Annihilation of single-species charged particles based on the Dyson gas dynamics

TL;DR

This paper studies the long-term decay of charged particle density in a Dyson gas model, deriving analytical expressions for the decay rate that depend on temperature and confirming results with simulations.

Contribution

It provides an analytical description of the asymptotic decay behavior of charged particles in Dyson gas dynamics, linking the decay exponent to the inverse temperature parameter.

Findings

Decay follows a power law with exponent $(eta +1)^{-1}$ for large $eta$

For small $eta$, decay exponent is 1/2, matching uncharged particle annihilation

Analytical results are supported by computational simulations

Abstract

We analyze the annihilation of equally-charged particles based on the Brownian motion model built by F. Dyson for $N$ particles with charge $q$ interacting via the log-Coulomb potential on the unitary circle at a reduced inverse temperature $\beta$, defined as $\beta=q^2/(k_B T)$. We derive an analytical approach in order to describe the large-$t$ asymptotic behaviour for the number density decay, which can be described as a power law, i.e., $n\sim t^{-\nu}$. For a sufficiently large $\beta$, the power law exponent $\nu$ behaves as $(\beta +1)^{-1}$, which was corroborated through several computational simulations. For small $\beta$, in the diffusive regime, we recover the exponent of 1/2 as predicted by single-species uncharged annihilation.