Stochastic foundations of undulatory transport phenomena: Generalized Poisson-Kac processes - Part II Irreversibility, Norms and Entropies

TL;DR

This paper investigates the dissipation and entropy properties of Generalized Poisson-Kac processes, emphasizing the importance of primitive variables for energy and entropy functions, with applications in chaotic advection and stochastic field equations.

Contribution

It introduces a detailed analysis of dissipation and entropy in GPK processes, highlighting the significance of primitive variables over overall probability densities.

Findings

Energy dissipation depends on primitive variables

Entropy functions based on primitive variables satisfy monotonicity

Chaotic advection examples illustrate dissipation phenomena

Abstract

In this second part, we analyze the dissipation properties of Generalized Poisson-Kac (GPK) processes, considering the decay of suitable $L^2$-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 $\{ p_\alpha({\bf x},t) \}_{\alpha=1}^N$, while the corresponding energy dissipation and entropy functions based on the overall probability density $p({\bf x},t)$ do not satisfy monotonicity requirements as a function of time. 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.