particle in a Newtonian system in thermal equilibrium [@Sp], and the propagation of lattice vibrations in insulating crystals [@BOS]. Since the evolution equations are linear, their analysis is simpler. From one side, these equations have been derived from an underlying microscopic dynamics globally in time [@Ga; @vBLLS; @BGS-R; @BOS]. From the other side, several results on the asymptotic behavior of the one-particle distribution have been obtained. In particular, by considering non degenerate scattering rates, under a diffusive rescaling the linear Boltzmann equation converges to the heat equation [@LK; @BLP; @BSS; @EP].

In the present paper, inspired by the general theory in [@AGS], we propose a formulation of linear Boltzmann equations in terms of gradient flows. Recently there have been some attempts to formulate the Fokker-Planck equation associated to continuous time reversible Markov chains, equivalently homogeneous linear kinetic equations, as gradient flows [@Ma; @Mi; @Er], essentially in terms of energy variational inequalities, and the potential applications of this approach have yet to be fully investigated. The case of homogeneous non linear Boltzmann equations is considered in [@Er2]. The present approach is based on an entropy dissipation inequality and can be applied naturally to the inhomogeneous case, that appears novel. In perspective, 