Some exact results for Boltzmann's annihilation dynamics

TL;DR

This paper derives exact analytical results for the time evolution of particle density in ballistic annihilation systems, revealing different decay behaviors depending on velocity distributions, and confirms predictions with molecular dynamics simulations.

Contribution

It provides new exact solutions for Boltzmann's annihilation dynamics with specific velocity distributions, including static traps, and compares these with simulations.

Findings

Power law decay with non-universal exponents

Exponential decay depending on velocity ratios

Excellent agreement with molecular dynamics simulations

Abstract

The problem of ballistic annihilation for a spatially homogeneous system is revisited within Boltzmann's kinetic theory in two and three dimensions. Exact analytical results are derived for the time evolution of the particle density for some isotropic discrete bimodal velocity modulus distributions. According to the allowed values of the velocity modulus, different behaviors are obtained: power law decay with non-universal exponents depending continuously upon the ratio of the two velocities, or exponential decay. When one of the two velocities is equal to zero, the model describes the problem of ballistic annihilation in presence of static traps. The analytical predictions are shown to be in excellent agreement with the results of two-dimensional molecular dynamics simulations.