{"title":"Prigogine–Resibois master equation with power-law kernel: quantum dynamics with memory","authors":"Vasily E. Tarasov","doi":"10.1140/epjp/s13360-025-06106-5","DOIUrl":null,"url":null,"abstract":"<div><p>The Prigogine–Resibois master equations is valid in the general case, regardless of the initial correlations, and make it possible to describe exactly strongly coupled systems. In this paper, exact solutions of the Prigogine–Resibois master equation with power-law kernel (PLK) are derived. To solve this equation we use methods of fractional calculus (FC) and theory of equations with derivatives with non-integer orders. Using the first fundamental theorem of FC, the Prigogine–Resibois master equation with PLK is represented as a fractional differential equation with the Caputo fractional derivative of arbitrary order greater than one. Then, solution of the Cauchy problem with this fractional differential equation is derived. Some properties of these solutions, which are represented via the two-parameter Mittag–Leffler function, are considered. Asymptotic behaviors of the solutions are described.</p></div>","PeriodicalId":792,"journal":{"name":"The European Physical Journal Plus","volume":"140 4","pages":""},"PeriodicalIF":2.8000,"publicationDate":"2025-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"The European Physical Journal Plus","FirstCategoryId":"4","ListUrlMain":"https://link.springer.com/article/10.1140/epjp/s13360-025-06106-5","RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"PHYSICS, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 0
Abstract
The Prigogine–Resibois master equations is valid in the general case, regardless of the initial correlations, and make it possible to describe exactly strongly coupled systems. In this paper, exact solutions of the Prigogine–Resibois master equation with power-law kernel (PLK) are derived. To solve this equation we use methods of fractional calculus (FC) and theory of equations with derivatives with non-integer orders. Using the first fundamental theorem of FC, the Prigogine–Resibois master equation with PLK is represented as a fractional differential equation with the Caputo fractional derivative of arbitrary order greater than one. Then, solution of the Cauchy problem with this fractional differential equation is derived. Some properties of these solutions, which are represented via the two-parameter Mittag–Leffler function, are considered. Asymptotic behaviors of the solutions are described.
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