Linear Boltzmann equations: a gradient flow formulation

I present some results obtained together with D. Benedetto and L. Bertini on a gradient flow formulation of linear kinetic equations, in terms of an entropy dissipation inequality. The setting includes the current as a dynamical variable. As an application I discuss the diffusive limit of linear Boltzmann equations and show that the rescaled entropy inequality asymptotically provides the corresponding inequality for heat equation.