Dynamics of ballistic annihilation

TL;DR

This paper develops a comprehensive analytical and numerical framework for understanding the dynamics of ballistic annihilation across various dimensions, deriving decay exponents and velocity distributions from the Boltzmann equation.

Contribution

It introduces a systematic perturbative solution to the Boltzmann equation for ballistic annihilation, providing explicit expressions for decay exponents and velocity distributions in arbitrary dimensions.

Findings

Analytical expressions for decay exponents nd re derived.

Monte Carlo and molecular dynamics simulations confirm analytical predictions in 2D.

The approach highlights the importance of the non-linear Boltzmann equation for higher dimensions.

Abstract

The problem of ballistically controlled annihilation is revisited for general initial velocity distributions and arbitrary dimension. An analytical derivation of the hierarchy equations obeyed by the reduced distributions is given, and a scaling analysis of the corresponding spatially homogeneous system is performed. This approach points to the relevance of the non-linear Boltzmann equation for dimensions larger than one and provides expressions for the exponents describing the decay of the particle density n(t) ~ t^{-\xi} and the root mean-square velocity ${\bar v} ~ t^{-\gamma}$ in term of a parameter related to the dissipation of kinetic energy. The Boltzmann equation is then solved perturbatively within a systematic expansion in Sonine polynomials. Analytical expressions for the exponents $\xi$ and $\gamma$ are obtained in arbitrary dimension as a function of the parameter $\mu$ characterizing the small velocity behavior of the initial velocity distribution. Moreover, the leading non-Gaussian corrections to the scaled velocity distribution are computed. These expressions for the scaling exponents are in good agreement with the values reported in the literature for continuous velocity distributions in $d=1$. For the two dimensional case, we implement Monte-Carlo and molecular dynamics simulations that turn out to be in excellent agreement with the analytical predictions.