Stochastic foundations of undulatory transport phenomena: generalized Poisson–Kac processes—part II Irreversibility, norms and entropies

Abstract

In this second part, we analyze the dissipation properties of generalized Poisson–Kac (GPK) processes, considering the decay of suitable L2-norms and the definition of entropy functions. In both cases, consistent energy dissipation and entropy functions depend on the whole system of primitive statistical variables, the partial probability density functions , while the corresponding energy dissipation and entropy functions based on the overall probability density do not satisfy monotonicity requirements as a function of time. These results provide new insights on the theory of Markov operators associated with irreversible stochastic dynamics. Examples from chaotic advection (standard map coupled to stochastic GPK processes) illustrate this phenomenon. Some complementary physical issues are also addressed: the ergodicity breaking in the presence of attractive potentials, and the use of GPK perturbations to mollify stochastic field equations.