Microscopic theory of Brownian motion revisited: The Rayleigh model
We investigate three force autocorrelation functions 〈F(0)·F(t)〉, 〈F+(0)·F+(t)〉, and 〈F0(0)·F0(t)〉 and the friction coefficient γ 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 〈F0(0)·F0(t)〉 in terms of the one-particle trajectory and corresponding expressions for γ by the time integration of these expressions. Performing MD simulations of the near-Brownian-limit (NBL) regime, we investigate the convergence of 〈F(0)·F(t)〉 and 〈F+(0)·F+(t)〉 and compare them with 〈F0(0)·F0(t)〉. We show that for a purely repulsive potential between the Brownian particle and a solvent particle, both expressions for 〈F0(0)·F0(t)〉 produce 〈F+(0)·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 γ disappears, and hence we confirm that both the ensemble-averaged expression and the ray representation expression for γ are valid even if the potential contains an attractive component. We also obtain a closed-form expression of γ for the square-well potential. Finally, we discuss theoretical and practical aspects for the evaluation of 〈F0(0)·F0(t)〉 and γ.