynamics of n hard spheres. It is defined by means of the phase trajectories of a hard sphere system with inelastic collisions almost everywhere on the phase space R3n×R3n\Wn, namely, outside the set Mn0 of the zero Lebesgue measure, as follows [14]:

Sntbnx1xnSnt1nbnx1xnbnX1tXnt,ifx1xnR3n\Wn×R3n,0,ifq1qnWn,E1

where the function XitXitx1xn is a phase trajectory of ith particle constructed in [7] and the set Mn0 consists from phase space points-specified initial data x1,,xn that generate multiple collisions during the evolution.

On the space Cn one-parameter mapping (1) is a bounded -weak continuous semigroup of operators, and SntCn<1.

The infinitesimal generator Ln of the semigroup of operators (1) is defined in the sense of a -weak convergence of the space Cn, and it has the structure Ln=j=1nLj+j1<j2=1nLintj1j2,, and the operators Lj and Lintj1j2 are defined by formulas:

Ljpjqj,E2

and

Lintj1j2bsσ2S+2ηpj1pj2(bsx1xj1xj2xsbsx1xs)δqj1qj2+ση,E3

respectively. In (2) and (3) the following notations are used: xjqjpj, the symbol means a scalar product, δ is the Dirac measure, S+2ηR3η=1ηpj1pj20, and the post-collision momenta are determined by the expressions:

pj1=pj11εηηpj1pj2,pj2=pj2+1εηηpj1pj2,E4

where ε=1e2012 and e01 is a restitution coefficient [6].

Let Lα1=n=0αnLn1 be the space of sequences f=f0f1fn of integrable functions fnx1xn defined on the phase space of n hard spheres that are symmetric with respect to the permutations of the arguments x1,,xn, equal to zero on the set of forbidden configurations Wn and equipped with the norm: fLα1=n=0αndx1dxnfnx1xn, where α>1 is a real number. We denote by L01Lα1 the everywhere dense set in Lα1 of finite sequences of continuously differentiable functions with compact supports.

On the space of integrable functions, the semigroup of operators Snt,t0, adjoint to semigroup of operators (1) in the sense of the continuous linear functional is defined (the functional of mean values of observables):

bf=n=01n!R3×R3ndx1dxnbnx1xnfnx1xn.

The adjoint semigroup of operators is defined by the Duhamel equation:

Snt1n=i=1nS1ti+0ti=1nS1tτij1<j2=1nLintj1j2Snτ1n,E5

where for t0 the operator Lintj1j2 is determined by the formula

Lintj1j2fsσ2S+2ηpj1pj2(112ε2fn(x1,,xj1,,xj2,,xn)δqj1qj2+σηfnx1xnδqj1qj2ση).E6

In (6) the notations similar to formula (3) are used, xjqjpj, and the pre-collision momenta (solutions of equations (4)) are determined as follows:

pj1=pj11ε12εηηpj1pj2,pj2=pj2+1ε12εηηpj1pj2.E7

Hence an infinitesimal generator of the adjoint semigroup of operators Snt is defined on L0,n1 as the operator, Ln=j=1nLj+j1<j2=1nLintj1j2, where the operator adjoint to free motion operator (2) Ljpjqj was introduced.

On the space Ln1 the one-parameter mapping defined by Eq. (5) is a bounded strong continuous semigroup of operators.

## 3. The dual hierarchy of evolution equations for observables

It is well known [7] that many-particle systems are described by means of states and observables. The functional for mean value of observables determines a duality of states and observables, and, as a consequence, there exist two equivalent approaches to describing the evolution of systems of many particles. Traditionally, the evolution is described in terms of the evolution of states by means of the BBGKY hierarchy for marginal distribution functions. An equivalent approach to describing evolution is based on marginal observables governed by the dual BBGKY hierarchy. In the same framework, the evolution of particles with the dissipative interaction, namely, hard spheres with inelastic collisions, is described [14].

Within the framework of observables, the evolution of a system of hard spheres is described by the sequences Bt=B0B1tx1Bstx1xsCγ of the marginal observables Bstx1xs defined on the phase space of s1 hard spheres that are symmetric with respect to the permutations of the arguments x1,,xn, equal to zero on the set Ws, and for t0 they are governed by the Cauchy problem of the weak formulation of the dual BBGKY hierarchy [14]:

tBstx1xs=j=1sLjBst+j1<j2=1sLintj1j2Bstx1xs+j1j2=1sLintj1j2Bs1tx1xj11xj1+1xs,E8
Bstx1xst=0=Bs0x1xs,s1,E9

where on the set Cs,0Cs the free motion operator Lj and the operator of inelastic collisions Lintj1j2 are defined by formulas (2) and (3), respectively. We refer to recurrence evolution equation (8) as the dual BBGKY hierarchy for hard spheres with inelastic collisions.

The solution Bt=B0B1tx1Bstx1xs of the Cauchy problem (8),(9) is determined by the expansions [10]:

Bstx1xs=n=0s11n!j1jn=1sA1+ntY\ZZBsn0x1xj11xj1+1xjn1xjn+1xs,E10

where the 1+nth-order cumulant of semigroups of operators (1) of hard spheres with inelastic collisions is defined by the formula

A1+ntY\XXP:Y\XX=iXi1P1P1!XiPSθXitθXi,E11

and Y1s,Zj1jnY, Y\Z is the set consisting of one element Y\Z=1j11j1+1jn1jn+1s, i.e., this set is a connected subset of the partition P such that P=1; the mapping θ is a declusterization operator defined by the formula: θY\Z=Y\Z.

We note that one component sequences of marginal observables correspond to observables of certain structure, namely, the marginal observable B1=0b1x10 corresponds to the additive-type observable, and the marginal observable Bk=(0,,0,bk(x1,,xk),0,) corresponds to the k-ary-type observable. If as initial data (9) we consider the marginal observable of additive type, then the decomposition structure of solution (10) is simplified and takes the form

Bs1tx1xs=Ast1sj=1sb1xj,s1.

On the space Cγ for abstract initial-value problem (8) and (9), the following statement is true. If B0=B0B10Bs0Cγ0Cγ is finite sequence of infinitely differentiable functions with compact supports, then the sequence of functions (10) is a classical solution, and for arbitrary initial data B0Cγ, it is a generalized solution.

We remark that expansion (10) can be also represented in the form of the weak formulation of the perturbation (iteration) series as a result of the applying of analogs of the Duhamel equation to cumulants of semigroups of operators (11).

The mean value of the marginal observable Bt=B0B1tBstCγ in initial state specified by a sequence of marginal distribution functions F0=1F10Fs0Lα1=s=0αsLs1 is determined by the following functional:

BtF0=s=01s!R3×R3sdx1dxsBstx1xsFs0x1xs.E12

In particular, functional (12) of mean values of the additive-type marginal observables B10=0B110x10 takes the form:

B1tF0=B10Ft=R3×R3dx1B110x1F1tx1,

where the one-particle marginal distribution function F1tx1 is determined by the series expansion [10]

F1tx1=n=01n!R3×R3ndx2dxn+1A1+ntF1+n0x1xn+1,

and the generating operator A1+nt of this series is the 1+nth-order cumulant of adjoint semigroups of hard spheres with inelastic collisions. In the general case for mean values of marginal observables, the following equality is true:

BtF0=B0Ft,

where the sequence Ft=1F1tFst is a solution of the Cauchy problem of the BBGKY hierarchy of hard spheres with inelastic collisions [14]. The last equality signifies the equivalence of two pictures of the description of the evolution of hard spheres by means of the BBGKY hierarchy [7] and the dual BBGKY hierarchy (8).

Hereinafter we consider initial states of hard spheres specified by a one-particle marginal distribution function, namely,

Fscx1xs=i=1sF10xiXR3s\Ws,s1,E13

where XR3s\WsXsq1qs is a characteristic function of allowed configurations R3s\Ws of s hard spheres and F10L1R3×R3. Initial data (13) is intrinsic for the kinetic description of many-particle systems because in this case all possible states are described by means of a one-particle marginal distribution function.

## 4. The non-Markovian Enskog kinetic equation

In the case of initial states (13), the dual picture of the evolution to the picture of the evolution by means of observables of a system of hard spheres with inelastic collisions governed by the dual BBGKY hierarchy (8) for marginal observables is the evolution of states described by means of the non-Markovian Enskog kinetic equation and a sequence of explicitly defined functionals of a solution of such kinetic equation.

Indeed, in view of the fact that the initial state is completely specified by a one-particle marginal distribution function on allowed configurations (13), for mean value functional (12), the following representation holds [14, 15]:

BtFc=B0FtF1t,

where Fc=1F1cFsc is the sequence of initial marginal distribution functions (13) and the sequence FtF1t=1F1tF2tF1tFstF1t is a sequence of the marginal functionals of the state Fstx1xsF1t represented by the series expansions over the products with respect to the one-particle marginal distribution function F1t:

Fstx1xsF1tn=01n!R3×R3ndxs+1dxs+nV1+ntYX\Yi=1s+nF1txi,s2.E14

In series (14) we used the notations Y1s,X1s+n, and the n+1th-order generating operator V1+nt,n0 is defined as follows [15]:

V1+ntYX\Yk=0n1km1=1nmk=1nm1mk1n!nm1mk!×Â1+nm1mktYs+1s+nm1mkj=1kk2j=0mj
knm1mj+sj=0knm1mj+s1jij=1s+nm1mj1knm1mj+s+1ijjknm1mj+s+2ijj!×Â1+knm1mj+s+1ijjknm1mj+s+2ijj(t,ij,s+nm1mj+1+ks+nm1mj+2ijj,,s+nm1mj+ks+nm1mj+1ijj),

where it means that k1jmj,knm1mj+s+1j0, and we denote the 1+nth-order scattering cumulant by the operator Â1+nt:

Â1+ntYX\YA1+ntYX\YXR3s+n\Ws+ni=1s+nA1ti1,

and the operator A1+nt is the 1+nth-order cumulant of adjoint semigroups of hard spheres with inelastic collisions.

We emphasize that in fact functionals (14) characterize the correlations generated by dynamics of a hard sphere system with inelastic collisions.

The second element of the sequence FtF1t, i.e., the one-particle marginal distribution function F1t, is determined by the following series expansion:

F1tx1=n=01n!R3×R3ndx2dxn+1A1+ntXR31+n\W1+ni=1n+1F10xi,E15

where the generating operator A1+ntA1+nt1n+1 is the 1+nth-order cumulant of adjoint semigroups of hard spheres with inelastic collisions.

For t0 the one-particle marginal distribution function (15) is a solution of the following Cauchy problem of the non-Markovian Enskog kinetic equation [14, 15]:

tF1tq1p1=p1q1F1tq1p1+σ2R3×S+2dp2ηp1p2112ε2F2tq1p1q1σηp2F1tF2tq1p1q1+σηp2F1t,E16
F1tt=0=F10,E17

where the collision integral is determined by the marginal functional of the state (14) in the case of s=2, and the expressions p1 and p2 are the pre-collision momenta of hard spheres with inelastic collisions (7), i.e., solutions of Eq. (4).

We note that the structure of collision integral of the non-Markovian Enskog equation for granular gases (16) is such that the first term of its expansion is the collision integral of the Boltzmann–Enskog kinetic equation and the next terms describe all possible correlations which are created by hard sphere dynamics with inelastic collisions and by the propagation of initial correlations connected with the forbidden configurations.

We remark also that based on the non-Markovian Enskog equation (16), we can formulate the Markovian Enskog kinetic equation with inelastic collisions [14].

For the abstract Cauchy problem of the non-Markovian Enskog kinetic equation (16), (17) in the space of integrable functions , the following statement is true [14]. A global in time solution of the Cauchy problem of the non-Markovian Enskog equation (16) is determined by function (15). For small densities and F10L01R3×R3, function (15) is a strong solution, and for an arbitrary initial data F10L1R3×R3, it is a weak solution.

Thus, if initial state is specified by a one-particle marginal distribution function on allowed configurations, then the evolution, describing by marginal observables governed by the dual BBGKY hierarchy (8), can be also described by means of the non-Markovian kinetic equation (16) and a sequence of marginal functionals of the state (14). In other words, for mentioned initial states, the evolution of all possible states of a hard sphere system with inelastic collisions at arbitrary moment of time can be described by means of a one-particle distribution function without any approximations.

## 5. The Boltzmann kinetic equation for granular gases

It is known [7, 8] the Boltzmann kinetic equation describes the evolution of many hard spheres in the Boltzmann–Grad (or low-density) approximation. In this section the possible approaches to the rigorous derivation of the Boltzmann kinetic equation from dynamics of hard spheres with inelastic collisions are outlined.

One approach to deriving the Boltzmann kinetic equation for hard spheres with inelastic collisions, which was developed in [10] for a system of hard spheres with elastic collisions, is based on constructing the Boltzmann–Grad asymptotic behavior of marginal observables governed by the dual BBGKY hierarchy (8). A such scaling limit is governed by the set of recurrence evolution equations, namely, by the dual Boltzmann hierarchy for hard spheres with inelastic collisions [14]. Then for initial states specified by a one-particle distribution function (13), the evolution of additive-type marginal observables governed by the dual Boltzmann hierarchy is equivalent to a solution of the Boltzmann kinetic equation for granular gases [12], and the evolution of nonadditive-type marginal observables is equivalent to the property of the propagation of initial chaos for states [10].

One more approach to the description of the kinetic evolution of hard spheres with inelastic collisions is based on the non-Markovian generalization of the Enskog equation (16).

Let the dimensionless one-particle distribution function F1ϵ,0, specifying initial state (13), satisfy the condition, F1ϵ,0x1ceβ2p12, where ϵ>0 is a scaling parameter (the ratio of the diameter σ>0 to the mean free path of hard spheres), β>0 is a parameter, and c< is some constant, and there exists the following limit in the sense of a weak convergence: wlimϵ0ϵ2F1ϵ,0x1f10x1=0. Then for finite time interval the Boltzmann–Grad limit of dimensionless solution (15) of the Cauchy problem of the non-Markovian Enskog kinetic equation (16) and (17) exists in the same sense, namely, wlimϵ0ϵ2F1tx1f1tx1=0, where the limit one-particle distribution function is a weak solution of the Cauchy problem of the Boltzmann kinetic equation for granular gases [6, 12]:

tf1tx1=p1q1f1tx1+R3×S+2dp2ηp1p2112ε2f1tq1p1f1tq1p2f1tx1f1tq1p2,E18
f1tx1t=0=f10x1,E19

where the momenta p1 and p2 are pre-collision momenta of hard spheres with inelastic collisions (7).

As noted above, all possible correlations of a system of hard spheres with inelastic collisions are described by marginal functionals of the state (14). Taking into consideration the fact of the existence of the Boltzmann–Grad scaling limit of a solution of the non-Markovian Enskog kinetic equation (16), for marginal functionals of the state (14), the following statement holds:

wlimϵ0ϵ2sFstx1xsF1tj=1sf1txj=0,

where the limit one-particle distribution function f1t is governed by the Boltzmann kinetic equation for granular gases (18). This property of marginal functionals of the state (14) means the propagation of the initial chaos [7].

It should be emphasized that the Boltzmann–Grad asymptotics of a solution of the non-Markovian Enskog equation (16) in a multidimensional space are analogous of the Boltzmann–Grad asymptotic behavior of a hard sphere system with the elastic collisions [10]. Such asymptotic behavior is governed by the Boltzmann equation for a granular gas (18), and the asymptotics of marginal functionals of the state (14) are the product of its solution (this property is interpreted as the propagation of the initial chaos).

## 6. One-dimensional granular gases

As is known, the evolution of a one-dimensional system of hard spheres with elastic collisions is trivial (free motion or Knudsen flow) in the Boltzmann–Grad scaling limit [7], but, as it was taken notice in paper [16], in this approximation the kinetics of inelastically interacting hard spheres (rods) is not trivial, and it is governed by the Boltzmann kinetic equation for one-dimensional granular gases [16, 17, 18, 19]. Below the approach to the rigorous derivation of Boltzmann-type equation for one-dimensional granular gases will be outlined. It should be emphasized that a system of many hard rods with inelastic collisions displays the basic properties of granular gases inasmuch as under the inelastic collisions only the normal component of relative velocities dissipates in a multidimensional case.

In case of a one-dimensional granular gas for t0 in dimensionless form, the Cauchy problem (16),(17) takes the form [20]:

tF1tq1p1=p1q1F1tq1p1+0dPP(112ε2F2tq1p1p1Pq1ϵp2p1PF1tF2tq1p1q1ϵp1+PF1t)+E20
0dPP(112ε2F2tq1p˜1p1Pq1+ϵp˜2p1PF1tF2tq1p1q1+ϵp1PF1t),
F1tt=0=F1ϵ,0,E21

where ϵ>0 is a scaling parameter (the ratio of a hard sphere diameter (the length) σ>0 to the mean free path), the collision integral is determined by marginal functional (14) of the state F1t in the case of s=2, and the expressions:

p1p1P=p1P+ε2ε1P,p2p1P=p1ε2ε1P

and

p˜1p1P=p1+Pε2ε1P,p˜2p1P=p1+ε2ε1P,

are transformed pre-collision momenta in a one-dimensional space.

If initial one-particle marginal distribution functions satisfy the following condition: F1ϵ,0x1Ceβ2p12, where β>0 is a parameter, C< is some constant, then every term of the series

F1ϵtx1=n=01n!R×Rndx2dxn+1A1+nti=1n+1F1ϵ,0xiXR1+n\W1+n,E22

exists, for finite time interval function (23) is the uniformly convergent series with respect to x1 from arbitrary compact, and it is determined a weak solution of the Cauchy problem of the non-Markovian Enskog equation (20), (22). Let in the sense of a weak convergence there exist the following limit:

wlimϵ0F1ϵ,0x1f10x1=0,

then for finite time interval there exists the Boltzmann–Grad limit of solution (23) of the Cauchy problem of the non-Markovian Enskog equation for one-dimensional granular gas (20) in the sense of a weak convergence:

wlimϵ0F1ϵtx1f1tx1=0,E23

where the limit one-particle marginal distribution function is defined by uniformly convergent arbitrary compact set series:

f1tx1=n=01n!R×Rndx2dxn+1A1+n0ti=1n+1f10xi,E24

and the generating operator A1+n0tA1+n0t1n+1 is the n+1th-order cumulant of adjoint semigroups Sn,0t of point particles with inelastic collisions. An infinitesimal generator of the semigroup of operators Sn,0t is defined as the operator:

Ln,0fnx1xn=j=1npjqjfnx1xn+j1<j2=1npj2pj1112ε2fnx1xj1xj2xnfnx1xnδqj1qj2,

where xjqjpj and the pre-collision momenta pj1,pj2 are determined by the following expressions:

pj1=pj2+ε2ε1pj1pj2,pj2=pj1ε2ε1pj1pj2.

For t0 the limit one-particle distribution function represented by series (25) is a weak solution of the Cauchy problem of the Boltzmann-type kinetic equation of point particles with inelastic collisions [20]

tf1tqp=pqf1tqp++dp1pp1×112ε2f1tqpf1(tqp1)f1(tqp)f1(tqp1)+n=1I0n.E25

In kinetic equation (26) the remainder n=1I0n of the collision integral is determined by the expressions

I0n1n!0dPPR×Rndq3dp3dqn+2dpn+2V1+nt(112ε2f1tqp1pP×f1tqp2pPf1tqpf1tqp+P)i=3n+2f1tqipi+0dPPR×Rndq3dp3dqn+2dpn+2V1+nt(112ε2f1tqp˜1pP×f1tqp˜2pPf1tqpf1tqpP)i=3n+2F1tqipi,

where the generating operators V1+ntV1+nt123n+2,n0, are represented by expansions (15) with respect to the cumulants of semigroups of scattering operators of point hard rods with inelastic collisions in a one-dimensional space:

Ŝn0t1nSn,0t1si=1nS1,0ti1.E26

In fact, the series expansions for the collision integral of the non-Markovian Enskog equation for a granular gas or solution (23) are represented as the power series over the density so that the terms I0n,n1, of the collision integral in kinetic equation (18) are corrections with respect to the density to the Boltzmann collision integral for one-dimensional granular gases stated in [17, 21].

Since the scattering operator of point hard rods is an identity operator in the approximation of elastic collisions, namely, in the limit ε0, the collision integral of the Boltzmann kinetic equation (26) in a one-dimensional space is identical to zero. In the quasi-elastic limit [21], the limit one-particle distribution function (25)

limε0εf1tqv=f0tqv,

satisfies the nonlinear friction kinetic equation for granular gases of the following form [16, 21]:

tf0tqv=vqf0tqv+vdv1v1vv1vf0tqv1f0tqv.

Taking into consideration result (24) on the Boltzmann–Grad asymptotic behavior of the non-Markovian Enskog equation (16), for marginal functionals of the state (14) in a one-dimensional space, the following statement is true [20]:

wlimϵ0Fstx1xsF1ϵtfstx1xsf1t=0,s2,

where the limit marginal functionals fstf1t,s2, with respect to limit one-particle distribution function (25) are determined by the series expansions with the structure similar to series (14) and the generating operators represented by expansions (15) over the cumulants of semigroups of scattering operators (27) of point hard rods with inelastic collisions in a one-dimensional space.

As mentioned above, in the case of a system of hard rods with elastic collisions, the limit marginal functionals of the state are the product of the limit one-particle distribution functions, describing the free motion of point hard rods.

Thus, the Boltzmann–Grad asymptotic behavior of solution (23) of the non-Markovian Enskog equation (20) is governed by the Boltzmann kinetic equation for a one-dimensional granular gas (18). Moreover, the limit marginal functionals of the state are represented by the appropriate series with respect to limit one-particle distribution function (25) that describe the propagation of initial chaos in one-dimensional granular gases.

## 7. Conclusions

In this chapter the origin of the kinetic description of the evolution of observables of a system of hard spheres with inelastic collisions was considered.

It was established that for initial states (13) specified by a one-particle distribution function, solution (10) of the Cauchy problem of the dual BBGKY hierarchy (8) and (9) and a solution of the Cauchy problem of the non-Markovian Enskog equation (16) and (17) together with marginal functionals of the state (14), give two equivalent approaches to the description of the evolution of states of a hard sphere system with inelastic collisions. In fact, the rigorous justification of the Enskog kinetic equation for granular gases (16) is a consequence of the validity of equality (14).

We note that the developed approach is also related to the problem of a rigorous derivation of the non-Markovian kinetic-type equations from underlying many-particle dynamics which make it possible to describe the memory effects of granular gases.

One more advantage also is that the considered approach gives the possibility to construct the kinetic equations in scaling limits, involving correlations at initial time which can characterize the condensed states of a hard sphere system with inelastic collisions [10].

Finally, it should be emphasized that the developed approach to the derivation of the Boltzmann equation for granular gases from the dynamics governed by the non-Markovian Enskog kinetic equation (16) also allows us to construct higher-order corrections to the collision integral compared to the Boltzmann–Grad approximation.

## References

1. 1. Cercignani C. The Boltzmann equation approach to the shear flow of a granular material. Philosophical Transactions of the Royal Society of London. 2002;360(1792):407-414. DOI: 10.1098/rsta.2001.0939
2. 2. Goldhirsch I. Scales and kinetics of granular flows. Chaos. 1999;9:659-672. DOI: 10.1063/1.166440
3. 3. Mehta A. Granular Physics. Cambridge, UK: Cambridge University Press; 2007. 362 p. DOI: 10.1017/CBO9780511535314
4. 4. Pöschel T, Brilliantov NV, editors. Granular gas dynamics. In: Lecture Notes in Phys. 624. Berlin, Heidelberg: Springer-Verlag; 2003. 369 p. ISBN 978-3-540-20110-6
5. 5. Brilliantov NV, Pöschel T. Kinetic Theory of Granular Gases. Oxford: Oxford University Press; 2004. 329 p. ISBN-13 978-0199588138
6. 6. Brey JJ, Dufty JW, Santos A. Dissipative dynamics for hard spheres. Journal of Statistical Physics. 1997;87(5-6):1051-1066. DOI: 10.1007/BF02181270
7. 7. Cercignani C, Gerasimenko VI, Petrina DY. Many-Particle Dynamics and Kinetic Equations. The Netherlands: Springer; 2012. 247 p. ISBN 978-94-010-6342-5
8. 8. Gallagher I, Saint-Raymond L, Texier B. From Newton to Boltzmann: Hard Spheres and Short-range Potentials. Zürich Lectures in Advanced Mathematics: EMS Publ House; 2014. 146 p. ISBN-10: 3037191295
9. 9. Pulvirenti M, Simonella S. The Boltzmann–Grad limit of a hard sphere system: Analysis of the correlation error. Inventiones Mathematicae. 2017;207(3):1135-1237. DOI: 10.1007/s00222-016-0682-4
10. 10. Gerasimenko VI, Gapyak IV. Low-density asymptotic behavior of observables of hard sphere fluids. Advances in Mathematical Physics. 2018;2018:6252919. DOI: 10.1155/2018/6252919
11. 11. Pareschi L, Russo G, Toscani G, editors. Modelling and Numerics of Kinetic Dissipative Systems. N.Y.: Nova Science Publ. Inc.; 2006. 220 p. ISBN-13 978-1594545030
12. 12. Villani C. Mathematics of granular materials. Journal of Statistical Physics. 2006;124(2-4):781-822. DOI: 10.1007/s10955-006-9038-6
13. 13. Capriz G, Mariano PM, Giovine P, editors. Mathematical Models of Granular Matter. (Lecture Notes in Math. 1937). Berlin Heidelberg: Springer-Verlag; 2008. 228 p. ISBN 978-3-540-78276-6
14. 14. Gerasimenko VI, Borovchenkova MS. On the non-Markovian Enskog equation for granular gases. Journal of Physics A: Mathematical and Theoretical. 2014;47(3):035001. DOI: 10.1088/1751-8113/47/3/035001
15. 15. Gerasimenko VI, Gapyak IV. Hard sphere dynamics and the Enskog equation. Kinetic and Related Models. 2012;5(3):459-484. DOI: 10.3934/krm.2012.5.459
16. 16. Mac Namara S, Young WR. Kinetics of a one-dimensional granular medium in the quasielastic limit. Physics of Fluids A. 1993;5(1):34-45. DOI: 10.1063/1.858896
17. 17. Bellomo N, Pulvirenti M, editors. Modeling in applied sciences. In: Modeling and Simulation in Science, Engineering and Technology. Boston: Birkhäuser; 2000. 433 p. ISBN 978-1-4612-6797-3
18. 18. Williams DRM, MacKintosh FC. Driven granular media in one dimension: Correlations and equation of states. Physical Review E. 1996;R9(R):54. DOI: 10.1103/PhysRevE.54.R9
19. 19. Toscani G. One-dimensional kinetic models with dissipative collisions. Mathematical Modelling and Numerical Analysis. 2000;34(6):1277-1291. DOI: 10.1051/m2an:2000127
20. 20. Gerasimenko VI, Borovchenkova MS. The Boltzmann–Grad limit of the Enskog equation of one-dimensional granular gases. Reports of the NAS of Ukraine. 2013;10:11-17
21. 21. Toscani G. Kinetic and hydrodynamic models of nearly elastic granular flows. Monatschefte für Mathematik. 2004;142:179-192. DOI: 10.1007/s00605-004-0241-8

Written By

Viktor Gerasimenko

Submitted: May 18th, 2019 Reviewed: October 4th, 2019 Published: August 26th, 2020