TIME DEPENDENT DENSITY FUNCTIONAL THEORY ... - TDDFT.org

Quantum many-body dynamics in Lagrangian
frame: Geometric formulation of TDDFT
I.V. Tokatly
Lerhrstuhl für Theoretische Festkörperphysik, Universität Erlangen-Nürnberg, Staudtstrasse
7/B2, 91058 Erlangen, Germany
In this talk I formulate equations of time-dependent density functional theory (TDDFT) in the
co-moving Lagrangian reference frame. The main advantage of Lagrangian description of manybody
dynamics is that in the co-moving frame both the density of particles and the current density
become the exact integrals of motion. The current density is exactly zero in every point of Lagrangian
space, while the density of particles preserves its initial form. In other words, a local
co-moving observer will see no currents in the system, and a stationary density distribution – the
picture, which is very similar to that, seen in the equilibrium system from the laboratory frame.
Reformulation of the theory in Lagrangian frame shows that the most natural complete set of basic
variables in TDDFT consists of Lagrangian coordinate, ξ, a symmetric deformation tensor g µν ,
and a skew-symmetric vorticity tensor, F µν . These three quantities, one vector, one symmetric
tensor, and one skew-symmetric tensor, contain twelve independent numbers, which are required
for the complete local characterization of a deformed state of any continuum media. Namely, ξ,
g µν and F µν respectively describe the translation, deformation and rotation of an infinitesimal fluid
element. On the other hand, tensors g µν and F µν have a clear geometric meaning: g µν is a metric
tensor in Lagrangian space, while F µν determines a local angular velocity of the co-moving frame.
All above quantities are functionals of the fluid velocity in accordance with Runge-Gross mapping
theorem. Formulation of TDDFT in terms of new basic variables allows to regularly derive a local
nonadiabatic approximation for the xc potential. Stationarity of the density in the co-moving frame
makes the derivation to a large extent similar to that of the static LDA. In the last part of the talk
I will show a few explicit examples of nonlinear nonadiabatic xc functionals.
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