Solution of the time-dependent Boltzmann equation

The time-dependent Boltzmann equation, which describes the propagation of radiation from a point source in a random medium, is solved exactly in Fourier space. An explicit expression in real space is given in two and four dimensions. In three dimensions an accurate interpolation formula is found. The average intensity at a large distance $r$ from the source has two peaks, a ballistic peak at time $t=r/c$ and a diffusion peak at $t\simeq r^2/D$ (with $c$ the velocity and $D$ the diffusion coefficient). We find that forward scattering adds a tail to the ballistic peak in two and three dimensions, $\propto (ct-{r)}^{-1/2}$ and $\propto -\ln \left(ct-r\right)$, respectively. Expressions in the literature do not contain this tail.