Irreversible Reactions and Diffusive Escape: Stationary Properties

TL;DR

This paper analyzes the stationary properties of three diffusion-controlled reactions starting from a half-line filled with particles and the other half empty, revealing finite infiltration and deriving key distribution functions.

Contribution

It introduces a detailed analysis of inhomogeneous initial conditions for diffusion reactions and provides explicit results for particle infiltration and distributions, including a novel procedure for coalescence.

Findings

Finite number of infiltrating particles with stationary distribution

Explicit average total number of particles for annihilation

Computed probability distribution for coalescence case

Abstract

We study three basic diffusion-controlled reaction processes -- annihilation, coalescence, and aggregation. We examine the evolution starting with the most natural inhomogeneous initial configuration where a half-line is uniformly filled by particles, while the complementary half-line is empty. We show that the total number of particles that infiltrate the initially empty half-line is finite and has a stationary distribution. We determine the evolution of the average density from which we derive the average total number N of particles in the initially empty half-line; e.g., for annihilation \langle N\rangle = 3/16+1/(4\pi). For the coalescence process, we devise a procedure that in principle allows one to compute P(N), the probability to find exactly N particles in the initially empty half-line; we complete the calculations in the first non-trivial case (N=1). As a by-product we derive the distance distribution between the two leading particles.