Generalized Poisson-Kac processes and hydrodynamic modeling of systems of interacting particles I - Theory

TL;DR

This paper develops a theoretical framework linking lattice particle dynamics to continuous hyperbolic hydrodynamic models using Generalized Poisson-Kac processes, revealing complex behaviors like phase transitions.

Contribution

It introduces a novel connection between stochastic lattice dynamics and hyperbolic hydrodynamics via Generalized Poisson-Kac processes, applicable to interacting particle systems.

Findings

Hydrodynamic models can exhibit singularities and phase transitions.

The Kac limit leads to infinite propagation velocity models.

The approach applies to exclusion and potential-based particle interactions.

Abstract

This article analyzes the formulation of space-time continuous hyperbolic hydrodynamic models for systems of interacting particles moving on a lattice, by connecting their local stochastic lattice dynamics to the formulation of an associated (space-time continuous) Generalized Poisson-Kac process possessing the same local transition rules. The hyperbolic hydrodynamic limit follows naturally from the statistical description of the latter in terms of the system of its partial probability density functions. Several cases are treated, with particular attention to: (i) models of interacting particles satisfying an exclusion principle, and (ii) models defined by a given interparticle interaction potential. In both cases, the hydrodynamic models may display singularities, dynamic phase-transitions and bifurcations (as regards the flux/concentration-gradient constitutive equations), whenever the Kac limit of the model (infinite propagation velocity limit) is considered.