# Lattice gases, large deviations and the incompressible Navier-Stokes equations

J. Quastel and H.-T. Yau

We study the incompressible limit for a class of stochastic particle systems on the cubic lattice \$\ZZ^d,~d=3\$. For initial distributions corresponding to arbitrary macroscopic \$L^2\$ initial data the distributions of the evolving empirical momentum densities are shown to have a weak limit supported entirely on global weak solutions of the incompressible Navier-Stokes equations. Furthermore explicit exponential rates for the convergence (large deviations) are obtained. The probability to violate the divergence free condition decays at rate at least \$\exp\{-\e^{-d+1}\}\$ while the probability to violate the momentum conservation equation decays at rate \$\exp\{-\e^{-d+2}\}\$ with an explicit rate function given by an \$H_{-1}\$ norm.