Ballistic aggregation for one-sided Brownian initial velocity

TL;DR

This paper analyzes a one-dimensional ballistic aggregation process with Brownian initial velocities, deriving explicit distributions and profiles, revealing self-similar scaling and exponential decay in an out-of-equilibrium system.

Contribution

It provides explicit analytical expressions for velocity distribution, density, and current profiles in a novel out-of-equilibrium ballistic aggregation model with Brownian initial conditions.

Findings

Mean density remains constant on the right side.

Mean current grows linearly with time.

Relevant lengths and masses scale as t^2.

Abstract

We study the one-dimensional ballistic aggregation process in the continuum limit for one-sided Brownian initial velocity (i.e. particles merge when they collide and move freely between collisions, and in the continuum limit the initial velocity on the right side is a Brownian motion that starts from the origin $x=0$). We consider the cases where the left side is either at rest or empty at $t=0$. We derive explicit expressions for the velocity distribution and the mean density and current profiles built by this out-of-equilibrium system. We find that on the right side the mean density remains constant whereas the mean current is uniform and grows linearly with time. All quantities show an exponential decay on the far left. We also obtain the properties of the leftmost cluster that travels towards the left. We find that in both cases relevant lengths and masses scale as $t^2$ and the evolution is self-similar.