Lattice Boltzmann model for particle Brownian motion

A fluctuating lattice Boltzmann model for particle Brownian motion was established by incorporating a stochastic term into the lattice Boltzmann equation, which represents the thermally-induced fluctuations in the stress tensor. The conditions for the stochastic term were derived and the expressions of the stochastic term for the D2Q9 lattice model were also presented. The fluctuating hydrodynamic equations were derived from the lattice Boltzmann equation through Chapman-Enskog expansion. The Brownian motion of a single circular particle was numerically investigated by the newly developed lattice Boltzmann model. Numerical results including particle mean-square velocity, velocity autocorrelation function and angular velocity autocorrelation function were presented. The energy equipartition theorem was reproduced by the results of mean-square velocity, which indicated that the particle was in thermal equilibrium. The results showed that the velocity autocorrelation function and the angular velocity autocorrelation function decayed as a power law of t-1 and t-2 respectively, as theoretically stated. Numerical results showed the accuracy and robustness of the present model, which was proved to be an effective numerical method for the particle Brownian motion.