Search for universality in one-dimensional ballistic annihilation kinetics

TL;DR

This paper investigates the long-term behavior of one-dimensional ballistic annihilation, proposing a universal scaling law for various velocity distributions supported by numerical simulations, and finds a specific decay exponent for particle density.

Contribution

The study introduces a dynamical scaling theory for ballistic annihilation and conjectures universality across symmetric, regular velocity distributions with non-vanishing density at zero velocity.

Findings

All symmetric, regular velocity distributions with  do not vanish at zero tend to a Gaussian distribution over time.

Particle density decays as t^{-0.785 .005} in the long-time limit.

Numerical simulations support the universality conjecture for different velocity distributions.

Abstract

We study the kinetics of ballistic annihilation for a one-dimensional ideal gas with continuous velocity distribution. A dynamical scaling theory for the long time behavior of the system is derived. Its validity is supported by extensive numerical simulations for several velocity distributions. This leads us to the conjecture that all the continuous velocity distributions \phi(v) which are symmetric, regular and such that \phi(0) does not vanish, are attracted in the long time regime towards the same Gaussian distribution and thus belong to the same universality class. Moreover, it is found that the particle density decays as n(t)~t^{-\alpha}, with \alpha=0.785 +/- 0.005.