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

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.

Lattice random walk: an old problem with a future ahead

“…by assuming some form of exclusion principle, or by introducing interparticle potentials). This extension is developed elsewhere [53], where it is shown that novel dynamic properties may occur, associated with bifurcations in the constitutive equations, the qualitative and quantitative analysis of which is open to further investigation. Moreover, the extension of the hyperbolic approach to heterogeneous lattices, introduced in [49,50] for two-phase lattice systems, can be transferred to a variety of disordered systems in any spatial dimension.…”

“…Within the theory of GPK processes, the description of the interaction amongst particles involves essentially the structure of the transition probability matrix A defined in equation (51) that, in the case of interacting particle systems, depends explicitly on the partial probability densities p ± (x, t) As addressed in [53], owing to the hyperbolic structure of the balance equations and to the nonlinearities associated with the functional form of A equation (73), bifurcations in the structure of the constitute equations, expressing the flux in terms of the gradient of the overall probability density, can occur in the Kac limit, and this may lead to new classes of phase-transitions and hysteretic behavior in system of interacting particles.…”

Age representation of Lévy walks: partial density waves, relaxation and first passage time statistics

“…The hyperbolic formulation of LWs is particularly suited for modelling more complex situations, which account for the occurrence of interparticle interactions, exclusion effects, etc, that in a mean-field modeling can be described by allowing the velocity b and the transition rate λ(τ ) to depend on the partial wave densities. This extension has been initiated in [77,78] for LWs and in [79] for Poisson-Kac processes. (t) versus t for the LW defined by equation ( 13), ξ = 1.5, with with b = 1, x 0 = 1 and π + 0 = 0.5.…”