Brownian motion in the environment of the thermal fluctuations is a long-study issue in nonequilibrium statistical physics. In recent years, the directed transport properties of Brownian ratchets attract the widespread attention of scholars. When a ratchet system possesses the spatio-temporal symmetry-breaking feature, the directed transport can be produced. Although the breakthrough progress in the directed transport of the Brownian ratchet has been made, the energy conversion efficiency of feedback ratchet is not clear. Therefore, the center-of-mass mean velocity and the energy conversion efficiency of coupled ratchet under the influences of the time asymmetry of external force and the spatial asymmetry of external potential are discussed in detail. The overdamped coupled Brownian particles are investigated. Nevertheless, the optimized control of the coupled ratchet is the important for directed transport. Therefore, the closed-loop control which depends on the state of the system is adopted. The dynamic behavior of coupled particles can be described by the overdamped Langevin equation, and the equation is numerically solved by using the stochastic Runge-Kutta algorithm. Some properties of the directed transport can be obtained through this method, such as the center-of-mass mean velocity, the energy conversion efficiency, etc. It is interesting to find that the center-of-mass mean velocity can reach a maximum as the amplitude of external force increases. However, the mean velocity can show the quasi-periodic oscillations with the increase of the period of external force for different values of the spatial asymmetry of external potential. In addition, it can be found that the feedback ratchet needs strong noise to make the directed transport of the ratchet reach the maximum as the coupled strength increases. On the other hand, the energy conversion efficiencies of the feedback ratchet can achieve their corresponding maximum values with the increase of the amplitude of external force for different values of the time asymmetry, and the maximum increases as the time asymmetry increases. However, the efficiency can also show the quasi-periodic oscillations with the increase of the period of the external force for different values of the spatial asymmetry of external potential. Moreover, the energy conversion efficiency can achieve the maximum as the noise strength increases, but the maximum of the efficiency will decrease with the increase of coupling strength. From the discussion above, the optimal values of the time asymmetry, the spatial asymmetry, the period of the external force and the noise strength can promote the directed transport of the feedback coupled Brownian ratchet. These conclusions can provide some guidance in the enhancement of the energy conversion efficiency of a nanomachine.

Molecular motors in life activities of cell are known to operate efficiently.They could convert molecular-scale chemical energy into macroscopic-scale mechanical work with high efficiency.In order to acquire the transport mechanism of the molecular motor,the Brownian ratchet has been proposed to explore the property of directed transport and energy conversion.There are different kinds of Brownian ratchet models like flashing ratchets,rocking ratchets,and time-asymmetric ratchets and so on.Through investigating the performance of Brownian ratchet moving in periodic potential,the directed transport of ratchet could be explained,and the effective usage of ratchet energy for directed transport could also be improved.Recently,optimizing the transport of Brownian ratchet has aroused the interest of researchers.It is found that the viscous resistance could reinforce the directed transport of the Brownian particle in damping liquid.Meanwhile,a large number of conclusions indicate that the transport of Brownian ratchets would be enhanced if the asymmetry of the potential is changed.Those results show that the influences of the external potential and the damping force on the particle flow cannot be neglected.Hence in this paper,the effects of the potential structure and the temperature of heat bath on transport are discussed. Furthermore,how to use the ratchet energy effectively has been investigated in recent years.When the Brownian motor operates with load,the input energy is reduced.More importantly,the energy transformation efficiency defined as the ratio of the useful work done against the load to the input energy is assumed to be a zero value in the absence of load.With the help of stochastic energetic theory proposed by Sekimoto,the Stokes efficiency has been used to explore the performance of the Brownian ratchet.So far,the numerical solution has been used extensively in most theoretical investigations.Nevertheless,in our work,the Stokes efficiency is discussed analytically for explaining the mechanism of directed transport.We consider the transport performance of the Brownian ratchet described by the Fokker Planck equation which is corresponding to the Langevin equation under time-varying external force and thermal noise.Mainly, the effects of potential asymmetry,external force,height of the barrier,and intensity of the thermal noise on transport are discussed in detail.It is found that the transport direction of Brownian ratchet will be reversed under the condition of appropriate potential structures,and the probability current can reach a maximal value by changing the asymmetry of potential.It is worthwhile to point out that the performance of directed transport of the ratchet can be improved when an appropriate amplitude of the external force is applied.Meanwhile,there is an optimal value of the barrier height at which the Stokes efficiency reaches a maximal value and the directed transport of ratchet is enhanced.Through our conclusions,the ratchets of different structures could be designed for improving the transport property of Brownian motor.And the results have helpful theoretical guidance not only in the aspect of medical delivery but also in the control of nano-devices.