{"title":"Ergodic and Non-Ergodic Phenomena in One-Dimensional Random Processes: Exploring Unconventional State Transitions","authors":"A. D. Ramos, C. S. Sousa, L. P. Cavalcanti","doi":"10.1007/s00023-025-01554-2","DOIUrl":null,"url":null,"abstract":"<div><p>Traditionally, the evolution of interacting particle systems has been based on an assumption that only the individual components undergo state changes. However, this rigid assumption is not the only possibility. This research explored a class of one-dimensional random processes that evolved in discrete time. During each time step, components in the state zero exhibited the following transitions. First, they could change to one with a probability <span>\\(\\beta _0\\)</span>. Second, they could be replaced by a sequence of <i>k</i> consecutive zeros with a probability <span>\\(\\beta _k\\)</span> (where <span>\\(k=1,\\ldots ,n\\)</span>). Moreover, these transitions occurred independent of the events occurring elsewhere in the involved system. Notably, this study revealed an unexpected phenomenon—the occurrence of a first-order phase transition between ergodic and non-ergodic behaviors within this system. Furthermore, in the non-ergodic regime, the existence of an invariant measure distinct from the trivial one was demonstrated.</p></div>","PeriodicalId":463,"journal":{"name":"Annales Henri Poincaré","volume":"26 8","pages":"3055 - 3073"},"PeriodicalIF":1.3000,"publicationDate":"2025-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annales Henri Poincaré","FirstCategoryId":"4","ListUrlMain":"https://link.springer.com/article/10.1007/s00023-025-01554-2","RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
引用次数: 0
Abstract
Traditionally, the evolution of interacting particle systems has been based on an assumption that only the individual components undergo state changes. However, this rigid assumption is not the only possibility. This research explored a class of one-dimensional random processes that evolved in discrete time. During each time step, components in the state zero exhibited the following transitions. First, they could change to one with a probability \(\beta _0\). Second, they could be replaced by a sequence of k consecutive zeros with a probability \(\beta _k\) (where \(k=1,\ldots ,n\)). Moreover, these transitions occurred independent of the events occurring elsewhere in the involved system. Notably, this study revealed an unexpected phenomenon—the occurrence of a first-order phase transition between ergodic and non-ergodic behaviors within this system. Furthermore, in the non-ergodic regime, the existence of an invariant measure distinct from the trivial one was demonstrated.
期刊介绍:
The two journals Annales de l''Institut Henri Poincaré, physique théorique and Helvetica Physical Acta merged into a single new journal under the name Annales Henri Poincaré - A Journal of Theoretical and Mathematical Physics edited jointly by the Institut Henri Poincaré and by the Swiss Physical Society.
The goal of the journal is to serve the international scientific community in theoretical and mathematical physics by collecting and publishing original research papers meeting the highest professional standards in the field. The emphasis will be on analytical theoretical and mathematical physics in a broad sense.