and Kapusta [@evol] solved the resulting evolution equations and found that the invariant distribution function in this model is given by $$\begin{aligned}
H(x,N) = x \sum_{n=1}^N \left( \begin{array}{c} N\\n \end{array} \right)
w^n (1-w)^{N-n}\,\, \frac{(-\ln{x})^{n-1}}{(n-1)!} +(1-w)^N \, \delta (x-1)
\, .\end{aligned}$$ The $\delta$-function represents elastic and soft inelastic contributions to the evolution of the nucleon through the matter. The probability $w$ is the ratio of inelastic to total nucleon– nucleon cross section. It corresponds to the probability that the nucleon scatters inelastically and therefore drops out of the evolution described by $H$; it is approximately 0.8 in free space. Csernai and Kapusta found that it reduces to about 0.5 for nucleons propagating through a nucleus. This value allowed them to obtain a good representation of data with beam energies in the range of 6-405 GeV. In our case the nucleon is propagating through hot and dense hadronic matter. We keep $w$ as a free parameter since we don’t know how the value of $w$ changes due to the thermal excitations and the increased density.

We are interested in the number of pion-producing nucleon-nucleon collisions with a relatively high center-of-momentum energy squared $s$. Our basic result from this section is $$\begin{aligned}
\frac{dN^{\rm 