A viscous quantum hydrodynamics model based on dynamic density functional theory

Dynamic density functional theory (DDFT) is emerging as a useful theoretical technique for modeling the dynamics of correlated systems. We extend DDFT to quantum systems for application to dense plasmas through a quantum hydrodynamics (QHD) approach. The DDFT-based QHD approach includes correlations in the the equation of state self-consistently, satisfies sum rules and includes irreversibility arising from collisions. While QHD can be used generally to model non-equilibrium, heterogeneous plasmas, we employ the DDFT-QHD framework to generate a model for the electronic dynamic structure factor, which offers an avenue for measuring hydrodynamic properties, such as transport coefficients via x-ray Thomson scattering.

Our viscous QHD formulation requires knowledge of both the direct correlation function C ee (q) and the electronic longitudinal viscosity η l . In this appendix we describe the methodology used to obtain these quantities.
The radial distribution function for a classical system with pair interactions u(r) can be found by solving the equations h(r) = c(r) + n 0 c(|r − r |)h(r )dr , which are the closure and the Ornstein-Zernicke equation (OZE), respectively. While these are exact, the bridge function B(r) is not known for a limited number of systems 1 . However, the electron-electron coupling is modest in most physical regimes of interest 2 , and we can neglect B(r), which is the so-called hypernetted chain (HNC) approximation, a closed set of equations for c(r) and h(r) = g(r) − 1. Here, β is the inverse temperature in energy units, n 0 is the mean density, and c(r) is the direct correlation function. Fast algorithms for solving the HNC equations over a wide range of parameters have been discussed elsewhere 3,4 . When constructing such algorithms, it is essential to remove the long-range character of the functions, choose an iteration cycle beginning with a reasonable initial guess, and solve the OZE in Fourier space. From the self-consistent solution, we can find the direct correlation function through We can examine the role of strong coupling by comparing the numerical solution of the HNC equations with the analytical solution of their mean-field approximations, obtained with c m f (r) = −β u(r).
Of course, we require the correlation functions for the quantum system, and under conditions of modest to large degeneracy. The classical HNC scheme is readily modified to treat the quantum case if quantum statistical potentials (QSPs) are used in place of u(r) 5 . Such an approach guarantees that g(r) > 0, ∀r. Dutta and Dufty 6 have compared the modified Kelbg QSP with the gold standard of PIMC 1 ; their results show that over an extremely wide range of physical conditions the QSP and PIMC results are in near perfect agreement, except at very low densities where there is a quite modest deviation. Here, we employ both diffraction and Pauli exclusion potentials, using the Hansen and McDonald 7 form where the DeBroglie thermal wavelength is defined as Here a is the Wigner-Seitz radius, Γ = β /a is the coupling parameter, r s = a/a B is the density parameter, a B is the Bohr radius, and θ is the quantum degeneracy defined as β F /β where β F is the inverse of the Fermi temperature.
Results from the HNC-QSP model can be obtained analytically in the mean-field approximation. Numerical results are shown in Fig. 1, where we compare the HNC and mean-field approximations for a range of Γ and r s = 2; as expected, the mean-field result differs from the HNC result when Γ 1. Moreover, the difference is quite small, further justifying the neglect of B(r). Figure 2 displays the spatial profile of HM, Kelgb, modified-Kelgb and pure Coulomb potentials. The comparison between these different QSPs show good agreement except at small r for some cases. For r > 1, we see some lack of sensitivity of the choice of the potential, suggesting that most QSPs will yield results close to PIMC results. However, we compare our model with PIMC 8 results in Fig. 3; except for very strong degeneracy θ = 0.5, our model is in quite reasonable agreement with PIMC, confirming the results of Dutta and Dufty. Next, to determine the longitudinal viscosity η l = (4η/3 + ξ ) which contains the shear η and bulk ξ viscosities of the electron fluid, we proceed as follows. In the zero-temperature limit, Conti and Vignale 10 suggested that the electron shear 1 It is worth emphasizing that g(r) here refers to the electron-electron g ee (r), a quantity not readily obtainable from DFT approaches.

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viscosity in the units of na 2 ω p can be approximated by where c 0 = 60, c 1 = 80, c 2 = −40, and c 3 = 62. In the classical limit, Stanton where the coupling parameter is defined as Γ = β /a, and Λ η is the effective Coulomb logarithm for viscosity, for which a numerical solution is given in Ref. 11 . In our calculation, we use the Thomas-Fermi length as the screening length. We obtain the viscosity in both the zero-and finite-temperature limits by interpolating the two formulas (6) and (7) as where the degeneracy parameter is θ = T /T F , and T F is the Fermi temperature. Note that the bulk viscosity ξ has been shown to vanish identically both in quantum 12 and classical fluids 13 . The result of the interpolation for the viscosity is shown in Fig. 4 as a function of the coupling parameter.