{"title":"Trend to equilibrium and Newtonian limit for the relativistic Langevin equation with singular potentials","authors":"Manh Hong Duong, Hung Dang Nguyen","doi":"arxiv-2409.05645","DOIUrl":null,"url":null,"abstract":"We study a system of interacting particles in the presence of the\nrelativistic kinetic energy, external confining potentials, singular repulsive\nforces as well as a random perturbation through an additive white noise. In\ncomparison with the classical Langevin equations that are known to be\nexponentially attractive toward the unique statistically steady states, we find\nthat the relativistic systems satisfy algebraic mixing rates of any order. This\nrelies on the construction of Lyapunov functions adapting to previous\nliterature developed for irregular potentials. We then explore the Newtonian\nlimit as the speed of light tends to infinity and establish the validity of the\napproximation of the solutions by the Langevin equations on any finite time\nwindow.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"264 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Probability","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.05645","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
We study a system of interacting particles in the presence of the
relativistic kinetic energy, external confining potentials, singular repulsive
forces as well as a random perturbation through an additive white noise. In
comparison with the classical Langevin equations that are known to be
exponentially attractive toward the unique statistically steady states, we find
that the relativistic systems satisfy algebraic mixing rates of any order. This
relies on the construction of Lyapunov functions adapting to previous
literature developed for irregular potentials. We then explore the Newtonian
limit as the speed of light tends to infinity and establish the validity of the
approximation of the solutions by the Langevin equations on any finite time
window.