Chemical Continuous Time Random Walks under Anomalous Diffusion

TL;DR

This paper introduces a novel chemical continuous time random walk model for anomalous diffusion in heterogeneous systems, extending the chemical master equation and Gillespie algorithm to account for arbitrary waiting time distributions.

Contribution

It develops a systematic stochastic theory for reaction-diffusion processes with non-exponential waiting times, including generalized equations and simulation methods.

Findings

Demonstrates the model with exponential and power-law waiting times

Shows strong fractional memory effects in power-law cases

Extends classical reaction-diffusion theory to anomalous diffusion

Abstract

Chemical master equation plays an important role to describe the time evolution of homogeneous chemical system. In addition to the reaction process, it is also accompanied by physical diffusion of the reactants in complex system that is generally not homogeneous, which will result in non-exponential waiting times for particle reactions and diffusion. In this paper we shall introduce a chemical continuous time random walk under anomalous diffusion model based on renewal process to describe the general reaction-diffusion process in the heterogeneous system, where the waiting times are arbitrary distributed. According to this model, we will develop the systematic stochastic theory including generalizing the chemical diffusion master equation, deriving the corresponding mass action law, and extending the Gillespie algorithm. As an example, we analyze the monomolecular $A\leftrightarrow B$ reaction-diffusion system for exponential and power-law waiting times respectively, and show the strong fractional memory effect of the concentration of the reactants on the history of the concentration in power-law case.