Dynamical generalization of nonequilibrium work relation

The evolution of systems in contact with thermal, chaotic, or turbulent surroundings-often modeled with stochastic equations of motion-can be particularly complex when these equations of motion are nonautonomous, that is, when external parameters of the surroundings are varied with time. In this paper we establish a rigorous equality relating the nonautonomous behavior of such a system, to solutions of the corresponding autonomous equations of motion, for arbitrary initial conditions. If the system is initially in thermal equilibrium, we recover previously known results relating nonequilibrium work values to equilibrium probability distributions. We discuss specific examples of our result, and suggest an experimental setting in which it might be verified.