Mixing time of A + B -> 0 in 1D

TL;DR

This paper introduces a density measure for the mixing time of the reaction A + B -> 0 in one dimension, providing exact solutions and extending analysis to complex structures like fractals.

Contribution

It presents an exact generating function for the mixing time density in 1D and extends the method to multiple fronts and fractal geometries.

Findings

Derived an exact expression for the mixing time generating function.

Extended the analysis to multiple reaction fronts.

Applied the method to finitely ramified fractals.

Abstract

A mixing time density of $A + B \to 0$ on a finite one dimensional domain is defined for general initial and boundary conditions in which $A$ and $B$ diffuse at the same rate. The density is a measure of the number of $A$ and $B$ particles that mix through the center of the reaction zone. It also corresponds to the reaction density for the special case in which $A$ and $B$ annihilate upon contact. An exact expression is found for the generating function of the mixing time. The analysis is extended to multiple reaction fronts and finitely ramified fractals. The full method involves using the kernel of the Laplace transform integral operator to map and analyze a moving homogeneous Dirichlet interior point condition.