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Application of First Principles Langiven
Molecular Dynamics to Hot Dense Plasmas
Jianmin YuanDepartment of Physics, National University of Defense Technology
Changsha 410073, China
Workshop: Computational Challenges in Hot Dense Plasmas March 29, 2012, IPAM,UCLA
Outline
• Introduction
• Quantum Langevin Molecular Dynamics (QLMD)
• Applications– Electronic structure and EOS of Iron – Electric and thermo conductivity– Carbon embedded in dense hydrogen– EOS of solar materials– EOS of D, T and the mixture
• Conclusion
A large Brownian particle with mass M immersed in a fluid of much smaller and lighter particles.
Brownian Motion: Langevin Equation
According to Einstein’s theory, the diffusion constant
Brownian Motion: Langevin Equation
According Newton’s law
Langevin Equation
aTk
aNRTD B
A πηπη 66==
)()( tFdt
tvdmrr
=
)(1)()(
)()(
tm
tvmdt
tvd
tvdt
txd
ξγ rrr
rr
+−=
=
γτ
γ
τ
mvetv
and
tvmdt
tvd
B
t B
=
=
−=
− )0()(
)()(
/ rr
rr
∫ −−− +=
−==
t stt sdsem
vetv
ttgttt
BB
0
/)(/
2121
)(1)0()(
)()()(,0)(
ξ
δξξξ
ττ
ξξ
rrr
rrr
For a Gaussian Process
Brownian Motion: Langevin Equation
TkTmkg
mge
mgvtv
BB
B
BtBeqeq
B
γτ
ττ τ
ξ
2222
)( 2/2
220
2
==
+⎥⎦⎤
⎢⎣⎡ −= −
Fluctuation-Dissipation (Langevin) Theorem: The equilibrium is brought about by a dissipation force (“friction”) between the particle and the bath. It is the same process that produces the random, fluctuating force on the particle. Both processes are uniquely determined by the statistical nature of the microscopic forces .
( )∑⎥⎥⎦
⎤
⎢⎢⎣
⎡−+++=
jj
jj
j
j xxm
mp
xUMpH 2
222
)(22
)(2
αω
∑ −+∂
∂−=
=
jjjj xxm
xxUp
pxM
)()( 2ω&
&
)(2 xxmp
pxm
jjjj
jjj
−−=
=
ω&
).()](sin[
)](sin[)(
)](cos[)()(
0
00
00
sxstds
ttm
tptttxtx
t
t jj
jjj
jjjj
∫ −+
−+−=
ωω
ωω
ω
Classical Independent Oscillator Model
See: P. Hänggi, Lecture Notes in Physics 484, 15-22 (1997)
According to Hamilton Equation:
∑
∫
−=−
+−−−−∂
∂−=
jjjj
t
t
stmM
st
tFtxttMsxstdsMxxUxM
)](cos[1)()(
)()()()()()(
22
000
ωωαγ
γγ &&&
)()()(
0)(
stMkTsFtF
tF
B
B
−=
=
γρ
ρ
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛+−= ∑−
jj
jj
j
jB x
mmp
Z 222
1
22exp
ωβρ
Classical Independent Oscillator Model
∑⎥⎥⎦
⎤
⎢⎢⎣
⎡−+−=
jj
jj
jjjjj tt
mtp
tttxmtF )](sin[)(
)](cos[)()()( 00
002 ω
ωωωα
( )∑⎥⎥⎦
⎤
⎢⎢⎣
⎡−+−−=
−−≡
jj
jj
jjj tt
mtp
tttxtx
txttMtFt
)](sin[)(
)](cos[)()()()(
)()()()(
00
000
00
ωω
ωαα
γξ
( )⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−+−=
=
∑−
jj
jj
j
j
jj
xxm
mp
Z
xtxxp
22
2
21
0
)(22
exp
))(|},({ˆ
αω
β
ρ
)()()()()(
0)(
ˆ00ˆ
ˆ
stMkTttstst
t
−=+−=
=
γξξξξ
ξ
ρρ
ρ
Classical Independent Oscillator Model
)()]()([)(' tsvstdsxUdtvdM
tξγrr
r
+−−= ∫ ∞−
Classical Independent Oscillator Model
Quantum corresponding Eqs. can also be obtained in Heisenberg picture [see for example: Ford and Kac, J. Stat. Phys. 46, 803(1987)]
)()()('
)()]()([
)()(
ttvxUdtvdM
tvsvstds
ststt
ξγ
γγ
γδγ
rrr
rr
+−=
−→−−
−=−
∫ ∞−
When the bath has dense infinite spectra distribution, with random phase approximation one has
Open Quantum System-Wavefunction
Partitioning into system and environment
∑=
⊗=
+⊗+⊗=
m
jjjSB
SBBSBS
tBtStH
tHtHIItHtH
1)(ˆ)(ˆ)(ˆ
)(ˆ)(ˆˆˆ)(ˆ)(ˆ α
∑=Ψ
=
Ψ=Ψ
qBqBSqSBS
BnBnBnBB
xtxtxx
xxH
ttHtdtdi
)(),(),,(
)()(ˆ
)()(ˆ)(
,,
,,
rrrr
rr
ϕψ
ϕεϕ
∑≠
⊗=
⊗=
qppBpBSq
qBqBSq
IQ
IP
,,
,,
ˆˆ
ˆˆ
ϕϕ
ϕϕ
P. Gaspard and M. Nagaoka,J. Chem. Phys. 111 (1999) 5676-5690
∫−
−−
−
−
Ψ+=∂
t
SqSq
t
qtQHQi
txF
qtQHQi
qqSqSSSqSt
dxPHQeQHPi
QeQHPtxHtxi
S
0 ,
)(ˆ
)(ˆˆˆ
)0,,(
ˆˆˆ
,,
),(ˆˆˆˆˆˆ
)0(ˆˆˆˆ),(ˆ),(
ττψ
ψψτγ
τ r4444 84444 76
444 8444 76rr
r
∑≠
=Ψ
=Ψ
qpBpBSpSBSq
BqBSqSBSq
xxxxQ
xtxtxxP
)()0,()0,,(ˆ)(),(),,(ˆ
,,
,,
rrrr
rrrr
ϕψ
ϕψ
Open Quantum System-Wavefunction
By making an expansion about α and summing over the thermo distributed bath and system initial states:
)()(ˆˆ)(
)(ˆ)()(ˆ)(
3)(ˆ2 αοττψτα
ψαψψ
τ +−−
+=∂
∑
∑−−+
pqSq
tHippq
qSqqSSSt
dSeStCi
tStltHti
S
)'()'()(
,0)'()(* ttCtltl
tltl
pqqp
qp
−=
=
)'()()(* tttltl pqqp −∝ δδFor Markovin
)()(ˆˆ2
)(ˆ)()(ˆ)( 32
αοτψαψαψψ +−+=∂ ∑∑ +
pqSqq
qSqqSSSt SSitStltHti
Open Quantum System-Wavefunction
Application in a Variety of Problem
P. W. Anderson: J. Phys. Soc. Jpn. 9 (1954) 316.R. Kubo: J. Phys. Soc. Jpn. 9 (1954) 935.R. Kubo: in Fluctuation, Relaxation, and Resonance in MagneticSystems, ed. D. TerHaar (Oliver and Boyd, Edinburgh, 1962) p. 23.Yoshitaka TANIMURA, J. Phys. Soc. Jpn. 75 (2006) 082001
Nuclear magnetic resonance spectroscopy
In the fast modulation limit: γγγ /', 2Δ=Δ>>
20
2 )(''2)(ωωγ
γω−+
=I
In the slow modulation limit:
⎥⎦
⎤⎢⎣
⎡Δ−
−Δ
= 2
20
2)(exp22)( ωωωI
Δ<<γ
]exp[)0()( tt γ−=ΩΩ
StHIˆ)(ˆ ΔΩ=
Stochastic Current Density Functional Theory, and Stochastic Quantum Molecular Dynamics
M DiVentra and D’Agosta PRL 98, 226403 (2007) H Appel and Di Ventra, PRB 80, 212303 2009;
Chem. Phys. 391, 27(2011)
Application in a Variety of Problem
Dynamical adsorption of atoms on the suface
M Evstigneev* and P Reimann, PRB 82, 224303 2010
Laser cooling and crystallization of electron-ion plasma
A. P. Gavriliuk et al, PRE 80, 056404 2009
Hot dense plasmas
Frank R Graziani et al,HEDP 8, 105(2012)
Introduction: Molecular dynamics
Classical molecular dynamics:large scale computationpotential is empiricaltemperature and density effect is complicated
Quantum molecular dynamicsmore accurate for degenerate electrons and ion coupling normally for relatively low temperaturesmall sizes
Introduction: Quantum Molecular Dynamics
QMD:– electrons are described using DFT– ions’ moving on smooth potential surface is described
by Newton’s equationLangevin molecular dynamics in condensed
matter and material sciences– ions in Langevin equation
represents the contribution of thermostat for controlling the temperature of the system.
tγ
IIItII MM NRFR +−= &&& γ
Quantum Langevin Molecular dynamics• Langevin equation combining DFT can accelerate the calculation
of QMD.
represents the contribution from numerical error.erγ
PRL, 98, 066401 (2007); EPL,88, 20001 (2009)
IIIertII MM NRFR ++−= &&& )( γγ
QLMD-normally difficult for high temperatures
• Computational cost
• Partially degenerate electrons to Boltzmann distribution
• Accurate potentials
• Numerical problems in self-consistent field calculations at high temperature
} Supercomputer
Efficient QMD ( EPL, 88 (2009) 20001)
Full electrons(semi-core states)
The lost physical effects
The electron-ion interactions are central to numerous phenomena
Ions in a “Sea of Hot Electrons”
QLMD-extended to Warm (Hot) Dense Matters
QLMD--Unified first principles model• Brown motion• Electron-ion collisions induced frinction (EI-CIF)• Langevin equation:
Whitenoise
Friction coefficient
1/3* 42 0.00001 0.01 . .
3e i B
BI e
m n k TZ a uM m
πγ π ⎛ ⎞= → −⎜ ⎟⎝ ⎠
EI-CIF:
DFT
QLM
D
Bγ represents the electron-ion collisions induced friction (EICIF)
Phys. Rev. Lett. 104: 245001 (2010)
IIIertBII MM NRFR +++−= &4434421
&& )( γγγ
What can we do based on QLMD?
• Electronic structures based on DFT.
• Ionic structures
• Equation of state
• Transport properties for WDM: diffusion, viscosity, conductivity, x-ray absorption, opacity etc.
Electronic structure and EOS of Iron
• Earth core
• Hydrodynamical process
• DAC (relatively low temperature)
• Shock-wave experiments (temperature is difficult to measure)
EOS of Iron
Influence of γ on pressure for Fe at a temperature of 100 eV. [Phys. Rev. Lett. 104: 245001 (2010)]
−100 0 100
DO
S
−50 0 50
DO
S
−100 0 100 200Energy (eV)
DO
S
3s3p3d4s4p
−1000 0 1000 2000
DO
S−1000 1000 3000 5000
0
1
Pop
ulat
ion
−1100 −900
(a) (0.1eV,10g/cm3) (b) (10eV,22.5g/cm
3)
(c) (100eV,34.5g/cm3)
(d) (1000eV,39.65g/cm3)
Energy (eV)
2s2p
(e) (f)
EF
EF
EF
Electronic structures transition from cold condensed matter to ideal ionized gas plasma
Electronic structures of Iron along the Hugoniot
Phys. Rev. Lett. 104: 245001 (2010)
Electronic structures of Iron along the Hugoniot
Hugoniot for Iron: the present calculation
• Rankine–Hugoniot relations:
U: Internal Energy (U0 is important)
P: Pressure
V: Volume (V0 is very important)
02/))(()( 000 =−+−− VVPPUU
3342.73389.83519.63816.24206.84548.95552.98885.011311.116348.618560.821549.925790.132228.064941.0
131123.9265089.8
1554.61667.71868.62165.42593.92968.23800.26809.98941.8
14213.616422.319591.323589.530749.163375.4131413.1261539.5
1560.1a
1681.8a
1878.6a
2217.0a
2608.7a
2941a
3890a
6640b
8780b
13687b
15920b
18969b
23334b
29972b
63676b
130941b
264847b
-1.2956-1.2689-1.2278-1.0876-1.0209-0.9467-0.7356-0.03600.45981.60952.13292.83193.85285.259912.81227.35655.878
-1.29739a
-1.27401a
-1.2313a
-1.1449a
-1.0456a
-0.9554a
-0.7276a
-0.0380b
0.4717b
1.6229b
2.1410b
2.8429b
3.8485b
5.3367b
12.775b
27.326b
56.052b
500010000200004000062500800001250002500003333335000005714286666678000001×106
2×106
4×106
8×106
PAA (GPa)PQLMD (GPa)PM (GPa)EQLMD/Ne(hartree)
EM /Ne (hartree)
T (K)
Comparison of QLMD results with PIMC results (PRE,79: 155105 (2009) ). He: 5.35g/cm3，Internal Energy of Militzer (EM )，Pressure of Militzer (PM), Internal Energy of QLMD
(EQLMD), Pressure of QLMD (PQLMD), Pressure of AA (PAA )。Ne is the number of electrons。
Test for the Pseudopotential
4.5 5 5.5 6 6.5crystal lattice (a.u.)
−329.32
−329.3
−329.28
−329.26
−329.24
Ene
rgy
(eV
)
E−calE−fit
5.33 a.u.
FLAPW: 5.268-5.342 ( PRB, 82, 132409 (2010) )
Bulk modulus: 169-200 GPa
Bulk modulus:
B=170 GPa
Hugoniot point example at 0.5 eV
10 11 12 13 14 15 16 17ρ (g/cm
3)
−20
−15
−10
−5
0
5
10
(U−
U0)−
P(V
0−V
)/2
(a.u
.)12.125g/cm
3
Hugoniot: Comparison with SESAME
666.7659.933.834.5100
33.5833.0820.522.510
16.915.2817.518.715
3.842 3.56813.2013.261
0.760.62510.1100.1
PQ (Mbar)PS (Mbar)ρQ (g/cm3)ρS (g/cm3)Temperature (eV)
Hugoniot: Ab initio benchmark from the cold normal, across the warm dense, to the hot dense states
Hugoniot of Iron: Density-Pressure
Ionic short ordered structures
There are medium or short ordered structures in hot dense matter
Along Hugoniot – ionic structures
Radial distribution and the orientation order number
6
61
exp 11,
exp 1
ijij i
Nj avi iav av av
iiji
j av
dd
dd d d
Ndd
=
⎡ ⎤⎛ ⎞⎢ ⎥− ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦= =
⎡ ⎤⎛ ⎞⎢ ⎥− ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
∑∑
∑
6
1 1
1 1 exp 1N N
iji i
i i j i av
dECN ECN
N N d= = ≠
⎡ ⎤⎛ ⎞⎢ ⎥= = −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
∑ ∑∑
Effective coordination numbers (ECN)
Along Hugoniot – ionic structures
Along Hugoniot – ionic structures
Along Hugoniot – ionic structures
Self diffusion coefficient - ionic transport properties
33.8
20.5
17.5
13.20
10.1
ρQ (g/cm3)
7.0×10-3
3.5×10-4
2.3×10-4
7.5×10-5
3.0×10-6
DQLMD (cm2/s)
6.4×10-37.0×10-334.5100
1.0×10-41.1×10-422.510
4.9×10-44.2×10-418.715
----13.261
----100.1
DOCP (cm2/s)
DOFMD (cm2/s)
ρS (g/cm3)Tempature(eV)
F. Lambert et al., Europhys. Lett. 75, 681 (2006).
Conductivity
Kubo-Greenwood formula:
)(ˆ
)(3
2)(
2
,
2
ωδψψ
ωπωσ
h
r
−−×
−= ∫ ∑
nmmn
mnmn
EE
ffkde
v
Electrical Conductivity
电导率
Lambert et al. POP, 18, 056306 (2011)
1 10 100 1000T (eV)
107
108
109
σ (1
/(Ω.m
))
QMD+OFMDQLMD
160 g/cm3
80 g/cm3
10 g/cm3
Conductivity of Hydrogen
Thermal Conductivity
热导率
Lambert et al. POP, 18, 056306 (2011)
1 10 100 1000T (eV)
104
105
106
107
κ (W
/(m
.K)
QMD+OFMDQLMD
160 g/cm3
80 g/cm3
10 g/cm3
Conductivity of Hydrogen
Carbon in Dense Hydrogen (10g/cm3,10eV)
0 0.5 1 1.5 20
0.5
1
1.5
H−HC−H
The size of the Coulomb hole depends on the ionic charge
Application: EOS for solar-interior
ApJ. 721: 1158 (2010)
Application: ICF target
How to diagnose the state and the electronic, ionic structures in the process of laser-target interactions.
How the carbon atoms affect the states (may induce instability)?
What’s the influence of local structures on physical properties?
High demand for the uniformities
OMEGA LMJ
NJP, 12 (2010) 043037
Application: EOS table for ICF Capsule
• Wide temperature and wide density
• Important for the simulation of hydrodynamics
• Dominant Ingredients: D-T (gas, ice, solid, warm dense, and hot dense)Plastic (outside face)
EOS table for Deuterium-Tritium mixture
EOS table for Plastic (C8H8)
Conclusions
• QLMD can do simulations from warm dense matter to hot dense plasmas.
• EOS, transport properties can be shown within the framework of QLMD.
• Principal Hugoniot of Iron is calculated from first principles up to 100 eV. The results are in agreement with SESAME table and most experiments, and more consistent with experiments than SESAME at high temperature.
• The basic idea of Langevin equation provide the flexibility of cutting the system (or degree of freedom) from the environment and reduce the complexity to be manageable.
• Reliability of the results depends on a adequate fluctuation-dissipation model.
Acknowledgements:
Colleagues:
Jiayu Dai, Yong Hou, Dongdong Kang
Zengxiu Zhao, Jiaolong Zeng
Supported by:
National Natural Science Foundation of China
