Asymptotic orderings and approximations of the Master kinetic equation for large hard spheres systems

Abstract

In this paper the problem is posed of determining the physically-meaningful asymptotic orderings holding for the statistical description of a large N −body system of hard spheres, i.e., formed by N ≡ 1 ε ≫ 1 particles, which are allowed to undergo instantaneous and purely elastic unary, binary or multiple collisions. Starting point is the axiomatic treatment recently developed [Tessarotto et al., 2013 and the related discovery of an exact kinetic equation realized by Master equation which advances in time the 1−body probability density function (PDF) for such a system. As shown in the paper the task involves introducing appropriate asymptotic orderings in terms of ε for all the physically-relevant parameters. The goal is that of identifying the relevant physically-meaningful asymptotic approximations applicable for the Master kinetic equation, together with their possible relationships with the Boltzmann and Enskog kinetic equations, and holding in appropriate asymptotic regimes. These correspond either to dilute or dense systems and are formed either by small-size or finite-size identical hard spheres, the distinction between the various cases depending on suitable asymptotic orderings in terms of ε. -INTRODUCTIONIn a series of papers [1][2][3][4][5][6][7] (see also Refs. [8,9]) a new kinetic equation has been established for hard sphere systems subject to elastic instantaneous collisions, denoted as Master kinetic equation. Its basic features are that, unlike the Boltzmann and Enskog kinetic equations [11][12][13], the same equation and its corresponding Master collision operator are exact, i.e., they hold for an arbitraryN −body hard-sphere system S N which is isolated, namely for which the number of particles (N ) is constant, while furthermore that its solutions are entropy-preserving [6] and globally defined [7]. Concerning, in particular, the first feature this means that in such a context S N is allowed to have an arbitrary finite number N of finite-size and finite-mass hard spheres, namely each one characterized by finite diameter σ > 0 and mass m > 0. In addition, by assumption S N is immersed in a bounded configuration domain Ω, subset of the Euclidean space R 3 which has a finite canonical measure L 3 o ≡ µ(Ω) > 0 (L o denoting a finite configuration-domain characteristic scale length) and is endowed with a stationary and rigid boundary ∂Ω. However, the total volume occupied by hard-spheres cannot exceed the configuration-space volume. Hence, the parameters N, L o and σ must necessarily satisfy the inequalitydenoting the volume of a single hard sphere and ∆ the global diluteness parameter. For such an equation the particle correlations appearing through the 2−body probability density function (PDF) are exactly taken into account by means of suitably-prescribed 1− and 2−body occupation coefficients which are position-dependent only. These peculiar features follow uniquely as a consequence of the new approach to classical statistical mechanics developed in Refs. [1][2][3] and referred to as "ab initio...

Macroscopic Irreversibility and Decay to Kinetic Equilibrium of the 1-Body PDF for Finite Hard-Sphere Systems

“…However, both Boltzmann and Grad theories are actually specialized to the treatment of the so-called Boltzmann-Grad limit obtained introducing, first, the dilute-gas ordering σ ∼ O(ε 1/2 ) with ε ≡ 1 N ≪ 1 and, then, taking the continuum limit ε → 0 (for a review of the topic see again Ref. [10]). Nevertheless, the possible realization of either PMI or DKE depends critically on the prescription of the functional class ρ (N ) (x 1 , t) , so that their occurrence is actually non-mandatory.…”

“…The Boltzmann equation can be recovered in an asymptotic sense when the so-called dilute-gas asymptotic ordering is introduced in the Master kinetic equation (see Refs. [7,10]).…”

“…The next step is to prove the monotonic time-decreasing behavior of the MKI functional, which involves No.#3 and No.#4 MKI Prescriptions and consequently also the validity of No.#2 MKI Prescription. The first two refer respectively to the validity of the time derivative inequality (7) and the conditions of existence of kinetic equilibrium (10), while the latter one concerns the right-hand inequality I M (ρ (N ) 1 (t)) ≤ 1. In order to prove these properties let us preliminarily determine the variation of the total directional kinetic energy M (v 1 , v 2 , b) (see Eq.…”

“…

In this paper the conditions are investigated for the occurrence of the so-called macroscopic irreversibility property and the related phenomenon of decay to kinetic equilibrium which may characterize the 1−body probability density function (PDF) associated with hard-sphere systems. The problem is set in the framework of the axiomatic "ab initio" approach to classical statistical mechanics recently developed [Tessarotto et al, -2017 and the related establishment of an exact kinetic equation realized by Master equation for the same kinetic PDF. As shown in the paper the task involves the introduction of a suitable functional of the 1−body PDF here identified with the Master kinetic information.

…”

On the Boltzmann-Grad limit of the Master Kinetic Equation

“…This is about the prescription of the so-called Boltzmann-Grad limit (BG-limit) first explicitly introduced by Grad [2] but actually set at the basis of Boltzmann's construct of his namesake kinetic equation [1]. In this letter we intend to point out crucial aspects involved in its prescription in the context of the "ab initio" axiomatic approach for hard-sphere systems [15,[27][28][29][30][31][32][33][34] and the related discovery of the Master kinetic equation for hard spheres undergoing elastic instantaneous mutual collisions [28,[30][31][32]. The problem is in fact physically relevant for two main reasons, i.e., to establish a rigorous connection with the Master kinetic equation itself and in order to ascertain whether and under which conditions the Boltzmann equation can be exactly recovered in an appropriate asymptotic limit.…”