Caricature of Hydrodynamics for Lattice Dynamics

The lattice dynamics in $\mathbb{Z}^d$, $d\ge1$, is considered. The initial data are supposed to be random function. We introduce the family of initial measures $\{\mu_0^\epsilon,\epsilon>0\}$ depending on a small scaling parameter $\epsilon$. We assume that the measures $\mu_0^\epsilon$ are locally homogeneous for space translations of order much less than $\epsilon^{-1}$ and nonhomogeneous for translations of order $\epsilon^{-1}$. Moreover, the covariance of $\mu_0^\epsilon$ decreases with distance uniformly in $\epsilon$. 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 $t=\tau/\epsilon^\kappa$ and at lattice points close to $[r/\epsilon]\in\mathbb{Z}^d$. The main goil is to study the asymptotics of these distributions as $\epsilon\to0$ 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 $\mathbb{Z}^d_+$.