The ballistic annihilation threshold is 1/4

TL;DR

This paper analyzes a one-dimensional system of particles with random speeds that annihilate upon collision, establishing a precise threshold at a 1/4 initial density for the survival of stationary particles, supported by explicit probability formulas.

Contribution

It proves the exact annihilation threshold for stationary particles in a Poisson particle system with speeds {-1,0,+1}, confirming prior predictions and providing explicit survival probabilities.

Findings

Particles with speed 0 vanish almost surely if initial density ≤ 1/4.

Explicit formula for stationary particle survival probability.

Threshold at 1/4 initial density for annihilation of stationary particles.

Abstract

We consider a system of annihilating particles where particles start from the points of a Poisson process on the line, move at constant i.i.d. speeds symmetrically distributed in {-1,0,+1} and annihilate upon collision. We prove that particles with speed 0 vanish almost surely if and only if their initial density is smaller than or equal to 1/4, and give an explicit formula for the probability of survival of a stationary particle, which is in accordance with the predictions of [Droz et al. 1995]. The present proof relies essentially on an identity proved in a recent paper by J. Haslegrave.