Explicit asymptotic velocity of the boundary between particles and antiparticles

TL;DR

This paper derives explicit asymptotic formulas for the boundary position between particles and antiparticles on a line, considering infinite initial distributions with specified velocities, and analyzing collision dynamics over time.

Contribution

It provides the first explicit asymptotic characterization of the boundary between particles and antiparticles in a one-dimensional collision system.

Findings

Explicit asymptotic velocity of the boundary is derived.

The boundary's position over time is characterized mathematically.

Results apply to systems with multiple velocity classes for particles and antiparticles.

Abstract

On the real line initially there are infinite number of particles on the positive half-line., each having one of $K$ negative velocities $v_{1}^{(+)},...,v_{K}^{(+)}$. Similarly, there are infinite number of antiparticles on the negative half-line, each having one of $L$ positive velocities $v_{1}^{(-)},...,v_{L}^{(-)}$. Each particle moves with constant speed, initially prescribed to it. When particle and antiparticle collide, they both disappear. It is the only interaction in the system. We find explicitly the large time asymptotics of $\beta(t)$ - the coordinate of the last collision before $t$ between particle and antiparticle.