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Continuum and thermodynamic limits for a simple random-exchange model

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  • Bertram During
  • Nicos Georgiou
  • Sara Merino-Aceituno
  • Enrico Scalas

Abstract

We discuss various limits of a simple random exchange model that can be used for the distribution of wealth. We start from a discrete state space - discrete time version of this model and, under suitable scaling, we show its functional convergence to a continuous space - discrete time model. Then, we show a thermodynamic limit of the empirical distribution to the solution of a kinetic equation of Boltzmann type. We solve this equation and we show that the solutions coincide with the appropriate limits of the invariant measure for the Markov chain. In this way we complete Boltzmann's program of deriving kinetic equations from random dynamics for this simple model. Three families of invariant measures for the mean field limit are discovered and we show that only two of those families can be obtained as limits of the discrete system and the third is extraneous. Finally, we cast our results in the framework of integer partitions and strengthen some results already available in the literature.

Suggested Citation

  • Bertram During & Nicos Georgiou & Sara Merino-Aceituno & Enrico Scalas, 2020. "Continuum and thermodynamic limits for a simple random-exchange model," Papers 2003.00930, arXiv.org.
    • Handle: RePEc:arx:papers:2003.00930
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    References listed on IDEAS

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    1. Federico Bassetti & Giuseppe Toscani, 2010. "Explicit equilibria in a kinetic model of gambling," Papers 1002.3689, arXiv.org.
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    6. Bertram Düring & Lorenzo Pareschi & Giuseppe Toscani, 2018. "Kinetic models for optimal control of wealth inequalities," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 91(10), pages 1-12, October.
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