Browninan motion in a fluctuating periodic potential

A classical Brownian particle is considered in a periodic potential field with a rapidly oscillating phase. The concept of effective potential is used for describing a slow averaged motion of a particle. It is shown that there exists a certain region in which a particle performs a stationary random motion without appreciable drift. By analogy with the ideal case, this region can be called an effective locking region. The situation described is valid for stationary fluctuations of the phase of a potential function, provided that they have a sufficiently small but finite correlation time. The study of the problem is reduced to the analysis of a stochastic system with external noise whose spectral density is zero at zero frequency (“green” noise 1]). The analysis of the first-and second-approximation equations of the averaging method exhibits the high stability of the locking phenomenon. This result has been verified by the numerical solution of appropriate stochastic equations. In this case, a predictor-corrector algorithm was used that allowed one to carry out a numerical simulation to a sufficiently high degree of accuracy. The result of the simulation is in good agreement with the theoretical results. The effective locking bandwidth calculated analytically by the averaging method actually coincides with the value obtained by the simulation.