Relative entropy and efficiency measure for diffusion-mediated transport processes

We propose an efficiency measure for diffusion-mediated transport processes including molecular-scale engines such as Brownian motors (BMot) moving in ratchet potentials acting as mechanical 'rectifiers'. The efficiency measure is based on the concept of 'minimal energy required to complete a task' and is defined via a class of stochastic optimal control problems. The underlying objective function depends on both the external force field (i.e. the fluctuation rectifier in the case of BMot) and the amplitude of the environmental noise. Ultimately, the efficiency measure can be directly interpreted as the relative entropy between two probability distributions, namely: the distribution of the particles in presence of the external rectifying force field and a reference distribution describing the behaviour in the absence of the rectifier.