observation that for $p,q\in {{\mathbb R}}_+$ we have $-2\sqrt{pq} =\sup_{a>0}\big\{ -
a p - a^{-1} q \big\}$. We thus let ${{\mathcal R}} \colon {{\mathcal S}}_\mathrm{be} \to [0,+\infty]$ be the functional defined by $$\label{varR}
  \begin{split}
    {{\mathcal R}} (P,\Theta)
    &:= \sup_{\zeta,\alpha} \bigg\{
    \Theta (\zeta) - 
    \int_0^T\!{\mathop{}\!\mathrm{d}}t \iiint \! P_t({\mathop{}\!\mathrm{d}}x,{\mathop{}\!\mathrm{d}}v)\pi({\mathop{}\!\mathrm{d}}v') \, 
    \sigma(v,v') 
    \\ & \qquad \qquad
    \times \big[ {\mathop{\rm ch}\nolimits}\zeta(t,x,v,v') -1 \big] \big[ \alpha(t,x,v,v') +
    \alpha(t,x,v',v)^{-1} \big]
    \bigg\},
\end{split}$$ where the supremum is carried out over the continuous functions $\zeta
\colon [0,T]\times {{\mathbb T}}^d\times {{\mathcal V}}\times {{\mathcal V}} \to {{\mathbb R}}$ with compact support and antisymmetric with respect to the exchange of the last two variables and the bounded continuous functions $\alpha \colon
[0,T]\times {{\mathbb T}}^d\times {{\mathcal V}}\times {{\mathcal V}} \to (0,+\infty)$ uniformly bounded away from zero. As before, the basic entropy inequality implies that ${{\mathcal R}}$ is well defined. At this point the gradient flow formulation of the linear Boltzmann equations is simply specified by the following entropy dissipation inequality.

\[t:dlbe\] Let $Q\in{{\mathcal P}}({{\mathbb T}}^d\times {{\mathcal V}})$ with ${{\mathcal H}}(Q)<+\infty$. An element $(P, \Theta)\in {{\mathcal S}}_\mathrm{be}$ is a solution to the linear Boltzmann equation with initial condition $Q$ iff $P_0=Q$ and $$\begin{aligned}
\label{beq}
&&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\displaystyle{  
\int_0^T \!{\mathop{}\!\mathrm{d}}t \,P_t(\partial_t\phi + b\cdot \nabla_x \phi) 
