Symmetry relations for trajectories of a Brownian motor

A Brownian motor is a nanoscale or molecular device that combines the effects of thermal noise, spatial or temporal asymmetry, and directionless input energy to drive directed motion. Because of the input energy, Brownian motors function away from thermodynamic equilibrium and concepts such as linear response theory, fluctuation dissipation relations, and detailed balance do not apply. The generalized fluctuation-dissipation relation, however, states that even under strongly thermodynamically nonequilibrium conditions the ratio of the probability of a transition to the probability of the time reverse of that transition is the exponential of the change in the internal energy of the system due to the transition. Here, we derive an extension of the generalized fluctuation dissipation theorem for a Brownian motor for the ratio between the probability for the motor to take a forward step and the probability to take a backward step.

Figure 1

Depiction of symmetry-related trajectories of a Brownian particle in a periodic ratchet potential. (a) Snapshot of a “ratchet” potential with a particle at $\alpha =0$. (b) With an external forcing $\psi \left(t\right)$ the four trajectories $F$, $F_R$, $B$, and $B_R$ are distinct from one another.