Microscopic solutions of the Boltzmann-Enskog equation in the series representation

Abstract

The Boltzmann-Enskog equation for a hard sphere gas is known to have so called microscopic solutions, i.e., solutions of the form of time-evolving empirical measures of a finite number of hard spheres. However, the precise mathematical meaning of these solutions should be discussed, since the formal substitution of empirical measures into the equation is not well-defined. Here we give a rigorous mathematical meaning to the microscopic solutions to the Boltzmann-Enskog equation by means of a suitable series representation. R 6 f (x, v, t) dxdv = 1, then f can be understood as a probability density of an arbitrary single hard sphere. Further, S 2 + = {ω ∈ S 2 | (v − v 1 ) · ω ≥ 0}, S 2 is the unit sphere in R 3 (with the surface measure dω), (v, v 1 ) is a pair of velocities in incoming collision configuration and (v , v 1 ) is the corresponding pair of outgoing velocities 2010 Mathematics Subject Classification. Primary: 82C05, 82C40; Secondary: 35Q20.

Propagation processes of correlations of hard spheres

“….) is a sequence of reduced correlation functions at initial instant, then by means of mappings (13) the evolution of all possible states, i.e. the sequence of the reduced correlation functions G s (t), s ≥ 1, is determined by the following series expansions:…”

Pointwise decay of truncated correlations in chaotic states at low density

“…In the previous reference it is shown (restricting to hard spheres) that, if the ansatz of maximal chaoticity is made at any time (thus disregarding dynamical correlations), then the density evolves according to an Enskog equation, and an H−Theorem can be deduced for the associated entropy; see also [2,14]. These remarks have not been object of rigorous investigation so far; see however [10,19,21] for related discussions.…”