Deriving hydrodynamic equations for lattice systems

We study the dynamics of lattice systems in $\mathbb Z^d$, $d\ge1$. We assume that the initial data are random functions. We introduce the system of initial measures $\{\mu_0^{\varepsilon},\;\varepsilon>0\}$. The measures $\mu_0^{\varepsilon}$ are assumed to be locally homogeneous or “slowly changing” under spatial shifts of the order $o(\varepsilon^{-1})$ and inhomogeneous under shifts of the order $\varepsilon^{-1}$. Moreover, correlations of the measures $\mu_0^{\varepsilon}$ decrease uniformly in $\varepsilon$ at large distances. For all $\tau\in\mathbb R\setminus0$, $r\in\mathbb R^d$, and $\kappa>0$, we consider distributions of a random solution at the instants $t=\tau/\varepsilon^{\kappa}$ at points close to $[r/\varepsilon]\in\mathbb Z^d$. Our main goal is to study the asymptotic behavior of these distributions as $\varepsilon\to0$ and to derive the limit hydrodynamic equations of the Euler and Navier–Stokes type.