this approach could be adapted to the non linear, non homogeneous Boltzmann equation.

We consider a linear Boltzmann equation of the form $$\label{BE}
  (\partial_t +b(v)\cdot \nabla_x )
  f(t,x,v)=
  \int\pi({\mathop{}\!\mathrm{d}}v')\sigma(v,v')\big[f(t,x,v')- f(t,x,v)   \big]$$ where $\pi({\mathop{}\!\mathrm{d}}v)$ is a reference probability measure on the velocity space, $b$ is the drift, $\sigma(v,v')\geq 0$ is the scattering kernel and $f$ is the density of the one-particle distribution with respect to ${\mathop{}\!\mathrm{d}}x\,\pi({\mathop{}\!\mathrm{d}}v)$. We assume the detailed balance condition, i.e. $\sigma(v,v')=\sigma(v',v)$. The entropy $\mathcal{H}(f)=\int {\mathop{}\!\mathrm{d}}x\int\pi({\mathop{}\!\mathrm{d}}v)f\log f$ is a Lyapunov functional for the evolution , and the transport term do not affect its rate of decrease. This observation will allow to formulate as the following entropy dissipation inequality $$\label{EDI}
 \mathcal{H}\big(f(T)\big) +\int_0^T {\mathop{}\!\mathrm{d}}t\, \mathcal{E}\big( f(t)\big) + \mathcal R_0 (f)\leq \mathcal H \big(f(0)\big),$$ where $\mathcal E\big( f\big)=\displaystyle\int {\mathop{}\!\mathrm{d}}x\,\iint \pi({\mathop{}\!\mathrm{d}}v)\pi({\mathop{}\!\mathrm{d}}v')\sigma(v,v')\big[\sqrt{f(x,v')}- \sqrt{f(x,v)} \big]^2$ is the Dirichlet form of the square root of $f$ and $\mathcal R_0\geq
0$ is a kinematic term that will be defined later.

As an application of we discuss the diffusive limit of the linear Boltzmann equation. This is a classical result, but the gradient flow formulation provides a transparent proof and allows to consider more general initial conditions, which are only required to satisfy an entropy bound. More precisely, 