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

TL;DR

This paper explores the derivation of an H-theorem for models of random wealth exchange, revealing conditions under which entropy-like functionals increase and interpreting the process as an irreversible relaxation.

Contribution

It introduces a reformulation of wealth exchange dynamics as a combination of linear transformations and feedback, leading to a Clausius-type inequality and new insights into entropy evolution.

Findings

Clausius-type inequality suggests irreversibility in wealth exchange models

H-theorem holds when equilibrium is a gamma distribution

Evolution often exhibits relaxation to non-equilibrium steady states

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.