Diffusive limit for a Boltzmann-like equation with non-conserved momentum

We consider a kinetic model whose evolution is described by a Boltzmann-like equation for the one-particle phase space distribution $f(x,v,t)$. There are hard-sphere collisions between the particles as well as collisions with randomly fixed scatterers. As a result, this evolution does not conserve momentum but only mass and energy. We prove that the diffusively rescaled $f^{\epsilon }(x,v,t)=f({\epsilon }^{-1}x,v,{\epsilon }^{-2}t)$, as $\epsilon \to 0$ tends to a Maxwellian $M_{\rho, 0, T}=\frac{\rho}{(2\pi T)^{3/2}}\exp[{-\frac{|v|^2}{2T}}]$,where$\rho$and$T$ are solutions of coupled diffusion equations and estimate the error in $L_{x,v}^2$.