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Navier-Stokes equations for stochastic particle systems on the lattice. (English) Zbl 0868.60079

Summary: We introduce a class of stochastic models of particles on the cubic lattice \(\mathbb{Z}^d\) with velocities and study the hydrodynamical limit on the diffusive space-time scale. Assuming special initial conditions corresponding to the incompressible regime, we prove that in dimension \(d\geq 3\) there is a law of large numbers for the empirical density and the rescaled empirical velocity field. Moreover, the limit fields satisfy the corresponding incompressible Navier-Stokes equations, with viscosity matrices characterized by a variational formula, formally equivalent to the Green-Kubo formula.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
35Q30 Navier-Stokes equations
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
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