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Heterogeneous Cattaneo-Vernotte equation connection to the noisy voter model

K. Górska, A. Horzela, D. Jankov Maširević, T. Pietrzak, 1T. K. Pogány, T. Sandev

TL;DR

This work develops a heterogeneous diffusion framework grounded in a position-dependent diffusion coefficient ${\cal D}(x)$ and its Cattaneo-Vernotte (CV) generalization to account for finite propagation speed. By incorporating stochastic interpretations via a parameter $\alpha$ and a delayed constitutive relation, the authors derive and solve the heterogeneous CV equation for diffusion laws ${\cal D}_{\lambda,\beta}(x)$, yielding exact PDFs and moment expressions that display short-time $t^{2\beta}$ and long-time $t^{\beta}$ scaling and a memory-driven crossover. They explicitly treat $\lambda=0$ and $\lambda>0$ (including $\nu=1/2$ and $\nu=3/2$ cases), provide MSDS in terms of Prabhakar and Mittag-Leffler functions, and analyze the $\tau\to0$ limit to connect with the heterogeneous diffusion equation. A key finding is the weak ergodicity breaking in TA-MSD, highlighting differences between ensemble and time-averaged statistics in these heterogeneous, finite-speed diffusion processes. The paper also draws a formal link to noisy voter models, connecting diffusion in heterogeneous media to opinion dynamics and financial market models through drift and diffusion mappings.

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.

Heterogeneous Cattaneo-Vernotte equation connection to the noisy voter model

TL;DR

This work develops a heterogeneous diffusion framework grounded in a position-dependent diffusion coefficient and its Cattaneo-Vernotte (CV) generalization to account for finite propagation speed. By incorporating stochastic interpretations via a parameter and a delayed constitutive relation, the authors derive and solve the heterogeneous CV equation for diffusion laws , yielding exact PDFs and moment expressions that display short-time and long-time scaling and a memory-driven crossover. They explicitly treat and (including and cases), provide MSDS in terms of Prabhakar and Mittag-Leffler functions, and analyze the limit to connect with the heterogeneous diffusion equation. A key finding is the weak ergodicity breaking in TA-MSD, highlighting differences between ensemble and time-averaged statistics in these heterogeneous, finite-speed diffusion processes. The paper also draws a formal link to noisy voter models, connecting diffusion in heterogeneous media to opinion dynamics and financial market models through drift and diffusion mappings.

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.
Paper Structure (17 sections, 1 theorem, 86 equations, 3 figures)

This paper contains 17 sections, 1 theorem, 86 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Heterogeneous diffusion equation
  3. Different interpretations of heterogeneous diffusion
  4. Connection to voter models of opinion dynamics and financial markets
  5. Heterogeneous CV equation
  6. Solution of heterogeneous CV equation
  7. Solution for $\lambda = 0$
  8. Solution for $\lambda > 0$ and $\nu = 1/2$ and $\nu = 3/2$
  9. Moments and MSD
  10. Limit approach to the heterogeneous diffusion equation
  11. TA-MSD
  12. Summary
  13. Special functions
  14. Solution $\widehat{P}_{\lambda, \beta}(\alpha; x, s)$
  15. Normalization of $P_{\lambda, \beta}(\alpha; x, t)$
  16. ...and 2 more sections

Key Result

Theorem 1

$f(t)$ is nonnegative if and only if its Laplace transform is a completely monotonic function (CMF) $\widehat{f}(s)$, $s > 0$.

Figures (3)

  • Figure 1: PDF $P_{0, \beta}(\alpha; x, t)$ for $\beta = 1/3$, $\tau = 0.1$, $v = 1$, $t=1$, and different values of $\alpha$: $\alpha=0$ (black curve), $\alpha = 1$ (blue curve) and $\alpha = 1/2$ (red curve).
  • Figure 2: PDF $P_{0, \beta}(\alpha; x, t)$ for $\beta = 1.9$, $\tau = 0.1$, $v = 1$, $t=1$, and different values of $\alpha$: $\alpha=0$ (black curve), $\alpha = 1$ (blue curve) and $\alpha = 1/2$ (red curve).
  • Figure 3: Plot of $P_{\rm CV}(x, t) = P_{0, 1}(1/2; x, t)$ (the black dashed curve), $P_{1/4, 1/2}(1/2; x, t)$ (the red solid curve), and $P_{1/4, 3/2}(1/2; x, t)$. PDFs are plotted for $\tau = 0.1$, $\upsilon = 1$, and $t =1$.

Theorems & Definitions (2)

  • Definition 1
  • Theorem 1