{"title":"与克莱因-戈登场耦合的谐波晶体的传输方程","authors":"T.V. Dudnikova","doi":"10.1134/S1061920823040076","DOIUrl":null,"url":null,"abstract":"<p> We consider the Hamiltonian system consisting of the Klein–Gordon field coupled to an infinite harmonic crystal. The dynamics of the coupled system is translation-invariant with respect to the space translations in <span>\\(\\mathbb{Z}^d\\)</span>, <span>\\(d\\ge1\\)</span>. We study the Cauchy problem and assume that the initial date is a random function. We introduce the family of initial probability measures <span>\\(\\{\\mu_0^\\varepsilon,\\varepsilon >0\\}\\)</span> slowly varying on the linear scale <span>\\(1/\\varepsilon\\)</span>. For times of order <span>\\(\\varepsilon^{-\\kappa}\\)</span>, <span>\\(0<\\kappa\\le1\\)</span>, we study the distribution of a random solution and prove the convergence of its covariance to a limit as <span>\\(\\varepsilon\\to0\\)</span>. If <span>\\(\\kappa<1\\)</span>, then the limit covariance is time stationary. In the case when <span>\\(\\kappa=1\\)</span>, the covariance changes in time and is governed by a semiclassical transport equation. We give an application to the case of the Gibbs initial measures. </p><p> <b> DOI</b> 10.1134/S1061920823040076 </p>","PeriodicalId":763,"journal":{"name":"Russian Journal of Mathematical Physics","volume":"30 4","pages":"501 - 521"},"PeriodicalIF":1.7000,"publicationDate":"2023-12-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Transport Equation for the Harmonic Crystal Coupled to a Klein–Gordon Field\",\"authors\":\"T.V. Dudnikova\",\"doi\":\"10.1134/S1061920823040076\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p> We consider the Hamiltonian system consisting of the Klein–Gordon field coupled to an infinite harmonic crystal. The dynamics of the coupled system is translation-invariant with respect to the space translations in <span>\\\\(\\\\mathbb{Z}^d\\\\)</span>, <span>\\\\(d\\\\ge1\\\\)</span>. We study the Cauchy problem and assume that the initial date is a random function. We introduce the family of initial probability measures <span>\\\\(\\\\{\\\\mu_0^\\\\varepsilon,\\\\varepsilon >0\\\\}\\\\)</span> slowly varying on the linear scale <span>\\\\(1/\\\\varepsilon\\\\)</span>. For times of order <span>\\\\(\\\\varepsilon^{-\\\\kappa}\\\\)</span>, <span>\\\\(0<\\\\kappa\\\\le1\\\\)</span>, we study the distribution of a random solution and prove the convergence of its covariance to a limit as <span>\\\\(\\\\varepsilon\\\\to0\\\\)</span>. If <span>\\\\(\\\\kappa<1\\\\)</span>, then the limit covariance is time stationary. In the case when <span>\\\\(\\\\kappa=1\\\\)</span>, the covariance changes in time and is governed by a semiclassical transport equation. We give an application to the case of the Gibbs initial measures. </p><p> <b> DOI</b> 10.1134/S1061920823040076 </p>\",\"PeriodicalId\":763,\"journal\":{\"name\":\"Russian Journal of Mathematical Physics\",\"volume\":\"30 4\",\"pages\":\"501 - 521\"},\"PeriodicalIF\":1.7000,\"publicationDate\":\"2023-12-25\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Russian Journal of Mathematical Physics\",\"FirstCategoryId\":\"101\",\"ListUrlMain\":\"https://link.springer.com/article/10.1134/S1061920823040076\",\"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}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Russian Journal of Mathematical Physics","FirstCategoryId":"101","ListUrlMain":"https://link.springer.com/article/10.1134/S1061920823040076","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}
Transport Equation for the Harmonic Crystal Coupled to a Klein–Gordon Field
We consider the Hamiltonian system consisting of the Klein–Gordon field coupled to an infinite harmonic crystal. The dynamics of the coupled system is translation-invariant with respect to the space translations in \(\mathbb{Z}^d\), \(d\ge1\). We study the Cauchy problem and assume that the initial date is a random function. We introduce the family of initial probability measures \(\{\mu_0^\varepsilon,\varepsilon >0\}\) slowly varying on the linear scale \(1/\varepsilon\). For times of order \(\varepsilon^{-\kappa}\), \(0<\kappa\le1\), we study the distribution of a random solution and prove the convergence of its covariance to a limit as \(\varepsilon\to0\). If \(\kappa<1\), then the limit covariance is time stationary. In the case when \(\kappa=1\), the covariance changes in time and is governed by a semiclassical transport equation. We give an application to the case of the Gibbs initial measures.
期刊介绍：
Russian Journal of Mathematical Physics is a peer-reviewed periodical that deals with the full range of topics subsumed by that discipline, which lies at the foundation of much of contemporary science. Thus, in addition to mathematical physics per se, the journal coverage includes, but is not limited to, functional analysis, linear and nonlinear partial differential equations, algebras, quantization, quantum field theory, modern differential and algebraic geometry and topology, representations of Lie groups, calculus of variations, asymptotic methods, random process theory, dynamical systems, and control theory.
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