The rule for a subdiffusive particle in an extremely diverse environment

TL;DR

This paper investigates how a subdiffusive particle's behavior in a highly heterogeneous environment transitions between different scaling laws, depending on the interplay of microscopic waiting times and environmental variability.

Contribution

It introduces a model showing a phase transition in the subdiffusive scaling law based on the distribution of waiting times and environment heterogeneity, revealing two competing mechanisms.

Findings

Identifies a transition in the effective waiting time distribution's scaling law.

Shows the transition depends on the relationship between distribution parameters.

Demonstrates the transition is dimension-independent and disappears under certain conditions.

Abstract

The dynamics of a subdiffusive continuous time random walker in an inhomogeneous environment is analyzed. In each microscopic jump, a random time is drawn from a waiting time probability density function (WT-PDF) that decays as a power law: phi(t;k)~k/(1+kt)^(1+beta), 0<beta<1. The parameter k, which is the diffusion coefficient for the jump, is a random quantity also; in each jump, it is drawn from a PDF, p(k)~1/k^gamma (0<gamma<1). We show that this system exhibits a transition in the scaling law of its effective WT-PDF, psi(t), which is obtained when averaging phi(t;k) with p(k). psi(t) decays as a power law, psi(t)~1/t^(1+mu), and mu is given by two different formula. When 1-gamma> beta;, mu=beta, but when 1-gamma<beta, mu=1-gamma. The transition in the scaling of psi(t) reflects the competition between two different mechanisms for subdiffusion: subdiffusion due to the heavily tailed phi(t;k) for microscopic jumps, and subdiffusion due to the collective effect of an environment made of many slow local regions. These two different mechanisms for subdiffusion are not additive, and compete each other. The reported transition is dimension independent, and disappears when the power beta is also distributed, in the range, 0<bate<1. Simulations exemplified the transition, and implications are discussed.