thermodynamic properties of nonideal strongly
TRANSCRIPT
Path integral Monte Carlo simulations of equation of state
of strongly coupled quark-gluon plasma
1Joint Institute for High Temperatures, RAS, Moscow, Russia2Institut für Theoretische Physic und Astrophysik, Kiel, Germany3Gesellschaft fur Schwerionenforschung, Darmstadt, Germany
4Russian Research Center ``Kurchatov Institute'', Moscow, Russia5Bogoliubov Lab. of Theoretical Physics, Joint Institue for Nuclear Reseach,
Dubna, Moscow region Russia
V.S. Filinov1,2,M. Bonitz2 , V.E. Fortov1
Y.B. Ivanov3,4, V.V. Skokov4,5, P.R. Levashov1
Joint Institute for High Temperatures, Moscow, RAS
OUTLINE
•States of matter at high energy concentration
•Formation of quark-gluon plasma
•Conclusions from Lattice QCD simulations
•Simulation of quantum many-particle systems
• Path integral Monte Carlo method
•Applications to the quasiparticle models of the
quark-gluon plasma
Matter transformation at high energy concentration
34 /10 cmg
314 /510.2 cmg
Atom
Atomic nucleus
Nucleon
Plasma
Nuclear Matter
Quark plasma
3/1 cmg
Color
deconfinement
Phase transition
Phase transition
Relativistic collisions heavy ions, elliptic flows
Initial space anisotropy Final space anisotropy
INTRODUCTION
f
f
f mDFg
L )(Tr 1 _
2
2
Main Problems: starting from Lagrangian
(1)obtain hadron spectrum,
(2)calculate various matrix elements,
(3) describe phase transitions, and phase
diagram
(4) explain confinement of color
http://www.claymath.org/millennium/
INTRODUCTIONMethods
Imaginary time t→it
Space-time discretization
Thus we get from functional integral thepartition function for statistical theory in fourdimensions
]}[exp{ SiDZ ]}[exp{ SDZ
x
xdxD )( ]}[exp{ SdZ x
x
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INTRODUCTIONThree limits
0
0
qm
L
aLattice spacing
Lattice size
Discrete quark mass
L
a
For typical lattice L=48, L4=5,308,416
We calculate integrals of dimension 32L4 =169,869,312
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Conclusions from
Lattice simulations
Computer simulations a) reproduce well known hadron
properties b) predict new phenomena c) help to create new
theoretical ideas.
Low dimensional objects (regions or quasiparticles - dressed quarks and
gluons) are responsible for most interesting nonperturbative effects: chiralsymmetry breaking, topological susceptibility and confinement.
The era of traditional quantum field theory (Feynman graphs, perturbation
theory) is over, nonperturbative field theory is close in spirit to solid state
theory; we have to study dislocations, fractals, phase transitions etc.
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Interaction and quantum effects in dense 3D and 2D plasma mediawith different mass ratio of charges.
Nonideality boundary:
KinCoul EU
atomes, molecules, clusters
Inside: Strong Coulomb interaction,
Many-body effects
Degeneracy boundary
re
Below: overlapping electron
Wave functions,
Quantum and spin effects
Coulomb interaction: reerU baab /)(
Schocks
Pressure dissociation and
ionization, Mott effect
CTCP
QTCP
COCP
QOCP
Classical one-component
plasma - COCP
Quantum one-component
plasma - QOCP
Classical two-component
plasma - CTCP
Quantum two-component
plasma - QTCP
13 45 3~10 , ~10T K n m
HYDROGEN, PIMC-SIMULATION, n = 1021 cm-3
Ne = Ni = 56, n = 20
T = 10 kK
= 2.7
n3 = 0.41
T = 50 kK
= 1.6710-3 g/cm3
= 0.54
n3 = 3.7 10-2
- proton
- electron
- electron
HYDROGEN, PIMC-SIMULATION, n = 1025 cm-3, T=10 000 Ko
Crystallization of protonsBloch oscillation of electron density
Theretical model of quark – gluon plasmais based on lattice simulations allowing consideration of
quark-gluon plasma as system of dressed quark, antiquark
and gluon quasiparticles (Phys. Rev. C, 74, 044909, (2006))
• All quasiparticles are heavy (m > T) and move non-relastivistically
•All quasiparticle masses are the same.
•We do not distiguish between quark flavors.
•Interparticle interaction is domonated by a color Coulomb potential.
•The color operators are substitured by their average values
– classical color vectors.
•Numbers of quarks, antiquarks and gluons are equal.
The model input requires :•The temperature dependence of the quasiparticle mass.
•The temperature dependence of the quasiparticle density.
•The temperature dependence of the coupling constant.
All the input quantiries should be deduced
from lattice QCD calculations.
Quark-gluon plasma in canonical ensemble
•NEW:Mixture of quasiparticles: Nq quarks, Nq’ antiquarks, and Ng gluons,
• Q are color coordinates of each quasiparticle - vector on unit 3D sphere
!!!,,,,, ''' gqqgqqgqq NNNNNNQVNNNZ
V
gqq QQ rdrdNNNQ ;,,,,, '
•Our modification of [ Phys.Rev. C, 74, 044909, (2006)] :
Temperature decomposition of N-particle density matrix:
HHH ˆexpˆexpˆexp
n+1
1 nkT1
||4
|
2
ˆˆ22
rr
QQgC
m
pH
ab
PATH INTEGRALS
MONTE-CARLO METHOD
gqq
P
gqq
PPPP V
nn
N
g
N
q
N
q
dQdQdrdrQr,,
11
33
'
3
'
'1
1;,,
PSQPrP;QrQrQr nnnn ˆ;ˆ,ˆ,;,;, 11)()1(1
spinmatrix
parity ofpermutations
q’b
Qb ,rb
antiquark
gqqgqq m ,',,', 2 qa
quark, antiquark, gluon
Qa,ra,
q
r
gqqm ,',2
r(1) = r + (1)
r(2)
r(n)
r(n+1) r’
Qc,rc
gluon
gc
),;,()(),;,( )1()1()()()1()()1()1()()( llllllllll QrQrQQQrQr
Density matrix
q q g
gqq
N
s
N
s
N
s
sssNNNrQQr
0 0' 0''
'''333
'
',
1,,
n
l
lgqq
l
n
QrUrQU
0
)(
1
,,
gqq
g
g
q
q
q
q
N
ps
n
ab
l
pp
N
ps
n
ab
l
pp
n
l
N
ps
n
ab
l
pp
N
s
N
N
s
N
N
s
N
sss
per
rQUCCC
rQ
1''
1,
1'
1,
1 1
1,
'''
'''
~~~~~det~
det
,exp222
,'
'
Color Kelbgpotentials
1
2,1 1 |
exp2
nn n a aab a b as
s
Q Qr r y
Color Coulomb and Color Kelbg potential
abababx rabab 22
ab
ba
r
ee
abab r
abab
x
abab
baabab
ab xerfxex
QQgCx ab
11
4
|,
22
ab
baee
0abr
Richardson, Gelman, Shuryak, Zahed, Harmann, Donko, Leval, Kalman
Objects Q are color coordinates
of quarks and gluons
There is no divergence at small
interparticle distances and
it has a true asymptotics (T, xab )
Our suggestion: Color Kelbg potential
Ha -> kBTc , Tc =175 Mev,
Tc < T, ma~5kBTc/c2 ,
Lo ~ hc/kBTc , rs= <r>/Lo<0.1,
Lo~7 10-15 m, xab ~1
EQUATION OF STATE
17
Line 1 – Lattice calculations.
Two models – lines 2, 3: g2(T)=6.28, Mall (T) [Liao,Shuryak]:
Line 2 – PIMC, density variation n/T3=0.244 [Shuryak]
Line 3 – PIMC, fixed density n=3 fm-3,
Upper rhomb relates to density n=0.37 fm-3
Pair distribution functions
Color correlation functions
!!!,,,,, ''' gqqgqqgqq NNNNNNQVNNNZ
V
gqq QQ rdrdNNNQ ;,,,,, '
||4
|
2
ˆˆ22
rr
QQgC
m
pH
ab
),;,,()()(
),,(
1),()(
2211
'
2121
QQ rrRrRdrd
NNNQRRgRRg
V
ba
gqq
abab
);,,(|)()(
),,(
1)(
212211
'
21
QQQ
Q
rrRrR
drdNNNQ
RRc
baba
Vgqq
Defab
PAIR DISTRIBUTION FUNCTIONS
COLOR CORRELATION FUNCTIONS
e-e
h-h
e-h
(e-e) r2
(e-h) r2
n=3 fm-3, T = 3 Tc
19
Canonical ensemble
1
1 1 2 2 1 1 2 22
1 1 2 2
1
1 1 2 2
ˆ( ) { exp( ) exp( )};
1, , ,
2
{exp( )}
1( ) ( , ) ( , )*
(2 )
( , ; , ; ; ),
( , ) exp( ) | |2 2
( , ; , ; ; )
c cFA
c
FA
Ht HtC t Z Tr F i A i
h h
hH K V t t i
kT
Z Tr H
C t dp dq dp dq F p q A p qh
W p q p q t i h
pA p q d i q A q
h
W p q p q t i h Z
1 1 2 21 2
1 2 2 11 2 2 1
exp( )exp( )*
| exp( ) | | exp( ) |2 2 2 2
c c
p pd d i i
h h
Ht Htq i q q i q
h h
Green-Kubo relations
Integral equation
222222
222222
111111
111111
2121
212211
0
20
20
10
10
2211
),,(,2
1
),,(),(2
1
),,(,2
1
),,(),(2
1
)()()
'2()'('
)2(
4),(
)()()},(),()(),({2
1),;,(
),,;,();;,;,(
);0;,;,();;,;,(
qqptqpmdt
dq
pqptpqFdt
dp
qqptqpmdt
dq
pqptpqFdt
dp
ds
sdqF
h
sqSinqqVdq
hhq
qVqFsqqsqqs
qqshiqpqspWddsd
hiqpqpWhitqpqpW
ttt
ttt
ttt
ttt
t
[
[
- positive time direction
- negative time direction
Iterations
Initial conditions
,2
2),
2
~|
~
exp()~,,(
),~,(|)exp(|~))~(exp())...~(exp(
~|)exp(|~))~(exp(~|)exp(|
*)~,,(|)exp(|))(exp())...(exp(
|)exp(|))(exp(|)exp(|
};~...~,~;...,;,;,{
},;~...~,~;...,;,;,{
*~...~~...);0;,;,(
)...,2
],[exp()exp()exp()exp(
2/),exp()...exp()exp()2
exp(
22
1112
21112
1222
21111
1
21212211
21212211
21212211
2
Mm
h
qqi
h
pqqi
h
p
qqp
qqpqKqqVqV
qKqqVqKq
qqpqKqqVqV
qKqqVqKqZ
hiqqqqqqqpqp
hiqqqqqqqpqp
qdqdqdqdqdqdhiqpqpW
VKVKH
MHHHH
MMM
MMM
MM
MM
MM
Schematic snapshot
Mqqq ...21
e
p1q1
p2q2
exp(-K) exp(-V)
Mqqq ~...~~21
~
t=0
-t/2
+t/2
<p(-t/2)p(t/2)>
0h
positive time direction
negative time direction
How to add color dynamics ?
Classical color molecular dynamics
( , , )
( , , )c
q
aabc b
Q
dq p
dt m
dpU p q Q
dt
dQf Q U p q Q
dt
( (0), (0), (0)) ~ exp( ( (0), (0), (0))W p q Q H p q Q
( ) (0)
( ) (0)xy xy
p t p
t
(0), (0), (0)p q Q
( ), ( ), ( )p t q t Q t
H(p,q,Q,t)=const, Energy is integral of motion
[ , ]a b abc c
c
Q Q f Q
f – structure constants
of the color group ,
U - sum of the color
interparticle potentials
Phys. Rev. C, 74, 044909, (2006)
Schematic snapshot
for color Wigner dynamics
Mqqq ...21
e
Qp1q1
Qp2q2
exp(-K) exp(-V)
Mqqq ~...~~21
~
t=0
-t/2
+t/2
<p(-t/2)p(t/2)>
0h
positive time direction
negative time direction
Temporal momentum-momentum
correlation functions of
quark –gluon plasma
1/3
15
/
3( )4
~ 1 10
s
B c
B
r r
rn
Units
hcfm m
k T
h
k T
Thank you for attention.
Contact E-mails:
Electrodynamical plasma
Quantum Monte Carlo method
Canonical ensemble
• Binary mixture of Ne electrons and Ni protons
!!,,,,, ieieie NNNNQVNNZ
• Partition function:
V
ie rdrNNQ ;,,,
• N-particle density matrix:
HHH expexpexp
n+1
1 nkT1
ACCURACY OF DIRECT PIMC
HHH eeeˆˆˆ
ˆ
n+1UKH ˆˆˆ TkB1 1 n
UKUKH eeee
ˆ,ˆ2ˆˆˆ
2
ˆ
Error ~ ( / n)2 for every multiplier
Total error ~ 2 / n 0 at n
potfreeHigh-temperature
pseudopotential
1+O(1/ n)2 for every multiplier
at n
PATH INTEGRAL
MONTE-CARLO METHOD
P V
nΝN
i
drdrrq P
ei
133
11
;,,
,;,,;,, 11 PSrPrqrrq nn
permutationoperator
spinmatrix
thermalwavelengths ofelectron and ion
parity ofpermutation
Pa
qa
proton
ee m 2eb
electron
rb
e
r
em 2
r(1) = r + (1)
r(2)
r(n)
r(n+1) r’
RELATIVE VARIATION IN PRESSURE
FAR AND NEAR THE CONFINEMENT PHASE TRANSITION
e-e
h-h
e-h
(e-e) r2
(e-h) r2
31
Canonical ensemble
2|)exp(|
22|)exp(|
2
*)exp()exp();;,;,(
2||
2)exp(),(
),;;,;,(
*),(),()2(
1)(
)}{exp(
,1
,2
,
)};exp()exp({)(
11
22
22
11
221121
1
2211
2211
221122112
1
qh
Htiqq
h
Htiq
h
pi
h
piddZhitqpqpW
qAqh
pidqpA
hitqpqpW
qpAqpFdqdpdqdph
tC
HTrZ
kT
hittVKH
h
HtiA
h
HtiFTrZtC
cc
FA
c
ccFA
Classical dynamics in phase space
m
tp
dt
qd
tqFdt
pd
)(
))((
))0(),0((exp(~))0(),0(( qpHqpW
)0()( ptp
)0(),0( qp
)(),( tqtp
Wigner approach to quantum
mechanics
)(2
~
);()0,(
);,(~),(
2
QVm
hH
tQHt
tQih
o
',2
'
)exp(),2
(),2
(),,(
),(),(),,(
*
'*'
QQQQ
q
dh
pitqtqtpqf
tQtQtQQ
N-particle functions
Wigner – Liouville equation
dh
pitqtqtpqf
qsqspfsddqpftqpf
pqptpqFdt
pd
qqptqm
p
dt
qd
VFs
sFqd
h
qsSinqqV
hhqs
hsdqsspfq
fF
q
f
m
p
t
f
t
t
t
)exp(),2
(),2
(),,(
),(),,()0,,(),,(
),,(),(
),,(,
,)(
')'2
()'()(
4),(
0,),()(
*
000
0
000
t
p
qf
0f
p
q
s
0f
Klimontovich equation
or Tatarskii condition
3N-dimensional vectors
Iteration series. Quantum averages.
100
0
00
0
000
|!
`|),()|,,(|
....),()|,,()0,,(),,(
n
nt
t
n
Qtqsqspfsdd
qsqspfsddqpftqpf
s),( 000 qpf
0),,,( htqpf
2||
2)exp(),(
),,(),(||
qAq
h
pidqpA
tqpfqpdpdqAA tt
t
q
p
0 ),( 000 qpf
f
SU(2) glue SU(3) glue 2qQCD (2+1)QCD
Quark-Gluon Plasma at Finite Temperature
Plasma thermodynamics, example: pressure
F. Karsch (2001-2005)
APPROACHES TO QUANTUM
THERMODYNAMICS
• Thomas-Fermi method
• Hartree-Fock(-Slater) method
• Density Functional Theory
• Monte Carlo methods (variational, diffusion, path-ingegral)
Temporal momentum-momentum
correlation functions and
frequency dependent conductivity
T = 100 000 K
0 20 40 60 80 100 120 140-5
-4
-3
-2
-1
0
1
p(t
)p(0
)/[p
(0)p
(0)]
t
T=1 105 K
0
rs=10
rs=6
rs=3
0 5 101
10
100
1
05
Re
{
p}
p
T=1 105 K
0 r
s=30
1
2
3
0 2 4 6 8 10 12 141
10
100
T=1 105 K
0 r
s=20
1
2
3
1
05
Re
{
p}
p
0 5 10 15 20
1
10
100
T=1 105 K
0 r
s=10
1
2
3
p
10
5 R
e{
p}
rs = 30
rs = 20
rs = 10
1 – Quantum Dynamics2 – Bornath T. et al., LPB, 18, 535 (2000)3 – Silin V.P., ZhETF 47 2254 (1964)
0 100 200 300 400
162
163
164
165
Т
, М
эВ
B, МэВ
Quark-gluon plasma
Hadronization
Phase transition of 1st order
Critical point
Phase diagram of quark-gluon plasma