Fluctuating Lattice Boltzmann for Diffusive Systems

Lattice Boltzmann methods continue to increase in popularity due to their simplicity and computational efficiency. Since lattice Boltzmann techniques were derived as ensemble averages of lattice gas automata, fluctuations are not present. Fluctuations play a large role in the behavior of multiphase systems, especially near the critical point. The question of how to re-introduce fluctuations into lattice Boltzmann methods has received much attention. We present a lattice Boltzmann approach to fluctuations in an ideal diffusive system. We then examine the fluctuation behavior of non-ideal van der Waals systems. We recover the equations of motion in the hydrodynamic limit for both cases. In certain situations, the method requires correction terms to better match the desired physical system. In these cases, we introduce fourth-order corrections to the non-ideal van der Waals system to recover proper phase behavior. We conclude with an outlook for extending the ideal and non-ideal systems to full hydrodynamical systems.