Fluctuation-Dissipation Equation and Incompressible Navier-Stokes Equations

J. Quastel and H.-T. Yau

We review some recent results concerning the hydrodynamical limits of lattice gases, in particular, lattice gases on the cubic lattice $\ZZ^d,~d=3$, with the incompressible Navier-Stokes equations as the hydrodynamical limit. We shall state precisely the law of large numbers theorem stating that for initial distributions corresponding to arbitrary macroscopic $L^2$ initial data the distributions of the evolving empirical momentum densities are supported entirely on global weak solutions of the incompressible Navier-Stokes equations. Furthermore the rate function for the large deviation will be described. The viscosity will be characterized by the Green-Kubo formula and variational principles. The key analytic inputs are a method solving the fluctuation-dissipation equation of the lattice gases and multiscale estimates based on the logarithmic Sobolev inequality. A precise formulation of the fluctuation-dissipation equation will be sketched. Some discussions of its relation to the Green-Kubo formula and the variational formulas will also be sketched.