Microscopic theory of Brownian motion: Effects of memory and confinement
Changho Kim and George Em Karniadakis
Brownian motion is a phenomenon where the motion of a tracer particle is significantly affected by fluctuations from the surrounding fluid. The motion is effectively described by mesoscopic methods such as the Langevin/Brownian dynamics and dissipative particle dynamics. In principle, parameters needed in these methods, such as friction coefficient, can be calculated through the Green- Kubo formulas by molecular dynamics (MD) simulations of the corresponding microscopic system. However, there are some subtle issues when one attempts to estimate them from MD simulations. One is related with the plateau problem and the finite-size effects of the MD system, which cause systematic errors, and the other is statistical errors present in the time correlation functions estimated from MD simulations. In addition, the time scale separation assumption and the Markovian assumption, on which these methods are based, are not exactly satisfied unless the particle has infinite mass. In other words, the time scale of microscopic memory renders the dynamics of the Brownian particle non- Markovian.
To address these issues, we consider Brownian motion in the near-Brownian- limit. We adopt the generalized Langevin equation approach and investigate the memory function of the Brownian particle. As the mass ¡em¿M¡/em¿ of the Brownian particle increase, the memory function converges to the limit mem- ory function, the time integral of which gives the friction coefficient. The limit memory function can be obtained from the following two types of infinite-mass dynamics. From the frozen dynamics, where the Brownian particle is held fixed, the plateau problem is investigated, whereas the constant-velocity dynamics, where the Brownian particle moves at an infinitesimal velocity, the macroscopic definition of the friction coefficient is investigated. For finite but sufficiently large M, we analyze asymptotic behaviors of the memory function and obtain the asymptotic expansions of the velocity/force autocorrelation functions with respect to M. Using these results, we analyze numerical methods which evalu- ate the friction coefficient from the long-time behaviors of the time correlation functions.
In order to demonstrate our results, we apply them to the Rayleigh gas model, where a Brownian particle is suspended in an ideal gas. A systematic MD simulation study is performed with well-controlled statistical errors. MD simulation results are compared with analytic expressions for the limit memory function, which we obtain from the two types of the infinite-mass dynamics. All theoretical predictions are confirmed and the finite-volume and finite-mass effects are separately investigated.
In addition, we demonstrate that our generalized Langevin equation ap- proach can be applied to confined Brownian motion. A Rayleigh gas confined by two walls is investigated and a power-law decaying tail occurring in the memory function is analyzed. The effects of confinement and coupling with stochastic thermal character of the walls are discussed.
 Kim and Karniadakis, ”Microscopic theory of Brownian motion revisited: The Rayleigh model”, Phys. Rev. E 87, 032129 (2013).
 Kim and Karniadakis, ”Time Correlation Functions of Brownian Motion and Evaluation of Friction Coefficient in the near-Brownian-limit regime”, Multiscale Model. Simul. 12, 225 (2014).