On the conditions of validity of the Boltzmann equation and Boltzmann H-theorem

TL;DR

This paper develops an axiomatic framework for the statistical description of classical N-body systems, examining the conditions under which the Boltzmann equation and H-theorem hold, and discussing implications for deterministic PDFs and paradoxes.

Contribution

It introduces a generalized axiomatic approach with modified collision boundary conditions, analyzing their impact on the validity of the Boltzmann kinetic theory.

Findings

The axiomatic approach is consistent with exact H-theorems for N-body and 1-body PDFs.

Both the Boltzmann equation and H-theorem fail when the N-body PDF is deterministic.

Modified boundary conditions influence the applicability of classical kinetic results.

Abstract

In this paper the problem is posed of the formulation of the so-called "ab initio" approach to the statistical description of the Boltzmann-Sinai N-body classical dynamical system (CDS) formed by identical smooth hard spheres. This amounts to introducing a suitably-generalized version of the axioms of Classical Statistical Mechanics. The latter involve a proper definition of the functional setting for the N-body probability density function (PDF), so that it includes also the case of the deterministic N-body PDF. In connection with this issue, a further development concerns the introduction of modified collision boundary conditions which differ from the usual ones adopted in previous literature. Both features are proved to be consistent with the validity of exact H-theorems for the N-body and 1-body PDFs respectively. Consequences of the axiomatic approach which concern the conditions of validity of the Boltzmann kinetic equation and the Boltzmann H-theorem are investigated. In particular, the role of the modified boundary conditions is discussed. It is shown that both theorems fail in the case in which the N-body PDF is identified with the deterministic PDF. Finally, the issue of applicability of the Zermelo and Loschmidt paradoxes to the "ab initio" approach presented here is discussed.