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Dynamical density functional theory for "dry" and "wet" active matter

In the last 50 years, equilibrium density functional theory (DFT) has been proven to be a powerful, versatile and predictive approach for the statics and structure of classical particles. This theory can be extended to the nonequilibrium dynamics of completely overdamped Brownian colloidal particles towards so-called dynamical density functional theory (DDFT). The success of DDFT makes it a promising candidate for a first-principle description of active matter. In this lecture, we shall first recapitulate classical DDFT for passive colloidal particles typically described by Smoluchowski equation. After a basic derivation of DDFT from the Smoluchowski equation, we discuss orientational degrees of freedom and the effect of hydrodynamic interactions for passive particles. This brings us into an ideal position to generalize DDFT towards active matter. In particular we distinguish between "dry active matter" which is composed of self-propelled particles that contain no hydrodynamic flow effects of a surrounding solvent and "wet active matter" where the hydrodynamic flow fields generated by the microswimmers are taken into account. For the latter, DDFT is a tool which unifies thermal fluctuations, direct particle interactions, external driving fields and hydrodynamic effects arising from internal self-propulsion discriminating between "pushers" and "pullers". A number of recent applications is discussed including transient clustering of self-propelled rods and the spontaneous formation of a hydrodynamic pump in confined microswimmers.