Transport properties of fractional coupled Brownian motors in ratchet potential with feedback

Abstract

Based on the theory of fractional integration, direct transport behaviors of coupled Brownian motors with feedback control in viscoelastic media are investigated. The mathematical model of fractional overdamped coupled Brownian motors is established by adopting the power function as damping kernel function of general Langevin equation due to the power-law memory characteristics of cytosol in biological cells. Numerical solution is observed by fractional difference method and the influence of model parameters on cooperative direct transport of the coupled Brownian motors is discussed in detail by numerical simulation. The research shows that the memory of the fractional dynamical system can affect the direct transport phenomenon of the coupled Brownian motors through changing the on-off switching frequency of the ratchet potential with feedback control. To be more specific, in a proper range of the fractional order, the memory of the dynamical system can increase the on-off switching frequency of the ratchet potential, which can lead to the velocity increase of the direct transport. Furthermore, in the case of small fractional order, since the coupled Brownian motors move under the competition between the damping force with memory and the potential force with feedback control, the resultant force exerted on the coupled particles is always positive when the ratchet potential with feedback control is on although the fractional damping force is large, which leads to the result that the coupled Brownian motors move in the positive direction in the mass. On the contrary, in the case of large fractional order, the on-off switching frequency of potential with feedback control becomes small, as a result of which the main influential factor of the direct transport becomes the potential depth. Therefore the coupled Brownian motors are more likely to stay in the potential wells for a long time because the probability that describes the possibility that the coupled Brownian motors surmount the potential barriers becomes small. Finally, with the parameters of the fractional dynamical system (e.g. potential depth, noise intensity) fixed, the direct transport velocity of the coupled Brownian motors shows the generalized stochastic resonant phenomenon while the fractional order varies.