IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v149y2022icp248-277.html
   My bibliography  Save this article

Continuum and thermodynamic limits for a simple random-exchange model

Author

Listed:
  • Düring, Bertram
  • Georgiou, Nicos
  • Merino-Aceituno, Sara
  • Scalas, Enrico

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 while the third is extraneous.

Suggested Citation

  • Düring, Bertram & Georgiou, Nicos & Merino-Aceituno, Sara & Scalas, Enrico, 2022. "Continuum and thermodynamic limits for a simple random-exchange model," Stochastic Processes and their Applications, Elsevier, vol. 149(C), pages 248-277.
    • Handle: RePEc:eee:spapps:v:149:y:2022:i:c:p:248-277
    DOI: 10.1016/j.spa.2022.03.015
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414922000783
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spa.2022.03.015?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to look for a different version below or search for a different version of it.

    Other versions of this item:

    References listed on IDEAS

    as
    1. Federico Bassetti & Giuseppe Toscani, 2010. "Explicit equilibria in a kinetic model of gambling," Papers 1002.3689, arXiv.org.
    2. Düring, B. & Toscani, G., 2007. "Hydrodynamics from kinetic models of conservative economies," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 384(2), pages 493-506.
    3. Aoki,Masanao & Yoshikawa,Hiroshi, 2011. "Reconstructing Macroeconomics," Cambridge Books, Cambridge University Press, number 9781107634206, September.
    4. Düring, Bertram & Toscani, Giuseppe, 2008. "International and domestic trading and wealth distribution," CoFE Discussion Papers 08/02, University of Konstanz, Center of Finance and Econometrics (CoFE).
    5. Garibaldi,Ubaldo & Scalas,Enrico, 2010. "Finitary Probabilistic Methods in Econophysics," Cambridge Books, Cambridge University Press, number 9780521515597, September.
    6. Robert Lee Taylor & Tien-Chung Hu, 1987. "Strong laws of large numbers for arrays of rowwise independent random elements," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 10, pages 1-10, January.
    7. Düring, Bertram & Matthes, Daniel & Toscani, Giuseppe, 2008. "Kinetic equations modelling wealth redistribution: A comparison of approaches," CoFE Discussion Papers 08/03, University of Konstanz, Center of Finance and Econometrics (CoFE).
    8. 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.
    9. Pareschi, Lorenzo & Toscani, Giuseppe, 2013. "Interacting Multiagent Systems: Kinetic equations and Monte Carlo methods," OUP Catalogue, Oxford University Press, number 9780199655465.
    10. Grosskinsky, Stefan & Jatuviriyapornchai, Watthanan, 2019. "Derivation of mean-field equations for stochastic particle systems," Stochastic Processes and their Applications, Elsevier, vol. 129(4), pages 1455-1475.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Marco Torregrossa & Giuseppe Toscani, 2017. "Wealth distribution in presence of debts. A Fokker--Planck description," Papers 1709.09858, arXiv.org.
    2. G. Dimarco & L. Pareschi & G. Toscani & M. Zanella, 2020. "Wealth distribution under the spread of infectious diseases," Papers 2004.13620, arXiv.org.
    3. Gualandi, Stefano & Toscani, Giuseppe, 2018. "Pareto tails in socio-economic phenomena: A kinetic description," Economics - The Open-Access, Open-Assessment E-Journal (2007-2020), Kiel Institute for the World Economy (IfW Kiel), vol. 12, pages 1-17.
    4. Gualandi, Stefano & Toscani, Giuseppe, 2019. "Size distribution of cities: A kinetic explanation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 524(C), pages 221-234.
    5. Max Greenberg & H. Oliver Gao, 2024. "Twenty-five years of random asset exchange modeling," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 97(6), pages 1-27, June.
    6. Giuseppe Toscani, 2016. "Kinetic and mean field description of Gibrat's law," Papers 1606.04796, arXiv.org.
    7. Xia Zhou & Shaoyong Lai, 2023. "The mutual influence of knowledge and individual wealth growth," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 96(6), pages 1-22, June.
    8. Toscani, Giuseppe, 2016. "Kinetic and mean field description of Gibrat’s law," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 461(C), pages 802-811.
    9. Lorenzo Pareschi & Giuseppe Toscani, 2014. "Wealth distribution and collective knowledge. A Boltzmann approach," Papers 1401.4550, arXiv.org.
    10. Kayser, Kirk & Armbruster, Dieter, 2019. "Social optima of need-based transfers," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 536(C).
    11. Düring, Bertram & Toscani, Giuseppe, 2008. "International and domestic trading and wealth distribution," CoFE Discussion Papers 08/02, University of Konstanz, Center of Finance and Econometrics (CoFE).
    12. G. Toscani & C. Brugna & S. Demichelis, 2012. "Kinetic models for the trading of goods," Papers 1208.6305, arXiv.org.
    13. Trimborn, Torsten & Frank, Martin & Martin, Stephan, 2018. "Mean field limit of a behavioral financial market model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 505(C), pages 613-631.
    14. Giacomo Dimarco & Giuseppe Toscani & Mattia Zanella, 2024. "A multi-agent description of the influence of higher education on social stratification," Journal of Economic Interaction and Coordination, Springer;Society for Economic Science with Heterogeneous Interacting Agents, vol. 19(3), pages 493-521, July.
    15. Torsten Trimborn & Lorenzo Pareschi & Martin Frank, 2017. "Portfolio Optimization and Model Predictive Control: A Kinetic Approach," Papers 1711.03291, arXiv.org, revised Feb 2019.
    16. Andrea Medaglia & Andrea Tosin & Mattia Zanella, 2022. "Monte Carlo stochastic Galerkin methods for non-Maxwellian kinetic models of multiagent systems with uncertainties," Partial Differential Equations and Applications, Springer, vol. 3(4), pages 1-30, August.
    17. Giuseppe Toscani & Andrea Tosin & Mattia Zanella, 2019. "Multiple-interaction kinetic modelling of a virtual-item gambling economy," Papers 1904.07660, arXiv.org.
    18. Bertram During & Nicos Georgiou & Enrico Scalas, 2016. "A stylized model for wealth distribution," Papers 1609.08978, arXiv.org, revised Jul 2021.
    19. Maria Letizia Bertotti & Amit K Chattopadhyay & Giovanni Modanese, 2017. "Economic inequality and mobility for stochastic models with multiplicative noise," Papers 1702.08391, arXiv.org.
    20. Gualandi, Stefano & Toscani, Giuseppe, 2017. "Pareto tails in socio-economic phenomena: A kinetic description," Economics Discussion Papers 2017-111, Kiel Institute for the World Economy (IfW Kiel).

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:spapps:v:149:y:2022:i:c:p:248-277. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/505572/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.