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

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  • Apenko, Sergey M.

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.

Suggested Citation

  • Apenko, Sergey M., 2014. "Clausius inequality and H-theorems for some models of random wealth exchange," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 414(C), pages 108-117.
    • Handle: RePEc:eee:phsmap:v:414:y:2014:i:c:p:108-117
    DOI: 10.1016/j.physa.2014.07.017
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    1. Chakrabarti,Bikas K. & Chakraborti,Anirban & Chakravarty,Satya R. & Chatterjee,Arnab, 2013. "Econophysics of Income and Wealth Distributions," Cambridge Books, Cambridge University Press, number 9781107013445.
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