{"title":"耦合于克莱因-戈登场的谐波晶体流体力学漫画","authors":"T.V. Dudnikova","doi":"10.1134/S1061920824040034","DOIUrl":null,"url":null,"abstract":"<p> We consider a Hamiltonian system consisting of the Klein–Gordon field coupled to an infinite harmonic crystal. The dynamics of the coupled system is 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> depending on a small parameter <span>\\(\\varepsilon\\)</span> and slowly varying on the linear scale <span>\\(1/\\varepsilon\\)</span>. For times of order <span>\\(\\varepsilon^{-\\kappa}\\)</span>, <span>\\(\\kappa>0\\)</span>, we study the asymptotics of the distributions of the random solution as <span>\\(\\varepsilon\\to0\\)</span>. In particular, we show that, for <span>\\(\\kappa=1\\)</span> and <span>\\(\\kappa=2\\)</span>, the limiting covariance is governed by the hydrodynamic equations of the Euler and Navier–Stokes type, respectively. </p><p> <b> DOI</b> 10.1134/S1061920824040034 </p>","PeriodicalId":763,"journal":{"name":"Russian Journal of Mathematical Physics","volume":"31 4","pages":"606 - 621"},"PeriodicalIF":1.7000,"publicationDate":"2025-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Caricature of Hydrodynamics for the Harmonic Crystal Coupled to a Klein–Gordon Field\",\"authors\":\"T.V. Dudnikova\",\"doi\":\"10.1134/S1061920824040034\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p> We consider a Hamiltonian system consisting of the Klein–Gordon field coupled to an infinite harmonic crystal. The dynamics of the coupled system is 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> depending on a small parameter <span>\\\\(\\\\varepsilon\\\\)</span> and slowly varying on the linear scale <span>\\\\(1/\\\\varepsilon\\\\)</span>. For times of order <span>\\\\(\\\\varepsilon^{-\\\\kappa}\\\\)</span>, <span>\\\\(\\\\kappa>0\\\\)</span>, we study the asymptotics of the distributions of the random solution as <span>\\\\(\\\\varepsilon\\\\to0\\\\)</span>. In particular, we show that, for <span>\\\\(\\\\kappa=1\\\\)</span> and <span>\\\\(\\\\kappa=2\\\\)</span>, the limiting covariance is governed by the hydrodynamic equations of the Euler and Navier–Stokes type, respectively. </p><p> <b> DOI</b> 10.1134/S1061920824040034 </p>\",\"PeriodicalId\":763,\"journal\":{\"name\":\"Russian Journal of Mathematical Physics\",\"volume\":\"31 4\",\"pages\":\"606 - 621\"},\"PeriodicalIF\":1.7000,\"publicationDate\":\"2025-02-09\",\"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/S1061920824040034\",\"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/S1061920824040034","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}
Caricature of Hydrodynamics for the Harmonic Crystal Coupled to a Klein–Gordon Field
We consider a Hamiltonian system consisting of the Klein–Gordon field coupled to an infinite harmonic crystal. The dynamics of the coupled system is 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 \(\{\mu_0^\varepsilon,\varepsilon >0\}\) depending on a small parameter \(\varepsilon\) and slowly varying on the linear scale \(1/\varepsilon\). For times of order \(\varepsilon^{-\kappa}\), \(\kappa>0\), we study the asymptotics of the distributions of the random solution as \(\varepsilon\to0\). In particular, we show that, for \(\kappa=1\) and \(\kappa=2\), the limiting covariance is governed by the hydrodynamic equations of the Euler and Navier–Stokes type, respectively.
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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