Though calculations based on density functional theory (DFT) are used remarkably widely in chemistry, physics, materials science, and biomolecular research and though the modern form of DFT has been studied for almost 60 years, some mathematical problems remain. From a physical science perspective, it is far from clear whether those problems are of major import. For context, we provide an outline of the basic structure of DFT as it is presented and used conventionally in physical sciences, note some unresolved mathematical difficulties with those conventional demonstrations, then pose several questions regarding both the time-independent and time-dependent forms of DFT that could benefit from attention in applied mathematics. Progress on any of these would aid in development of better approximate functionals and in interpretation of DFT.

Coulomb charges confined by a harmonic potential display a rich structure at strong coupling, both classical and quantum. A simple density functional theory is reviewed showing the essential role of correlations in forming shell structure and order within the shells. An overview of previous comparisons with molecular dynamics and Monte Carlo simulations is summarized and extended. It is shown that correlations for the fluid phase (shell structure only) are well approximated by those for the uniform one-component plasma even at very strong coupling. A corresponding representation of the correlations for the ordered phase is still an open question. The confirmed success for the classical density functional theory is important for the subsequent representation of the quantum case. Here, a mapping of the quantum description onto an equivalent classical description with effective potentials allows direct application of the classical methods, both theory and simulation. This is particularly relevant at low but finite temperatures where quantum simulation methods are compromised. The special case of Coulomb charges in a harmonic trap is the simplest example of more complex systems of experimental interest where confinement and strong coupling play an essential role (e.g., quantum dots, ions in complex traps, electrons on a helium surface, dusty Yukawa plasmas, ultracold neutral plasmas).

Definitions for a local pressure in an inhomogeneous fluid are considered for both equilibrium and local equilibrium states. Thermodynamic and mechanical (hydrodynamic) contexts are reconciled. Remaining problems and uncertainties are discussed.

Definitions for a local pressure in an inhomogeneous fluid are considered for both equilibrium and local equilibrium states. Thermodynamic and mechanical (hydrodynamic) contexts are reconciled. Remaining problems and uncertainties are discussed.

During the past decade, a number of attempts to formulate a continuum description of complex states of matter have been proposed to circumvent more cumbersome many-body and simulation methods. Typically, these have been quantum systems (e.g., electrons) and the resulting phenomenologies collectively often called "quantum hydrodynamics." However, there is extensive work from the past based in nonequilibrium statistical mechanics on the microscopic origins of macroscopic continuum dynamics that has not been exploited in this context. Although formally exact, its original target was the derivation of Navier-Stokes hydrodynamics for slowly varying states in space and time. The objective here is to revisit that work for the present interest in complex quantum states-possible strong degeneracy, strong coupling, and all space-time scales. The result is an exact representation of generalized hydrodynamics suitable for introducing controlled approximations for diverse specific cases and for critiquing existing work.

The quantum mechanical microscopic conservation laws for the number, momentum, and energy density operators are given in Refs. 1 and 2. However, the details of the derivation are not given and the results do not include an external force. For completeness the general derivation is given here. The time dependence of an operator A(t) which depends on the position and momenta of the system particles is given by

The calculation of dynamical properties for matter under extreme conditions is a challenging task. The popular Kubo-Greenwood model exploits elements from equilibrium density-functional theory (DFT) that allow a detailed treatment of electron correlations, but its origin is largely phenomenological; traditional kinetic theories have a more secure foundation but are limited to weak ion-electron interactions. The objective here is to show how a combination of the two evolves naturally from the short-time limit for the generator of the effective single-electron dynamics governing time correlation functions without such limitations. This provides a theoretical context for the current DFT-related approach, the Kubo-Greenwood model, while showing the nature of its corrections. The method is to calculate the short-time dynamics in the single-electron subspace for a given configuration of the ions. This differs from the usual kinetic theory approach in which an average over the ions is performed as well. In this way the effective ion-electron interaction includes strong Coulomb coupling and is shown to be determined from DFT. The correlation functions have the form of the random-phase approximation for an inhomogeneous system but with renormalized ion-electron and electron-electron potentials. The dynamic structure function, density response function, and electrical conductivity are calculated as examples. The static local field corrections in the dielectric function are identified in this way. The current analysis is limited to semiclassical electrons (quantum statistical potentials), so important quantum conditions are excluded. However, a quantization of the kinetic theory is identified for broader application while awaiting its detailed derivation.

Currently, the most common approach to calculate transport properties for materials under extreme conditions is based on the phenomenological Kubo-Greenwood method. The results of an inquiry into the justification and context of that model are summarized here. Specifically, the basis for its connection to equilibrium density functional theory (DFT) and the assumption of static ions are discussed briefly.

A recent description of an exact map for the equilibrium structure and thermodynamics of a quantum system onto a corresponding classical system is summarized. Approximate implementations are constructed by pinning exact limits (ideal gas, weak coupling), and illustrated by calculation of pair correlations for the uniform electron gas and shell structure for harmonically confined charges. A wide range of temperatures and densities are addressed in each case. For the electron gas, comparisons are made to recent path integral Monte Carlo simulations (PIMC) showing good agreement. Finally, the relevance for orbital free density functional theory for conditions of warm, dense matter is discussed briefly.

A quantum system of N Coulomb charges confined within a harmonic trap is considered over a wide range of densities and temperatures. A recently described construction of an equivalent classical system is applied in order to exploit the rather complete classical description of harmonic confinement via liquid-state theory. Here, the effects of quantum mechanics on that representation are described with attention focused on the origin and nature of shell structure. The analysis extends from the classical strong Coulomb coupling conditions of dusty plasmas to the opposite limit of low temperatures and large densities characteristic of "warm, dense matter."

A system of N classical Coulomb charges trapped in a harmonic potential displays shell structure and orientational ordering. The local density profile is well understood from theory, simulation, and experiment. Here, pair correlations are considered for this highly inhomogeneous system for both the fluid and ordered states. In the former, it is noted that there is a close relationship to pair correlations in the uniform one component plasma. For the ordered state, it is shown that the disordered tiling is closely related to the ground state Thomson sites for a single sphere ((c) 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

A theoretical description of the radial density profile for charged particles with Yukawa interaction in a harmonic trap is described. At strong Coulomb coupling shell structure is observed in both computer simulations and experiments. Correlations responsible for such shell structure are described here using a recently developed model based in density functional theory. A wide range of particle number, Coulomb coupling, and screening lengths is considered within the fluid phase. A hypernetted chain approximation shows the formation of shell structure, but fails to give quantitative agreement with Monte Carlo simulation results at strong coupling. Significantly better agreement is obtained within the hypernetted chain structure using a renormalized coupling constant, representing bridge function corrections.

A theoretical description of shell structure for charged particles in a harmonic trap is explored at strong coupling conditions of Γ = 50 and 100. The theory is based on an extension of the hypernetted chain approximation to confined systems plus a phenomenological representation of associated bridge functions. Predictions are compared to corresponding Monte Carlo simulations and quantitative agreement for the radial density profile is obtained (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A theoretical description for the radial density profile of a finite number of identical charged particles confined in a harmonic trap is developed for application over a wide range of Coulomb coupling (or, equivalently, temperatures) and particle numbers. A simple mean-field approximation neglecting correlations yields a density profile which is monotonically decreasing with radius for all temperatures, in contrast to molecular dynamics simulations and experiments showing shell structure at lower temperatures. A more complete theoretical description including charge correlations is developed here by an extension of the hypernetted chain approximation, developed for bulk fluids, to the confined charges. The results reproduce all of the qualitative features observed in molecular dynamics simulations and experiments. These predictions are then tested quantitatively by comparison with benchmark Monte Carlo simulations. Quantitative accuracy of the theory is obtained by correcting the hypernetted chain approximation with a representation for the associated bridge functions.

The dynamics of fluctuations is considered for electrons near a positive ion or for charges in a confining trap. The stationary nonuniform equilibrium densities are discussed and contrasted. The linear response function for small perturbations of this nonuniform state is calculated from a linear Markov kinetic theory whose generator for the dynamics is exact in the short time limit. The kinetic equation is solved in terms of an effective mean field single particle dynamics determined by the local density and dynamical screening by a dielectric function for the nonuniform system. The autocorrelation function for the total force on the charges is discussed.

A theoretical description of time correlation functions for electron properties in the presence of a positive ion of charge number Z is given. The simplest case of an electron gas distorted by a single ion is considered. A semi-classical representation with a regularized electron–ion potential is used to obtain a linear kinetic theory that is asymptotically exact at short times. This Markovian approximation includes all initial (equilibrium) electron–electron and electron–ion correlations through renormalized pair potentials. The kinetic theory is solved in terms of single-particle trajectories of the electron–ion potential and a dielectric function for the inhomogeneous electron gas. The results are illustrated by a calculation of the autocorrelation function for the electron field at the ion. The dependence on charge number Z is shown to be dominated by the bound states of the effective electron–ion potential. On this basis, a very simple practical representation of the trajectories is proposed and shown to be accurate over a wide range including strong electron–ion coupling. This simple representation is then used for a brief analysis of the dielectric function for the inhomogeneous electron gas.

The traditional theory of plasma line broadening is re-examined to correct for phenomenological assumptions regarding charge correlations. Conditions for static ions are assumed, and the ion microfield distribution is introduced without neglecting ion-electron correlations, and with a precise definition for the ion field at the radiator. Radiator and plasma subsystems are defined so as to make a second order calculation of electron broadening valid for the case of high Z radiators. The electron broadening operator is identified in terms of the fluctuation of the electron density at the radiator. averaged over the entire plasma constrained by a given value for the ion microfield.