Chaotic distributions for relativistic particles

We study a modified Kac model where the classical kinetic energy is replaced by an arbitrary energy function $\varphi (v)$, $v\in \mathbb{R}$. The aim of this paper is to show that the uniform density with respect to the microcanonical measure is $C{e}^{-z_0\varphi (v)}$-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.