Brown movement in complex asymmetric periodic potential under the influence of colored noise

The idea of the molecular motor in an asymmetric periodic potential is a well-known problem. The motion of a Brownian particle is often studied when the system is subjected to the action of white noise. In practical situations noise is colored (``red'') random process. The red noise is the Ornstein-Uhlenbeck process. In this work we consider noise when the spectral density of the external noise is equal to zero on the zeroth frequency. In our previous works such a noise is been called as ``green'' noise. For the analytical study of green noise action, we use an approach based on a Krylov-Bogoliubov averaging method which is modified to study the action of noise with arbitrary intensity. A certain effective potential can be built which determines the basic features of the system dynamics. Further, we compare two numerical cases. The first one is the time-derivative of the Ornstain-Uhlenbeck process (green noise). The complex potentials when the system does not work as a molecular motor in the case of red noise, i.e. the average motion of the particle does not exhibit a drift in a given direction. If green noise operates on the same system, it turn out the effective molecular motor. We demonstrate this fact by a histograms for realizations of these processes.