The directional transport phenomenon in fractional logarithm coupled system under a non-periodic external force

Abstract

Using the fractional calculus theory, we investigate the directional transport phenomenon in a fractional logarithm coupled system under the action of a non-periodic external force. When a Brownian particle moves in the media with memory such as viscoelastic media, the system should be modeled as a nonlinear fractional logarithm coupled one. Using the method of fractional difference, we can solve the model numerically and discuss the influences of the various system parameters on the average transport velocity of the particles. Numerical results show that: 1) The directional transport phenomenon in this fractional logarithmic coupled model appears only when the external force exists, and the value of the average transport velocity of the particles increases with increasing external force. 2) When the fractional order of the system is small enough, the damping memory has a significant impact on the average transport velocity of the particles. Furthermore, the average transport velocity of the particles has an upper bound (although it is very small), no matter how the external force, coupled force and the intensity of noise change, the average transport velocity of the particles is no more than the upper bound. When there is no external force and the damping force is big enough, the directional transport phenomenon disappears. 3) When the fractional order of the system and the external force are big enough, although the directional transport phenomenon appears, the coupled force and the intensity of noise have no impact on the system. 4) Only when the external force is small enough, could the coupled force and noise intensity influence the average transport velocity of the particles. In this situation, the directional transport phenomenon appears when the fractional order of the system is big enough, and the average transport velocity of the particles changes along with the change of the coupled force and the noise intensity.