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.

中文翻译：

---

与克莱因-戈登场耦合的调和晶格的大次数统计解的稳定性

摘要

我们考虑由克莱因-戈登场和无限调和晶体组成的哈密顿系统的柯西问题。耦合系统的动力学相对于\(\mathbb{R}^d\)的离散子群\(\mathbb{Z}^d\)是平移不变的。假设初始日期是一个随机函数，当\ (\pm x_1>a\)与一些\( a>0\)。我们研究了时间\(t\in\mathbb{R}\)时解的分布\(\mu_t \) 并证明了\(\mu_t\)与高斯测度\(\mu_\infty\ )的弱收敛性)为\(t\to\infty\)。此外，我们证明了相关函数收敛到极限，并推导了极限测度的协方差的显式公式\(\mu_\infty\)。我们对吉布斯测度进行了应用。