Velocity correlations of a thermally fluctuating Brownian particle: A novel model of the hydrodynamic coupling

A new derivation of the velocity correlations of a thermally fluctuating Brownian particle is shown by direct calculation from a stochastic hydrodynamic model in which the fluid-particle coupling is treated in a simple manner. The model significantly simplifies the calculation of statistics of a particle and has the virtue of being readily amenable to numerical simulation. To show that the model correctly captures physical features of a Brownian particle the diffusion coefficient in three dimensions is computed and shown to have the correct scaling in the physical parameters. The velocity correlation function for both short and long times scales is then discussed. It is found for short times that the velocity correlation of a particle satisfies an equipartition principle. For long times the autocorrelation function is shown to have non-exponential decay of algebraic order τ−3/2τ−3/2 capturing well-known hydrodynamic effects [B.J. Alder, T.E. Wainwright, Phys. Rev. A 1 (1) (1970) 18–21]. The results are then compared with numerical simulations of the hydrodynamic model using the computational method proposed in [P.J. Atzberger, P.R. Kramer, C.S. Peskin, A stochastic immersed boundary method for microscopic biological fluid dynamics, (2005), submitted for publication].