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

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