{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,19]],"date-time":"2026-08-19T17:16:56Z","timestamp":1787159816559,"version":"build-2736575974"},"reference-count":35,"publisher":"American Institute of Mathematical Sciences (AIMS)","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["NHM"],"published-print":{"date-parts":[[2024]]},"abstract":"<abstract><p>We present a novel reshuffling exchange model and investigate its long time behavior. In this model, two individuals are picked randomly, and their wealth $ X_i $ and $ X_j $ are redistributed by flipping a sequence of fair coins leading to a binomial distribution denoted $ B\\circ (X_i+X_j) $. This dynamics can be considered as a natural variant of the so-called uniform reshuffling model in econophysics. May refer to Cao, Jabin and Motsch (2023), Dragulescu and Yakovenko (2000). As the number of individuals goes to infinity, we derive its mean-field limit, which links the stochastic dynamics to a deterministic infinite system of ordinary differential equations. Our aim of this work is then to prove (using a coupling argument) that the distribution of wealth converges to the Poisson distribution in the $ 2 $-Wasserstein metric. Numerical simulations illustrate the main result and suggest that the polynomial convergence decay might be further improved.<\/p><\/abstract><\/jats:p>","DOI":"10.3934\/nhm.2024002","type":"journal-article","created":{"date-parts":[[2024,1,8]],"date-time":"2024-01-08T05:58:53Z","timestamp":1704693533000},"page":"24-43","source":"Crossref","is-referenced-by-count":8,"title":["From the binomial reshuffling model to Poisson distribution of money"],"prefix":"10.3934","volume":"19","author":[{"given":"Fei","family":"Cao","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, University of Massachusetts Amherst, 710 N. Pleasant St, MA 01003, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Nicholas F.","family":"Marshall","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Oregon State University, 368 Kidder Hall, Corvallis, OR 97331, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"2321","reference":[{"key":"key-10.3934\/nhm.2024002-1","doi-asserted-by":"publisher","unstructured":"F. Bassetti, G. Toscani, Mean field dynamics of interaction processes with duplication, loss and copy, Math. Models Methods Appl. Sci.<\/i>, 25<\/b> (2015), 1887\u20131925. https:\/\/doi.org\/10.1142\/S0218202515500487","DOI":"10.1142\/S0218202515500487"},{"key":"key-10.3934\/nhm.2024002-2","unstructured":"F. Cao, S. Reed, A biased dollar exchange model involving bank and debt with discontinuous equilibrium<\/i>, arXiv: 2311.07851, [Preprint], (2023) [cited 2024 Jan 08]. Available from: https:\/\/doi.org\/10.48550\/arXiv.2311.07851<\/ext-link>"},{"key":"key-10.3934\/nhm.2024002-3","doi-asserted-by":"publisher","unstructured":"F. Cao, S. Motsch, Derivation of wealth distributions from biased exchange of money, Kinet. Relat. Models<\/i>, 16<\/b> (2023), 764\u2013794. https:\/\/doi.org\/10.3934\/krm.2023007","DOI":"10.3934\/krm.2023007"},{"key":"key-10.3934\/nhm.2024002-4","doi-asserted-by":"publisher","unstructured":"F. Cao, PE. Jabin, S. Motsch, Entropy dissipation and propagation of chaos for the uniform reshuffling model, Math. Models Methods Appl. Sci.<\/i>, 33<\/b> (2023), 829\u2013875. https:\/\/doi.org\/10.1142\/S0218202523500185","DOI":"10.1142\/S0218202523500185"},{"key":"key-10.3934\/nhm.2024002-5","doi-asserted-by":"publisher","unstructured":"F. Cao, Explicit decay rate for the Gini index in the repeated averaging model, Math. Methods Appl. Sci.<\/i>, 46<\/b> (2023), 3583\u20133596. https:\/\/doi.org\/10.1002\/mma.8711","DOI":"10.1002\/mma.8711"},{"key":"key-10.3934\/nhm.2024002-6","unstructured":"F. Cao, PE. Jabin, From interacting agents to Boltzmann-Gibbs distribution of money<\/i>, arXiv: 2208.05629, [Preprint], (2022) [cited 2024 Jan 08]. Available from: https:\/\/doi.org\/10.48550\/arXiv.2208.05629<\/ext-link>"},{"key":"key-10.3934\/nhm.2024002-7","doi-asserted-by":"publisher","unstructured":"F. Cao, S. Motsch, Uncovering a two-phase dynamics from a dollar exchange model with bank and debt, SIAM J. Appl. Math.<\/i>, 83<\/b> (2023), 1872\u20131891. https:\/\/doi.org\/10.1137\/22M1518621","DOI":"10.1137\/22M1518621"},{"key":"key-10.3934\/nhm.2024002-8","doi-asserted-by":"publisher","unstructured":"E. A. Carlen, E. Gabetta, G. Toscani, Propagation of smoothness and the rate of exponential convergence to equilibrium for a spatially homogeneous Maxwellian gas, Comm. Math. Phys.<\/i>, 199<\/b> (1999), 521\u2013546. https:\/\/doi.org\/10.1007\/s002200050511","DOI":"10.1007\/s002200050511"},{"key":"key-10.3934\/nhm.2024002-9","unstructured":"J. A. Carrillo, G. Toscani, Contractive probability metrics and asymptotic behavior of dissipative kinetic equations, Riv. Mat. Univ. Parma<\/i>, 6<\/b> (2007), 75\u2013198. https:\/\/ddd.uab.cat\/record\/44032<\/ext-link>"},{"key":"key-10.3934\/nhm.2024002-10","doi-asserted-by":"publisher","unstructured":"L P. Chaintron, A. Diez, Propagation of chaos: A review of models, methods and applications. I. Models and methods, Kinet. Relat. Models<\/i>, 15<\/b> (2022), 895\u20131015. https:\/\/doi.org\/10.3934\/krm.2022017","DOI":"10.3934\/krm.2022017"},{"key":"key-10.3934\/nhm.2024002-11","doi-asserted-by":"publisher","unstructured":"A. Chakraborti, B. K. Chakrabarti, Statistical mechanics of money: how saving propensity affects its distribution, Eur. Phys. J. B<\/i>, 17<\/b> (2000), 167\u2013170. https:\/\/doi.org\/10.1007\/s100510070173","DOI":"10.1007\/s100510070173"},{"key":"key-10.3934\/nhm.2024002-12","doi-asserted-by":"publisher","unstructured":"A. Chatterjee, B. K. Chakrabarti, S. S. Manna, Pareto law in a kinetic model of market with random saving propensity, Physica A<\/i>, 335<\/b> (2004), 155\u2013163. https:\/\/doi.org\/10.1016\/j.physa.2003.11.014","DOI":"10.1016\/j.physa.2003.11.014"},{"key":"key-10.3934\/nhm.2024002-13","doi-asserted-by":"publisher","unstructured":"S. Chatterjee, P. Diaconis, A. Sly, L. Zhang, A phase transition for repeated averages, Ann. Probab.<\/i>, 50<\/b> (2022), 1\u201317. https:\/\/doi.org\/10.1214\/21-AOP1526","DOI":"10.1214\/21-AOP1526"},{"key":"key-10.3934\/nhm.2024002-14","doi-asserted-by":"publisher","unstructured":"R. Cortez, J. Fontbona, Quantitative propagation of chaos for generalized Kac particle systems, Ann. Appl. Probab.<\/i>, 26<\/b> (2016), 892\u2013916. https:\/\/doi.org\/10.1214\/15-AAP1107","DOI":"10.1214\/15-AAP1107"},{"key":"key-10.3934\/nhm.2024002-15","unstructured":"R. Cortez, Particle system approach to wealth redistribution<\/i>, arXiv: 1809.05372, [Preprint], (2018) [cited 2024 Jan 08]. Available from: https:\/\/doi.org\/10.48550\/arXiv.1809.05372<\/ext-link>"},{"key":"key-10.3934\/nhm.2024002-16","unstructured":"R. Cortez, F. Cao, Uniform propagation of chaos for a dollar exchange econophysics model<\/i>, arXiv: 2212.08289, [Preprint], (2022) [cited 2024 Jan 08]. Available from: https:\/\/doi.org\/10.48550\/arXiv.2212.08289<\/ext-link>"},{"key":"key-10.3934\/nhm.2024002-17","doi-asserted-by":"crossref","unstructured":"T. M. Cover, J.A. Thomas, Elements of information theory<\/i>, John Wiley & Sons, 1999. https:\/\/doi.org\/10.1002\/0471200611<\/ext-link>","DOI":"10.1002\/0471200611"},{"key":"key-10.3934\/nhm.2024002-18","doi-asserted-by":"crossref","unstructured":"G. Da Prato, An introduction to infinite-dimensional analysis<\/i>, Springer Science & Business Media, 2006. https:\/\/doi.org\/10.1007\/3-540-29021-4<\/ext-link>","DOI":"10.1007\/3-540-29021-4"},{"key":"key-10.3934\/nhm.2024002-19","doi-asserted-by":"publisher","unstructured":"A. Dragulescu, V. M. Yakovenko, Statistical mechanics of money, Eur. Phys. J. B<\/i>, 17<\/b> (2000), 723\u2013729. https:\/\/doi.org\/10.1007\/s100510070114","DOI":"10.1007\/s100510070114"},{"key":"key-10.3934\/nhm.2024002-20","doi-asserted-by":"publisher","unstructured":"G. Gabetta, G. Toscani, B. Wennberg, Metrics for probability distributions and the trend to equilibrium for solutions of the Boltzmann equation, J. Stat. Phys.<\/i>, 81<\/b> (1995), 901\u2013934. https:\/\/doi.org\/10.1007\/BF02179298","DOI":"10.1007\/BF02179298"},{"key":"key-10.3934\/nhm.2024002-21","doi-asserted-by":"publisher","unstructured":"B. T. Graham, Rate of relaxation for a mean-field zero-range process, Ann. Appl. Probab.<\/i>, 19<\/b> (2009), 497\u2013520. 10.1214\/08-AAP549","DOI":"10.1214\/08-AAP549"},{"key":"key-10.3934\/nhm.2024002-22","doi-asserted-by":"publisher","unstructured":"E. Heinsalu, P. Marco, Kinetic models of immediate exchange, Eur. Phys. J. B<\/i>, 87<\/b> (2014), 1\u201310. https:\/\/doi.org\/10.1140\/epjb\/e2014-50270-6","DOI":"10.1140\/epjb\/e2014-50270-6"},{"key":"key-10.3934\/nhm.2024002-23","doi-asserted-by":"publisher","unstructured":"P E. Jabin, B. Niethammer, On the rate of convergence to equilibrium in the Becker\u2013D\u00f6ring equations, J Differ Equ<\/i>, 191<\/b> (2003), 518\u2013543. https:\/\/doi.org\/10.1016\/S0022-0396(03)00021-4","DOI":"10.1016\/S0022-0396(03)00021-4"},{"key":"key-10.3934\/nhm.2024002-24","doi-asserted-by":"publisher","unstructured":"N. Lanchier, Rigorous proof of the Boltzmann\u2013Gibbs distribution of money on connected graphs, J. Stat. Phys.<\/i>, 167<\/b> (2017), 160\u2013172. https:\/\/doi.org\/10.1007\/s10955-017-1744-8","DOI":"10.1007\/s10955-017-1744-8"},{"key":"key-10.3934\/nhm.2024002-25","doi-asserted-by":"publisher","unstructured":"N. Lanchier, S. Reed, Rigorous results for the distribution of money on connected graphs, J. Stat. Phys.<\/i>, 171<\/b> (2018), 727\u2013743. https:\/\/doi.org\/10.1007\/s10955-018-2024-y","DOI":"10.1007\/s10955-018-2024-y"},{"key":"key-10.3934\/nhm.2024002-26","doi-asserted-by":"publisher","unstructured":"N. Lanchier, S. Reed, Rigorous results for the distribution of money on connected graphs (models with debts), J. Stat. Phys.<\/i>, 176<\/b> (2019), 1115\u20131137. https:\/\/doi.org\/10.1007\/s10955-019-02334-z","DOI":"10.1007\/s10955-019-02334-z"},{"key":"key-10.3934\/nhm.2024002-27","doi-asserted-by":"crossref","unstructured":"T. M. Liggett, Interacting particle systems<\/i>, New York: Springer, 1985.","DOI":"10.1007\/978-1-4613-8542-4"},{"key":"key-10.3934\/nhm.2024002-28","doi-asserted-by":"publisher","unstructured":"D. Matthes, G. Toscani, On steady distributions of kinetic models of conservative economies, J. Stat. Phys.<\/i>, 130<\/b> (2008), 1087\u20131117. https:\/\/doi.org\/10.1007\/s10955-007-9462-2","DOI":"10.1007\/s10955-007-9462-2"},{"key":"key-10.3934\/nhm.2024002-29","doi-asserted-by":"publisher","unstructured":"M. Merle, J. Salez, Cutoff for the mean-field zero-range process, Ann. Probab.<\/i>, 47<\/b> (2019), 3170\u20133201. https:\/\/doi.org\/10.1214\/19-AOP1336","DOI":"10.1214\/19-AOP1336"},{"key":"key-10.3934\/nhm.2024002-30","unstructured":"G. Naldi, L. Pareschi, G. Toscani, Mathematical modeling of collective behavior in socio-economic and life sciences<\/i>, Berlin: Springer Science & Business Media, 2010. https:\/\/doi.org\/10.1007\/978-0-8176-4946-3<\/ext-link>"},{"key":"key-10.3934\/nhm.2024002-31","doi-asserted-by":"publisher","unstructured":"G. Peyr\u00e9, M. Cuturi, Computational optimal transport: With applications to data science, Found. Trends Mach. Learn.<\/i>, 11<\/b> (2019), 355\u2013607. http:\/\/dx.doi.org\/10.1561\/2200000073","DOI":"10.1561\/2200000073"},{"key":"key-10.3934\/nhm.2024002-32","unstructured":"R. Pymar, N. Rivera, Mixing of the symmetric beta-binomial splitting process on arbitrary graphs<\/i>, arXiv: 2307.02406, [Preprint], (2023) [cited 2024 Jan 08]. Available from: https:\/\/doi.org\/10.48550\/arXiv.2307.02406<\/ext-link>"},{"key":"key-10.3934\/nhm.2024002-33","unstructured":"A. Santos, Showing convergence of an infinite ODE system, MathOverflow<\/i> 2022. Available from: https:\/\/mathoverflow.net\/q\/432268<\/ext-link>."},{"key":"key-10.3934\/nhm.2024002-34","doi-asserted-by":"crossref","unstructured":"A S. Sznitman, Topics in propagation of chaos, in Ecole d'\u00e9t\u00e9 de probabilit\u00e9s de Saint-Flour XIX\u20141989<\/i>, Berlin: Springer, (1991), 165\u2013251. https:\/\/doi.org\/10.1007\/BFb0085166<\/ext-link>","DOI":"10.1007\/BFb0085166"},{"key":"key-10.3934\/nhm.2024002-35","unstructured":"R. Temam, Infinite-dimensional dynamical systems in mechanics and physics<\/i>, Berlin: Springer Science & Business Media, 2012. https:\/\/doi.org\/10.1007\/978-1-4612-0645-3<\/ext-link>"}],"container-title":["Networks and Heterogeneous Media"],"original-title":[],"link":[{"URL":"http:\/\/www.aimspress.com\/article\/doi\/10.3934\/nhm.2024002?viewType=html","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,1,8]],"date-time":"2024-01-08T05:58:58Z","timestamp":1704693538000},"score":1,"resource":{"primary":{"URL":"http:\/\/www.aimspress.com\/article\/doi\/10.3934\/nhm.2024002"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024]]},"references-count":35,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2024]]}},"URL":"https:\/\/doi.org\/10.3934\/nhm.2024002","relation":{},"ISSN":["1556-1801"],"issn-type":[{"value":"1556-1801","type":"print"}],"subject":[],"published":{"date-parts":[[2024]]}}}