Phase space measure concentration for an ideal gas

TL;DR

This paper demonstrates that in a specific ideal gas model, the phase space volume concentrates around the equator as the system size grows, using geometric inequalities to quantify the convergence rate.

Contribution

It introduces a geometric perspective on phase space concentration in an ideal gas, applying isoperimetric inequalities to analyze the thermodynamic limit.

Findings

Phase space volume concentrates around the equator in the thermodynamic limit.

The rate of convergence to this concentration is explicitly determined.

The approach relies on spherical isoperimetric inequalities.

Abstract

We point out that a special case of an ideal gas exhibits concentration of the volume of its phase space, which is a sphere, around its equator in the thermodynamic limit. The rate of approach to the thermodynamic limit is determined. Our argument relies on the spherical isoperimetric inequality of L\'{e}vy and Gromov.