Convergence to equilibrium distribution. The Klein-Gordon equation coupled to a particle

TL;DR

This paper proves that the distribution of a coupled Klein-Gordon field and particle system converges to a translation-invariant Gaussian measure over time, under specific initial randomness and mixing conditions.

Contribution

It establishes the convergence of the system's distribution to a Gaussian measure, extending understanding of long-term behavior in coupled field-particle Hamiltonian systems.

Findings

Distribution $bla_t$ converges to a Gaussian measure as $t o .

The limiting measure is translation-invariant.

Initial conditions include mixing and translation-invariance properties.

Abstract

We consider the Hamiltonian system consisting of a Klein-Gordon vector field and a particle in $\R^3$. The initial date of the system is a random function with a finite mean density of energy which also satisfies a Rosenblatt- or Ibragimov-type mixing condition. Moreover, initial correlation functions are translation-invariant. We study the distribution $\mu_t$ of the solution at time $t\in\R$. The main result is the convergence of $\mu_t$ to a Gaussian measure as $t\to\infty$, where $\mu_\infty$ is translation-invariant.