are phase space densities normalized such that the total number of nucleons of type $i$ is $$N_i^{\rm tot} = \int \frac{d^3x d^3p}{(2\pi)^3} f_i({\bf x,p},t) \, .$$ A differential distribution in the variable $Y$ is obtained by replacing $\sigma$ with $d\sigma/dY$.

For our purpose it is reasonable to represent the colliding nuclei as cylinders with radius $R$ and thickness $L$. All the action is along the beam axis. We assume that the phase space distributions are independent of transverse coordinates $x$ and $y$ and of transverse momentum. Integrating over the cross sectional area of the nuclei, and counting only those collisions that occur within the hot zone, yields $$N_{1+2 \rightarrow X} = \pi R^2 \int_0^{t_0} dt \int_{-vt}^{vt} dz
\int \frac{dp_{1z} \, dp_{2z}}{(2 \pi)^2}
f_1 (z,p_{1z},t) \, f_2 (z,p_{2z},t) \, v_{12}
\, \sigma_{1+2 \rightarrow X}(s_{12}) \, .$$ Here there is a change in notation: $f_i(z,p_{iz},t)/2\pi$ is the probability per unit volume to find a nucleon $i$ with longitudinal momentum $p_{iz}$ at longitudinal position $z$ at time $t$. The integration limits on $z$ ensure that the collisions under consideration really occur in the hot zone; see Figure 1. The integration limits on $t$ mean that we only count those collisions which occur before 