Exact solution of the one-dimensional ballistic aggregation

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.