Exact solution of the one-dimensional ballistic aggregation

TL;DR

This paper derives an exact, long-time solution for the mass distribution in a one-dimensional ballistic aggregation model, revealing its scaling behavior and asymptotic forms, with implications for Burgers turbulence.

Contribution

It provides the first exact expression for the mass distribution in 1D ballistic aggregation, including its scaling form and asymptotic behaviors.

Findings

Mass distribution obeys a specific scaling form over time.

Scaling function exhibits power-law and exponential decay regimes.

Results are relevant to understanding Burgers turbulence.

Abstract

An exact expression for the mass distribution $\rho(M,t)$ of the ballistic aggregation model in one dimension is derived in the long time regime. It is shown that it obeys scaling $\rho(M,t)=t^{-4/3}F(M/t^{2/3})$ with a scaling function $F(z)\sim z^{-1/2}$ for $z\ll 1$ and $F(z)\sim \exp(-z^3/12)$ for $z\gg 1$. Relevance of these results to Burgers turbulence is discussed.