A viscous quantum hydrodynamics model based on dynamic density functional theory

Abstract

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, satis es 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 o ers an avenue for measuring hydrodynamic properties, such as transport coe cients via x-ray Thomson scattering.

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A viscous quantum hydrodynamics
model based on dynamic density
functional theory
Abdourahmane Diaw & Michael S. Murillo
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, satises 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 oers an avenue for measuring hydrodynamic properties, such as transport
coecients via x-ray Thomson scattering.
Access to high-power laser sources, such as the Linac Coherent Light Source (LCLS)1, National Ignition Facility
(NIF)2 and Omega Laser3, has opened the path to investigating essential properties of non-ideal plasmas such
as ionization potential depression4, transport coecients5 and ionization state6. Understanding the dynamical
properties of non-ideal plasmas is critical for modeling and designing high energy-density science experiments,
including inertial-connement fusion7, cluster explosions8, laser-produced ion beams9, hypervelocity impacts10,
in nanotechnology11,12 and astrophysics13.
Among all the approaches to modeling heterogeneous, non-equilibrium quantum systems, quantum hydro-
dynamics (QHD) is a computationally attractive approach with rich history in statistical mechanics. Shortly aer
the development of quantum mechanics, Bloch14 proposed the first QHD model by simply choosing the
omas-Fermi pressure for the electrons in an otherwise classical hydrodynamics model. In 1964, Hohenberg
and Kohn15 developed ground-state density functional theory for the inhomogeneous electron gas, which was
immediately generalized to nite temperature by Mermin16. Combining the ideas of Bloch with DFT, Ying17 pro-
posed a new quantum hydrodynamic model via an adiabatic generalization of the density functionals. In Ying’s
model, the pressure is represented by P[n(r,t)] with n(r,t) a time-dependent density described by the continuity
equation. Ying’s QHD model includes explicitly all correlation and exchange eects included in the chosen energy
functional. Using an alternate approach, Gasser and Jüngel18 derived QHD equations using the Schrodinger equa-
tion with Wentzel-Kramers-Brillouin (WKB) wave functions. is approach yields the classical momentum equa-
tion with the Bohm potential but it does not account for correlations. Correlations effects and quantum
degeneracy can be included in an ad hoc manner in this model by replacing the Bohmian potential with quantum
potentials12 or self-consistently through orbital-free density functional theory (OF-DFT)19,20. In yet another
approach, using the moment expansion of the Wigner-Boltzmann equation, Gardner21 proposed a QHD model
for semiconductor devices that extends the classical hydrodynamic model to include
O()
2

quantum corrections.
Similar results were obtained with the Wigner-Poisson system by Manfredi and Haas22 for a quantum electron
gas. Following Levermore23, Degond and Ringhofer24 used a non-commutative version of the entropy externali-
zation principle to build a QHD model starting from the quantum Liouville equation. e moment equations are
closed by a quantum Wigner distribution function that minimizes the entropy.
Despite these important advances, describing collisional processes in moderately coupled quantum plasmas
remains a challenge10,22,25. Here, we explore an alternate approach based on a new formulation of quantum hydro-
dynamics (QHD). QHD approaches have the advantage of including equation-of-state and transport quantities
more naturally than response-function approaches. Apart from these potential modeling advantages, QHD mod-
els of DSF therefore also provide access to experimental measurements of these quantities, thereby extending the
utility of DSF. We develop a QHD framework based on the extension of the classical dynamical density functional
Department of Computational Mathematics, Science and Engineering, Michigan State University East Lansing,
Michigan, 48823, USA. Correspondence and requests for materials should be addressed to A.D. (email: rahmane@
melix.org)
Received: 9 February 2017
Accepted: 11 October 2017
Published: xx xx xxxx
OPEN
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theory (DDFT)26–28, a variant of time-dependent density functional theory (TDDFT)29. DDFT provides a set of
hydrodynamics equations by taking the velocity moments of Liouville equation and closes the system using den-
sity functional theory26–28. A fundamental assumption of this theory is that the equilibrium energy functional of
the system can be used to guess the correlation energy functional when the system is out of equilibrium. While
DDFT has found wide use in many-body classical systems30–32, we extend its use in quantum systems17 to viscous
quantum systems, in general, and to DSF, specically.
We apply the DDFT-QHD model to stationary, homogeneous and isothermal plasmas for which the dynamic
structure factor (DSF) is well dened. While the DSF is of interest in its own right, it is also connected to x-ray
omson scattering (XRTS) experiments; XRTS yields much essential information about plasmas, including den-
sity, temperature and atomic physics information (e.g., ionization state6, ionization potential depression4, etc.).
Results
Dense strongly coupled plasmas are characterized by large collisional contributions and degenerate electrons.
ese features make the DDFT-QHD approach a reliable tool for accurately describing the dynamical properties
of these systems. For simplicity, here, we consider a quantum plasma comprising only electrons with density
distribution n and mass m interacting through a pairwise Coulomb potential
|−′|vrr()
. We use atomic units (i.e.,
πε====e m 41
0
) for the remainder of this work. e hydrodynamic equations for the electrons can be
written generally as
∂
∂
+∇⋅=
n
t
nu() 0,
(1)

∂
∂
+∇⋅=−∇ ⋅
n
t
n
u
uu
()
() ,
(2)
which are continuity and momentum equations written in terms of a generalized force tensor

. Note that the
continuity equation (1) and the le-hand side of (2) are generic, with the physical properties of the quantum
electron gas entering through terms on the right-hand side of (2). In the DDFT approach17,28,33, it is assumed that
the system is close enough to equilibrium that an adiabatic closure can be chosen for

; that is,
=nu[, ] 
. e
primary assumption of this model is that the system is near equilibrium, a condition well satised in highly colli-
sional plasmas. Further, the equilibrium density is forced to be consistent with the thermodynamic ground state
of the system by choosing the diagonal portion of the tensor to be of the form
δδΩnn[]/
, where  is the free
energy of the system. When
Ωn[]
is expressed using orbital-free density-functional theory (OFDFT), that portion
of

is closed. e o-diagonal portion of

can be written in its long-wavelength form to yield a generalized
Navier-Stokes equation of the form
δ
δ
ηξ
η
−∇ ⋅=−∇
Ω
+∇⋅∇ +



+




∇∇⋅n
n
n
uu,
[]
3
(),
(3)

where η is the shear viscosity, and ξ is the bulk viscosity; all other symbols have their usual meanings. Provided η
and ξ can be expressed in terms of
nu(, )
, the hydrodynamic equations are closed.
In DDFT, one writes the total free-energy functional as
Ω= +Ω +ΩnTnt nt ntrrr[] [(,)][]( ,) [(,)], (4)
Hxc
where
Tntr[(,)]
is the free energy of the noninteracting system,
Ωntr[](, )
H
is the Hartree free-energy functional,
and
Ωntr[](, )
xc
is the exchange-correlation (xc) functional. e Hartree term is exactly known and is an explicit
function of space and time.
A key advantage to the DDFT approach to QHD is that all thermodynamic properties are included
self-consistently through the total free energy , for which a wide range of approximations are available34–37.
In fact, this approach is very similar to the well-known generalized hydrodynamics, developed by Frenkel38,
that extends the classical Navier-Stokes equation to describe the properties of both solid and liquid bodies.
Furthermore, our DDFT-QHD approach can be connected with other approaches based on Bohmian dynam-
ics. If we set the viscous terms equal to zero in (3) and choose the gradient-corrected Thomas-Fermi (TF)
functional for T[n], one recovers the well-known Bohmian QHD20 form; again, however, the DDFT approach
enforces self-consistency of its form with the other terms in the free energy. e connection between DFT and the
Bohmian potential will be briey shown below.
Density uctuations are not readily available in density-functional theories, and our DDFT-QHD approach
suers from this limitation. However, in equilibrium, the uctuation-dissipation theorem allows us to connect the
linear response of the system to density uctuations. We write the DSF of the electrons as
ω
π
χω
=−
−
βω−
S
m
e
k
k
(, )
1I(,)
1
,
(5)
ee
ee
where β is the inverse electron temperature and
χ ωk(, )
ee
is the susceptibility of the free electrons. A large body of
literature39 focuses on the calculation of the system DSF
ωS k(, )
, with most work based on the Chihara40
decomposition
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∫
ωωωω ωω ω=| +| +′′−′+ .Sk fk qk Sk ZdSk Sk ZS k(, )()()(,) (, )(,) (, )
(6)
ii bs ce fe
2
e
e quantities
fk()
and
qk()
are the Fourier components of the density of bound and free electrons. e rst
term of (6) corresponds to low-frequency electron-density uctuations arising from ion dynamics and is propor-
tional to the ion-ion DSF
ωS k(, )
ii
. e factor
ωS k(, )
ce
in the second term describes the contribution from core
electrons41 and is modulated by the ion self-motion
ω′S k(, )
s
. e third term is the free-electron DSF
ωS k(, )
ee
in
the presence of a uniform ionic background. e quantity
ωS k(, )
ee
can be obtained from the standard Lindhard
dielectric function within the random-phase approximation (RPA), or extended to include collisions as proposed
by Mermin42. iele et al. generalized the Mermin form to include a dynamic collision frequency within the Born
approximation43, and Arkhipov et al. generalized the Mermin form to two-component plasmas, including sum
rules44.
e RPA results were also improved by including exchange and correlations through the local eld corrections45–52.
e ionic correlations contributions in a warm dense matter have been considered by Gregori and Gericke53. In
this scheme, the strongly coupled eects of the ions are included through the dierent components of the mem-
ory function constrained by the sum rules54. is phenomenological approach has been applied successfully in
Coulomb liquid54–57 community for systems where the memory functions have a Gaussian or exponential form.
However, for more complex systems, the form of the memory becomes mathematically intractable. Schmidt and
coworkers58 have proposed a hydrodynamic model that begins with moments of the Wigner-Poisson system with
a collision term added. In such an approach you cannot describe correlations properly since the resulting pressure
term is of an ideal gas. e DDFT-QHD approach we introduce here accounts for self-consistently many-body
physics eects and also non-local hydrodynamic eects through the choice of the free-energy functional.
The linear susceptibility associated with a weak external potential
δωvk(, )
ext
that induces a disturbance
δωnk(, )
in the electronic density
ωnk(, )
is dened as
χω
δω
δω
=.
n
v
k
k
k
(, )
(, )
(, )
(7)
e
ext
e
us, the susceptibility can be determined by linearizing the quantum hydrodynamics equation and using (7). To
do, we rst expand the density and velocities about a uniform mean as
δ=+nt nntrr(, )(,),(8)
0
δ=ttur ur(, )(,),(9)
which yields the linearized QHD equations in Fourier space:
ωδ δ−+ ⋅=nnku0, (10)
0
ωδ δ
δ
δξηδ δ−=−|+


+



+
∼
nn
V
n
niknvuk uk
4
3
,
(11)
xt00020e
where
δδ=ΩVnnr() []/
, and the tilde sign denotes the Fourier transform. By combining (10) and (11) and using
(7), we obtain an expression for the electron susceptibility:
χω ωδ
δ
ηω=−+|−
∼
kknknV
ni
k
n
k
(, )() ,
(12)
el
e2022
00
2
0
where
ηηξ=+(4 /3 )
l
is the longitudinal viscosity. To proceed, we need to choose specic forms for the dierent
contributions of the free-energy functional [n]. e free-energy functional is typically chosen to ensure that an
accurate equilibrium density is recovered, although exact analytical forms are generally not known. However, the
contributions of the excess free-energy functional to the free energy of the system,
Ω=Ω +Ωnnn[] [] []
ex Hxc
, can
be expressed formally in terms of the direct correlation function
′|− |crr()
ee
as follows59:
∫∫∫
μ
β
′′′Ω=Ω+ ∆− ∆∆ |− |+nn dn dd nncOnrr rr rrrr[] [] ()
1
2
() () ()(),
(13)
xxxeee0ee2
where
∆= −nnnrr() () 0
, and μex is the excess chemical potential. Once the pair potential
v q()
has been speci-
fied, the self-consistent contributions of the excess free-energy functional can be calculated using the
direct-correlation function via path integral quantum Monte Carlo (PIMC) simulations60,61, integral equations62
or analytical ts63. Let us now evaluate the free-energy functional of the non-interacting electron gas] T[n].
Many approximations for T[n]64–66, have been described in the literature- omas-Fermi (TF), Kirzhnits gra-
dient correction (TFK)65, von Weizsäcker funtional (vW)64, Perrot functional66 to name a few. Most of these
models are based on an extension of the TF functional
∫
πβ αα α=


−



Tn dI Ir[] 2() 2
3(),
(14)
FT25/2 1/23/2
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where
∫
=
+
α
∞
−
I
x
e
dx
1(15)
p
p
x
0
is the Fermi-Dirac integral of order p,
αr()
is the chemical potential normalized with the kBT and the electron
density is given by
πβ
α=.nIrr()
2
[()]
(16)
23/2 1/2
We consider here the functional form for an electron gas based on the TF functional with the nite-temperature
Kirzhnits gradient correction (TFK)65:
∫
γ
π
β
α
α
=+
′
|∇ |+ .
−
−
Tn Tn d
I
Inr[] []
32
8
()
()
(17)
FK FTT
2
3/21/2
1/2
22
Here, we introduced a coecient γ that allows to capture a variety of results. First, the systematic gradient expan-
sion of Kirzhnits yields the prefactor γ = 1/9. Second, the von Weizsäcker result follows by a partial integration
and γ = 1. e assumption in this gradient-correction expansion is that the error made by neglecting the third-
and higher-order terms is very small. For high-density plasmas, interface-mixing problems or shock structures
in which temperature and density gradients can be large, this expansion ceases to be valid. In such circum-
stances, it may be important to include higher-order terms for the thermal terms through higher gradient cor-
rections q in the TFK functional. e non-interacting free-energy functional T[n] can also be expressed in terms
of the Lindhard function, which is exactly known, instead of using OFDFT66,67. Let us now show the connection
between DFT and the Bohm description. e functional derivative of the kinetic energy functional (17) is given
by20
δ
δ
α
βγπβξαξα=+ ′|∇| +∇
T
n
r
nn
() 32
8
[()2() ],
(18)
FKT
2
3/22 2
where
ξααα=′
−−
II() ()/()
1/21/2
2
, its derivative with respect to the density is denoted
ξ α
′()
. e rst term of (18) is
the Fermi pressure while the second term corresponds to the generalized Bohm potential in the nite-temperature
regime, revealing the underlying connection between DFT and the Bohm description. e connection between
DFT and the Bohmian picture has recently been discussed in a paper by Stanton and Murillo20 where a Kirzhnits65
correction was used to get the “Bohm” term quite generally.
Aer linearizing and taking the Fourier transform of (13) and (17), the contributions of the non-interacting
and interacting electron gases to the susceptibility become
δ
δπλ πν λβ|= +−
∼
−
V
n
kCk
k
k
()
()
4(),
(19)
FFe0T
22
T
41
e
where the TF length and the parameter
ν
are given by
λπβ
ανβ
πα==
−
−
II
2
4()and
8
3(),
(20)
FT
2
1/20
3/20
respectively. Substituting (19) into (12), we nd
χω πωλ ληω
=−+ +− −.
ν
Γ
−−
qa
q
qq nC qiq
(, )
1
4()
(21)
e
FF
q
e
l
e2
2
22
T2
4
4T4
3
0e 2
2
Here the frequency ω is in units of the electron plasma frequency,
=| |qka
is the wave number,
π=an(3/4 )1/3
is
the Wigner-Seitz radius,
λ λ=a/
FFTT
‐
is the omas-Fermi length in the units of the Wigner-Seitz radius, the
viscosity is in units of
ωn a
p2
, and the coupling parameter, dened as the ratio of the potential energy to the average
kinetic energy, is given by
βΓ= a/
.
Substituting (21) into (5) yields the free-electron DSF
()
Sq
nr r
q
nC qq qq
(, )
1
3
1
1exp(3/) () ()
(22)
pe
ss
l
q
eFFl
e
4
230e 2T244T4222
2
ωω
πω
ηω
ωλληω
=−−Γ+−−+
.
Γ
ν
−−
Equation (22) is the main result of this work. e second factor is the usual Bose function. e denominator
of the third factor includes quantum degenerate plasma eects through the direct correlation function
C q()
ee
,
thermal eects with high-order gradient terms, and viscous damping through
ηl
. It is worth noting that when the
degeneracy parameter
θ∼Γr/
s
is very large, electrons can be considered to be in a non-degenerate, classical state.
If we then replace the exponential in the Bose function by its Taylor expansion, we recover the dynamic structure
of non-degenerate electrons given by the Navier-Stokes model68.
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e form of the DSF obtained from this theory, without the dissipative eects, is connected to the approaches
based on the local field corrections49. The direct correlation function which is the main ingredient of this
approach is related to the local eld correction (LFC)
G q()
as:
β
=− −.Gk
vk
Ck() 1
1
()
()
(23)
ee
In the random phase approximation (RPA), the direct correlation function is given by
β=−Gkvk() ()
ee
con-
sequently the LFC vanishes,
=G k() 0
. us, the direct correlation function describes the strongly Coulomb cor-
relation and exchange eects beyond the RPA. Several approximations for the LFC45,46,49,69, has been proposed
starting from the formulation of local eld corrections due to Coulomb correlations and exchange eects by
Hubbard45. Utsumi and Ichimaru49 formula has been widely used to investigated the static properties of systems
at metallic densities. Holas, Aravind and Singwi70 have suggested an expression for the dynamical LFC in strongly
coupled electron gas. Although, we can use existing analytical ts for the LFC to obtain the direct correlation
function, we choose here to computed this quantity directly using Ornstein-Zernike equations with the
hypernetted-chain approximation closure71. Furthermore, matter under extreme conditions of temperature and
pressure undergoes large spatial gradients (i.e., shocks structure, interface problems, etc.). e heterogeneity can
greatly altered the ompson spectra with respect to the uniform case as recently discussed by Kozlowski and
coworkers72. In our approach, the constitutive equations will relax to the correct DFT thermodynamic ground
state, which other methods cannot guarantee. is means we have the full non-local correlations absent from
most other approaches opening up the possibility of studying the cases for which the usual homogeneous and
isotropic forms like
ε ωk(, )
are not applicable. is can be done through the introduction of the inhomogeneous
direct correlation function
′crr(, )
ee
. In the next section we will focus on the characteristic features of our main
result (22).
Discussion
Computing the DSF (22) requires knowledge of the viscosity and the direct correlation function. In the literature,
the DSF is oen expressed in terms of the local eld correlation, and the latter quantity is evaluated using analyt-
ical ts52,73,74. e direct correlation function
C q()
ee
can also be obtained directly through numerical simulations;
that is the avenue pursued in this work, using hypernetted-chain calculations62,75, with a quantum statistical
potential (QSP)76–78. We use QSP approach here to merely have easy access to results for which we can illustrate
the DDFT method, which is the main point of the paper; other methods can be also used to get the structure
information for the DDFT model. In fact, we see a strength of QSPs in this regard: the key quantity is the
electron-electron
c r()
ee
, which is not accessible from DFT approaches. Electron-electron correlation functions are
available from PIMC, however, and that provides validation for our input quantities. Jones and Murillo79 have
shown the theoretical underpinnings of QSPs and reviewed their extension to fully degenerate quantum systems.
Dutta and Duy60 have compared compared QSP-based RDFs from the modied Kelbg QSP and PIMC and show
that over an extremely wide range of physical conditions the QSP predictions are nearly perfect; it is only at very
low densities that we can see a modest deviation. Here, we choose the QSP from the pioneering work of Hansen
and McDonald76 since they yield results similar to the more complicated Kelbg potentials. Comparisons between
Coulomb, HM and Kelbg potentials are shown in the online Supplementary Material. Next, we turn to the
Figure 1. Eects of the coupling parameter on the spectra. We show the variation in the DSF for dierent
values of the coupling parameter Γ and the normalized wavelengths
=| |qak
. e coupling parameter Γ ranges
from 0.2 to 0.8, and rs ranges from 1.0 to 4.0. e dynamic structure factor
ωS q(, )
ee
is normalized by its
maximum value. e two plasmon peaks are symmetric and the ratio of their amplitudes gives a measure of the
electrons temperature.
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electronic viscosity is needed, which is determined by both electron-electron and electron-ion collisions. e
electron viscosity is obtained by interpolating the zero-temperature viscosity proposed by Conti and Vignale80
and the nite-temperature viscosity for classical plasmas by Stanton and Murillo33. When building this t, we
considered contributions only from electron-electron interactions. However, the electron viscosity should also
take into account electron-ion81–83, contributions, which can be more signicant than the electron-electron vis-
cosity in the regimes of interest. Please see the online Supplementary Material for a detailed description of the
calculation of these two quantities.
Figure1 shows the spectra of the DSF for dierent values of the wave number q, the coupling strength Γ and
the density parameter rs. e free electron DSF is normalized by its maximum value. In Fig.1a, the positions of
the plasmon peaks (Stokes and anti-Stokes) and its amplitude remain almost unchanged when the quantum
parameter rs increases from 1.0 to 4.0. e reason is, in this regime
θ(1)
, the correlations and quantum degen-
eracy eects are negligible and consequently have no impact on the propagation of the plasmon. Furthermore, the
width of the plasmon peak reduces when q and rs increase owing to the fact that the viscosity which acts to
broaden the width of the peak is very sensitive to the density parameter rs. Figure1a–c show that the position and
width of the plasma peak vary strongly with Γ. ese gures also display some of the standard features of the
plasmons peaks; they are symmetric with respect to the zero frequency, and the dierence between their ampli-
tudes gives a measure of the electron temperature through the detailed balance relation. It is worth noting that for
large values of the coupling parameter and density, the plasmon peak is at a frequency smaller than the plasma
frequency
ωp
.
For a given value of the wave number q, the peak of
ωS q(, )
ee
corresponds to the dispersion relation of the
plasmon
ω ω=q()
q
. According to (22), the dispersion relation is approximately given by
ωω Γ
νη
≅





−+λ+λ−






.
‐‐
qnC qq qq
3() 42
(24)
qp eF Fl
22
2
0e T
22T
44
24
In the RPA limit, with a pure Coulomb potential, the direct correlation function is given by
β=−Cqvq() ()
ee
.
By substituting this result into (24) and setting the viscosity equal to zero, we recover the Bohm-Gross dispersion
relation84 for an isothermal plasma. e dispersion relation of the plasmon is shown in Fig.2 for
=.r186
s
and for
Γ = 1.0 and Γ = 0.7. e red triangles lines show the DDFT-QHD result (24), and the data points indicated with
blue line corresponds to (24), with the viscosity set to zero, ηl = 0. From our basic result (24) we can explore sev-
eral limits that yield other models. For example, because the direct correlation function is related to the local-eld
correction through the relation
β=− −cqvqGq() ()[1 ()]
ee ee
, we can neglect the three higher-order terms in (24)
to obtain the local-eld correction (LFC) result. We show this limit in Fig.2 as a green line. Next, we can retain
the second and third terms to example quantum corrections to the LFC result, and that model is shown as blue
triangles. e full result, including viscosity is shown as the red line; note that the viscous correction is large,
suggesting that the power series in q of (24) is probably not converged at the largest values of q in the plot.
e dispersion relation (24) implies that the width of the peak of the DSF is broadened by the viscosity.
erefore, by measuring the width of the peak of a spectrum, information about the electron viscosity can be
inferred. We can obtain a good estimate of the width in the following way. We know the location of the peak from
the dispersion relation (24). Near that peak, we know what ω is, and this information can be put into
ηωqlpeak
2
to
Figure 2. Plasmon dispersion relation. We show the frequency as a function of the wave number q for
=.r186
s
and (a) Γ = 1.0 and (b) Γ = 0.7. DDFT-QHD refers to (24), which accounts for strong correlations and viscosity.
e label “DDFT-QHD:
η=0
l
” corresponds to (24) with the viscosity set to zero. e green curve shows the
local eld correction dispersion relation.
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obtain the width of the DSF. is scaling gives the width in terms of both ηl and
C q()
ee
, which means that ηl cannot
be determined without knowledge of the direct correlation function. However, by tting the entire spectrum of
the DSF, all quantities can be obtained: the density, the temperature, and the viscosity. e DSF spectra suggest
that measurements at a few q values is best.
Concluding Remarks
A general framework for electron dynamics is provided with DDFT-QHD. Our model is the specic approxima-
tion of TDDFT in which the generalized-force functional is replaced by the equilibrium functional. We estab-
lished the connection between DDFT-QHD and DSF through the uctuation-dissipation theorem, allowing for
improved QHD models to be compared with experimental data. e predicted DSF spectrum exhibits strong cor-
relations and collisions that are built self-consistently into the model; this result diers from those obtained with
more common Lindhard approaches42–44, in which collisions enter through a dynamic collision frequency43,44, or
though local eld corrections85. Our result suggests that the electronic viscosity can be determined experimen-
tally by measuring the electron DSF.
Our approach is a full hydrodynamics model that can be used to simulate non-equilibrium, heterogeneous
dense plasmas72. For example, we could investigate shock physics, uid instabilities, and large-scale experiments.
Most other methods are based on stationary, homogeneous/isotropic approximations; this is explicit in functions
such as
χ ωk(, )
. us, while our DDFT formulation of QHD is signicantly beyond these simple linear response
functions, we show here that we are able to make contact with the XRTS community and connect the scattering
spectrum to transport coecients in a direct way with a hydrodynamic approach.
Finally, our model still lacks an equation for the energy uctuations. Past experience suggests54–57,86, that
energy uctuations might cause a zero frequency mode. is latter was experimentally measured in a liquid
lithium by Sinn and coworkers87 conrming molecular dynamics simulations results performed by Canales,
González, and Padró86. Our DDFT-QHD approach would miss any mode originating from thermal uctua-
tions54,58,88, because it is based on an isothermal assumption. We think this approach can incorporate an energy
equation, but this is work in progress. In future work, it would be useful to include other transport quantities,
such as the viscoelastic relaxation time and the thermal conductivity. Extension of this model to the full XRTS
form factor with an electron-ion generalization of the DDFT-QHD equations89,90, is le for the future.
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Acknowledgements
is work was supported by the Air Force Oce of Scientic Research (Grant No. FA9550-12-1-0344).
Author Contributions
M.M. conceived the project. A.D. and M.M. wrote the main manuscript text. A.A. prepared gures. Numerical
data were analysed by both authors.
Additional Information
Supplementary information accompanies this paper at https://doi.org/10.1038/s41598-017-14414-9.
Competing Interests: e authors declare that they have no competing interests.
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