= \frac 12 \int\!\Theta({\mathop{}\!\mathrm{d}}t,{\mathop{}\!\mathrm{d}}x,{\mathop{}\!\mathrm{d}}v, {\mathop{}\!\mathrm{d}}v') \, 
\big[\phi(t,x,v) -\phi(t,x,v')\big], }
\\
\label{ineq1-calore2}
&&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\displaystyle{  
  {{\mathcal H}}(P_T)+\int_0^T \!{\mathop{}\!\mathrm{d}}t\, {{\mathcal E}}(P_t)+ {{\mathcal R}} (P, \Theta) 
  \leq {{\mathcal H}}(Q).}\end{aligned}$$ for all continuous functions $\phi\colon (0,T)\times {{\mathbb T}}^d\times {{\mathcal V}}$ with compact support and continuously differentiable in the first two variables.

\[t:rem\] If $(P,\Theta)$ is a solution to the linear Boltzmann equation in the time interval $[0,T]$ then it solves the same problem in the time interval $[0,t]$, $t\le T$ as well. This follows from the fact that any element $(P,\Theta)\in {{\mathcal S}}_\mathrm{be}$ satisfies, for $0\le s < t
  \le T$, the inequality $$\label{jj1t}
      {{\mathcal H}}(P_t)+\int_s^t \!{\mathop{}\!\mathrm{d}}u\, {{\mathcal E}}(P_u)+ {{\mathcal R}}^{s,t} (P, \Theta_{[s,t]}) 
      \geq {{\mathcal H}}(P_s)$$ where $\Theta_{[s,t]}$ is the restriction of $\Theta$ to the interval $[s,t]$ and the kinematic term ${{\mathcal R}}^{s,t}$ is defined as in with the interval $[0,T]$ replaced by $[s,t]$. This inequality corresponds in fact to the trivial inequality ${{\mathcal I}}_{[s,t]}(P,\Theta)\ge 0$ where the action functional ${{\mathcal I}}_{[s,t]}$ is defined as in with the interval $[0,T]$ replaced by $[s,t]$. The actual proof of is detailed in Appendix \[app2\].

It is of course possible to obtain a formulation only in terms of the one particle distribution. More precisely, the formulation is obtained 