Caricature of Hydrodynamics for Lattice Dynamics

Abstract

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 $\{μ_0^ε,ε>0\}$ depending on a small scaling parameter $ε$. We assume that the measures $μ_0^ε$ 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 $μ_0^ε$ decreases with distance uniformly in $ε$. Given $τ\in\mathbb{R}\setminus 0$, $r\in\mathbb{R}^d$, and $κ>0$, we consider the distributions of random solution in the time moments $t=τ/ε^κ$ and at lattice points close to $[r/ε]\in\mathbb{Z}^d$. The main goil is to study the asymptotics of these distributions as $ε\to0$ 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 $\mathbb{Z}^d_+$.