Nonlinear lattice gas hydrodynamics

Abstract

Starting from a microdynamical description, we derive the equations governing the evolution of the hydrodynamic variables in a lattice gas automaton. The essential features are: (i) the local collision rules satisfy semi-detailed balance; this condition guarantees that a factorized local equilibrium distribution of the Fermi-Dirac form is invariant under the collision step, but not under propagation; (ii) particles entering a collision are uncorrelated (Boltzmann hypothesis); and (iii) the system can be arbitrarily far from global equilibrium; we do not not assume linear response, as usually imposed, to obtain the dissipative contributions. Linearization of the resulting hydrodynamic equations leads to Green-Kubo formulas for the transport coefficients. The main result is the set of fully nonlinear hydrodynamic equations for the automaton in the lattice Boltzmann approximation; these equations have a validity domain extending beyond the region close to equilibrium.