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We are examining a new kind of lattice gas that closely resembles modernlattice Boltzmann methods. This new kind of lattice gas, that we call a MonteCarlo Lattice Gas, has interesting properties that shed light on the origin ofthe multi-relaxation time collision operator and it derives the equilibriumdistribution for entropic lattice Boltzmann. Furthermore these lattice gasmethods have Galilean invariant fluctuations given by a Poisson statistics,giving further insight into the properties that we should expect forfluctuating lattice Boltzmann methods.

Integer Lattice Gas with Monte Carlo collision operator recovers the entropic lattice Boltzmann method with Poisson distributed fluctuations