Exact Solution of the One-Dimensional Ballistic Aggregation

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 $\rho (M;t){=t}^{-4\mathrm{}/3}F\left(M/t^{2/3}\right)$ with a scaling function $F\left(z\right)\sim z^{-1\mathrm{}/2}$ for $z\ll 1$ and $F\left(z\right)\sim \exp \left({-z}^3/12\right)$ for $z\gg 1$. Applications of these results to Burgers turbulence are discussed.