Diffusion equations from kinetic models with non-conserved momentum

We derive diffusive macroscopic equations for the particle and energy density of a system whose time evolution is described by a kinetic equation for the one particle position and velocity function f(r,v,t) that consists of a part that conserves energy and momentum such as the Boltzmann equation and an external randomization of the particle velocity directions that breaks the momentum conservation. Rescaling space and time by epsilon and epsilon square respectively and carrying out a Hilbert expansion in epsilon around a local equilibrium Maxwellian yields coupled diffusion equations with specified Onsager coefficients for the particle and energy density. Our analysis includes a system of hard disks at intermediate densities by using the Enskog equation for the collision kernel.The authors thank H Spohn, C Bernardin and especially R Esposito and D Gabrielli for very helpful correspondences. This work was supported in part by AFOSR [grant FA-9550-16-1-0037]. PLG was supported also by the Spanish governement project FIS2013-43201P. We thank the IAS System Biology divison for its hospitality during part of this work