{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T11:28:12Z","timestamp":1787225292058,"version":"build-2736575974"},"reference-count":32,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"5","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Appl. Math."],"published-print":{"date-parts":[[2023,10,31]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>We investigate the unbiased model for money exchanges with collective debt limit: agents give at random time a dollar to one another as long as they have at least one dollar or they can borrow a dollar from a central bank if the bank is not empty. Surprisingly, this dynamic eventually leads to an asymmetric Laplace distribution of wealth (conjectured in [N. Xi, N. Ding, and Y. Wang, Phys. A, 357 (2005), pp. 543\u2013555]\u00a0 and shown formally in a recent work [N. Lanchier and S. Reed, J. Stat. Phys., 176 (2019), pp. 1115\u20131137]). In this manuscript, we carry out a formal mean-field limit as the number of agents goes to infinity where we uncover a two-phase ODE dynamic. Convergence towards the unique equilibrium (two-sided geometric) distribution in the large time limit is also shown and the role played by the bank and debt (in terms of Gini index or wealth inequality) will be explored numerically as well.<\/jats:p>","DOI":"10.1137\/22m1518621","type":"journal-article","created":{"date-parts":[[2023,9,20]],"date-time":"2023-09-20T04:19:40Z","timestamp":1695183580000},"page":"1872-1891","source":"Crossref","is-referenced-by-count":13,"title":["Uncovering a Two-Phase Dynamics from a Dollar Exchange Model with Bank and Debt"],"prefix":"10.1137","volume":"83","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1367-8692","authenticated-orcid":true,"given":"Fei","family":"Cao","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, University of Massachusetts Amherst, MA 01003 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9135-5530","authenticated-orcid":true,"given":"S\u00e9bastien","family":"Motsch","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistical Sciences, Arizona State University, Tempe, AZ 85287 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2023,9,20]]},"reference":[{"key":"ref1","doi-asserted-by":"publisher","DOI":"10.1016\/j.crma.2019.09.007"},{"key":"ref2","unstructured":"F. 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Motsch  , Derivation of Wealth Distributions from Biased Exchange of Money, preprint, https:\/\/arxiv.org\/abs\/2105.07341, 2021."},{"key":"ref8","doi-asserted-by":"publisher","DOI":"10.1142\/S0218202513500115"},{"key":"ref9","doi-asserted-by":"publisher","DOI":"10.1007\/s100510070173"},{"key":"ref10","doi-asserted-by":"publisher","DOI":"10.1016\/j.physa.2003.11.014"},{"key":"ref11","doi-asserted-by":"publisher","DOI":"10.1007\/s10955-008-9667-z"},{"key":"ref12","unstructured":"R. Cortez  , Particle System Approach to Wealth Redistribution, preprint, https:\/\/arxiv.org\/abs\/1809.05372, 2018."},{"key":"ref13","unstructured":"R. Cortez  and \nF. Cao  , Uniform Propagation of Chaos for a Dollar Exchange Econophysics Model, preprint, https:\/\/arxiv.org\/abs\/2212.08289, 2022."},{"key":"ref14","doi-asserted-by":"publisher","DOI":"10.1214\/15-AAP1107"},{"key":"ref15","doi-asserted-by":"publisher","DOI":"10.1007\/s100510070114"},{"key":"ref16","doi-asserted-by":"publisher","DOI":"10.1016\/j.spa.2022.03.015"},{"key":"ref17","doi-asserted-by":"publisher","DOI":"10.1142\/S0218202520400023"},{"key":"ref18","doi-asserted-by":"publisher","DOI":"10.1214\/08-AAP549"},{"key":"ref19","doi-asserted-by":"publisher","DOI":"10.1140\/epjb\/e2014-50270-6"},{"key":"ref20","doi-asserted-by":"publisher","DOI":"10.1016\/j.physa.2018.10.019"},{"key":"ref21","doi-asserted-by":"publisher","DOI":"10.1007\/s10955-017-1744-8"},{"key":"ref22","doi-asserted-by":"publisher","DOI":"10.1007\/s10955-018-2024-y"},{"key":"ref23","doi-asserted-by":"publisher","DOI":"10.1007\/s10955-019-02334-z"},{"key":"ref24","doi-asserted-by":"publisher","DOI":"10.3934\/krm.2008.1.249"},{"key":"ref25","doi-asserted-by":"publisher","DOI":"10.1007\/s10955-007-9462-2"},{"key":"ref26","doi-asserted-by":"publisher","DOI":"10.1214\/19-AOP1336"},{"key":"ref27","doi-asserted-by":"publisher","DOI":"10.1007\/s00285-017-1143-4"},{"key":"ref28","volume-title":"Interacting Multiagent Systems: Kinetic Equations and Monte Carlo Methods","author":"Pareschi L.","year":"2013"},{"key":"ref29","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-4296-1"},{"key":"ref30","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0085169"},{"key":"ref31","unstructured":"M. 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A Fokker\u2013Planck Description, preprint, https:\/\/arxiv.org\/abs\/1709.09858, 2017."},{"key":"ref32","doi-asserted-by":"publisher","DOI":"10.1016\/j.physa.2005.04.014"}],"container-title":["SIAM Journal on Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/22M1518621","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T11:03:16Z","timestamp":1787223796000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/22M1518621"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,9,20]]},"references-count":32,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2023,10,31]]}},"alternative-id":["10.1137\/22M1518621"],"URL":"https:\/\/doi.org\/10.1137\/22m1518621","relation":{},"ISSN":["0036-1399","1095-712X"],"issn-type":[{"value":"0036-1399","type":"print"},{"value":"1095-712X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,9,20]]}}}