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1 Introduction Variational methods play a key role in physics. Density functional theory (DFT) is a special and important example of such a variational formulation. DFT relies on the fact that there is a functional of the one-particle density which gives access to the equilibrium thermodynamics when it is minimized with respect to the density. This important theory can be both applied to quantum-mechanical electrons and to classical systems. In this book chapter we shall consider nonequilibrium situations for completely overdamped Brownian dynamics of colloids. A dynamical version of DFT, the socalled dynamical density functional theory (DDFT), is available and makes dynamical predictions which are in good agreement with computer simulations. Here we shall derive DDFT for Brownian dynamics in a tutorial way from the microscopic Smoluchowski equation and mention some applications such as crystallization on a patterned substrate. The theory will then be generalized towards hydrodynamic interactions between the particles and to orientational degrees of freedom describing e.g. rod-like colloids. Finally some recent developments will be discussed. For parts of this tutorial we follow the ideas outlined in Ref. [60, 61].

H. Löwen (B) Institut für Theoretische Physik II: Weiche Materie, Heinrich-Heine-Universität Düsseldorf, Universitätsstrasse 1, D-40225 Düsseldorf, Germany e-mail: [email protected] © Springer Science+Business Media Singapore 2017 J. Wu (ed.), Variational Methods in Molecular Modeling, Molecular Modeling and Simulation, DOI 10.1007/978-981-10-2502-0_9

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2 Density Functional Theory (DFT) in Equilibrium 2.1 Basics We shall consider density functional theory (DFT) here for classical systems at finite temperature as opposed to DFT for quantum mechanical electrons. The cornerstone of classical density functional theory is an existence theorem combined with a basic variational principle [25, 55, 57, 58, 86]. In detail, there exists a unique grand canonical free energy-density-functional Ω(T, μ, [ρ]), which gets minimal for the equilibrium density ρ0 (r) and then coincides with the real grand canonical free energy, i.e.  δΩ(T, μ, [ρ])  = 0. (1)  δρ(r) ρ(r)=ρ0 (r)

Here, T is the imposed temperature and μ the prescribed chemical potential of the system. In particular DFT is also valid for systems which are inhomogeneous on a microscopic scale. In principle, all fluctuations are included if an imposed external potential Vext (r) breaks all symmetries. For interacting systems in three dimensions (3d), however, the functional Ω(T, μ, [ρ]) is not known. One can split the functional Ω(T, μ, [ρ]) exactly as follows  d 3 r ρ(r) (Vext (r) − μ)

Ω(T, μ, [ρ]) = F (T, [ρ]) +

(2)

V

where F (T, [ρ]) is a Helmholtz free energy functional and V denotes the system volume. Fortunately, there are few exceptions where the density functional is known exactly. First, for low density, the ideal-gas-limit is reached and the density functional can be constructed analytically (see below). Next leading orders for