Chaotic distributions for relativistic particles

Abstract

We study a modified Kac model where the classical kinetic energy is replaced by an arbitrary energy function $ϕ(v)$, $v \in \mathbb{R}$. The aim of this paper is to show that the uniform density with respect to the microcanonical measure is $Ce^{-z_0ϕ(v)}$-chaotic, $C,z_0 \in \mathbb{R}_+$. The kinetic energy for relativistic particles is a special case. A generalization to the case $v\in \mathbb{R}^d$ which involves conservation momentum is also formally discussed.