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A kinetic description of dense liquids is one of the-long-standing problems in statistical mechanics. We apply here a well-known Markovian approximation to the Mori-Fujisaka nonlinear generalized Langevin equation. This enables us to derive systematically and without further approximations the Smoluchowski equation for an interacting many-body system and the dynamic density functional theory (DDFT), which are playing important roles for the glass transition and dynamics in supercooled liquids. The free energy functional in our DDFT may be termed microcanonical as compared with the grand-canonical one widely used in the equilibrium theory of liquids.

Markovian approximation and dynamic density functional theory for classical dense liquids