Heterogeneous Cattaneo-Vernotte equation connection to the noisy voter model

TL;DR

This paper explores a heterogeneous Cattaneo-Vernotte equation derived from stochastic interpretations, analyzing its probability density, mean squared displacement, and ergodicity breaking effects.

Contribution

It introduces a heterogeneous diffusion model with position-dependent coefficients and provides exact analytical results for key statistical measures.

Findings

Exact probability density function derived

Mean squared displacement calculated

Ergodicity breaking observed in the model

Abstract

We consider a heterogeneous diffusion equation and its corresponding generalization to the Cattaneo-Vernotte equation. It is derived by a combination of the continuity equation and the constitutive relation in various stochastic interpretations of the heterogeneous diffusion process. The heterogeneity in the system is introduced by considering a position-dependent diffusion coefficient. Exact results for the probability density function and the mean squared displacement are provided. The limiting case of heterogeneous diffusion is analyzed in detail, and the corresponding time-averaged mean-squared displacement is calculated. From the obtained results, an ergodicity breaking is observed.