Ballistic Annihilation

Ballistic annihilation with continuous initial velocity distributions is investigated in the framework of Boltzmann equation. The particle density and the rms velocity decay as $c=t^{-\alpha }$ and $<v>=t^{-\beta }$, with the exponents depending on the initial velocity distribution and the spatial dimension. For instance, in one dimension for the uniform initial velocity distribution we find $\beta =\mathrm{0.230472...}$. We also solve the Boltzmann equation for Maxwell particles and very hard particles in arbitrary spatial dimension. These solvable cases provide bounds for the decay exponents of the hard sphere gas.