in $x$. Hence, by the antisymmetry of $\eta$, $$\int\! {\mathop{}\!\mathrm{d}}x 
  \iint\! \pi({\mathop{}\!\mathrm{d}}v)\pi({\mathop{}\!\mathrm{d}}v') \,
   \eta \frac{\partial}{\partial \xi} \Phi_\varkappa(f,f';0)  
  = \frac{d}{dt} {{\mathcal H}}(f(t)).$$ for any $\varkappa >0$. Setting $\varkappa = \sigma$, inserting and integrating in time we obtain that, for any $(f(t), \eta(t))$, $t\in[0,T]$ satisfying the balance equation, it holds $$\label{uguh}
  \begin{split}
    {{\mathcal H}}(f(T)) +  \int_0^T\!{\mathop{}\!\mathrm{d}}t \int\! {\mathop{}\!\mathrm{d}}x \iint\!
    \pi({\mathop{}\!\mathrm{d}}v) \pi({\mathop{}\!\mathrm{d}}v') \,
    \big[ \Phi_{\sigma}(f,f';0)+\Psi_{\sigma}(f,f';\eta)\big]
    \\
    = {{\mathcal H}}(f(0)) + \int_0^T\!{\mathop{}\!\mathrm{d}}t \int \!{\mathop{}\!\mathrm{d}}x \iint\!
    \pi({\mathop{}\!\mathrm{d}}v) \pi({\mathop{}\!\mathrm{d}}v') \,
    \Phi_{\sigma}(f,f';\eta).
  \end{split}$$ Gathering the above computations we conclude that can be rewritten as $$\label{jj2}
  {{\mathcal H}}(f(T)) + \int_0^T\!{\mathop{}\!\mathrm{d}}t \, {{\mathcal E}}({f(t)}) + {{\mathcal R}} (f,\eta)
  \le {{\mathcal H}}(f(0))$$ where $$\label{dfsr}
  {{\mathcal E}}({f}) = 
  \int \!{\mathop{}\!\mathrm{d}}x 
  \iint \!\pi({\mathop{}\!\mathrm{d}}v)\pi({\mathop{}\!\mathrm{d}}v')\, \sigma(v,v')
  \big[ \sqrt{f'} -\sqrt{f} \big]^2$$ and $$\label{metric}
  {{\mathcal R}} (f,\eta) = \int_0^T\!{\mathop{}\!\mathrm{d}}t \int \!{\mathop{}\!\mathrm{d}}x 
  \iint\!\pi(dv)\pi(dv')\,
  \Psi_\sigma (f,f'; \eta).$$ The inequality , formally analogous to , is the proposed gradient flow formulation of the linear Boltzmann equation .

We now discuss the precise formulation in which we introduce the measures ${\mathop{}\!\mathrm{d}}P = f(x,v) \, {\mathop{}\!\mathrm{d}}x\, \pi({\mathop{}\!\mathrm{d}}v)$ and $\Theta({\mathop{}\!\mathrm{d}}t,{\mathop{}\!\mathrm{d}}x,
{\mathop{}\!\mathrm{d}}v, {\mathop{}\!\mathrm{d}}v')= \eta(t,x,v,v') \, {\mathop{}\!\mathrm{d}}t\, {\mathop{}\!\mathrm{d}}x\, {\mathop{}\!\mathrm{d}}v\, {\mathop{}\!\mathrm{d}}v'$. We first specify the hypotheses on the scattering rate $\sigma$ and the drift $b$ that are assumed to hold throughout the whole paper.

\[t:asb\]$\phantom{i}$

