Paper detail

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

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 $μ_t$ of the solution at time $t\in\R$. The main result is the convergence of $μ_t$ to a Gaussian measure as $t\to\infty$, where $μ_\infty$ is translation-invariant.