equivalent to $$\label{jj1}
  {{\mathcal I}} (f,\eta) := \int_0^T\!{\mathop{}\!\mathrm{d}}t \int\! {\mathop{}\!\mathrm{d}}x
  \iint\! \pi({\mathop{}\!\mathrm{d}}v)\,\pi({\mathop{}\!\mathrm{d}}v')\,
  \Phi_\sigma (f,f';\eta) \le 0.$$ This functional is connected with the large deviations asymptotic of a Markov chain on ${{\mathcal V}}$ with transition rates $\sigma(v',v)\pi({\mathop{}\!\mathrm{d}}v')$, see [@BFG; @MPR].

We next write $$\label{psi}
  \Phi_\varkappa (p,q;\xi) = \Phi_\varkappa(p,q;0) 
  +\xi \frac{\partial}{\partial \xi}
  \Phi_\varkappa(p,q;0)  +\Psi_\varkappa(p,q;\xi).$$ By few explicit computations, $$\begin{split}
    &\Phi_\varkappa(p,q;0) = \varkappa \big(\sqrt{p} -\sqrt{q}\big)^2\\
    &\frac{\partial}{\partial \xi} \Phi_\varkappa(p,q;0) 
    = \frac 12 \log \frac qp\\
    &\Psi_\varkappa(p,q;\xi) =
    \xi {\mathop{\rm ash}\nolimits}\frac \xi{2 \varkappa\sqrt{pq}}
    - \Big[ \sqrt{ \xi^2 + 4 \varkappa^2 pq} - 2\varkappa \sqrt{pq} \Big].
  \end{split}$$ Observe that $\Psi_\varkappa$ has the variational representation $$\label{legpsi}
  \Psi_\varkappa(p,q;\xi) =\sup_{\lambda\in {{\mathbb R}}} \Big\{ \lambda \xi -
  2\varkappa \sqrt{pq} \big[ {\mathop{\rm ch}\nolimits}\lambda -1\big] \Big\}.$$ In particular, $\Psi_\varkappa\ge 0$. Moreover, while the map $(p,q;\xi)\mapsto \Psi_\varkappa(p,q;\xi)$ is convex, the map $\xi\mapsto 
\Psi_\varkappa(p,q;\xi)$ is strictly convex. Finally, $\Psi_\varkappa(p,q;\xi)\sim \xi^2$ for $\xi$ small and $\Psi_\varkappa(p,q;\xi)\sim |\xi|\log|\xi|$ for $\xi$ large.

Observe now that for the path $(f(t), \eta(t))$, $t\in[0,T]$ satisfying the balance equation in we have $$\begin{split}
    \frac{d}{dt} {{\mathcal H}}(f(t)) &= \int\! {\mathop{}\!\mathrm{d}}x \int\! \pi({\mathop{}\!\mathrm{d}}v) \, \log f 
    \Big[ - b(v)\cdot\nabla_x f
    - \int \pi({\mathop{}\!\mathrm{d}}v') \, \eta(t,x,v,v')\Big]
    \\
    &= - \int\!{\mathop{}\!\mathrm{d}}x \iint \!
    \pi({\mathop{}\!\mathrm{d}}v)\pi({\mathop{}\!\mathrm{d}}v')\, \eta\log f
  \end{split}$$ since the first term is a total derivative 