Entropy of continuous mixtures and the measure problem

TL;DR

This paper investigates the measure problem in continuous entropy functionals within collision processes, providing conditions for a measure-free entropy maximization in particle systems.

Contribution

It introduces a sufficient condition based on Jacobian factorization that ensures a well-defined entropy maximization without measure issues.

Findings

Identifies the measure problem in continuous entropy functionals.

Derives a Jacobian factorization condition for measure-free entropy maximization.

Applies to generic collision processes with conservation laws.

Abstract

In its continuous version, the entropy functional measuring the information content of a given probability density may be plagued by a "measure" problem that results from improper weighting of phase space. This issue is addressed considering a generic collision process whereby a large number of particles/agents randomly and repeatedly interact in pairs, with prescribed conservation law(s). We find a sufficient condition under which the stationary single particle distribution function maximizes an entropy-like functional, that is free of the measure problem. This condition amounts to a factorization property of the Jacobian associated to the binary collision law, from which the proper weighting of phase space directly follows.