Stochastic foundations of undulatory transport phenomena: Generalized Poisson-Kac processes - Part III Extensions and applications to kinetic theory and transport

TL;DR

This paper extends the Generalized Poisson-Kac process theory to nonlinear stochastic models and continuum states, providing a stochastic derivation of the nonlinear Boltzmann equation and addressing transport phenomena.

Contribution

It introduces nonlinear GPK models with state-dependent parameters and a continuum of states, advancing the stochastic foundations of kinetic and transport theories.

Findings

Derivation of nonlinear Boltzmann equation from GPK processes

Extension of GPK theory to nonlinear and continuum models

Numerical and physical examples illustrating the theory

Abstract

This third part extends the theory of Generalized Poisson-Kac (GPK) processes to nonlinear stochastic models and to a continuum of states. Nonlinearity is treated in two ways: (i) as a dependence of the parameters (intensity of the stochastic velocity, transition rates) of the stochastic perturbation on the state variable, similarly to the case of nonlinear Langevin equations, and (ii) as the dependence of the stochastic microdynamic equations of motion on the statistical description of the process itself (nonlinear Fokker-Planck-Kac models). Several numerical and physical examples illustrate the theory. Gathering nonlinearity and a continuum of states, GPK theory provides a stochastic derivation of the nonlinear Boltzmann equation, furnishing a positive answer to the Kac's program in kinetic theory. The transition from stochastic microdynamics to transport theory within the framework of the GPK paradigm is also addressed.