Discrete to Continuous-Time Crossover due to Anisotropy in Diffusion-Limited Two-Particle Annihilation Reactions

TL;DR

This paper provides an exact solution for anisotropic diffusion-limited A+A->inert reactions on a 1D lattice, revealing how anisotropy influences reaction dynamics and transitions from discrete to continuous time regimes.

Contribution

It offers an exact analysis of anisotropic hopping in a discrete-time setting, showing how anisotropy affects diffusion slowdown and continuum limits.

Findings

Diffusion slows down with increased anisotropy.

Continuum limits absorb anisotropy effects via time rescaling.

Discrete-time effects are small but nontrivial in certain regimes.

Abstract

Diffusion-limited reaction A+A->inert with anisotropic hopping on the d=1 lattice, is solved exactly for a simultaneous updating, discrete time-step dynamics. Diffusion-dominated processes slow down as the anisotropy increases. For large times or large anisotropy, one can invoke the appropriate continuum limits. In these limits the effects of the anisotropy on variation of particle density can be absorbed in time rescaling. However, in other regimes, when the discreteness of the time steps is nonnegligible, the anisotropy effects are nontrivial, although they are always quite small numerically.