{"title":"Condensation of the Invariant Measures of the Supercritical Zero Range Processes","authors":"Tiecheng Xu","doi":"10.1007/s10955-024-03287-8","DOIUrl":null,"url":null,"abstract":"<p>For <span>\\(\\alpha \\ge 1\\)</span>, let <span>\\(g:{\\mathbb {N}}\\rightarrow {\\mathbb {R}}_+\\)</span> be given by <span>\\(g(0)=0\\)</span>, <span>\\(g(1)=1\\)</span>, <span>\\(g(k)=(k/k-1)^\\alpha \\)</span>, <span>\\(k\\ge 2\\)</span>. Consider the homogeneous zero range process on a discrete set in which a particle jumps from a site <i>x</i>, occupied by <i>k</i> particles, to site <i>y</i> with rate <span>\\(g(k)p(y-x)\\)</span> for some fixed probability <span>\\(p:{\\mathbb {Z}}\\rightarrow [0,1]\\)</span>. Armendáriz and Loulakis (Probab Theory Relat Fields 145:175–188, 2009, https://doi.org/10.1007/s00440-008-0165-7) proved a strong form of the equivalence of ensembles for the invariant measure of the supercritical zero range process with <span>\\(\\alpha >2\\)</span>. We generalize their result to all <span>\\(\\alpha \\ge 1\\)</span>.</p>","PeriodicalId":667,"journal":{"name":"Journal of Statistical Physics","volume":null,"pages":null},"PeriodicalIF":1.3000,"publicationDate":"2024-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Statistical Physics","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.1007/s10955-024-03287-8","RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
引用次数: 0
Abstract
For \(\alpha \ge 1\), let \(g:{\mathbb {N}}\rightarrow {\mathbb {R}}_+\) be given by \(g(0)=0\), \(g(1)=1\), \(g(k)=(k/k-1)^\alpha \), \(k\ge 2\). Consider the homogeneous zero range process on a discrete set in which a particle jumps from a site x, occupied by k particles, to site y with rate \(g(k)p(y-x)\) for some fixed probability \(p:{\mathbb {Z}}\rightarrow [0,1]\). Armendáriz and Loulakis (Probab Theory Relat Fields 145:175–188, 2009, https://doi.org/10.1007/s00440-008-0165-7) proved a strong form of the equivalence of ensembles for the invariant measure of the supercritical zero range process with \(\alpha >2\). We generalize their result to all \(\alpha \ge 1\).
For \(α ge 1\), let \(g:{\mathbb {N}}\rightarrow {\mathbb {R}}_+\) be given by \(g(0)=0\), \(g(1)=1\), \(g(k)=(k/k-1)^\α \), \(k\ge 2\).考虑离散集合上的同质零范围过程,在这个过程中,一个粒子以某种固定概率(p:{/mathbb {Z}}\rightarrow [0,1])从一个被k个粒子占据的位置x跳到位置y,速率为(g(k)p(y-x))。Armendáriz和Loulakis(Probab Theory Relat Fields 145:175-188,2009,https://doi.org/10.1007/s00440-008-0165-7)为具有\(\alpha >2\)的超临界零范围过程的不变度量证明了集合等价的强形式。我们将他们的结果推广到所有的(\alpha \ge 1\ )。
期刊介绍:
The Journal of Statistical Physics publishes original and invited review papers in all areas of statistical physics as well as in related fields concerned with collective phenomena in physical systems.