A Solution to Driven Brownian Motion and its Application to the Optimal Control of Small Stochastic Systems

We present an exact, iterative solution to the Fokker-Planck equation for driven Brownian motion in the overdamped limit under some mild assumptions. Driven Brownian motion is a paradigmatic model for the physics of small nonequilibrium stochastic machines. Of central interest are the related problems of calculating average entropy production and predicting optimal driving protocols for such systems. We calculate the average entropy produced in the reservoir over the course of driving a Brownian system, and we find that this functional contains an integral over a positive semidefinite metric, which we identify as the metric, a quantity relevant to the computation of optimal protocols. Our method for solving Brownian motion therefore yields a new formula for the thermodynamic metric. We calculate the thermodynamic metric of a harmonic oscillator with time-dependent spring constant and reservoir temperature, and discuss the implications of our work for the prediction of optimal protocols.