to the propagation equation, so that $$\begin{aligned}
\label{eq:boltzmann}
\frac{\partial \tilde{n}(t,E)}{\partial t}
& = & \frac{\partial}{\partial E}(b \tilde{n}(t,E) ) + {\cal L}(t,E) \\ 
& - & c n_{t} \sigma \tilde{n}(t,E)
+ c n_{t} \int_{E}^{\infty} \! dE^{\prime} \, \tilde{n}(t,E^{\prime}) \sum_{i}\frac{d\sigma_{i}}{dE} \,. \nonumber\end{aligned}$$ We assume that the neutrino (or other particle) targets are non-relativistic; if they are not, their energy distribution needs to be taken into account in the propagation equation. The third term on the right accounts for attenuation at a given energy due to scattering with cross section $\sigma(E)$. The fourth term accounts for particle regeneration from one energy to another, including when an incoming particle of energy $E^\prime$ is down-scattered to a lower energy $E$ but not lost and when a target particle with their rest mass energy is up-scattered to energy $E$ to join the beam. The distributions of secondary particles are described by the differential cross sections $d\sigma_{i}(E,E^\prime)/dE$, where $i$ denotes each process. Here, we only include down-scattering and up-scattering with neutrino targets; this could be generalized. The net effect is therefore a distortion of the beam spectrum in a way that conserves energy but not particle number.

This propagation equation automatically takes into account the re-scattering of 