Ballistic Annihilation Kinetics: The Case of Discrete Velocity Distributions

TL;DR

This paper investigates the annihilation kinetics of particles with discrete velocities, revealing unique decay behaviors and phenomena through theoretical analysis and large-scale simulations.

Contribution

It introduces new analytical results and efficient algorithms to study ballistic annihilation with discrete velocity distributions, uncovering novel decay laws and phenomena.

Findings

Different velocity species decay with distinct power laws.

Special initial conditions lead to uniform decay rates across species.

Impurity survival probability exhibits a logarithmic squared decay.

Abstract

The kinetics of the annihilation process, $A+A\to 0$, with ballistic particle motion is investigated when the distribution of particle velocities is {\it discrete}. This discreteness is the source of many intriguing phenomena. In the mean field limit, the densities of different velocity species decay in time with different power law rates for many initial conditions. For a one-dimensional symmetric system containing particles with velocity 0 and $\pm 1$, there is a particular initial state for which the concentrations of all three species as decay as $t^{-2/3}$. For the case of a fast ``impurity'' in a symmetric background of $+$ and $-$ particles, the impurity survival probability decays as $\exp(-{\rm const.}\times \ln^2t)$. In a symmetric 4-velocity system in which there are particles with velocities $\pm v_1$ and $\pm v_2$, there again is a special initial condition where the two species decay at the same rate, $t^{-\a}$, with $\a\cong 0.72$. Efficient algorithms are introduced to perform the large-scale simulations necessary to observe these unusual phenomena clearly.