Deriving hydrodynamic equations for lattice systems

摘要

We study the dynamics of lattice systems in ? ~d, d ≥ 1. We assume that the initial data are random functions. We introduce the family of initial measures {μ 0 ~(e{open}), e{open} > 0}. The measures μ 0 ~(e{open}) are assumed to be locally homogeneous or "slowly changing" under spatial shifts of the order o(e{open} ~(-1)) and inhomogeneous under shifts of the order e{open} ~(-1). Moreover, correlations of the measures μ 0 ~(e{open}) decrease uniformly in e{open} at large distances. For all τ ∈ ? 0, r ∈ ? ~d, and κ > 0, we consider distributions of a random solution at the instants t = τ/e{open} κ at points close to [r/e{open}] ∈ ? ~d. Our main goal is to study the asymptotic behavior of these distributions as e{open} → 0 and to derive the limit hydrodynamic equations of the Euler and Navier-Stokes type.