Reactive dynamics on fractal sets: anomalous fluctuations and memory effects

TL;DR

This paper investigates how fractal initial conditions influence the dynamics of reactive systems, revealing exponential decay without diffusion and power-law decay with diffusion, where the decay exponent depends on the fractal dimension.

Contribution

It provides analytical and numerical analysis of reactive systems with fractal initial conditions, highlighting the impact of fractal geometry on decay dynamics and memory effects.

Findings

Without diffusion, particles decay exponentially to a steady state.

With diffusion, the particle number decays as a power law with exponent related to fractal dimension.

Initial fractal conditions influence long-term decay behavior and memory effects.

Abstract

We study the effect of fractal initial conditions in closed reactive systems in the cases of both mobile and immobile reactants. For the reaction $A+A\to A$, in the absence of diffusion, the mean number of particles $A$ is shown to decay exponentially to a steady state which depends on the details of the initial conditions. The nature of this dependence is demonstrated both analytically and numerically. In contrast, when diffusion is incorporated, it is shown that the mean number of particles $<N(t)>$ decays asymptotically as $t^{-d_f/2}$, the memory of the initial conditions being now carried by the dynamical power law exponent. The latter is fully determined by the fractal dimension $d_f$ of the initial conditions.