Diffusion-Limited One-Species Reactions in the Bethe Lattice

TL;DR

This paper investigates the kinetics of diffusion-limited one-species reactions on the Bethe lattice, revealing how correlations develop over time and affect the decay rate of particle density.

Contribution

It provides an analytical study of reaction kinetics on the Bethe lattice, highlighting the evolution of correlations and their impact on decay rates.

Findings

Particle density decays as 1/kt over time.

Correlation effects cause the reaction rate to decrease to an asymptotic constant.

Long-time behavior shows a transition from uncorrelated to correlated particle distributions.

Abstract

We study the kinetics of diffusion-limited coalescence, A+A-->A, and annihilation, A+A-->0, in the Bethe lattice of coordination number z. Correlations build up over time so that the probability to find a particle next to another varies from \rho^2 (\rho is the particle density), initially, when the particles are uncorrelated, to [(z-2)/z]\rho^2, in the long-time asymptotic limit. As a result, the particle density decays inversely proportional to time, \rho ~ 1/kt, but at a rate k that slowly decreases to an asymptotic constant value.