Ballistic annihilation with superimposed diffusion in one dimension

TL;DR

This paper studies a one-dimensional particle system with combined ballistic motion and diffusion, revealing new long-time behaviors, fractal structures, and complex dynamics when initial conditions are slightly imbalanced.

Contribution

It uncovers previously unnoticed long-time behaviors, fractal structures, and detailed dynamics for imbalanced initial conditions in a ballistic-annihilation-diffusion model.

Findings

Particle concentration decays faster than pure ballistic or diffusive cases.

Existence of a fractal structure in the system.

Complex crossover behaviors for small initial imbalances.

Abstract

We consider a one-dimensional system with particles having either positive or negative velocity, which annihilate on contact. To the ballistic motion of the particle, a diffusion is superimposed. The annihilation may represent a reaction in which the two particles yield an inert species. This model has been the object of previous work, in which it was shown that the particle concentration decays faster than either the purely ballistic or the purely diffusive case. We report on previously unnoticed behaviour for large times, when only one of the two species remains and also unravel the underlying fractal structure present in the system. We also consider in detail the case in which the initial concentration of right-going particles is $1/2+\varepsilon$, with $\varepsilon\neq0$. It is shown that a remarkably rich behaviour arises, in which two crossover times are observed as $\varepsilon \to 0$.