Transport Equation for the Harmonic Crystal Coupled to a Klein–Gordon Field

IF 1.7 3区物理与天体物理 Q2 PHYSICS, MATHEMATICAL Russian Journal of Mathematical Physics Pub Date : 2023-12-25 DOI:10.1134/S1061920823040076

T.V. Dudnikova

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Abstract

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.

DOI 10.1134/S1061920823040076

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与克莱因-戈登场耦合的谐波晶体的传输方程

摘要我们考虑了由克莱因-戈登场与无限谐波晶体耦合组成的哈密顿系统。耦合系统的动力学在 \(\mathbb{Z}^d\), \(d\ge1\) 空间平移方面是平移不变的。我们研究 Cauchy 问题，并假设初始日期是一个随机函数。我们引入在线性尺度（1//varepsilon）上缓慢变化的初始概率度量族（\{\mu_0^\varepsilon,\varepsilon >0/}）。对于阶次为\(\varepsilon^{-\kappa}\), \(0<\kappa\le1\)的时间，我们研究了随机解的分布，并证明了其协方差收敛到\(\varepsilon\to0\)的极限。如果（\kappa<1\），那么极限协方差是时间静止的。在 \(\kappa=1\) 的情况下，协方差随时间变化，并受半经典输运方程支配。我们给出了吉布斯初始量的应用。 doi 10.1134/s1061920823040076

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来源期刊

Russian Journal of Mathematical Physics 物理-物理：数学物理

CiteScore

3.10

自引率

14.30%

发文量

审稿时长

>12 weeks

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.