Entropy of continuous mixtures and the measure problem

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.

References in corpus (8)

• Exponential and power-law probability distributions of wealth and income in the United Kingdom and the United States
• Statistical mechanics of money
• Optimal packing of polydisperse hard-sphere fluids
• Condensation Transition in Polydisperse Hard Rods
• What do emulsification failure and Bose-Einstein condensation have in common?
• Algorithm for normal random numbers
• Continuous phase transition in polydisperse hard-sphere mixture
• Optimal packing of polydisperse hard-sphere fluids II

Cited by in corpus (7)

• Configurational entropy of polydisperse supercooled liquids
• Towards an $H$-theorem for granular gases
• About an H-theorem for systems with non-conservative interactions
• Kullback--Leibler Divergence of a Freely Cooling Granular Gas
• On the role of composition entropies in the statistical mechanics of polydisperse systems
• Clausius inequality and H-theorems for some models of random wealth exchange
• Relative entropy of freely cooling granular gases. A molecular dynamics study