Interrupted escape and the emergence of exponential relaxation

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Abstract
A simple statistical theory of irreversible processes in a subsystem coupled to (or “interrupted” by) a stochastic bath is formulated. The theory does not explicitly invoke time scale separation that underlies the standard description of nonequilibrium phenomena and is intrinsic to the concept of quasiequilibrium in the canonical ensemble.Arbitrary statistics and speed of bath fluctuations are straightforwardly treated by the theory. Except in the case of an extremely slow, nonequilibrium bath, the ultimate statistics of interrupted escape are shown to be Poisson, which is solely a consequence of the stationary nature of interactions in a sufficiently dense system. In the limit of a fast bath, the corresponding relaxation rate is shown to equal the initial rate of decay, thus validating a wide class of Golden Rate type expressions at long times. This true exponentiality thus appears when the time scale separation takes place. The theory also applies to a number of specific phenomena including transport in a fluctuating or disordered medium, gated reactions, the line shape theory, and the quantum Zeno effect. The general nature of motional narrowing phenomena is demonstrated and related to the bath mediated slowing down of a decay process with a nearly deterministic uninterrupted escape probability. The corresponding survival probability is shown also to exhibit discernible oscillations around the exponential background. Mathematical tools necessary for using the theory in specific applications are exposed in some detail.