Entropy Production in Stochastic Systems with Fast and Slow Time-Scales

The behavior of stochastic systems with interacting fast and slow degrees of freedom is investigated both for discrete and continuous processes. Effective equations that govern the process on the slow timescale are derived by asymptotic methods, both for the propagator and the entropy production of the systems. It is found that in general the result of the limiting procedure for entropy does not coincide with the one defined for the effective slow process and features an additional contribution. The specific conditions under which such a correction does or does not arise are stated and the general explicit form of this remnant entropy production is offered. Finally, the fluctuation theorems that are satisfied by this additional term are given.