Statistical Properties of the Final State in One-dimensional Ballistic Aggregation

TL;DR

This paper analyzes the long-term behavior of a one-dimensional ballistic aggregation model, revealing universal properties, exact cluster and energy distributions, and providing a comprehensive understanding of the final 'fan' state in a dissipative particle system.

Contribution

It provides a closed-form stationary measure and exact results for cluster and energy distributions, highlighting universal features independent of initial conditions.

Findings

Universal properties of the final state are identified.

Exact cluster size and energy distributions are derived.

Distributions do not follow classical extreme value universality classes.

Abstract

We investigate the long time behaviour of the one-dimensional ballistic aggregation model that represents a sticky gas of N particles with random initial positions and velocities, moving deterministically, and forming aggregates when they collide. We obtain a closed formula for the stationary measure of the system which allows us to analyze some remarkable features of the final `fan' state. In particular, we identify universal properties which are independent of the initial position and velocity distributions of the particles. We study cluster distributions and derive exact results for extreme value statistics (because of correlations these distributions do not belong to the Gumbel-Frechet-Weibull universality classes). We also derive the energy distribution in the final state. This model generates dynamically many different scales and can be viewed as one of the simplest exactly solvable model of N-body dissipative dynamics.