Stabilization of the statistical solutions for large times for a harmonic lattice coupled to a Klein–Gordon field

IF 1 4区物理与天体物理 Q3 PHYSICS, MATHEMATICAL Theoretical and Mathematical Physics Pub Date : 2024-02-27 DOI:10.1134/s0040577924020053

T. V. Dudnikova

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Abstract

We consider the Cauchy problem for the Hamiltonian system consisting of the Klein–Gordon field and an infinite harmonic crystal. The dynamics of the coupled system is translation-invariant with respect to the discrete subgroup \(\mathbb{Z}^d\) of \(\mathbb{R}^d\). The initial date is assumed to be a random function that is close to two spatially homogeneous (with respect to the subgroup \(\mathbb{Z}^d\)) processes when \(\pm x_1>a\) with some \(a>0\). We study the distribution \(\mu_t\) of the solution at time \(t\in\mathbb{R}\) and prove the weak convergence of \(\mu_t\) to a Gaussian measure \(\mu_\infty\) as \(t\to\infty\). Moreover, we prove the convergence of the correlation functions to a limit and derive the explicit formulas for the covariance of the limit measure \(\mu_\infty\). We give an application to Gibbs measures.

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与克莱因-戈登场耦合的谐波晶格的大时间统计解的稳定性

摘要我们考虑了由克莱因-戈登场和无限谐波晶体组成的哈密顿系统的考奇问题。耦合系统的动力学关于离散子群 \(\mathbb{Z}^d\) 是平移不变的。假定初始日期是一个随机函数，当 \(\pm x_1>a\) 与某个 \(a>0\) 时，它接近于两个空间同构（关于子群）过程。我们研究了在时间\(t\in\mathbb{R}\)时解的分布\(\mu_t\)，并证明了\(\mu_t\)在\(t\to\infty\)时向高斯量\(\mu_\infty\)的弱收敛性。此外，我们证明了相关函数向极限的收敛性，并推导出极限度量 \(\mu_\infty\) 的协方差的明确公式。我们给出了吉布斯度量的应用。

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来源期刊

Theoretical and Mathematical Physics 物理-物理：数学物理

CiteScore

1.60

自引率

20.00%

发文量

103

审稿时长

4-8 weeks

期刊介绍： Theoretical and Mathematical Physics covers quantum field theory and theory of elementary particles, fundamental problems of nuclear physics, many-body problems and statistical physics, nonrelativistic quantum mechanics, and basic problems of gravitation theory. Articles report on current developments in theoretical physics as well as related mathematical problems. Theoretical and Mathematical Physics is published in collaboration with the Steklov Mathematical Institute of the Russian Academy of Sciences.