Anomalous reaction-transport processes: the dynamics beyond the Mass Action Law

TL;DR

This paper explores how anomalous transport affects reversible reactions, revealing that traditional mass action laws do not hold and that transport can lead to particle extinction, with results supported by simulations.

Contribution

It introduces a mesoscopic framework for anomalous reaction-transport processes, showing transport influences reaction rates and can cause extinction, challenging classical mass action assumptions.

Findings

Effective reaction rates depend on transport characteristics.

Transport can induce particle extinction.

Mesoscopic densities show transient growth before decay.

Abstract

In this paper we reconsider the Mass Action Law (MAL) for the anomalous reversible reaction $A\rightleftarrows B$ with diffusion. We provide a mesoscopic description of this reaction when the transitions between two states $A$ and $B$ are governed by anomalous (heavy-tailed) waiting-time distributions. We derive the set of mesoscopic integro-differential equations for the mean densities of reacting and diffusing particles in both states. We show that the effective reaction rate memory kernels in these equations and the uniform asymptotic states depend on transport characteristics such as jumping rates. This is in contradiction with the classical picture of MAL. We find that transport can even induce an extinction of the particles such that the density of particles $A$ or $B$ tends asymptotically to zero. We verify analytical results by Monte Carlo simulations and show that the mesoscopic densities exhibit a transient growth before decay.