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 $L^2$-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 $\{p_{\alpha }(\mathbf{x},t){\}}_{\alpha =1}^N$, while the corresponding energy dissipation and entropy functions based on the overall probability density $p(\mathbf{x},t)$ 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.

References in corpus (2)

• A Wong-Zakai theorem for stochastic PDEs
• Stochastic foundations of undulatory transport phenomena: Generalized Poisson-Kac processes - Part I Basic theory

Cited by in corpus (6)

• Stochastic foundations of undulatory transport phenomena: Generalized Poisson-Kac processes - Part III Extensions and applications to kinetic theory and transport
• Extended Poisson-Kac theory: A unifying framework for stochastic processes with finite propagation velocity
• Relativistic analysis of stochastic kinematics
• Series representations for the characteristic function of the multidimensional Markov random flight
• Spectral properties of stochastic processes possessing finite propagation velocity
• Slow diffusion by Markov random flights