to a general scheme formalized in [@SaSe; @Se], a gradient flow formulation is particularly handy for analyzing asymptotic evolutions and it does not require a direct analysis of the dynamics. Indeed, by comparing Definition \[t:dlbe\] and Definition \[he\] we realize that the balance equation immediately leads to the continuity equation . Moreover, taking into account the convexity and lower semicontinuity of the entropy, in order to establish the diffusive limit we only need to prove two limiting variational inequalities comparing the Dirichlet form and the kinematic term for the linear Boltzmann equation with the corresponding ones for the heat flow. We shall prove these variational inequalities but, maybe surprisingly, the two terms exchange their role in the diffusive limit: the Dirichlet form ${{\mathcal E}}$ leads to $R$ while the kinematic term ${{\mathcal R}}$ leads to the Fisher information $E$.

To carry out the analysis of the diffusive limit of linear Boltzmann equations a few extra conditions, implying in particular homogenization of the velocity, are needed. As we show in the next section, this assumptions are satisfied for few natural models. To this end, recalling that the scattering rate $\lambda$ is defined by $\lambda(v)=\int\pi({\mathop{}\!\mathrm{d}}v')\sigma(v',v)$, let $\tilde\pi$ be the probability on ${{\mathcal 