Caricature of Hydrodynamics for the Harmonic Crystal Coupled to a Klein–Gordon Field

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