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SCiEntifiC REPORTS | 7: 15352 | DOI:10.1038/s41598-017-14414-9

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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, satises 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 oers an avenue for measuring hydrodynamic properties, such as transport

coecients 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 coecients5 and ionization state6. Understanding the dynamical

properties of non-ideal plasmas is critical for modeling and designing high energy-density science experiments,

including inertial-connement 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 aer

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 eects 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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SCiEntifiC REPORTS | 7: 15352 | DOI:10.1038/s41598-017-14414-9

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, specically.

We apply the DDFT-QHD model to stationary, homogeneous and isothermal plasmas for which the dynamic

structure factor (DSF) is well dened. 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 satised 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 briey shown below.

Density uctuations are not readily available in density-functional theories, and our DDFT-QHD approach

suers 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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SCiEntifiC REPORTS | 7: 15352 | DOI:10.1038/s41598-017-14414-9

∫

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 eects of the ions are included through the dierent 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 eects and also non-local hydrodynamic eects 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 dened 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 specic forms for the dierent

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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4

SCiEntifiC REPORTS | 7: 15352 | DOI:10.1038/s41598-017-14414-9

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 coecient 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.

Aer 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, dened 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 eects through the direct correlation function

C q()

ee

,

thermal eects 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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SCiEntifiC REPORTS | 7: 15352 | DOI:10.1038/s41598-017-14414-9

e form of the DSF obtained from this theory, without the dissipative eects, 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 eects 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 eects 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 oen 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 Duy60 have compared compared QSP-based RDFs from the modied 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. Eects of the coupling parameter on the spectra. We show the variation in the DSF for dierent

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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SCiEntifiC REPORTS | 7: 15352 | DOI:10.1038/s41598-017-14414-9

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 signicant 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.

Figure1 shows the spectra of the DSF for dierent 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 eects 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. Figure1a–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 dierence 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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SCiEntifiC REPORTS | 7: 15352 | DOI:10.1038/s41598-017-14414-9

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 specic 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 diers 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 signicantly 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 coecients 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 conrming molecular dynamics simulations results performed by Canales,

GonzÃ¡lez, 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.

References

1. Bostedt, C. et al. Linac coherent light source: e rst ve years. ev. Mod. Phys. 88, 015007 (2016).

2. Lindl, J., Landen, O., Edwards, J. & Moses, E. eview of the National Ignition Campaign 2009–2012. Phys. Plasmas 21, 020501

(2014).

3. Saunders, A. M. et al. X-ray omson scattering measurements from hohlraum-driven spheres on the OMEGA laser. ev. Sci.

Instrum. 87, 11E724 (2016).

4. Crowley, B. J. B. Continuum lowering - A new perspective. High Energy Density Physics 13, 84 (2014).

5. Mcelvey, A. et al. Thermal conductivity measurements of proton-heated warm dense matter. In APS Shoc Compression of

Condensed Matter Meeting Abstracts (2015).

6. Gregori, G. et al. Measurement of carbon ionization balance in high-temperature plasma mixtures by temporally resolved X-ray

scattering. J. Quant. Spectrosc. adiat. Transf. 99, 225 (2006).

7. Meezan, N. B. et al. Indirect drive ignition at the National Ignition Facility. Plasma Physics and Controlled Fusion 59, 014021 (2017).

8. omas et al. H. Explosions of Xenon Clusters in Ultraintense Femtosecond X-ay Pulses from the LCLS Free Electron Laser. Phys.

ev. Lett. 108, 133401 (2012).

9. Hegelich, B. M. et al. Laser acceleration of quasi-monoenergetic MeV ion beams. Nature 439, 441 (2006).

10. Fletcher, A., Close, S. & Mathias, D. Simulating plasma production from hypervelocity impacts. Physics of Plasmas 22, 093504

(2015).

11. Bigot, J.-Y., Halté, V., Merle, J.-C. & Daunois, A. Electron dynamics in metallic nanoparticles. Chemical Physics 251, 181–203 (2000).

12. Wang, Y. & Eliasson, B. One-dimensional rarefactive solitons in electron-hole semiconductor plasmas. Phys. ev. B 89, 205316

(2014).

13. D avis, P. et al. X-ray scattering measurements of dissociation-induced metallization of dynamically compressed deuterium. Nat.

Commun. 7, 11189 (2016).

14. Bloch, F. B. von Atomen mit mehreren Eletronen. Zeitschri fur Physi 81, 363–376 (1933).

15. Hohenberg, P. & ohn, W. Inhomogeneous electron gas. Phys. ev. 136, B864–B871 (1964).

16. Mermin, N. D. ermal Properties of the Inhomogeneous Electron Gas. Physical eview 137, 1441–1443 (1965).

17. Ying, S. C. Hydrodynamic response of inhomogeneous metallic systems. Nuovo Cimento B Serie 23, 270 (1974).

18. Gasser, I. & Jüngel, A. e quantum hydrodynamic model for semiconductors in thermal equilibrium. Zeitschri Angewandte

Mathemati und Physi 48, 45–59 (1997).

19. Michta, D., Graziani, F. & Bonitz, M. Quantum Hydrodynamics for Plasmas - a omas-Fermi eory Perspective. Contrib. Plasma

Phys. 55, 437 (2015).

20. Stanton, L. G. & Murillo, M. S. Unied description of linear screening in dense plasmas. Phys. ev. E 91, 033104 (2015).

21. Gardner, C. L. Quantum hydrodynamic model for semiconductor devices. SIAM Journal of Applied Mathematics 54, 409–427

(1994).

22. Manfredi, G. & Haas, F. Self-consistent uid model for a quantum electron gas. Phys. ev. B 64, 075316 (2001).

23. Levermore, C. D. Moment closure hierarchies for inetic theories. Journal of Statistical Physics 83, 1021–1065 (1996).

24. Degond, P. & inghofer, C. Quantum moment hydrodynamics and the entropy principle. Journal of Statistical Physics 112, 587–628

(2003).

25. Gardner, C. L. Quantum hydrodynamic model for semiconductor devices. SIAM J. Appl. Math. 54, 409 (1994).

26. Marini Bettolo Marconi, U. & Tarazona, P. Dynamic density functional theory of uids. J. Chem. Phys 110, 8032 (1999).

27. Lutso, J. F. Density functional theory of inhomogeneous liquids. III. Liquid-vapor nucleation. J. Chem. Phys. 129, 244501–244501

(2008).

28. Diaw, A. & Murillo, M. S. Generalized hydrodynamics model for strongly coupled plasmas. Phys. ev. E 92, 013107 (2015).

29. unge, E. & Gross, E. . U. Density-functional theory for time-dependent systems. Phys. ev. Lett. 52, 997 (1984).

30. Goddard, B. D., Nold, A., Savva, N., Pavliotis, G. A. & alliadasis, S. General dynamical density functional theory for classical uids.

Phys. ev. Lett. 109, 120603 (2012).

31. Marconi, U. M. B. & Tarazona, P. Dynamic density functional theory of uids. J. Chem. Phys. 110, 8032 (1999).

Content courtesy of Springer Nature, terms of use apply. Rights reserved

www.nature.com/scientificreports/

8

SCiEntifiC REPORTS | 7: 15352 | DOI:10.1038/s41598-017-14414-9

32. ex, M. & Löwen, H. Inuence of hydrodynamic interactions on lane formation in oppositely charged driven colloids. Eur. Phys. J.

E 26, 143 (2008).

33. Stanton, L. G. & Murillo, M. S. Ionic transport in high-energy-density matter. Phys. ev. E 93, 043203 (2016).

34. Perdew, J. P., Bure, . & Ernzerhof, M. Generalized gradient approximation made simple. Phys. ev. Lett. 77, 3865 (1996).

35. Malone, F. D. et al. Accurate exchange-correlation energies for the warm dense electron gas. Phys. ev. Lett. 117, 115701 (2016).

36. arasiev, V. V., Sjostrom, T., Duy, J. & Tricey, S. B. Accurate homogeneous elect ron gas exchange-correlation free energy for local

spin-density calculations. Phys. ev. Lett. 112, 076403 (2014).

37. Huang, C. & Carter, E. A. Nonlocal orbita l-free inetic energy density functional for semiconductors. Phys. ev. B 81, 045206 (2010).

38. Frenel, J. inetic eory of Liquids (Clarendon, Oxford, 1946).

39. Glenzer, S. H. & edmer, . X-ray thomson scattering in high energy density plasmas. ev. Mod. Phys. 81, 1625 (2009).

40. Chihara, J. Interaction of photons with plasmas and liquid metals - photoabsorption and scattering. J. Phys. Condens. Matter 12, 231

(2000).

41. Sahoo, S., Gribain, G. F., Shabbir Naz, G., ohano, J. & iley, D. Compton scatter proles for warm dense matter. Phys. ev. E 77,

046402 (2008).

42. Mermin, N. D. Lindhard dielectric function in the relaxation-time approximation. Phys. ev. B 1, 2362 (1970).

43. iele, . et al. omson scattering on inhomogeneous targets. Phys. ev. E 82, 056404 (2010).

44. Arhipov, Y. V. & Davletov, A. E. Screened pseudopotential and static structure factors of semiclassical two-component plasmas.

Physics Letters A 247, 339–342 (1998).

45. Hubbard, J. e Description of Collective Motions in Terms of Many-Body Perturbation eory. Proceedings of the oyal Society of

London Series A 240, 539–560 (1957).

46. Singwi, . S., Tosi, M. P., Land, . H. & Sjölander, A. Electron correlations at metallic densities. Phys. ev. 176, 589–599 (1968).

47. Vashishta, P. & Singwi, . S. Electron Correlations at Metallic Densities. V. Phys. ev. B 6, 875–887 (1972).

48. Vaishya, J. S. & Gupta, A. . Dielectric esponse of the Electron Liquid in Generalized andom-Phase Approximation: A Critical

Analysis. Phys. ev. B 7, 4300–4303 (1973).

49. Utsumi, . & Ichimaru, S. Dielectric formulation of strongly coupled electron liquids at metallic densities. II. Exchange eects and

static properties. Phys. ev. B 22, 5203–5212 (1980).

50. Geldart, D. J. W. & Voso, S. H. e screening function of an interacting electron gas. Canadian Journal of Physics 44, 2137 (1966).

51. Dharma-wardana, M. W. C. & Perrot, F. Simple classical mapping of the spin-polarized quantum electron gas: Distribution funct ions

and local-eld corrections. Phys. ev. Lett. 84, 959–962 (2000).

52. Gregori, G., avasio, A., Höll, A., Glenzer, S. H. & ose, S. J. Derivation of the static structure factor in strongly coupled non-

equilibrium plasmas for X-ray scattering studies. High Energy Density Physics 3, 99 (2007).

53. Gregori, G. & Gerice, D. O. Low frequency structural dynamics of warm dense mattera). Physics of Plasmas 16, 056306 (2009).

54. Boon, J. P. & Yip, S. Molecular hydrodynamics (Dover Publications, New Yor, 1991).

55. Pines, D. & Nozières, P. e eory of Quantum Liquids (W. A. Benjamin, New Yor, 1989).

56. ugler, A. A. Collective modes, damping, and the scattering function in classical liquids. Journal of Statistical Physics 8, 107–153

(1973).

57. Hansen, J. P., McDonald, I. . & Polloc, E. L. Statistical mechanics of dense ionized matter. iii. dynamical properties of the classical

one-component plasma. Phys. ev. A 11, 1025–1039 (1975).

58. Schmidt, ., Crowley, B. J. B., Mithen, J. & Gregori, G. Quantum hydrodynamics of strongly coupled electron uids. Phys. ev. E 85,

046408 (2012).

59. Hansen, J. & McDonald, I. inetic eory of Liquids (Academic, London, 1986).

60. Dutta, S. & Duy, J. Uniform electron gas at warm, dense matter conditions. EPL (Europhysics Letters) 102, 67005 (2013).

61. Brown, E. W., Clar, B. ., DuBois, J. L. & Ceperley, D. M. Path-Integral Monte Carlo Simulation of the Warm Dense Homogeneous

Electron Gas. Phys. ev. Lett. 110, 146405 (2013).

62. Xu, H. & Hansen, J.-P. Density-functional theory of pair correlations in metallic hydrogen. Phys. ev. E 57, 211 (1998).

63. Groth, S., Dornheim, T. & Bonitz, M. Free Energy of the Uniform Electron Gas: Testing Analytical Models against First Principle

esults. ArXiv e-prints (2016).

64. Weizsäcer, C. F. V. Zur eorie der ernmassen. Zeitschri fur Physi 96, 431 (1935).

65. irzhnits, D. Quantum Corrections to the omas-Fermi Equation. ZSoviet Phys. JETP 5, 64 (1957).

66. Perrot, F. Hydrogen-hydrogen interaction in an electron gas. J. Phys.: Cond. Mat. 6, 431 (1994).

67. Wang, L.-W. & Teter, M. P. inetic-energy functional of the electron density. Phys. ev. B 45, 13196 (1992).

68. Murillo, M. S. X-ray thomson scattering in warm dense matter at low frequencies. Phys. ev. E 81, 036403 (2010).

69. Farid, B., Heine, V., Engel, G. E. & obertson, I. J. Extremal properties of the harris-foules functional and an improved screening

calculation for the electron gas. Phys. ev. B 48, 11602–11621 (1993).

70. Holas, A., Aravind, P. . & Singwi, . S. Dynamic correlations in an electron gas. I. First-order perturbation theor y. Phys. ev. B 20,

4912–4934 (1979).

71. Wünsch, ., Hilse, P., Schlanges, M. & Gerice, D. O. Structure of strongly coupled multicomponent plasmas. Phys. ev. E 77,

056404 (2008).

72. ozlowsi, P. M., Crowley, B. J. B., Gerice, D. O., egan, S. P. & Gregori, G. eory of omson scattering in inhomogeneous media.

Scientic eports 6, 24283 (2016).

73. Ichimaru, S. Nuclear fusion in dense plasmas. ev. Mod. Phys. 65, 255 (1993).

74. Nagao, ., Bonev, S. A. & Ashcro, N. W. Cusp-condition constraints and the thermodynamic properties of dense hot hydrogen.

Phys. ev. B 64, 224111 (2001).

75. Chihara, J. Unied description of metallic and neutral liquids and plasmas. J. Phys. Condens. Matter 3, 8715 (1991).

76. Hansen, J. P. & McDonald, I. . Microscopic simulation of a strongly coupled hydrogen plasma. Phys. ev. A 23, 2041 (1981).

77. Schwarz, V. et al. Static ion structure factor for dense plasmas: Semi-classical and ab initio calculations. High Energ. Dens. Phys. 6,

305 (2010).

78. Lado, F. Eective Potential Description of the Quantum Ideal Gases. J. Chem. Phys. 47, 5369–5375 (1967).

79. Jones, C. S. & Murillo, M. S. Analysis of semi-classical potentials for molecular dynamics and Monte Carlo simulations of warm

dense matter. High Energy Density Physics 3, 379–394 (2007).

80. Conti, S. & Vignale, G. Elasticity of an electron liquid. Phys. ev. B 60, 7966 (1999).

81. Murillo, M. S. Viscosity estimates of liquid metals and warm dense matter using the Yuawa reference system. High Energ. Dens.

Phys. 4, 49 (2008).

82. Clérouin, J. e viscosity of dense hydrogen: from liquid to plasma behaviour. J. Phys. Condens. Matter 14, 9089 (2002).

83. Faussurier, G., Libby, S. B. & Silvestrelli, P. L. e viscosity to entropy ratio: From string theory motivated bounds to warm dense

matter transport. High Energ. Dens. Phys. 12, 21 (2014).

84. Gouedard, C. & Deutsch, C. Dense electron-gas response at any degeneracy. Journal of Mathematical Physics 19, 32–38 (1978).

85. Ichimaru, S. & Tanaa, S. Generalized viscoelastic theory of the glass transition for strongly coupled, classical, one-component

plasmas. Phys. ev. Lett. 56, 2815 (1986).

86. Canales, M., González, L. E. & Padró, J. À. Computer simulation study of liquid lithium at 470 and 843 . Phys. ev. E 50, 3656–3669

(1994).

Content courtesy of Springer Nature, terms of use apply. Rights reserved

www.nature.com/scientificreports/

9

SCiEntifiC REPORTS | 7: 15352 | DOI:10.1038/s41598-017-14414-9

87. Sinn, H. et al. Coherent dynamic structure factor of liquid lithium by inelastic x-ray scattering. Phys. ev. Lett. 78, 1715–1718 (1997).

88. Mountain, . D. Spectral distribution of scattered light in a simple uid. ev. Mod. Phys. 38, 205–214 (1966).

89. Fu, Z.-G. et al. Dynamic properties of the energy loss of multi-mev charged particles traveling in two-component warm dense

plasmas. Phys. ev. E 94, 063203 (2016).

90. Barriga-Carrasco, M. D. Target electron collision eects on energy loss straggling of protons in an electron gas at any degeneracy.

Physics of Plasmas 15, 033103 (2008).

Acknowledgements

is work was supported by the Air Force Oce of Scientic 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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