Heterogeneous continuous time random walks

TL;DR

This paper introduces a heterogeneous continuous time random walk (HCTRW) model that provides a unified framework for analyzing diffusion in complex, heterogeneous environments across multiple scientific disciplines.

Contribution

The paper develops the HCTRW formalism, deriving exact propagators and analyzing how heterogeneities affect diffusion and first passage times in various media.

Findings

Exact form of the propagator derived

Heterogeneities significantly alter diffusion dynamics

Unified framework applicable across sciences

Abstract

We introduce a heterogeneous continuous time random walk (HCTRW) model as a versatile analytical formalism for studying and modeling diffusion processes in heterogeneous structures, such as porous or disordered media, multiscale or crowded environments, weighted graphs or networks. We derive the exact form of the propagator and investigate the effects of spatio-temporal heterogeneities onto the diffusive dynamics via the spectral properties of the generalized transition matrix. In particular, we show how the distribution of first passage times changes due to local and global heterogeneities of the medium. The HCTRW formalism offers a unified mathematical language to address various diffusion-reaction problems, with numerous applications in material sciences, physics, chemistry, biology, and social sciences.