Microscopic theory of Brownian motion revisited: The Rayleigh model

We investigate three force autocorrelation functions < F(0) center dot F(t)>, < F+(0)> center dot F+(t)>, and < F-0(0) center dot F-0(t)> and the friction coefficient gamma for the Rayleigh model (a massive particle in an ideal gas) by analytic methods and molecular-dynamics (MD) simulations. Here, F and F+ are the total force and the Mori fluctuating force, respectively, whereas F0 is the force on the Brownian particle under the frozen dynamics, where the Brownian particle is held fixed and the solvent particles move under the external potential due to the presence of the Brownian particle. By using ensemble averaging and the ray representation approach, we obtain two expressions for < F-0(0) center dot F-0(t)> in terms of the one-particle trajectory and corresponding expressions for gamma by the time integration of these expressions. Performing MD simulations of the near-Brownian-limit (NBL) regime, we investigate the convergence of < F(0) center dot F(t)> and < F+(0) center dot F+(t)> and compare them with < F-0(0) center dot F-0(t)>. We show that for a purely repulsive potential between the Brownian particle and a solvent particle, both expressions for < F-0(0) center dot F-0(t)> produce < F+(0) center dot F+(t)> in the NBL regime. On the other hand, for a potential containing an attractive component, the ray representation expression produces only the contribution of the nontrapped solvent particles. However, we show that the net contribution of the trapped particles to gamma disappears, and hence we confirm that both the ensemble-averaged expression and the ray representation expression for gamma are valid even if the potential contains an attractive component. We also obtain a closed-form expression of. for the square-well potential. Finally, we discuss theoretical and practical aspects for the evaluation of < F-0(0) center dot F-0(t)> and gamma. DOI: 10.1103/PhysRevE. 87.032129