Uniformity of the phase space and fluctuations in thermal equilibrium

TL;DR

This paper establishes general relations between phase space uniformity, fluctuations, and correlations in systems near thermal equilibrium, using Renyi entropy, with analytical and simulation validation for quantum fermion and boson systems.

Contribution

It introduces new analytical relations linking phase space uniformity and fluctuations, and develops unbiased estimators for Renyi entropies applicable to quantum systems.

Findings

Analytical relations between uniformity and correlations are verified.

Unbiased estimators for Renyi entropies are proposed and validated.

Simulations confirm theoretical predictions for quantum fermions and bosons.

Abstract

General relations are found between the measure of the uniformity of distributions on the phase space and the first moments and correlations of extensive variables for systems close to thermal equilibrium. The role played by the parameter of the Renyi entropy for the analysis of their fluctuations and correlations is studied. Analytical results are verified and illustrated by direct simulations of quantum systems of ideal fermions and bosons. Problems of finite statistics, usual in experiments and simulations, are addressed and discussed and solved by finding unbiased estimators for Renyi entropies and uniformities.