Caricature of Hydrodynamics for Lattice Dynamics

The lattice dynamics in ${\mathbb{Z}}^d$, $d\ge 1$, is considered. The initial data are supposed to be random function. We introduce the family of initial measures $\{{\mu }_0^{ϵ},ϵ>0\}$ depending on a small scaling parameter $ϵ$. We assume that the measures ${\mu }_0^{ϵ}$ are locally homogeneous for space translations of order much less than ${ϵ}^{-1}$ and nonhomogeneous for translations of order ${ϵ}^{-1}$. Moreover, the covariance of ${\mu }_0^{ϵ}$ decreases with distance uniformly in $ϵ$. Given $\tau \in \mathbb{R}\setminus 0$, $r\in {\mathbb{R}}^d$, and $\kappa >0$, we consider the distributions of random solution in the time moments $t=\tau \mathrm{/}{ϵ}^{\kappa }$ and at lattice points close to $[r\mathrm{/}ϵ]\in {\mathbb{Z}}^d$. The main goil is to study the asymptotics of these distributions as $ϵ\to 0$ and derive the limit hydrodynamic equations of the Euler or Navier-Stokes type. The similar results are obtained for lattice dynamics in the half-space ${\mathbb{Z}}_{+}^d$.