Clausius inequality and H-theorems for some models of random wealth exchange

Authored by Sergey M. Apenko

Date Published: 2014-11-15

DOI: 10.1016/j.physa.2014.07.017

Sponsors: Russian Foundation for Basic Research

Platforms: No platforms listed

Model Documentation: Other Narrative Mathematical description

Model Code URLs: Model code not found

Abstract

We discuss a possibility of deriving an H-theorem for nonlinear discrete time evolution equation that describes random wealth exchanges. In such kinetic models economical agents exchange wealth in pairwise collisions just as particles in a gas exchange their energy. It appears useful to reformulate the problem and represent the dynamics as a combination of two processes. The first is a linear transformation of a two-particle distribution function during the act of exchange while the second one corresponds to new random pairing of agents and plays a role of some kind of feedback control. This representation leads to a Clausius-type inequality which suggests a new interpretation of the exchange process as an irreversible relaxation due to a contact with a reservoir of a special type. Only in some special cases when equilibrium distribution is exactly a gamma distribution, this inequality results in the H-theorem with monotonically growing `entropy' functional which differs from the Boltzmann entropy by an additional term. But for arbitrary exchange rule the evolution has some features of relaxation to a non-equilibrium steady state and it is still unclear if any general H-theorem could exist, (C) 2014 Elsevier B.V. All rights reserved.

Tags

Agent based models Entropy H-theorem Kinetic equation