Note on a one-dimensional system of annihilating particles

We consider a system of annihilating particles where particles start from the points of a Poisson process on either the full-line or positive half-line and move at constant i.i.d. speeds until collision. When two particles collide, they annihilate. We assume the law of speeds to be symmetric. We prove almost sure annihilation of positive-speed particles started from the positive half-line, and existence of a regime of survival of zero-speed particles on the full-line in the case when speeds can only take 3 values.


Introduction
Let us first define informally the model that we are working on. Particles are released from the locations of a Poisson point process on either the full-line R or the half-line R`with i.i.d. velocities sampled from a distribution µ with bounded support. Each particle moves at constant velocity, and when two particles collide, they annihilate. We are interested in the possible survival of some particles forever.
In this note, we treat the case of a symmetric distribution µ (i.e. v pdq "´v if v has law µ) for particles starting from the half-line, and the case of a symmetric distribution µ on t´1, 0, 1u for particles starting from the full-line. For symmetric distributions µ, and particles starting on the full-line, only 0-speed particles may survive. The question on the half-line is more intriguing. In Section 3, we present an argument based on symmetry to conclude that positive-velocity particles annihilate almost surely. We believe, but could not prove, that negative-velocity particles have positive chance to survive. On the full-line, we are specifically interested in the symmetric discrete velocity case µ " 1´p 2 δ´1`pδ 0`1´p 2 δ 1 , where p is a parameter in r0, 1s. In this case, numerical evidence and predictions from physicists (see below) suggest that there is a phase transition: for p small (approximately for p ă 0.25), every particle annihilates almost surely but every site is still crossed by some particles at arbitrarily large times, while for larger values of p, 0-speed particles manage to survive with positive probability. In Section 4, we present a simple proof of survival at p ą 1{3 and discuss some extensions. We could not prove that particles die at small p, and leave it as another open question (see the discussion below regarding this question). Although this particle system was little known by mathematicians until recently, it turns out to have been introduced in the physics community in the 1990's (cf. [3]), where it is known as ballistic annihilation, and has been an active subject of research at the time. Note however that the special case of two speeds was considered earlier ( [7,12]), cf. also, for a mathematical treatment [2]. In the case of a continuous speed distribution (with Poisson initial distribution on the full-line), several heuristics and numerical simulations suggest a polynomial decay of the density of particles and of their average speed, and universal relations between their exponents (cf. [3,11,16]). For discrete velocity distributions, beyond the case of two speeds, the cases of three speeds (as in the present paper) and of four speeds have been considered. In the symmetric three-speed case µ " 1´p 2 δ´1p δ 0`1´p 2 δ 1 , Krapivsky, Redner and Leyvraz [10] infered the critical value p " 1 4 from heuristic and numerical arguments, together with exponents for the decay of density c v ptq of each speed at time t: at p ă 1 4 , one should have c 0 ptq « t´1 and c`1ptq « t´1 {2 ; at p " 1 4 , c 0 ptq « c`1ptq « t´2 {3 ; and at p ą 1 4 , c 0 ptq Ñ C and c˘ptq should decay faster than polynomially. Essentially at the same time, Droz, Frachebourg, Piasecki and Rey [5], using computation of Piasecki [14], strongly supported these asymptotics by showing that exact computations can be done in this and possibly other discrete cases, furthermore giving explicit prefactors: where A and u are also explicit functions of p. These very precise results come from the exact resolution of an involved differential equation satisfied by the probability density function of the interdistance between neighbor particles at time t conditioned on having given speeds. The derivation and resolution of this equation is not entirely written in a rigorous way but it is not unlikely it might be turned to a formal proof. Unfortunately though, as Krapivsky et al. [10] already pointed out, the previous explicit resolution is "a formidable task" and there is still need for methods "that would provide better intuitive insights into the intriguing qualitative features of ballistic annihilation". In particular, this computation, even if indeed formally correct, does not seem to give an intuitive proof of survival for any small a value of p. On a side note, coalescing processes also have interesting ballistic counterparts, cf. for instance the model from [8], where speeds are resampled at collision, or the physically relevant model of ballistic aggregation, where particles have a speed and mass, and coalescence occurs with conservation of mass and momentum (i.e. particles with mass and speed respectively given by pm 1 , v 1 q and pm 2 , v 2 q produce pm 1`m2 , pm 1 v 1`m2 v 2 q{pm 1`m2 qq), cf. [4], [13]. An account of related models can be found in the review [15].
The interest for ballistic annihilating particles has known a recent revival, among the mathematical community, after a closely related "bullet problem" appeared as a challenge by LTK software engineer David Wilson on IBM website [9]. In the "bullet" model, particles are released from 0 at times of a Poisson point process, with i.i.d. positive velocities pw k q kě1 , and annihilate on collision. It was conjectured that, if the law of speeds is uniform on r0, 1s, there is a velocity v c P p0, 1q such that bullets slower than v c annihilate almost surely, while faster ones may survive. This question was apparently not considered in physics papers. Compared to ballistic annihilation started on R`, the "bullet" model corresponds to switching time with space, which results in replacing speeds w k by v k " 1{w k . Our results on the half-line can therefore be rephrased for this other model. Note that the present version of the model enjoys invariance by linear transforms of speeds and symmetry properties that make it more suitable for study than the "bullet" version.
We learned before publishing that other authors [6] independently obtained results similar to ours. The physics literature on ballistic annihilation mentioned above, in particular Reference [5], was pointed to us after we finished the first version of this paper.

Definitions and basic properties
Let px k q kPZ be a Poisson point process with intensity 1 on R under Palm measure, i.e. conditioned on containing 0:¨¨¨ă which is equivalent to px k q kě1 and p´x´kq kě1 being independent Poisson point processes on R`. The points x k are meant as starting locations of particles.
Let µ be a distribution on R with bounded support. Let pv k q kPZ be i.i.d. random variables with law µ, independent of px k q kPZ , standing for the velocities of the particles.
Let us denote by P µ the law of px k q k , pv k q k . We shall write P p in the particular case when µ " 1´p 2 δ´1`pδ 0`1´p 2 δ 1 , for p P r0, 1s. The process is defined as follows. From each location x k , a particle is released at time 0, with speed v k (hence going to the right if v k ą 0, to the left otherwise). Particles move at constant speed until they collide with another particle, at which point both annihilate and therefore don't take part in later collisions. Due to interdistances between starting locations having an atomless distribution and being independent of speeds, almost surely no triple collision happens. The model is obviously well-defined if only finitely many particles are considered, for there is a chronologically first collision to be dealt with. In the case when particles initially lie on R`, well-definedness is still ensured if speeds are lower bounded: for i, j P Z`, given x i , x j , v i , v j , the collision of i and j can indeed be checked by only considering the system formed by the finitely many particles on the left of the rightmost location where a triple collision with i and j could be triggered. Using symmetry, collisions are then well-defined on R since speeds are assumed bounded.
For k, l P Z belonging to an interval I Ă R, we denote by k ÐÑ I l the property that the particles released from x k and x l mutually annihilate in the system restricted to particles departing from tx i : i P Iu. We denote by k ÐÑ I 8 the property that the particle from x k survives, i.e. is not annihilated by another particle, in this restricted system.
Note that, for given starting locations, collision events ti ÐÑ I ju only depend on ratios of differences of speeds, so that adding the same amount to every speed (i.e. shifting µ), or multiplying them by a constant, does not change the probability of any event relating to collisions. We shall refer to this simple fact as the linear speed-change invariance property. In particular, several "bullet" models (see Introduction) are in correspondence with a given µ, through w k Þ Ñ v k " 1{pα`βw k q for any α, β such that α`βw k ą 0 a.s.. Also, a symmetry property obviously holds with respect to the reflection x Þ Ñ´x, namely that the law of the model with particles starting from I Ă R is the same as that of the reflection of the model with particles starting from´I with speeds sampled from µp´dvq.

Extinction for symmetric distributions
where the conditioning here is understood as setting v 0 " v and letting pv k q k‰0 be i.i.d. with law µ. b) If P µ p0 ÐÑ R 8q " 0, then almost surely infinitely many particles cross 0 in the system restricted to particles starting from R`. This Proposition will follow from the two lemmas below.
ju denote the number of couples of particles that meet over x. If P µ p0 ÐÑ R 8q " 0, then a.s., for all x, N x " 8.
Proof of Lemma 1. We use a parity argument inspired by [1, page 43]. First note that the event tN x " 8u does not depend on x, is translation invariant, and therefore has probability 0 or 1 due to ergodicity. Assume that P µ p0 ÐÑ R 8q " 0 and, by contradiction, 5 that N x 1 {2 ă 8 almost surely. Let us view the trajectories of particles as space-time curves (actually, line segments) in the upper half-plane H (cf. Figures 1 and 2). Since P µ p0 ÐÑ R 8q " 0, almost surely for every i P Z, there is j P Z such that i ÐÑ R j, and the trajectories of i and j split H into one finite and one infinite components. Note also that all these pairs of trajectories are disjoint of each other. Since furthermore N x 1 {2 ă 8 a.s., the union of these curves delimitates exactly one infinite component in H. This implies that the event tN x 1 {2 is evenu is invariant by even shifts, i.e. by replacing px k , v k q kPZ by px k`2´x2 , v k`2 q kPZ . N x 1 {2 is indeed the number of curves that separate x 1 2 from the infinite component, and a shift amounts to crossing a boundary and thus increases or decreases by 1 the number of curves separating from the infinite component, hence changing parity. By ergodicity, this event therefore has to have probability 0 or 1. However, its probability is 1{2 due to shift invariance and alternance of parities. The conclusion follows from this contradiction.
Proof of Lemma 2. If P µ pv 0 ą 0, 0 ÐÑ R 8q ą 0 then, by ergodicity, almost surely a positive density of particles survive and have a positive speed, and by symmetry the same holds for negative speeds, contradicting their mutual survival.
Proof of Proposition 2. Let us first prove a). If P µ p0 ÐÑ R 8q ą 0, then by Lemma 2 surviving particles have 0 speed, and by ergodicity a positive density of them survive, which prevents any positive speed particle from surviving. We can thus now suppose that P µ p0 ÐÑ R 8q " 0. Assume by contradiction that there is v ą 0 such that P µ p0 ÐÑ

R8
| v 0 " vq ą 0. Since this probability is non-increasing in v, we may furthermore choose v small enough so that µppv,`8qq ą 0. We have by symmetry that P µ p0 ÐÑ

R´´8
| v 0 " vq ą 0 and thus, by independence, with positive probability we may have (cf. Figure 3) that, at the same time, v 0 ą v, v 1 ă 0, a p´vq-speed particle at 0 would survive R´and a v-speed particle at x 1 would survive R`. On this event, both an additional v-speed particle and an additional p´vq-speed particle launched at 0 (at time 0`) would survive, counting with the particle already at 0. Since this has positive probability, by ergodicity we conclude that almost surely there is a positive density of Poisson points where this happens. This implies that N 0 ă 8, for if j, k are indices of such points, with j ă 0 ă k, then the surviving trajectories of a p´vq particle at k and of a v-particle at j bound the couples straddling 0 to lie between j and k and thus be finitely few (cf. Figure 3). Lemma 1 yields a contradiction.
Let us now consider b). Suppose P µ p0 ÐÑ R 8q " 0. Let us denote by N`(resp. N´) the number of particles that ever cross 0 in the system restricted to particles on R`(resp. R´). Assume by contradiction that P µ pN`ă 8q ą 0. One may notice that N`" N´ă 8 doesn't automatically imply that N 0 ă 8. However, there is k P N and rational locations u 1 ă v 1 ă u 2 ă v 2 ă¨¨¨ă u k ă v k such that, with positive probability, N`" k and these k particles cross 0 at times τ 1 P ru 1 , v 1 s,..., τ k P ru k , v k s. By symmetry, the same event relative to R´has the same positive probability. We can then produce a positive probability event such that (cf. Figure 3) the above happens for the process on the right of x k and on the left of x´k`1 respectively, and such that the particles from 1 to  k (resp. from 0 to´k`1) meet the k particles that arrive from the right (resp. from the left). However we then have N 0 " 0 on this event, contradicting Lemma 1.
Let us note that the following weaker result, namely the extinction of a positive-speed particle at 0 when the law of its speed is µ, is easier: Proof of the proposition. By contradiction, assume P µ p0 ÐÑ R`8 | v 0 ą 0q ą 0. Then we may find v ą 0 such that (1) P µ pv 0 P p0, vsq ą 0 and P µ p0 ÐÑ where the conditioning here is merely understood as letting v 0 " v and letting pv k q k‰0 be i.i.d. with law µ. Indeed, one may for instance take v to be the median or any quantile of P µ pv 0 P¨| 0 ÐÑ R`8 , v 0 ą 0q, for then P µ p0 ă v 0 ď vq ą 0 and P µ pv 0 ě v, 0 ÐÑ R`8 q ą 0, and the last probability is smaller than P µ p0 ÐÑ R`8 | v 0 " vqµppv, 8qq because replacing the speed v 0 at x 0 by v ď v 0 preserves survival on R`. Then (1), with its symmetric P µ p0 ÐÑ
We may remark that Lemma 2 ensures that the assumption of Proposition 2 b) is satisfied in particular if µpt0uq " 0. Also, the arguments of the proof of this lemma show that surviving particles on R have the same speed, which is deterministic and therefore has to be an atom of µ: 7 Lemma 3. If P µ p0 ÐÑ R 8q ą 0, then there is v P R such that P µ pv 0 " v | 0 ÐÑ R 8q " 1.
Proof. Assume P µ p0 ÐÑ R 8q ą 0. Let ν denote the law of v 0 given t0 ÐÑ R 8u. Let w P R. If P µ pv 0 ą w, 0 ÐÑ R 8q ą 0 and P µ pv 0 ă w, 0 ÐÑ R 8q ą 0, then, by ergodicity, almost surely a positive density of particles survive and have a speed ą w, and similarly with speed ă w, which is contradicting their mutual survival. Therefore, either νppw,`8qq " 0 or νpp´8, wqq " 0. Hence ν has to be a Dirac measure, for otherwise taking w to be its median yields a contradiction.