On the convergence to a statistical equilibrium for the wave equations coupled to a particle

TL;DR

This paper proves that a coupled particle-field system with random initial data converges to a Gaussian equilibrium over time, with applications to Gibbs measures and analysis of mixing properties.

Contribution

It establishes the convergence of the distribution of solutions to a Gaussian measure for a coupled wave-particle system with random initial conditions, extending understanding of statistical equilibria.

Findings

Distribution  solutions converges to Gaussian as t  .

Limit measures exhibit mixing properties.

Results apply to Gibbs initial measures.

Abstract

We consider a linear Hamiltonian system consisting of a classical particle and a scalar field describing by the wave or Klein-Gordon equations with variable coefficients. The initial data of the system are supposed to be a random function which has some mixing properties. We study the distribution \mu_t of the random solution at time moments t\in\R. The main result is the convergence of \mu_t to a Gaussian probability measure as t\to\infty. The mixing properties of the limit measures are studied. The application to the case of Gibbs initial measures is given.