content_image/349032.jpg content_image/349035.jpg content_image/349038.jpg
model looks like 

<img src='content_image/349032.jpg'>

where, α is the force constant of the spring connecting the particles sitting at x$_{1}$ and x$_{2}$ ( x$_{1}$ > x$_{2}$ say). The η ’s are the thermal noises experienced by the system where < η$_{i}$ > = 0 and < η$_{i}$ ( t$_{1}$ ) η$_{j}$ ( t$_{2}$ ) > = 2 T δ$_{ij}$δ ( t$_{1}$ − t$_{2}$ ) which is the fluctuation-dissipation relation. Note that, the α and the Boltzmann constant κ$_{B}$ have been absorbed by adjusting the scale of the temperature in the fluctuation-dissipation relation which is a very standard practice. 

Such a system would soon attain equilibrium and one naturally cannot expect having di- rectional transport. So, we put a drive to one of the particles which will be driving the system out of equilibrium by breaking the fluctuation-dissipation relation. Throughout this paper, we will consider an over damped dynamics of the system since this over damped regime is physically relevant in most of the biological and other cases. Now, in the overdamped regime, where there is no inertial term present, the velocity of a particle is proportional to the force acting on it. Taking this into consideration we will use a drive which is equivalent to feeding back the instantaneous velocity to one of the particles and that would invalidate the otherwise existing fluctuation dissipation relation. This not only is a very general way of driving the system out of equilibrium, but also it would enable us to solve the system exactly and calculate the average velocity and efficiency of the system in exact analytic forms. So, with this driving, our model looks like 

<img src='content_image/349035.jpg'>

where β is the velocity feedback constant or the strength of the external drive. In what follows we will keep − 1 ≤ β ≤ 1 in order to remain safely within the over damped regime. A large forcing would eventually require the inertial term be taken into consideration which we would not like to move into at the moment. 

In order to understand where this modified situation with the velocity feedback actually differs from the model in Eq.1 let us write it down in an analogous from as 

<img src='content_image/349038.jpg'>

