Quantum simulation of thermodynamic and transport properties of - - PowerPoint PPT Presentation

Quantum simulation of thermodynamic and transport properties of quark – gluon plasma

1Joint Institute for High Temperatures, RAS, Moscow, Russia 2Institut für Theoretische Physic und Astrophysik, Kiel, Germany 3Gesellschaft fur Schwerionenforschung, Darmstadt , Germany

• V. Filinov1, M. Bonitz2 , Y. Ivanov3,
• P. Levashov1, V. Fortov1

Joint Institute for High Temperatures, Moscow, RAS

Outlook Outlook

• Path integral approach to quark

Path integral approach to quark-

• gluon plasma

gluon plasma

• Quantum effects in particle interactions and

Quantum effects in particle interactions and Kelbg potentials Kelbg potentials

• Thermodynamic quantities and pair distribution

Thermodynamic quantities and pair distribution functions functions

• Wigner formulations of quantum mechanics

Wigner formulations of quantum mechanics

• Integral form of the color Wigner

Integral form of the color Wigner – – Liouville Liouville equation equation

• Quantum dynamics and kinetic properties

Quantum dynamics and kinetic properties

Matter transformation at high density Matter transformation at high density and energy concentration and energy concentration

14 3

10 / g cm ρ =

15 3

2.510 / g cm ρ = Atom Atomic nucleus Nucleon Electromagnetic plasma Nuclear Matter Quark-gluon plasma

3

/ 1 cm g < ρ

I nteraction and I nteraction and quantum quantum effects effects in in dense dense 3D and 2D 3D and 2D plasma plasma media media with with different different mass mass ratio ratio of

• f charges

charges. .

Nonideality boundary:

> >=< <

Kin Coul

E U

atomes, molecules, clusters Inside: Strong Coulomb interaction, Many-body effects

Degeneracy boundary

r

e =

λ

Below: overlapping electron Wave functions, Quantum and spin effects

Coulomb interaction: r e e r U

b a ab

/ ) ( =

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

3 45 13

10 ~ , 10 ~

−

cm n K T

Semi Semi-

• classical approximation for non

classical approximation for non-

• Abelian

Abelian plasmas plasmas

Phase diagram

(F.Karsch)

T µ early universe ALICE

<ψψ> > 0

SPS

quark-gluon plasma hadronic fluid nuclear matter vacuum

RHIC

Tc ~ 170 MeV µ ∼

• <ψψ> > 0

n = 0 <ψψ> ∼ 0 n > 0 922 MeV

phases ? quark matter

neutron star cores

crossover

CFL

B B

superfluid/superconducting

2SC

crossover

?

massive dressed quarks and soft gauge fields

Feinberg, Litim, Manuel, Stoecker,Bleicher,, Richardson, Bonasera,Maruyama, Hatsuda, Shuryak,….

Confinement Chiral restoration

In restricted part of phase diagram results of resummation technique and lattice simulations allow for consideration of quark-gluon plasma as system of dressed quarks, antiquarks and gluons which can be presented by massive color Coulomb quasiparticles with T-dependent dispersion curves and width (at least at μ=0 at T~Td

• r

above Td and below Tc if Td <Tc)

Basic asumptions

• f the

semi-classical quasiparticle model

• f quark

– gluon plasma

is based on resummation technique and lattice simulations allowing for consideration of quark-gluon plasma as system of dressed quark, antiquark and gluon presented by color Coulomb quasiparticles with T-dependent dispersion curves and width.

(Shuryak , Phys.Lett.B478,161(2000), Phys. Rev. C, 74, 044909, (2006))

• We consider relativistic color quasiparticles representing gluons and the most

stable quarks of three flavors (up, down and strange).

• Up, down and strange quasiparticles have the same masses
• Interparticle interaction is domonated by a color Coulomb potential with

distance dependent coupling constant.

• The color operators are substituted by their average values

– classical color vectors in SU(3) (8D vectors with 2 Casimirs conditions.).

The model input requires :

• The temperature dependence of the quasiparticle masses.
• The temperature dependence of the coupling constant.

All input quantities should be deduced from lattice QCD calculations

• r

experimental data and substitued in quantum Hamiltonian.

Thermodynamics

• f quark
• gluon

plasma in grand canonical ensemble within Feynman formulation

• f quantum

mechanics

( )

( )

, , , , , ,

, 0, , exp( ( )) , , , / ! ! ! ! ! ! ! ;

u d s g d u s

g q q N N N N N N N q g u d s g q u d s q u d s q u d s

V N N Z N N N N N N N N N N N N N N N N N N μ μ β βμ β Ω = = − × × = + + = + +

∑

( ) ( )

, , , , , ;

q q g V

Z N N N drd Q r Q

σ

β μ ρ σ β =∑∫

• (

) ( ) ( )

exp ( ) exp ( ) exp ( ) H H H ρ β β β β β β = − = −Δ × × −Δ …

n+1

( )

1 n β β Δ = +

kT 1 = β

Grand canonical partition function

2 2 2 2 2 ,

( ) (| |, ) | ( ) 4 | |

C a a C a a b a b a a a a b a b

H K U p m U g r r Q Q p m r r

β β

β β β π = + = + + = − < > = + + −

∑ ∑ ∑

• SU(3) Haar measure

with two Casimirs !!!!

PATH INTEGRAL MONTE-CARLO METHOD

( )

( )

( ) ( ) ( ) ( )

1 1 3 3 3 , ,

1 , , ; 1

P q q g q g q

n n N N N P P P P V q g q

r Q dr dr d Q d Q

κ

ρ σ β μ μ λ λ λ

= Δ Δ Δ

= ± ×

∑ ∫

• …

…

( )

( )

( ) ( ) ( )

( ) (

)

(2) ( ) 1 1 1

ˆ ˆ ˆ , ; , ; , , ;

n n n n

r Q r Q r Q ;Pr PQ S P ρ β ρ β σ σ

+ +

′ Δ Δ ,

• …

spin matrix

parity of permutations

q’b Qa ,rb

antiquark

, ', , ',

1/

q q g q q g

m λ =

qa

quark, antiquark, gluon

Qa ,ra,

λq r

3 2 3 , , ', , ', , ',

2 ( / ( 1) )

q q g q q g q q g

m n T λ π λ

Δ

= +

r(1) = r + λΔ ξ(1) r(2) r(n)

r(n+1) ≡ r σ’ ≡ σ

Qc ,rc

gluon

gc

) , ; , ( ) ( ) , ; , (

) ( ) 1 ( ) ( ) ( ) 1 ( ) ( ) 1 ( ) 1 ( ) ( ) ( l l l l l l l l l l

Q r Q r Q Q Q r Q r

+ + + +

− ≈ ρ δ ρ

Density matrix

( )

( )

3 3 3

1 , , ,

q q g

N N N

r Q rQ

σ σ

ρ σ β ρ β λ λ λ

Δ Δ Δ

⎡ ⎤ ; = ⎣ ⎦

∑ ∑

• [

]

( )

( )

( )

, , 1

l n l l

U r Q U rQ n β β

=

⎡ ⎤ ⎣ ⎦ = +

∑

[ ]

( )

[ ]

( )

{ }

,1 ,1 ,1 1 1 1 1

, exp , det det

q g q g q

N N N q n l n l n l n pp ab pp ab pp ab N N N l p p p

rQ U rQ per ρ β β β ϕ ψ ϕ ψ ϕ ψ

= = = =

= − × ×∏∏

∏ ∏

• (

)

2 (0) ( ) ,1 2 , , 2

( / (( 1) ))

a b a b

n a a n ab f f a s a

r r K m n T

σ σ

ψ δ δ λ 2 ⎧ ⎫ − ⎪ ⎪ ≡ + + ⎨ ⎬ Δ ⎪ ⎪ ⎩ ⎭

Exchange matrix Pairwise sum of Kelbg potentials for each l=0,…,n Relativistic measure instead of Gaussian one

Color Color Kelbg Kelbg potential potential

ab ab ab

x λ = r

• 2

2

ab ab

λ β μ = Δ

• 2

| 4

a b ab ab

Q Q g x πλ < >

• ab

ab

λ >> r

• (

) ( )

{ }

2

2

| , 1 1 4

ab

x ab a b ab ab ab ab ab

Q Q g x e x erf x x β π πλ

−

< > Φ Δ = − + − ⎡ ⎤ ⎣ ⎦

• →

ab

r

Richardson, Gelman, Shuryak, Zahed, Harmann, Donko, Levai, Kalman (r=0 ?)

Objects Q are color coordinates

• f quarks and gluons

There is no divergence at small interparticle distances and it has a true asymptotics (T, xab )

Ha -> kB Tc , Tc =175 Mev, Tc < T, ma ~ kB Tc /c2 , Lo ~ hc/kB Tc , rs = <r>/Lo ~0.3, Lo ~1.2 10-15 m

2

| ~ 4

a b ab

Q Q g π πλ < >

Input quantities 1) Coupling constant

( ) ~ / ~5 T U K Γ

2

( ) ( ) / 4 1 T g T α π = <

Density from grand canonical ensemble rs - Wigner-Seitz radis 2) Quasiparticle masses: mq , mq, mg

Ratio of potential to kinetic energy per quasiparticle

B

μ =

Equation of State. Equation of State. The entropy density. The trace anomaly. Comparison path integral results with lattice (2+ 1) QCD Comparison path integral results with lattice (2+ 1) QCD

The QCD equation

• f

state with dynamical quarks Szabolcs Borsanyi, Gergely Endrodi, Zoltan Fodor, Antal Jakovac, Sandor D. Katz, Stefan Krieg, Claudia Ratti, Kalman K. Szabo, JHEP 11 (2010) 077

1.0 1.5 2.0 2.5 3.0 8 9 10 11 12 13 PIMC lattice (2010)

ε/T4

T/Tc

Pair distribution functions in canonical emsemble

1 2 1 2 1 1 2 2

1 (| |) ( , ) ( , , ) ( ) ( ) ( , , ; ),

a b a b q g q a b V

g R R g R R Z N N N d rd Q R r R r r Q

σ

δ δ ρ σ β − = = × − −

∑ ∫

2 2 2 ,

(| |, ) | ( ) 4 | |

a b ab a b a a a b a b

g r r C Q Q H m p r r

β α

β β π − < > = + + −

∑ ∑

PAIR DISTRIBUTION AND COLOR CORRELATION FUNCTIONS Similar quasiparticles Different quasiparticles

Classical dynamics in phase space

m t p dt q d t q F dt p d ) ( )) ( ( = =

( (0), (0)) ~ exp( ( (0), (0))

C

W p q H p q β −

> < ) ( ) ( p t p

) ( ), ( q p

) ( ), ( t q t p

Wigner approach to quantum mechanics

2

( , ) ( , ); ( ,0) ( ); ( ) 2

• q t

ih H q t t q q h H V q m ∂Ψ = Ψ ∂ Ψ = Ψ = − Δ +

• '

* ' *

( , , ) ( , ) ( , ) ( , , ) ( , ) ( , )exp( ) 2 2 ', ' 2

L

q q t q t q t p W q p t q t q t i d h q q q q q ρ ξ ξ ξ ξ ξ = Ψ Ψ = Ψ − Ψ + + = = −

∫

WL - Wigner-Louville function

QUANTUM DYNAMI CS I N QUANTUM DYNAMI CS I N WI GNER REPRESENTATI ON WI GNER REPRESENTATI ON

[ ]

ρ ρ , ˆ H t i = ∂ ∂

( ) ( ) ∫

−

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ 2 − 2 + 2 1 = ξ ξ ξ ρ π

ξd

e q q p q W

ip Nd L

, ,

( )

( )

∫

′ ′ − ′

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ 2 ′ ′ + ′ = ′ ′ ′ dp e p q q W q q

p q q i L

, , ρ

( ) ( )

∫

− = ∂ ∂ + ∂ ∂ ds q s t q s p dsW q W m p t W

L L L

, , , ω

( ) ( ) ( )

∫

⎥ ⎦ ⎤ ⎢ ⎣ ⎡2 ′ − ′ 2 2 =

• '

sin , sq q q U q d q s

Nd

π ω = ∂ ∂ ∂ ∂ − ∂ ∂ + ∂ ∂ p W q U q W m p t W

L L L

m p q =

• q

U p ∂ ∂ − =

• Density matrix:

Wigner function: Evolution equation: Classical limit ħ → 0: Characteristics (Hamilton equations):

Quasi-distribution function in phase space for the quantum case

( ) ( ) ( )

q q q q ′ ′ ′ = ′ ′ ′ ψ ψ ρ

*

, C ∈ ψ R W L ∈

SOLUTION OF THE WIGNER SOLUTION OF THE WIGNER EQUATION IN INTEGRAL FORM EQUATION IN INTEGRAL FORM

, , p q t

( ) ( ) ( ) ( ) ( ) ( )

' ' ' ' ' ' '

, , , , ; , ,0 , ' , , ; , , ' , , ' ,

L W t W L

W p q t p q t p q W p q dp dq d dp dq p q t p q dsW p s q s q

τ τ τ τ τ τ τ

τ τ τ ω

∞ −∞

= Π × + Π −

∫ ∫ ∫∫ ∫

( )

( )

( )

' ' ' '

, | ; , , '

t t

dp F q t q t p q q dt

τ τ τ τ

τ

=

= =

( ) ( )

' ' ' '

, | ; , , '

t t

dq p t m p t p q p dt

τ τ τ τ

τ

=

= =

Dynamical trajectories:

' , ,

' '

τ

τ τ q

p

p q

( ) ( )

' , , ; ( ) ' , , ; ( ) ' , , ; , , (

' ' ' ' ' '

τ δ τ δ τ

τ τ τ τ τ τ

q p t q q q p t p p q p t q p

t t W

− − = Π

Propagator:

p q

, , p q t

, , p q t

' , ,

' '

τ

τ τ q

p

Итерационный ряд. Квантовые средние.

1

( , , ) ( , ,0) ( , , ) ( , ) .... ` | ( , , ) ( , ) | | !

t L L L t n L n

W p q t W p q d dsW p s q s q t Q d dsW p s q s q n

τ τ τ τ τ τ

τ τ ω τ τ ω

=

= + − + − ≤

∫ ∫ ∫ ∫

s ( , )

L

W p q

( , , )

L

W p q t

( )

( )

* '

ˆ | | ( , ) ( , , ) ( , ) exp( ) | | 2 2 ( , ) ( , ) , , 2

L t t i q q p L

A dpdqA p q W p q t p A p q d i q A q h q q q t q t q q W p e dp ξ ξ ξ ξ ρ

′ ′′ −

< Ψ Ψ >= = < − + > ′ ′′ + ⎛ ⎞ ′ ′′ Ψ Ψ = = ⎜ ⎟ ⎝ ⎠

∫ ∫ ∫

t q p τ ( , )

L

W p q

L

W ω

Klimontovich equation

• r Tatarskii condition

Quantum dynamics in phase space

m t p dt q d t q F dt p d ) ( )) ( ( = =

~ ( (0), (0))

L

P W p q

> < ) ( ) ( p t p

) ( ), ( q p

) ( ), ( t q t p

( )

sign ( (0), (0))

L

weight W p q =

random momentum jumps

Averaged Weil symbols

• f operators

periodic boundary conditions

WL - Wigner-Louville function

Vertual classical trajectories

Vertual quantum trajectories

Kinetic properties of quark – gluon plasma in canonical ensemble

1 * 1 1 1 1 2 2 2 1 1 1 2 2 2

( ) {exp( ) exp( ) exp( )} ( ) { exp( ) exp( )}; ( ) exp( ) ( ) 2 1 ( ), , , 2 {exp( )} 1 ( ) ( , , ) ( , , (2 )

FA c c FA FA FA c FA

Ht Ht G t Z Tr H F i A i h h Ht Ht h C t Z Tr F i A i C G h h h H K V qQ t t i kT Z Tr H C t d Q dp dq d Q dp dq F p q Q A p q Q h

ν

β β ω ω ω β β β μ μ π

− −

= − − = − = − = + = − = = − =

2 1 1 1 2 2 2 1 1 1 2 2 1 1 1 2 2 2 1 2 * 1 2 2 1 1 2 2 1

) ( , , ; , , ; ; ), ( , , ) exp( ) | | 2 2 ( , , ; , , ; ; ) exp( )exp( ) | exp( ) | | exp( ) | 2 2 2 2

c c

W p q Q p q Q t i h p A p q Q d i q A q h p p W p q Q p q Q t i h Z d d i i h h Ht Ht q i q q i q h h β ξ ξ ξ ξ ξ ξ β ξ ξ ξ ξ ξ ξ

−

× = − < − + > = × < + − >< + − − >

∫∫ ∫∫ ∫∫

Integral color Wigner – Liouville equation

1 1 1 2 2 2 1 1 1 2 2 2 1 2 1 1 1 2 2 2 1 1 2 2 1 1 2 2 1 1 2 2

( , , ; , , ; ; ) ( , , ; , , ;0; ) ( ) ( , , ; , , ; ; ) ( , , ; , , ), 1 ( , , ; , , ) { ( , , ) ( ) ( , , ) ( 2

t

W p q Q p q Q t i h W p q Q p q Q i h Q Q d ds d W p s q Q p q Q i h s q Q q Q s q Q q Q s q Q q Q

τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ

β β δ τ η η τ β γ η γ η ω δ η ω η δ = − + + − − = −

∫ ∫∫ ∫∫

1,

1 1 1 1 1 2 2 1 , 1, 1, 1 1 , 1 1 1

)}, ( , ) ( , ) 4 2 ' ( ) ( , , ) ' ( ', ) ( ) ( , ) (2 ) 1 1 , ( , ), 2 2 ( ) 1 ( , ), 2 ( , , ,

c i

q t t t t t t t a i abc b t t i Q b c t

s F q Q V q Q sq d s q Q dq V q q Q Sin F q Q h h h ds dq p dp F q Q dt dt m p dQ f Q V q Q dt p t p q Q

ν

δ ω η π = −∇ = − + − − − − − − − − − − − − − − − − − − − − − − − − − − = = + = ∇

∫∫ ∑

i

2,

1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 , 2, 2, 2 2 , 2 2 2 1 2 2 2 2 1 2 2 2 2

) , ( , , , ) , ( , , , ) 1 1 , ( , ), 2 2 ( ) 1 ( , ), 2 ( , , , ) , ( , , , ) , ( , ,

c i

t t t t t t t t t a i abc b t t i Q b c t t t

p q t p q Q q Q t p q Q Q dq p dp F q Q dt dt m p dQ f Q V q Q dt p t p q Q p q t p q Q q Q t p q = = = − − − − − − − − − − − − − − − − − − − − = − = − + = − ∇ = =

∑

2 2

, ) Q Q =

Positive time direction Negative time direction

Color Wong dynamics in SU(3)

Initial conditions Hamiltonian eqations

Initial conditions

1 1 1 2 2 1 1 1 2 2 2 1 2 1 2 2 1 1 2 2 1 1 2 2

( , , ; , , ;0; ) exp( )exp( ) | exp( ) | | exp( ) | ( ) 2 2 2 2 2 2 exp( ) exp( )exp( )...exp( ), / 2 , 2 [ exp( ) exp( )exp( )exp( p p W p q Q p q Q i h Z d d i i h h H H q q q q Q Q H H H H M t H K V ξ ξ β ξ ξ ξ ξ ξ ξ β β δ β ε ε ε ε β ε ε ε ε

−

= × < + − − >< + − − > − − = − − − = = − = − − −

∫∫

1 1 1 2 2 2 1 2 1 2 1 1 1 2 2 2 1 2 1 2 1 1 1 2 2 2 1 2 1 2 1 1 1 1 1

, ] )..., 2 ( , , ; , , ;0; ) ... ... { , , ; , , ; , ... ; , ... ; }, { , , ; , , ; , ... ; , ... ; } | exp( ) | exp( ( , )

M M M M M M

K V W p q Q p q Q i h dq dq dq dq dq dq p q Q p q Q q q q q q q i h p q Q p q Q q q q q q q i h Z q K q V q Q β β β ε ε

−

≈ × Ψ Ψ = < − > −

∫∫

• 1

2 2 1 1 2 2 1 2 1 1 2 1 2 2 2 2 1 1 1

) | exp( ) | exp( ( , ))...exp( ( , )) | exp( ) | ( , , ) | exp( ) | exp( ( , )) | exp( ) | exp( ( , ))...exp( ( , )) | exp( ) | ( , , ) ( , ,

M M M M M M

q K q V q Q V q Q q K q p q q q K q V q Q q K q V q Q V q Q q K q p q q p q ε ε ε ε φ ε ε ε ε ε ε φ φ < − > − − < − > × < − > − < − > − − < − >

• 2

2

| 2 ) ~ exp( ), , 2 2 p q q p q q i i h h h q Mm

ν

λ λ π π π β λ λ λ λ π − − < + + > =

Schematic snapshot for color phase space dynamics

M

q q q ...

2 1

e p1 q1 Q p2 q2 Q ϕ ϕ exp(-εK) exp(-εV)

M

q q q ~ ... ~ ~

2 1

∼λ t=0

• t/2

+t/2

<p(-t/2)p(t/2)>

→ h

positive time direction negative time direction

Velocity autocorrelation function and diffusion constant QGP

1

lim ( ) lim ( ) ( ) ( / 2) ( / 2) 1 ( / 2) ( / 2) 3

t t t N i i i

D D t d D D v v v v N τ τ τ τ τ τ τ

→∞ →∞ =

= = =< − >= = <

• −

>

∫ ∑

Time autocorrelation function of the stress energy tensor and shear viscosity of quark –gluon plasma

, , 1

lim ( ) , ( ) ( / 2) ( / 2) 3 1 1 ( ) / 2

t t XY XY X Y B N XY ix iy i ij x ij y i i j

n d k T p p m r F N η η τ τ η τ σ τ σ τ σ τ

→∞ < = ≠

= = − ⎛ ⎞ = + ⎜ ⎟ ⎝ ⎠

∑ ∫ ∑ ∑

Diffusion coefficient and Diffusion coefficient and shear viscosity shear viscosity

CONCLUSIONS CONCLUSIONS

• Path integral Monte Carlo is a reliable and very

fast method of simulation thermodynamic properties in a wide range of plasma parameters

• Results of simulations agree with available

theoretical and experimental data.

• Combination of path integral MC with Wigner and

Wong dynamics can be applied to treatment transport properties of QGP.

Thank you for attention Thank you for attention. .

Contact E Contact E-

• mails:

mails: v vladimir_filinov@mail.ru ladimir_filinov@mail.ru