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

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.

References in corpus (6)

• Stochastic foundations of undulatory transport phenomena: Generalized Poisson-Kac processes - Part I Basic theory
• Stochastic foundations of undulatory transport phenomena: Generalized Poisson-Kac processes - Part II Irreversibility, Norms and Entropies
• Relativistic analysis of stochastic kinematics
• Markovian nature, completeness, regularity and correlation properties of Generalized Poisson-Kac processes
• Propagation of chaos for the thermostatted Kac master equation
• Kac's chaos and Kac's program

Cited by in corpus (6)

• Stochastic foundations of undulatory transport phenomena: Generalized Poisson-Kac processes - Part I Basic theory
• 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