# Relationship between Liouvile's theorem and Diffusion equation

Consider a Hamiltonian system. According to the Liouville's theorem there exists a probability density function $\rho(q^a,p_a,t)$ in the phase space whose evolution is given by $$\frac{\partial \rho }{\partial t} + \{\rho, H\} = 0 \, .$$ Now, assume that the Hamiltonian system is a sort of Lorentz gas that presents a diffusive behavior, i.e. its evolution satisfies the Diffusion equation, $$\frac{\partial \tilde{\rho} }{\partial t} = D \nabla^2 \tilde{\rho} \, ,$$ being $D$ the diffusion coefficient, $\nabla^2$ the Laplacian operator and $\tilde{\rho}(\bf{r},t)$ the probability density in the physical (configuration) space.

Does anyone know some works where a relationship between $\rho$ and $\tilde{\rho}$ is set up? Any bibliography will be appreciated.

• Mori-Zwanzig theory connects the phase space to a generalized Langevin equation (which, if you write out with Fokker-Planck, should give you a diffusion equation). – alarge Nov 24 '14 at 17:49