Boltzmann–Grad asymptotic behavior of collisional dynamics

Abstract

<jats:p> This paper discusses some of the latest advances in the mathematical understanding of the nature of kinetic equations that describe the collective behavior of many-particle systems with collisional dynamics. </jats:p>

On the cardinality of collisional clusters for hard spheres at low density

“…(12) is a rough bound, plausible for short times only. In fact the argument reminds Lanford's proof of the local validity of the Boltzmann equation [15] (see also [13,25,6,9,19,20,7,3,10]). The estimate is too pessimistic, because it ignores that the particles not belonging to BC(1) do not interact with it, which produces exponential damping.…”

Propagation processes of correlations of hard spheres

“…, x s ) into two nonempty mutually disjoint subsets X 1 and X 2 , the symbol X i means the set of indexes of the set X i of phase space coordinates and the operator L * s is defined on the subspace L 1 0 ⊂ L 1 by formula (4). It should be noted that the Liouville hierarchy (15) is the evolution recurrence equations set.…”

“…We note that because the Liouville hierarchy ( 15) is the recurrence evolution equations set, we can construct a solution of the Cauchy problem ( 15),( 16), integrating each equation of the hierarchy as the inhomogeneous Liouville equation. For example, as a result of the integration of the first two equations of the Liouville hierarchy (15), we obtain the following equalities:…”

“…As is known, an equivalent approach adapted to describing the evolution of states of systems of both finite and infinite number of hard spheres is to describe a state by means of a sequence of so-called reduced distribution functions (marginals) governed by the BBGKY hierarchy [1]. In what follows, we outline an approach to describing the evolution of a state using both reduced distribution functions and reduced correlation functions, based on the dynamics of correlations in a system of hard spheres governed by the Liouville hierarchy for correlation functions (15). For a hard-sphere system of a non-fixed, i.e.…”

“…In consequence of definition (18) and the cumulant structure of representation of a solution (9) for the Liouville hierarchy (15), if initial state specified by the sequence of reduced distribution functions F (0) = (1, F 0 1 (x 1 ), . .…”