On the Boltzmann-Grad Limit for the classical hard-spheres system

TL;DR

This paper investigates the existence of a strong Boltzmann-Grad limit for the BBGKY hierarchy of classical hard spheres, using Klimontovich's approach to clarify aspects of the Lanford conjecture in kinetic theory.

Contribution

It introduces a new approach to analyze the Boltzmann-Grad limit by seeking strong local convergence, contrasting previous methods based on w*-convergence.

Findings

Identifies conditions for the strong Boltzmann-Grad limit in phase space.

Provides explicit representation of reduced distribution functions via Klimontovich density.

Clarifies the mathematical structure underlying the Lanford conjecture.

Abstract

Despite the progress achieved by kinetic theory, the search of possible exact kinetic equations remains elusive to date. This concerns, specifically, the issue of the validity of the conjecture proposed by Grad (Grad, 1972) and developed in a seminal work by Lanford (Lanford, 1974) that kinetic equations - such as the Boltzmann equation for a gas of classical hard spheres - might result exact in an appropriate asymptotic limit, usually denoted as Boltzmann-Grad limit. The Lanford conjecture has actually had a profound influence on the scientific community, giving rise to a whole line of original research in kinetic theory and mathematical physics. Nevertheless, certain aspects of the theory remain to be addressed and clarified. The purpose of this paper is to investigate the possible existence of the strong Boltzmann-Grad limit for the BBGKY hierarchy. Contrary to previous approaches in which the w*-convergence was considered for the definition of the Boltzmann-Grad limit functions, based on their construction in terms of time-series expansions obtained from the BBGKY hierarchy, here we look for the possible existence of strong limit functions in the sense of local convergence in phase space. The result is based on the adoption of the Klimontovich approach to statistical mechanics, permitting the explicit representation of the $s$-body reduced distribution functions in terms of the Klimontovich probability density.