Abstract
We consider a Hamiltonian system consisting of the Klein–Gordon field coupled to an infinite harmonic crystal. The dynamics of the coupled system is invariant with respect to the space translations in \(\mathbb{Z}^d\) , \(d\ge1\) . We study the Cauchy problem and assume that the initial date is a random function. We introduce the family of initial probability measures \(\{\mu_0^\varepsilon,\varepsilon >0\}\) depending on a small parameter \(\varepsilon\) and slowly varying on the linear scale \(1/\varepsilon\) . For times of order \(\varepsilon^{-\kappa}\) , \(\kappa>0\) , we study the asymptotics of the distributions of the random solution as \(\varepsilon\to0\) . In particular, we show that, for \(\kappa=1\) and \(\kappa=2\) , the limiting covariance is governed by the hydrodynamic equations of the Euler and Navier–Stokes type, respectively.
DOI 10.1134/S1061920824040034
