Phase transition in annihilation-limited processes

TL;DR

This paper investigates a particle system with type-change and multi-particle annihilation, revealing a phase transition in long-term behavior depending on the annihilation rate and initial conditions, especially under a double scaling limit.

Contribution

It demonstrates that under double scaling, the system remains multi-Poisson with evolving parameters and identifies a dynamical phase transition related to the annihilation order and critical value k_c.

Findings

System remains multi-Poisson with evolving parameters under double scaling.

Existence of a dynamical phase transition depending on the annihilation order.

For k > k_c, annihilation does not affect relaxation; for k < k_c, it dominates relaxation.

Abstract

A system of particles is studied in which the stochastic processes are one-particle type-change (or one-particle diffusion) and multi-particle annihilation. It is shown that, if the annihilation rate tends to zero but the initial values of the average number of the particles tends to infinity, so that the annihilation rate times a certain power of the initial values of the average number of the particles remain constant (the double scaling) then if the initial state of the system is a multi-Poisson distribution, the system always remains in a state of multi-Poisson distribution, but with evolving parameters. The large time behavior of the system is also investigated. The system exhibits a dynamical phase transition. It is seen that for a k-particle annihilation, if k is larger than a critical value k_c, which is determined by the type-change rates, then annihilation does not enter the relaxation exponent of the system; while for k < k_c, it is the annihilation (in fact k itself) which determines the relaxation exponent.