rewrite the linear Boltzmann equation in the form $$\label{continuita}
  \begin{cases}
  \big(\partial_t +b(v)\cdot \nabla_x \big) f(t,x,v)
  +\int\! \pi(dv')\, \eta(t,x,v,v')=0\\
  \eta = \eta^f
  \end{cases}$$ We understand that the first equation has to be satisfied weakly and we shall refer to it as the *balance* equation. We are going to rewrite the condition $\eta=\eta^f$ as an inequality that expresses the decrease of the relative entropy ${{\mathcal H}}$. To this end, given $\varkappa\geq 0$ let $\Phi_\varkappa\colon {{\mathbb R}}_+\times{{\mathbb R}}_+\times {{\mathbb R}}\to [0,+\infty)$ be the convex function defined by $$\Phi_\varkappa (p,q;\xi) :=\sup_{\lambda\in{{\mathbb R}}}\Big\{\lambda \xi -
  \varkappa p \big(e^\lambda -1) - \varkappa q \big(e^{-\lambda}
  -1\big)\Big\}$$ observing that given $p,q\in{{\mathbb R}}_+$ the map $\xi\mapsto
\Phi_\varkappa(p,q;\xi)$ is positive (take $\lambda=0$), and equal to zero iff $\xi=\varkappa (p-q)$. Explicitly, as few computations shows, $\Phi_\varkappa$ reads $$\label{Phi}
  \begin{split}
    \Phi_\varkappa (p,q;\xi)  &=
        \xi \Big[ {\mathop{\rm ash}\nolimits}\frac \xi{2 \varkappa\sqrt{pq}}
      -{\mathop{\rm ash}\nolimits}\frac {\varkappa (p-q)}{2 \varkappa\sqrt{pq}} \Big]
    \\
    & - \Big[ \sqrt{ \xi^2 + 4 \varkappa^2 pq} - 
      \sqrt{ \big[\varkappa(p-q)\big]^2 + 4 \varkappa^2 pq}\Big]
  \end{split}$$ where we recall that ${\mathop{\rm ash}\nolimits}(z) = \log(z+\sqrt{1+z^2})$. We note that if $\varkappa=0$ then $\Phi_0(p,q;0)=0$ while $\Phi_0(p,q;\xi)=+\infty$ if $\xi\neq 0$.

Fix a path $(f(t),\eta(t))$, $t\in[0,T]$ satisfying $\eta(t,x,v,v')=-\eta(t,x,v',v)$ and the balance equation in . The condition $\eta(t)= \eta^{f(t)}$, $t\in[0,T]$ is 