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

Joint Institute for High Temperatures, Moscow, RAS Quantum simulation of thermodynamic and transport properties of quark – gluon plasma V. Filinov 1 , M. Bonitz 2 , Y. Ivanov 3 , P. Levashov 1 , V. Fortov 1 1 Joint Institute for High Temperatures, RAS, Moscow, Russia 2 Institut für Theoretische Physic und Astrophysik, Kiel, Germany 3 Gesellschaft fur Schwerionenforschung, Darmstadt , Germany

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 ρ < 3 1 g / cm Electromagnetic plasma Atom ρ = 14 3 10 g cm / Atomic nucleus Nuclear Matter ρ = 15 3 2.510 g cm / Quark-gluon plasma Nucleon

I nteraction and quantum quantum effects effects Classical one-component I nteraction and plasma - COCP in dense dense 3D and 2D 3D and 2D plasma plasma media media in with different different mass mass ratio ratio of of charges charges. . with Quantum one-component = plasma - QOCP U ( r ) e e / r Coulomb interaction: ab a b Classical two-component − 13 45 3 plasma - CTCP T ~ 10 K , n ~ 10 cm Quantum two-component plasma - QTCP Nonideality boundary: CTCP < >=< > U E Coul Kin Inside: Strong Coulomb interaction, Many-body effects Schocks QTCP atomes, molecules, clusters QOCP Degeneracy boundary λ e = r COCP Below: overlapping electron Wave functions, Pressure dissociation and Quantum and spin effects ionization, Mott effect

Semi- -classical approximation for non classical approximation for non- -Abelian Abelian plasmas plasmas Semi 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~T d or above T d and below T c if T d <T c ) Feinberg, Litim, Manuel, Stoecker,Bleicher,, Richardson, Bonasera,Maruyama, Hatsuda, Shuryak,…. early universe ALICE quark-gluon plasma RHIC <ψψ> ∼ 0 crossover Tc ~ 170 MeV massive dressed quarks Chiral restoration T and soft gauge fields SPS quark matter Confinement ? <ψψ> > 0 crossover hadronic fluid superfluid/superconducting phases ? n > 0 Phase diagram n = 0 2SC <ψψ> > 0 B B CFL vacuum nuclear matter neutron star cores (F.Karsch) µ ∼ 922 MeV o µ

Basic asumptions of the semi-classical quasiparticle model of 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 or experimental data and substitued in quantum Hamiltonian.

Thermodynamics of quark - gluon plasma in grand canonical ensemble within Feynman formulation of quantum mechanics ∑ = + = + β + = 2 2 ( ) H K U p m U β β C a a C a � � − β < > 2 (| |, ) | g r r Q Q ∑ ∑ = + β + 2 2 a b a b ( ) p m π − a a 4 | | r r , a a b a b Grand canonical partition function ( ) ∑ Ω μ μ = β = βμ − × , 0, , exp( ( )) V N N g q q N , N , N , N , N , N , N u d s g d u s ( ) × β , , , / ! ! ! ! ! ! ! Z N N N N N N N N N N q g u d s g q u d s SU(3) Haar measure = + + = + + ; N N N N N N N N � � � � ( ) ( ) q u d s q u d s with two Casimirs !!!! = ∑∫ β μ ρ σ β , , , , , ; Z N N N drd Q r Q q q g σ ( ) ( V ) ( ) ρ = − β β = −Δ β β × × −Δ β β … exp ( ) exp ( ) exp ( ) H H H n +1 ( ) Δ β = β + 1 n β = 1 kT

PATH INTEGRAL MONTE-CARLO METHOD quark, antiquark, gluon λ = π λ + 3 2 3 q a 2 ( / ( 1) ) m n T Δ r (2) , , ', , ', , ', q q g q q g q q g q’ b λ = 1/ m , ', , ', q q g q q g λ q r (n+1) ≡ r r (1) = r + λ Δ ξ (1) σ ’ ≡ σ r (n) Q a ,r b antiquark r g c Q a ,r a, Q c ,r c gluon parity of � � ��� � � � ( ) permutations ( ) ( ) 1 ∑ ( ) ( ) ( ) κ 1 ∫ n ρ σ β = ± μ μ × 1 … n … , , ; 1 P r Q dr dr d Q d Q 3 N λ 3 λ λ 3 N N q q g = , , ) ( ( P P P P Δ ) Δ ( V Δ � � � � � � � � q g q g q q ) ( ) + ( ) ( ) ( ) (2) ( ) 1 n n + ′ ρ Δ β ρ ˆ ˆ Δ β σ , ˆ σ 1 1 … n n , ; , ; , , ; r Q r Q r Q ;Pr PQ S P + + + + spin ρ ≈ δ − ρ ( ) ( ) ( 1 ) ( 1 ) ( ) ( 1 ) ( ) ( ) ( 1 ) ( ) l l l l l l l l l l ( , ; , ) ( ) ( , ; , ) r Q r Q Q Q r Q r Q matrix

Density matrix � � � � ( ) ( ) 1 ∑ ∑ ⎡ ⎤ ρ σ β ; = ρ β , , , r Q rQ ⎣ ⎦ 3 N λ 3 λ λ 3 N N q q g σ σ Δ Δ Δ { } ( ) ( ) [ ] [ ] ρ β = − β β × , exp , rQ U rQ N N N q n q g × ∏∏ ∏ ∏ � � ϕ ψ ϕ ψ ϕ ψ � � � � ,1 ,1 ,1 l n l n l n det det per pp ab pp ab pp ab N N N = = q = = g q 1 1 1 1 l p p p Relativistic measure ( ) ⎡ ⎤ β Pairwise sum of instead of Gaussian one ( ) l , U r Q ⎣ ⎦ n ( ) [ ] ∑ l β = Kelbg potentials , U rQ + for each l=0,…,n 1 n = Exchange 0 l ⎧ ⎫ matrix ( ) 2 − (0) ( ) n ⎪ ⎪ r r ψ ≡ δ δ + + a a ,1 2 n ⎨ ⎬ ( / (( 1) )) K m n T λ 2 Δ σ σ , , 2 ab f f a s a b a b ⎪ ⎪ a ⎩ ⎭

Color Kelbg Kelbg potential Color potential Richardson, Gelman, Shuryak, Zahed, Harmann, Donko, Levai, Kalman (r=0 ?) � = r λ � λ = Δ β μ x � 2 2 ab ab ab ab ab � � { } < > 2 | Q Q g ( ) ( ) − 2 Φ Δ β = − + π ⎡ − ⎤ ab x a b , 1 1 x e x ⎣ erf x ⎦ ab � πλ ab ab ab 4 x ab ab Objects Q are color coordinates of quarks and gluons → There is no divergence at small 0 r ab interparticle distances and � >> λ it has a true asymptotics (T, x ab ) r < > π 2 ab ab | Q Q g a b ~ � πλ 4 Ha -> k B T c , T c =175 Mev, ab /c 2 , T c < T, m a ~ k B T c L o ~ hc/k B T c , r s = <r>/L o ~0.3, < > 2 | Q Q g ~1.2 10 -15 m L o a b � πλ 4 x ab ab

Input quantities α = π < 2 ( ) ( ) / 4 1 T g T 1) Coupling constant μ = 0 B 2) Quasiparticle masses: m q , m q, m g Ratio of potential to kinetic energy per quasiparticle Γ ( ) ~ / ~5 T U K Density from grand canonical ensemble r s - Wigner-Seitz radis

Equation of State. The entropy density. The trace anomaly. Equation of State. Comparison path integral results with lattice (2+ 1) QCD Comparison path integral results with lattice (2+ 1) QCD 13 ε / T 4 12 11 PIMC 10 lattice (2010) 9 8 1.0 1.5 2.0 2.5 3.0 T/Tc The QCD equation of 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

Pair distribution functions in canonical emsemble � � − β < > 2 (| |, ) | g r r C Q Q ∑ ∑ = β + + 2 2 a b ab a b ( ) H m p β α π − a 4 | | r r , a a b a b 1 − = = × (| |) ( , ) g R R g R R 1 2 1 2 a b a b ( , , ) Z N N N q g q ∑ ∫ δ − δ − ρ σ β a b ( ) ( ) ( , , ; ), d rd Q R r R r r Q 1 1 2 2 V σ

PAIR DISTRIBUTION AND COLOR CORRELATION FUNCTIONS Different quasiparticles Similar quasiparticles