{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:30:05Z","timestamp":1760236205883,"version":"build-2065373602"},"reference-count":24,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2021,10,26]],"date-time":"2021-10-26T00:00:00Z","timestamp":1635206400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100000038","name":"Natural Sciences and Engineering Research Council","doi-asserted-by":"publisher","award":["Discovery Grants"],"award-info":[{"award-number":["Discovery Grants"]}],"id":[{"id":"10.13039\/501100000038","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>A family of heterogeneous mean-field systems with jumps is analyzed. These systems are constructed as a Gibbs measure on block graphs. When the total number of particles goes to infinity, the law of large numbers is shown to hold in a multi-class context, resulting in the weak convergence of the empirical vector towards the solution of a McKean\u2013Vlasov system of equations. We then investigate the local stability of the limiting McKean\u2013Vlasov system through the construction of a local Lyapunov function. We first compute the limit of adequately scaled relative entropy functions associated with the explicit stationary distribution of the N-particles system. Using a Laplace principle for empirical vectors, we show that the limit takes an explicit form. Then we demonstrate that this limit satisfies a descent property, which, combined with some mild assumptions shows that it is indeed a local Lyapunov function.<\/jats:p>","DOI":"10.3390\/e23111407","type":"journal-article","created":{"date-parts":[[2021,10,26]],"date-time":"2021-10-26T23:49:52Z","timestamp":1635292192000},"page":"1407","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Local Stability of McKean\u2013Vlasov Equations Arising from Heterogeneous Gibbs Systems Using Limit of Relative Entropies"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7263-6241","authenticated-orcid":false,"given":"Donald A.","family":"Dawson","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Carleton University, 1125 Colonel by Drive, Ottawa, ON K1S 5B6, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3962-2398","authenticated-orcid":false,"given":"Ahmed","family":"Sid-Ali","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Carleton University, 1125 Colonel by Drive, Ottawa, ON K1S 5B6, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7530-8064","authenticated-orcid":false,"given":"Yiqiang Q.","family":"Zhao","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Carleton University, 1125 Colonel by Drive, Ottawa, ON K1S 5B6, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,10,26]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"2199","DOI":"10.1142\/S0217979208039423","article-title":"Phase transitions in social sciences: Two-population mean field theory","volume":"22","author":"Contucci","year":"2008","journal-title":"Internat. 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