A Fluctuating Lattice Boltzmann Model with Independent Directional Stochastic Fluctuations for Brownian Motion

Accurate simulation of Brownian motion is essential for understanding particle dynamics at micro- and nanoscales. The fluctuating lattice Boltzmann method provides a mesoscopic framework for modeling thermal fluctuations, but conventional formulations impose identical fluctuations on opposite lattice directions, limiting independent stochastic variation. Here, we relax this symmetry by allowing independent fluctuations in each direction. Formulations are derived for the D3Q19 and D3Q27 models and applied to spherical Brownian particles, with comparisons against the conventional D3Q15 model. The simulated translational and rotational mean-square velocities are consistent with the energy equipartition theorem, confirming thermal equilibrium. The resulting diffusion coefficients demonstrate that the D3Q19 and D3Q27 models reduce the relative error by approximately 3% compared with the D3Q15 model when reproducing the Stokes-Einstein prediction. To quantify inherent errors, a correction factor α is introduced to rescale the fluctuation intensity so that the Stokes-Einstein diffusion coefficient is recovered within statistical uncertainty. The smaller α values required for the D3Q19 and D3Q27 models indicate more physically faithful stochastic behavior. The proposed methods therefore provide a more flexible and accurate representation of thermal fluctuations for Brownian motion and related microscale simulations.