The relationship of the thermodynamics for non-uniform systems and density functional theory is reviewed. It is observed that the relevant universal functionals are the same in both cases under appropriate conditions for thermodynamics. The variational context arises from the convexity properties of the functionals as represented by equilibrium statistical mechanics. Within thermodynamics a choice of local density or local chemical potential for the local variable can be made. In the latter case a "dual" form for density functional theory applies ("potential functional" theory). The special case of electrons and ions is considered, and the important condition of charge neutrality for thermodynamics is noted. These general results are extended to include the interatomic pair potential as an independent variable. Its conjugate is the equilibrium pair correlation function. As an application, the construction of a classical system to represent a given quantum system in terms of effective potentials is shown to result from the Euler equations of this extended variational formulation.

The data that support the findings of this study are available on request from the corresponding author. The data are not publicly available due to privacy or ethical restrictions.

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.

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

Renewed interest in the homogeneous electron gas (HEG) has been stimulated by recent accurate simulations of it over a wide range of densities and temperatures. Those data, combined with known theoretical limits, have led to analytical representations of the free energy. Such a representation is, at least in principle, the complete HEG equation of state. The initial objective here is to establish that the two best representations [“corrKSDT,” Phys. Rev. Lett. 112, 076403 (2014), Phys. Rev. Lett. 120, 076401 (2018), and “GDB” Phys. Rev. Lett. 119, 135001 (2017)] of the simulation data and constraints are effectively the same in both functional form and accuracy of representation. The second objective is to disclose and delineate a significant difficulty. Despite their expected accuracy for the free energy, the underlying functional form is not adequate for derived thermodynamic properties. As an example, the specific heats obtained from the representations exhibit anomalies suggesting the need first for additional simulation data in critical regimes, then for refined fitting functions. The existing representations are, however, almost certainly adequate for applications based on the free energy alone (e.g., density-functional theory for warm dense matter). The third objective is to show that, despite their inability to provide a complete thermodynamic description of the HEG, the best analytical representations do provide a fully adequate exchange-correlation local density approximation for free-energy density-functional calculations.

The density linear response function for an inhomogeneous system of electrons in equilibrium with an array of fixed ions is considered. Two routes to its evaluation for extreme conditions (e.g., warm dense matter) are considered. The first is from a recently developed short-time kinetic equation; the second is from time-dependent density functional theory. The result from the latter approach agrees with that from kinetic theory in the “adiabatic approximation,” providing support and contextual clarity for each. Both provide a connection to the phenomenological Kubo-Greenwood method for calculating transport properties.

Realizing the potential for predictive density functional calculations of matter under extreme conditions depends crucially upon having an exchange-correlation (XC) free energy functional accurate over a wide range of state conditions. Unlike the ground-state case, no such functional exists. We remedy that with systematic construction of a generalized gradient approximation XC free-energy functional based on rigorous constraints, including the free energy gradient expansion. The new functional provides the correct temperature dependence in the slowly varying regime and the correct zero-T, high-T, and homogeneous electron gas limits. Its accuracy in the warm dense matter regime is attested by excellent agreement of the calculated deuterium equation of state with reference path integral Monte Carlo results at intermediate and elevated T. Pressure shifts for hot electrons in compressed static fcc Al and for low density Al demonstrate the combined magnitude of thermal and gradient effects handled well by this functional over a wide T range.

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.

The potential for density functional theory calculations to address, reliably, the extreme conditions of warm dense matter is predicated upon having an accurate representation for the free energy functional over a wide range of state conditions. Distinct from the ground-state situation, no such exchange-correlation functional exists. We remedy that with a systematic, constraint-based construction of a non-empirical finite-temperature generalized gradient approximation exchangecorrelation functional, based on the free energy gradient expansion and other formal limits. The new functional provides the correct temperature dependence in the slowly varying regime and the correct zero-T, high-T, and homogeneous electron gas limits. Its application in Kohn-Sham calculations for hot electrons in a static fcc Aluminum lattice demonstrates the combined magnitude of thermal and gradient effects accounted for by this functional. Its accuracy in the warm dense matter regime is attested by excellent agreement of the calculated deuterium equation of state with reference path integral Monte Carlo results at intermediate and elevated temperatures.

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.

V.V. Krasiev, 2, ∗ James W. Dufty, and S.B. Trickey Quantum Theory Project, Department of Physics and Department of Chemistry, P.O. Box 118435, University of Florida, Gainesville FL 32611-8435, USA Laboratory for Laser Energetics, University of Rochester, 250 East River Road, Rochester NY 14623, USA Department of Physics, P.O. Box 118435, University of Florida, Gainesville FL 32611-8435, USA Quantum Theory Project, Department of Physics and Department of Chemistry, P.O. Box 118435, University of Florida, Gainesville FL 32611-8435, USA

Approximations for the many-fermion free-energy density functional that include the Thomas-Fermi (TF) form for the noninteracting part lead to singular densities for singular external potentials (e.g., attractive Coulomb). This limitation of the TF approximation is addressed here by a formal map of the exact Euler equation for the density onto an equivalent TF form characterized by a modified Kohn-Sham potential. It is shown to be a "regularized" version of the Kohn-Sham potential, tempered by convolution with a finite-temperature response function. The resulting density is nonsingular, with the equilibrium properties obtained from the total free-energy functional evaluated at this density. This new representation is formally exact. Approximate expressions for the regularized potential are given to leading order in a nonlocality parameter, and the limiting behavior at high and low temperatures is described. The noninteracting part of the free energy in this approximation is the usual Thomas-Fermi functional. These results generalize and extend to finite temperatures the ground-state regularization by R. G. Parr and S. Ghosh [Proc. Natl. Acad. Sci. U.S.A. 83, 3577 (1986)] and by L. R. Pratt, G. G. Hoffman, and R. A. Harris [J. Chem. Phys. 88, 1818 (1988)] and formally systematize the finite-temperature regularization given by the latter authors.

The potential for density functional theory (DFT) calculations to address, reliably, the extreme conditions of warm dense matter (WDM) is predicated upon having an accurate representation for the exchange-correlation (XC) free energy functional. To that end, we give a systematic, constraint-based construction of a non-empirical finite-temperature (T) generalized gradient approximation (GGA), based on the XC free energy gradient expansion. The new functional provides the correct T-dependence in the slowly varying regime and the correct zero-T, high-T, and homogeneous electron gas limits. Kohn-Sham calculations with finite-T XC functionals (the new GGA and the local density approximation (LDA)), with ground state (LDA and GGA) functionals, and with a previously published mixed ground state GGA and finite-T LDA functional on a static fcc Aluminum lattice with hot electrons demonstrate the combined magnitude of thermal and gradient effects accounted for by the finite-T GGA functional. Accuracy of the new GGA functional in the WDM regime is illustrated by deuterium equation of state calculations in excellent agreement with reference path integral Monte Carlo results at intermediate and elevated T.