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

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 $β$, defined as $β=q^2/(k_B T)$. 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., $n\sim t^{-ν}$. For a sufficiently large $β$, the power law exponent $ν$ behaves as $(β+1)^{-1}$, which was corroborated through several computational simulations. For small $β$, in the diffusive regime, we recover the exponent of 1/2 as predicted by single-species uncharged annihilation.

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