# Dynamic Density Functional Theory

Our group has developed a variant of time-dependent density functional theory (TD-DFT) for theoretical and computational studies of dense plasmas. While the powerful theorems of TD-DFT provide some guidance on the structure of the time dependent equations, most of the important details are left to the user. One approach, which predates the development of the theorems of TD-DFT, is to find the unknown functional from forms that are well known in some limit. For dense plasmas, which are highly collisional, functionals inspired by near equilibrium forms provide what is known as “Dynamic Density Functional Theory” or DDFT. In DDFT we take the effective forces in diffusion or hydrodynamic equations from their thermodynamic form as a gradient in the free energy. DDFT is related to a wide range of other approaches in statistical physics that employ similar approaches, such as Landau, Ginzburg-Landau and Cahn-Hilliard theories. The forces take the form

$\nabla \frac{\delta F[n,T]}{\delta n({\bf r})}$,

which is the gradient of the functional derivative of the free energy $F[n,T]$ with respect to the local density $n({\bf r},t)$. The implicit time dependence of the free energy is evolved through the usual continuity equation

$\frac{\partial }{\partial t}n({\bf r},t) = -\nabla \cdot \left[n({\bf r},t){\bf u}({\bf r},t) \right]$,

and an appropriate momentum equation for  ${\bf u}({\bf r},t)$  containing the force term above.

The work of our group has taken four main thrusts:

1. We employ DDFT formally in the context of closures of the BBGKY hierarchy to obtain hydrodynamic equations that retain pair correlation information. Our goal is to extend moment-based approaches to plasma hydrodynamics without starting with a kinetic equation (an equation that is only aware of a one-particle distribution function). Generalized forces are obtained from isothermal Helmholtz free energy functionals in the momentum equation of a two-moment hydrodynamic description.
2. Our basic DDFT formulation for dense plasmas has been extended to transport in white dwarf and neutron stars.  For the strongly coupled plasma mixtures in these stars we formulate a DDFT that includes fluctuations around the mean field described by the free energy functional of the mixture via a collision operator of the BGK form. The correlation portion of the free energy provides strong coupling corrections beyond the Hartree (electric field) terms that correctly describe strong correlations. We have shown that the central quantity for transport in such stars is the direct correlation function and we have provided a scheme for calculating the correlation functions for binary mixtures through a modified hypernetted chain approach.
3. Our current research extends DDFT to higher moments that allow for non-isothermal fluctuations associated with less dense plasmas.
4. Our work to date has focussed on ionic fluctuations and transport and has therefore employed the classical variant of DDFT; however, TD-DFT approaches are also applicable to quantum and mixed quantum-classical systems. We are extending DDFT to dense electron-ion plasmas to allow for the dynamical response of quantum electrons self-consistently with the ionic response. Such theoretical models can be applied to many laboratory plasmas, such as laser-matter interactions.

This work is led by Dr. Abdu Diaw, who is now at Los Alamos National Laboratory.

Publications

A viscous quantum hydrodynamics model based on dynamic density functional theory
Scientific Reports 7, 15352 (2017)
A. Diaw and M. S. Murillo

A Dynamic Density Functional Theory Approach to Diffusion in White Dwarfs and Neutron Star Envelopes
Astrophysical Journal 829, (2016)
A. Diaw and M. S. Murillo

A Generalized Hydrodynamics Model for Strongly Coupled Plasmas
Physical Review E 92, 013107 (2015)
A. Diaw and M. S. Murillo