Integer lattice gas with Monte Carlo collision operator recovers the entropic lattice Boltzmann method with MRT collision operator and exact fluctuations

We present a new kind of lattice gas that closely resembles modern lattice Boltzmann methods. This new kind of lattice gas, that we call a Monte Carlo Lattice Gas, has interesting properties that shed light on the origin of the multi-relaxation time (MRT) collision operator and it derives the equilibrium distribution for entropic lattice Boltzmann. Furthermore these lattice gas methods have Galilean invariant fluctuations given by a Poisson statistics, giving further insight into the properties that we should expect for fluctuating lattice Boltzmann methods. We report on our progress in making our collision operator more computationally efficient by factorizing the collisions into one dimensional components. We also examine the physical justification for over-relaxation through a scaling analysis of our lattice gas.