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.

  • 2
    $\begingroup$ 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). $\endgroup$ – alarge Nov 24 '14 at 17:49

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