Caricature of Hydrodynamics for Lattice Dynamics

We consider the lattice dynamics in the half-space, with zero boundary condition. The initial data are supposed to be random function. We introduce the family of initial measures {mu(epsilon)(0),epsilon > 0} depending on a small scaling parameter epsilon. We assume that the measures mu(epsilon)(0) 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(epsilon)(0) decreases with distance uniformly in E. Given tau is an element of R \ 0, r is an element of 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] is an element of Z(+)(d). The main goal is to study the asymptotic behavior of these distributions as epsilon -> 0 and to derive the limit hydrodynamic equations of the Euler or Navier-Stokes type.