Chaotic transport of fractional over-damped ratchet with fluctuation and periodic drive

The fractional over-damped ratchet model with thermal fluctuation and periodic drive is introduced by using the damping kernel function of general Langevin equation in the form of power law based on the assumption that cytosol in biological cells has characteristics of power-law memory. On basis of the Grunwald-Letnikov definition of fractional derivative, the numerical solution of this ratchet model is obtained. And furthermore, according to the numerical solution, the transport behaviors of stochastic ratchet and corresponding deterministic ratchet (especially when the deterministic ratchet has chaotic trajectory) are investigated, based on which we try to analyze how chaotic properties of the deterministic ratchet and the actions of noise influence the transport properties of molecular motors and moreover find the possible mechanism of current reversal of fractional molecular motor. Numerical results show that, as barrier height, barrier asymmetry and memorability of model change, the current reversal in deterministic ratchet is not necessarily required to appear when happening indeed in corresponding stochastic ratchet; moreover, with the decrease of order p, there exists a chaotic regime in deterministic ratchet model before current reversal, but with the disturbance of noise, current reversal will happen more earlier, namely, chaotic current direction in deterministic ratchet model can be reversed when disturbance of noise exists. This also demonstrates that noise can essentially change the transport behavior of a ratchet; current can change from chaotic state in a ratchet with no noise to directed transport with noise. This is a possible mechanism of current reversal of a fractional stochastic ratchet, and also a reflection that noise plays an active role in directed transport.