Three-speed ballistic annihilation: phase transition and universality

Abstract

We consider ballistic annihilation, a model for chemical reactions first introduced in the 1980’s physics literature. In this particle system, initial locations are given by a renewal process on the line, motions are ballistic—i.e. each particle is assigned a constant velocity, chosen independently and with identical distribution—and collisions between pairs of particles result in mutual annihilation. We focus on the case when the velocities are symmetrically distributed among three values, i.e. particles either remain static (with given probability p) or move at constant velocity uniformly chosen among $\pm 1$. We establish that this model goes through a phase transition at $p_c=1/4$ between a subcritical regime where every particle eventually annihilates, and a supercritical regime where a positive density of static particles is never hit, confirming 1990s predictions of Droz et al. (Phys Rev E 51(6):5541–5548, 1995) for the particular case of a Poisson process. Our result encompasses cases where triple collisions can happen; these are resolved by annihilation of one static and one randomly chosen moving particle. Our arguments, of combinatorial nature, show that, although the model is not completely solvable, certain large scale features can be explicitly computed, and are universal, i.e. insensitive to the distribution of the initial point process. In particular, in the critical and subcritical regimes, the asymptotics of the time decay of the densities of each type of particle is universal (among exponentially integrable interdistance distributions) and, in the supercritical regime, the distribution of the “skyline” process, i.e. the process restricted to the last particles to ever visit a location, has a universal description. We also prove that the alternative model introduced in [7], where triple collisions resolve by mutual annihilation of the three particles involved, does not share the same universality as our model, and find numerical bounds on its critical probability.