On the non-Markovian Enskog Equation

for Granular Gases

M.S. Borovchenkova^{1}^{1}1E-mail:
and V.I. Gerasimenko ^{2}^{2}2E-mail:

Taras Shevchenko National University of Kyiv,

Department of Mechanics and Mathematics,

2, Academician Glushkov Av.,

03187, Kyiv, Ukraine

Institute of Mathematics of NAS of Ukraine,

3, Tereshchenkivs’ka Str.,

01601, Kyiv-4, Ukraine

Abstract. We develop a rigorous formalism for the description of the kinetic evolution of many-particle systems with the dissipative interaction. The relationships of the evolution of a hard sphere system with inelastic collisions described within the framework of marginal observables governed by the dual BBGKY hierarchy and the evolution of states described by the Cauchy problem of the Enskog kinetic equation for granular gases are established. Moreover, we consider the Boltzmann–Grad asymptotic behavior of the constructed non-Markovian Enskog kinetic equation for granular gases in a one-dimensional space.

Key words: granular gas; inelastic collision; dual BBGKY hierarchy; Enskog equation; kinetic evolution

2010 Mathematics Subject Classification: 82C05; 82C40; 82D99; 35Q20; 35Q82

###### Contents

## 1 Introduction

A granular gas is a dynamical system of significant interest not only in view of its applications but also as a many-particle system displaying a collective behavior that differs from the statistical behavior of usual gases; for example, it is related to typical macroscopic properties [1]-[9].

As is known, the collective behavior of many-particle systems can be effectively described within the framework of a one-particle marginal distribution function governed by the kinetic equation in a suitable scaling limit of underlying dynamics [10]-[14]. At present, the considerable advances are being in the rigorous derivation of the Boltzmann kinetic equation of a system of hard spheres in the Boltzmann–Grad scaling limit [15]-[17]. At the same time, many recent papers [18]-[20] (and see references therein) are considering the Boltzmann-type and the Enskog-type kinetic equations for inelastically interacting hard spheres, modelling granular gases, as the original evolution equations and the rigorous derivation of such kinetic equations remain a problem [21]-[26].

The goal of this paper is to develop an approach based on the dynamics of particles with the dissipative interaction to properly justify the kinetic equations that previous works have already applied a priori to the description of granular gases. In the paper, we consider the problem of potentialities inherent in the description of the evolution of states of a hard sphere system with inelastic collisions in terms of a one-particle distribution function. We established that, in fact, if the initial state is completely specified by a one-particle marginal distribution function, then all possible states at an arbitrary moment of time can be described within the framework of a one-particle distribution function without any approximations.

To outline the structure of the paper and the main results, in section 2, we develop an approach to the description of the kinetic evolution of hard spheres with inelastic collisions within the framework of the evolution of marginal observables. Then, in section 3, the main results related to the origin of the kinetic evolution of granular gases are stated. We prove that underlaying dynamics governed by the dual BBGKY hierarchy for marginal observables can be completely described within the framework of the one-particle marginal distribution function governed by the non-Markovian Enskog kinetic equation with inelastic collisions. In this case, we prove that all possible correlations, creating by hard sphere dynamics, are described by the explicitly defined marginal functionals with respect to the solution of the established kinetic equation. In section 4, we consider a one-dimensional granular gas. The Boltzmann–Grad asymptotic behavior of the constructed non-Markovian Enskog kinetic equation with inelastic collisions in a one-dimensional space is outlined. Finally, in section 5, we conclude with some observations and perspectives for future research.

## 2 Hierarchies of evolution equations for granular gases

It is well known that many-particle systems are descried in terms of two sets of objects: observables and states. The functional of the mean value of observables defines a duality between observables and states, and as a consequence two approaches to the description of the evolution exist. Usually, the evolution of many-particle systems is described within the framework of the evolution of states by the BBGKY hierarchy for marginal distribution functions. An equivalent approach to this description is given in terms of the marginal observables governed by the dual BBGKY hierarchy. In the same framework, the systems of particles with the dissipative interaction, namely hard spheres with inelastic collisions, can be described.

### 2.1 The dual BBGKY hierarchy and semigroups of operators of 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, interacting as hard spheres with inelastic collisions. Every hard sphere with the diameter is characterized by the phase coordinates

Let be the space of sequences of bounded continuous functions defined on the phase space of hard spheres that are symmetric with respect to the permutations of the arguments , equal to zero on the set of forbidden configurations for at least one pair and equipped with the norm

Within the framework of observables, the evolution of a system of an arbitrary but finite average number of hard spheres is described by the sequences of the marginal (-particle) observables defined on the phase space of hard spheres that are symmetric with respect to the permutations of the arguments , equal to zero on the set , and for they are governed by the Cauchy problem of the weak formulation of the dual BBGKY hierarchy [27]

(1) | |||

(2) |

where on the set the free motion Liouville operator and the operator of inelastic collisions are defined by the following formulas

(3) |

and

(4) | |||

respectively. In (3),(4) the following notations are used: the symbol means a scalar product, is the Dirac measure, and the post-collision momenta are determined by

(5) | |||

where is a restitution coefficient [7]. and

We give explicit examples of recurrence evolution equations (1):

We refer to recurrence evolution equations (1) as the dual BBGKY hierarchy for hard spheres with inelastic collisions or for granular gases.

The first term of a generator of the dual BBGKY hierarchy (1) is the Liouville operator , which is an infinitesimal generator of the semigroup of operators of a system of inelastically interacting hard spheres. The semigroup of operators is defined almost everywhere on the phase space , namely, outside the set of the Lebesgue measure zero, as a shift operator of the arguments of functions from the space along the phase trajectories of hard spheres with inelastic collisions

(6) | |||

where the function is a phase trajectory of the hard sphere constructed in [26] and the set consists from phase space points specified in the initial data that generate multiple collisions during the evolution [14].

On the space , one-parameter mapping (6) is a bounded -weak continuous semigroup of operators [28], and .

Let be the space of sequences of integrable functions defined on the phase space of hard spheres that are symmetric with respect to the permutations of the arguments , equal to zero on the set of forbidden configurations , and equipped with the norm , where is a real number. We denote by the everywhere dense set in of finite sequences of continuously differentiable functions with compact supports.

On the space of integrable functions , the semigroup of operators is defined adjoint to the semigroup of operators (6) in the sense of the continuous linear functional (the functional of mean values of observables)

The adjoint semigroup of operators is defined by the Duhamel equation

(7) |

where the operator is given by the formula

(8) | |||

In expression (8), the notations similar to formula (4) are used, and the pre-collision momenta, i.e. solutions of equations (5), are determined as follows:

(9) | |||

Hence, an infinitesimal generator of the semigroup of operators adjoint to semigroup (4) is defined on as the operator: , where we introduce the operator adjoint to free motion operator (3): and the operator is defined by formula (8).

On the space the one-parameter mapping defined by equation (7) is a bounded strong continuous semigroup of operators.

We remark that, according to the definition of the marginal observables [29]

in terms of a sequence of the observables , the dual BBGKY hierarchy (1) for hard spheres with inelastic collisions can be rigorously derived due to the properties of semigroups (6) of a hard sphere system with inelastic collisions.

### 2.2 A solution expansion of the dual BBGKY hierarchy for granular gases

A solution of the Cauchy problem (1),(2) is determined by the expansions [29]

(10) | |||

where the -order cumulant of semigroups of operators (6) of hard spheres with inelastic collisions is defined by the formula [30]

(11) |

and we used the notations: , the set is a set consisting of one element , i.e. this set is a connected subset of the partition such that , the mapping is a declusterization operator defined by the formula .

The simplest examples of marginal observables (10) are given by the following expansions:

Under the condition , for the sequence of marginal observables (10) the estimate holds

(12) |

For initial data of finite sequences of infinitely differentiable functions with compact supports, the sequence of functions (10) is a classical solution and for arbitrary initial data it is a generalized solution.

We note that single-component sequences of marginal observables correspond to observables of a certain structure; namely, the marginal observable corresponds to the additive-type observable, and the marginal observable corresponds to the -ary-type observable [27]. If in capacity of initial data (2) we consider the additive-type marginal observable, then the structure of solution expansion (10) is simplified and attains the form

(13) |

### 2.3 A functional of mean values of marginal observables

The mean value of the marginal observables in the initial state described by the sequence of marginal distribution functions is determined by the functional

(14) |

Owing to estimate (12), functional (14) exists under the condition that .

In particular, functional (14) of mean values of the additive-type marginal observables takes the form

where the one-particle marginal distribution function is determined by the series [30]

and the generating operator of this series expansion is the -order cumulant of adjoint semigroups of operators of hard spheres with inelastic collisions, i.e.

(15) |

where it was used notations accepted in formula (11).

In the general case for mean values of marginal observables, the following equality is true:

where the sequence is a solution of the Cauchy problem of the BBGKY hierarchy of hard spheres with inelastic collisions [26]. This equality signifies the equivalence of two pictures of the description of the evolution of hard spheres by means of the BBGKY hierarchy and the dual BBGKY hierarchy (1).

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

(16) |

where is a characteristic function of allowed configurations of hard spheres and . Initial data (16) is intrinsic for the kinetic description of many-particle systems because in this case all possible states are characterized by means of a one-particle marginal distribution function.

## 3 The non-Markovian Enskog equation for granular gases

In view of the fact that initial state is completely specified by a one-particle marginal distribution function, the evolution of states can be described within the framework of the sequence of the marginal functionals of the state , which are explicitly defined with respect to the solution of the kinetic equation. We refer to such a kinetic equation of inelastically interacting hard spheres as the non-Markovian Enskog kinetic equation for granular gases.

### 3.1 A description of the collective behavior of granular gases by means of kinetic equations

In the case of initial states (16), the dual picture to the Heisenberg picture of the evolution of a system of hard spheres with inelastic collisions described in terms of the dual BBGKY hierarchy (1) for marginal observables is the evolution of states described within the framework of the non-Markovian Enskog kinetic equation and a sequence of explicitly defined functionals of the solution of this kinetic equation.

In fact, for mean value functional (14), the following equality holds:

(17) |

where is the sequence of initial marginal distribution functions (16), and the sequence is a sequence of the marginal functional of the state represented by the series expansion over the products with respect to the one-particle distribution function :

(18) | |||

In series (18) we used the notations ; and the -order generating operator , is defined as follows [31]:

(19) | |||

where it means that , and we denote the -order scattering cumulant by the operator :

(20) |

and the operator is the -order cumulant of adjoint semigroups of hard spheres with inelastic collisions (15). We give several examples of expansions (19):

We emphasize that, in fact, functionals (18) characterize the correlations generated by dynamics of a hard sphere system with inelastic collisions.

If , then for arbitrary series (18) converges in the norm of the space .

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

(21) |

where the generating operator is the -order cumulant of adjoint semigroups of hard spheres with inelastic collisions defined by expansion (15).

For , the one-particle marginal distribution function (21) is a solution of the following Cauchy problem of the non-Markovian Enskog kinetic equation

(22) | |||

(23) |

where the collision integral is determined by the marginal functional of the state (18) in the case of , and the expressions and are the pre-collision momenta of hard spheres with inelastic collisions (9), i.e. solutions of equations (5).

Thus, if initial states are specified by a one-particle marginal distribution function on allowed configurations, then the evolution of marginal observables governed by the dual BBGKY hierarchy (1) can be also described within the framework of the non-Markovian kinetic equation (22) and a sequence of marginal functionals of the state (18). In other words, for mentioned initial states, the evolution of all possible states of a hard sphere system with inelastic collisions at an arbitrary moment of time can be described within the framework of a one-particle distribution function without any approximations.

### 3.2 The proof of the main results

In the particular case of initial data (2) specified by the -ary marginal observable , i.e. the marginal observable , equality (17) takes the form

(24) | |||

where the marginal functional of the state is represented by series expansion (18).

To verify the validity of equality (24), we transform the functional to the form

(25) | |||

where we used notations accepted in formula (18), and the operator is the -order cumulant of adjoint semigroups of hard spheres with inelastic collisions. For and obtained functional (25) exists under the condition that .

Then we expand the cumulants of adjoint semigroups of hard spheres in functional (25) over the new evolution operators into the kinetic cluster expansions [31]:

(26) | |||

where the following convention is assumed: , and the operator is defined by the formula

where is a characteristic function of allowed configurations of a system of hard spheres.