## 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

Let

We introduce the semigroup of operators

where the function * i*th particle constructed in [7] and the set

On the space

The infinitesimal generator _{,} and it has the structure _{,} and the operators

and

respectively. In (2) and (3) the following notations are used:

where

Let

On the space of integrable functions, the semigroup of operators

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

where for

In (6) the notations similar to formula (3) are used,

Hence an infinitesimal generator of the adjoint semigroup of operators

On the space

## 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

where on the set

The solution

where the

and

We note that one component sequences of marginal observables correspond to observables of certain structure, namely, the marginal observable

On the space

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

In particular, functional (12) of mean values of the additive-type marginal observables

where the one-particle marginal distribution function

and the generating operator

where the sequence

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

where

## 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]:

where

In series (14) we used the notations

where it means that

and the operator

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

where the generating operator

For

where the collision integral is determined by the marginal functional of the state (14) in the case of

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

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

where the momenta

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

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

where

and

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

If initial one-particle marginal distribution functions satisfy the following condition:

exists, for finite time interval function (23) is the uniformly convergent series with respect to

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:

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

and the generating operator

where

For

In kinetic equation (26) the remainder

where the generating operators

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

Since the scattering operator of point hard rods is an identity operator in the approximation of elastic collisions, namely, in the limit

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

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.
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.
Goldhirsch I. Scales and kinetics of granular flows. Chaos. 1999; 9 :659-672. DOI: 10.1063/1.166440 - 3.
Mehta A. Granular Physics. Cambridge, UK: Cambridge University Press; 2007. 362 p. DOI: 10.1017/CBO9780511535314 - 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.
Brilliantov NV, Pöschel T. Kinetic Theory of Granular Gases. Oxford: Oxford University Press; 2004. 329 p. ISBN-13 978-0199588138 - 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.
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.
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.
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.
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.
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.
Villani C. Mathematics of granular materials. Journal of Statistical Physics. 2006; 124 (2-4):781-822. DOI: 10.1007/s10955-006-9038-6 - 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.
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.
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.
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.
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.
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.
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.
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.
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