On the Relation between One-Species Diffusion-Limited Coalescence and Annihilation in One Dimension

TL;DR

This paper clarifies the relationship between coalescence and annihilation processes in one dimension, demonstrating that correlation hierarchies fully describe the systems and introducing a new hierarchy for particle configurations.

Contribution

It establishes the equivalence of correlation hierarchies for both processes and introduces a new hierarchy of probability densities for particle arrangements.

Findings

Correlation hierarchies uniquely determine the particle system.

The new hierarchy is computable via the empty interval method.

The approach clarifies the relation between coalescence and annihilation processes.

Abstract

The close similarity between the hierarchies of multiple-point correlation functions for the diffusion-limited coalescence and annihilation processes has caused some recent confusion, raising doubts as to whether such hierarchies uniquely determine an infinite particle system. We elucidate the precise relations between the two processes, arriving at the conclusion that the hierarchy of correlation functions does provide a complete representation of a particle system on the line. We also introduce a new hierarchy of probability density functions, for finding particles at specified locations and none in between. This hierarchy is computable for coalescence, through the method of empty intervals, and is naturally suited for questions concerning the ordering of particles on the line.