Multiphase Partitions of Lattice Random Walks

TL;DR

This paper demonstrates that a hyperbolic hydrodynamic model accurately describes phase transitions and transport properties of particles on a lattice, outperforming traditional parabolic models, especially near discontinuous transitions.

Contribution

It shows that the hyperbolic model derived by Giona (2018) correctly captures lattice phase dynamics and boundary conditions, unlike common parabolic models.

Findings

Hyperbolic model predicts effective diffusion coefficient accurately.

Parabolic models fail near phase transition points.

Numerical experiments confirm the superiority of hyperbolic transport theory.

Abstract

Considering the dynamics of non-interacting particles randomly moving on a lattice, the occurrence of a discontinuous transition in the values of the lattice parameters (lattice spacing and hopping times) determines the uprisal of two lattice phases. In this Letter we show that the hyperbolic hydrodynamic model obtained by enforcing the boundedness of lattice velocities derived by Giona (2018) correctly describes the dynamics of the system and permits to derive easily the boundary condition at the interface, which, contrarily to the common belief, involves the lattice velocities in the two phases and not the phase diffusivities. The dispersion properties of independent particles moving on an infinite lattice composed by the periodic repetition of a multiphase unit cell are investigated. It is shown that the hyperbolic transport theory correctly predicts the effective diffusion coefficient over all the range of parameter values, while the corresponding continuous parabolic models deriving from Langevin equations for particle motion fail. The failure of parabolic transport models is shown via a simple numerical experiment.