Kinetic Equations of Granular Media

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.


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.

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: Þ of bounded continuous functions b n ∈ C n defined on the phase space of n hard spheres that are symmetric with respect to the permutations of the arguments x 1 , … , x n , equal to zero on the set of forbidden configurations  n ≐ f q 1 , … , q n À Á ∈  3n kq i À q j | < σ for at least one pair We denote the set of continuously differentiable functions with compact supports by C n,0 ⊂ C n .
We introduce the semigroup of operators S n t ð Þ, t ≥ 0, 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  3n Â  3n n n À Á , namely, outside the set  0 n of the zero Lebesgue measure, as follows [14]: where the function X i t ð Þ X i t, x 1 , … , x n ð Þis a phase trajectory of ith particle constructed in [7] and the set  0 n consists from phase space points-specified initial data x 1 , … , x n that generate multiple collisions during the evolution.
On the space C n one-parameter mapping (1) is a bounded * -weak continuous semigroup of operators, and ∥S n t ð Þ∥ C n < 1. The infinitesimal generator L n of the semigroup of operators (1) is defined in the sense of a * -weak convergence of the space C n, and it has the structure L n ¼ P n j¼1 L j ð Þ þ P n j 1 < j 2 ¼1 L int j 1 , j 2 À Á , , and the operators L j ð Þ and L int j 1 , j 2 À Á are defined by formulas: and Þdefined on the phase space of n hard spheres that are symmetric with respect to the permutations of the arguments x 1 , … , x n , equal to zero on the set of forbidden configurations  n and equipped with the norm: where α > 1 is a real number. We denote by L 1 0 ⊂ L 1 α the everywhere dense set in L 1 α of finite sequences of continuously differentiable functions with compact supports.
On the space of integrable functions, the semigroup of operators S * n t ð Þ, t ≥ 0, adjoint to semigroup of operators (1) in the sense of the continuous linear functional is defined (the functional of mean values of observables): The adjoint semigroup of operators is defined by the Duhamel equation: where for t ≥ 0 the operator L * int j 1 , j 2 À Á is determined by the formula In (6) the notations similar to formula (3) are used, Hence an infinitesimal generator of the adjoint semigroup of operators S * n t ð Þ is defined on L 1 0,n as the operator, L * n ¼ On the space L 1 n the one-parameter mapping defined by Eq. (5) is a bounded strong continuous semigroup of operators.

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 B t Þdefined on the phase space of s ≥ 1 hard spheres that are symmetric with respect to the permutations of the arguments x 1 , … , x n , equal to zero on the set  s , and for t ≥ 0 they are governed by the Cauchy problem of the weak formulation of the dual BBGKY hierarchy [14]: where on the set C s,0 ⊂ C s the free motion operator L j ð Þ and the operator of inelastic collisions L int j 1 , j 2 À Á 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 Þ of the Cauchy problem (8),(9) is determined by the expansions [10]: where the 1 þ n ð Þth-order cumulant of semigroups of operators (1) of hard spheres with inelastic collisions is defined by the formula and , this set is a connected subset of the partition P such that |P| ¼ 1; the mapping θ Á ð Þ is a declusterization operator defined by the formula: θ YnZ f g ð Þ¼YnZ. We note that one component sequences of marginal observables correspond to observables of certain structure, namely, the marginal observable Þ corresponds to the additive-type observable, and the marginal observable … , x k Þ, 0, … Þ 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 On the space C γ for abstract initial-value problem (8) and (9), the following statement is true.
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 B 0 ð Þ ∈ C γ , 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 B t In particular, functional (12) of mean values of the additive-type marginal where the one-particle marginal distribution function F 1 t, x 1 ð Þis determined by the series expansion [10] Þ 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, where X  3s n s X s q 1 , … , q s À Á is a characteristic function of allowed configurations  3s n s of s hard spheres and F 0 (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.

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 oneparticle marginal distribution function on allowed configurations (13), for mean value functional (12), the following representation holds [14,15]: is the sequence of initial marginal distribution functions (13) and the sequence F tjF 1 t ð Þ ð Þ¼ 1, Þ is a sequence of the marginal functionals of the state F s t, x 1 , … , x s jF 1 t ð Þ ð Þ represented by the series expansions over the products with respect to the one-particle marginal distribution function F 1 t ð Þ: In series (14) we used the notations Y 1, … , s ð Þ, X 1, … , s þ n ð Þ , and the n þ 1 ð Þth-order generating operator V 1þn t ð Þ, n ≥ 0 is defined as follows [15]:

Progress in Fine Particle Plasmas
where it means that k j 1 m j , k j nÀm 1 À … Àm j þsþ1 0, and we denote the 1 þ n ð Þthorder scattering cumulant by the operator b and the operator A * 1þn t ð Þ is the 1 þ n ð Þth-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 F tjF 1 t ð Þ ð Þ, i.e., the one-particle marginal distribution function F 1 t ð Þ, is determined by the following series expansion: where the generating operator A * 1þn t ð Þ A * 1þn t, 1, … , n þ 1 ð Þis the 1 þ n ð Þthorder cumulant of adjoint semigroups of hard spheres with inelastic collisions.
For t ≥ 0 the one-particle marginal distribution function (15) is a solution of the following Cauchy problem of the non-Markovian Enskog kinetic equation [14,15]: where the collision integral is determined by the marginal functional of the state (14) in the case of s ¼ 2, and the expressions p ⋄ 1 and p ⋄ 2 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].  (16) is determined by function (15). For small densities and (15) is a strong solution, and for an arbitrary initial data , 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.

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 F ϵ,0 1 , specifying initial state (13), satisfy the condition, |F ϵ,0 1 x 1 ð Þ| ≤ ce À β 2 p 2 1 , 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: w À lim ϵ!0 ϵ 2 F ϵ,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, w À lim 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]: where the momenta p ⋄ 1 and p ⋄ 2 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: where the limit one-particle distribution function f 1 t ð Þ 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).

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 t ≥ 0 in dimensionless form, the Cauchy problem (16), (17) takes the form [20]: satisfies the nonlinear friction kinetic equation for granular gases of the following form [16,21]: 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]: where the limit marginal functionals f s tj f 1 t ð Þ À Á , s ≥ 2, with respect to limit oneparticle 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 onedimensional granular gases.

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 manyparticle 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 higherorder corrections to the collision integral compared to the Boltzmann-Grad approximation.

Author details
Viktor Gerasimenko Institute of Mathematics of the NAS of Ukraine, Kyiv, Ukraine *Address all correspondence to: gerasym@imath.kiev.ua © 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.