Stochastic Entropy Production for Classical and Quantum Dynamical Systems with Restricted Diffusion

Modeling the evolution of a system using stochastic dynamics typically implies increasing subjective uncertainty in the adopted state of the system and its environment as time progresses, and stochastic entropy production has been developed as a measure of this change. In some situations, the evolution of stochastic entropy production can be described using an Itô process, but mathematical difficulties can emerge if diffusion in the system phase space happens to be restricted to a subspace of a lower dimension. This situation can arise if there are constants of the motion, for example, or more generally when there are functions of the coordinates that evolve without noise. More simply, difficulties can emerge if there are more coordinates than there are independent noises. We show how the problem of computing the stochastic entropy production in such a situation can be overcome. We illustrate the approach using a simple case of diffusion on an ellipse. We go on to consider an open three-level quantum system modeled within a framework of Markovian quantum state diffusion. We show how a nonequilibrium stationary state of the system, with a constant mean rate of stochastic entropy production, can be established under suitable environmental couplings.