Reaction-diffusion on the fully-connected lattice: $A+A\rightarrow A$

TL;DR

This paper analyzes the reaction-diffusion process $A+A\rightarrow A$ on a fully-connected lattice, deriving exact distributions and revealing universal, non-Gaussian fluctuation behaviors in the time to reach a finite number of particles.

Contribution

It provides exact analytical expressions for particle density and survival time distributions in a high-dimensional, mean-field setting, highlighting universal fluctuation phenomena.

Findings

Exact expressions for particle density distribution.

Universal non-Gaussian fluctuations in survival times.

Strong fluctuations characterized by extreme value statistics.

Abstract

Diffusion-coagulation can be simply described by a dynamic where particles perform a random walk on a lattice and coalesce with probability unity when meeting on the same site. Such processes display non-equilibrium properties with strong fluctuations in low dimensions. In this work we study this problem on the fully-connected lattice, an infinite-dimensional system in the thermodynamic limit, for which mean-field behaviour is expected. Exact expressions for the particle density distribution at a given time and survival time distribution for a given number of particles are obtained. In particular we show that the time needed to reach a finite number of surviving particles (vanishing density in the scaling limit) displays strong fluctuations and extreme value statistics, characterized by a universal class of non-Gaussian distributions with singular behaviour.