Population splitting, trapping, and non-ergodicity in heterogeneous diffusion processes

TL;DR

This paper investigates how spatially varying diffusivity affects anomalous diffusion, revealing phenomena like population splitting, trapping, and weak ergodicity breaking through analytical and simulation methods.

Contribution

It provides a comprehensive analysis of heterogeneous diffusion with different functional forms of diffusivity, highlighting their impact on ergodic properties and particle dynamics.

Findings

Weak ergodicity breaking observed in all cases

Population splitting into fast and slow diffusers identified

Particle trapping occurs with logarithmic diffusivity

Abstract

We consider diffusion processes with a spatially varying diffusivity giving rise to anomalous diffusion. Such heterogeneous diffusion processes are analysed for the cases of exponential, power-law, and logarithmic dependencies of the diffusion coefficient on the particle position. Combining analytical approaches with stochastic simulations, we show that the functional form of the space-dependent diffusion coefficient and the initial conditions of the diffusing particles are vital for their statistical and ergodic properties. In all three cases a weak ergodicity breaking between the time and ensemble averaged mean squared displacements is observed. We also demonstrate a population splitting of the time averaged traces into fast and slow diffusers for the case of exponential variation of the diffusivity as well as a particle trapping in the case of the logarithmic diffusivity. Our analysis is complemented by the quantitative study of the space coverage, the diffusive spreading of the probability density, as well as the survival probability.