# On collisions times of self-sorting interacting particles in one-dimension with random initial positions and velocities

## TL;DR

This paper analyzes the asymptotic behavior of collision times in a one-dimensional particle system with random initial conditions, comparing elastic and non-elastic interactions and their impact on system sorting.

## Contribution

It derives asymptotic distributions for collision times in elastic systems and explores the effects of non-elastic collisions through numerical simulations.

## Key findings

- Asymptotic distributions for collision times in elastic systems
- Impact of initial distributions on collision time behavior
- Numerical results on non-elastic collision effects

## Abstract

We investigate a one-dimensional system of $N$ particles, initially distributed with random positions and velocities, interacting through binary collisions. The collision rule is such that there is a time after which the $N$ particles do not interact and become sorted according to their velocities. When the collisions are elastic, we derive asymptotic distributions for the final collision time of a single particle and the final collision time of the system as the number of particles approaches infinity, under different assumptions for the initial distributions of the particles' positions and velocities. For comparison, a numerical investigation is carried out to determine how a non-elastic collision rule, which conserves neither momentum nor energy, affects the median collision time of a particle and the median final collision time of the system.