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Low Viscosity Matter at RHIC initial state pre-equilibrium QGP and hydrodynamic expansion hadronic phase freeze-out QGP-like phase at RHIC observed to behave very much like ideal fluid: success of ideal hydro description of bulk-evolution viscous hydro calculations of elliptic flow indicate low values of η/s (late) hadronic phase resembles dilute gas: large value of η/s expected separation of thermal and kinetic freeze-out: hadronic phase evolves out of chemical equilibrium low η/s large η/s A comprehensive analysis of the viscosity of QCD matter at RHIC requires a systematic calculation of η/s in the hadronic phase in and out of chemical equilibrium.
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η/s of a Relativistic Hadron Gas at RHIC: Approaching the AdS/CFT bound?
Nasser Demir in collaboration with Steffen A. Bass
Quark Matter 2009 : 21st International Conference on Ultrarelativistic Nucleus-Nucleus Collisions
April 3, 2009N. Demir and S.A. Bass: arXiv:0812.2422 [nucl-th]
Overview• Motivation: “Low Viscosity Matter” at RHIC
& Consequences• Theory: Kubo Formalism for Transport
Coefficients• Analysis/Results: Equilibriation: Thermal
vs. Chemical Freezeout, Results for η/s• Summary/Outlook
Low Viscosity Matter at RHIC
initial state
pre-equilibrium
QGP andhydrodynamic expansion
hadronic phase
freeze-out
•QGP-like phase at RHIC observed to behave very much like ideal fluid: success of ideal hydro description of bulk-evolution
•viscous hydro calculations of elliptic flow indicate low values of η/s• (late) hadronic phase resembles dilute gas: large value of η/s expected•separation of thermal and kinetic freeze-out: hadronic phase evolves out of chemical equilibrium
low η/s
large η/s
A comprehensive analysis of the viscosity of QCD matter at RHIC requires a systematic calculation of η/s in the hadronic phase in and out of chemical equilibrium.
Constraining η/s with viscous hydro
Csernai, Kapusta, McLerran:nucl-th/0604032 PRL 97. 152303 (2006)
Pert. Theory N/A here.
Viscous hydro needs η/s~(1-3)/4π , depending upon choice of initial condition.
M.Luzum & P. Romatschke: Phys. Rev. C78: 034915, 2008
NOTE: T- dependence of η/s neglected in vRFD.
What do we know thus far?
• Determining hadronic viscosity necessary to constrain viscosity of QGP.
• vRFD neglects T-dep. of η/s.• Perturbative methods not well trusted near Tc on hadronic side microscopic
transport model can help here!Next Question: How do we compute transport coefficients?
Phenomenological Transport Equation: thermodynamic/mechanical flux linearly proportional to applied field in small field limit.
Examples of transport coefficients: thermal conductivity, diffusion, shear viscosity.
y
x
y=a
y=0
Pyx
Vx= v1
Vx= v2
Shear Viscosity Coefficient:
Green-Kubo: compute linear transport coefficients by examining correlations near kinetic equilibrium!
Linear Transport Coefficients & Green-Kubo Relations
Modeling the Hadronic Medium:UrQMD (Ultrarelativistic Quantum Molecular Dynamics)
- Transport model based on Boltzmann Equation:
-Hadronic degrees of freedom.-Particles interact only through scattering. ( cascade )-Classical trajectories in phase space.
- Values for σ of experimentally measurable processes input from experimental data.
• 55 baryon- and 32 meson species, among those 25 N*, Δ* resonances and 29 hyperon/hyperon resonance species
• Full baryon-antibaryon and isospin symmetry:- i.e. can relate nn cross section to pp cross section.
“Box Mode” for Infinite Hadronic Matter & Equilibriation• Strategy: PERIODIC BOUNDARY
CONDITIONS! • Force system into equilibrium, and
PREVENT FREEZEOUT.Equilibrium Issues :- Kinetic Equilibrium: Compute TEMPERATURE by fitting to Boltzmann distribution!- Chemical equilibrium: DISABLE multibody decays/collisions. RESPECT detailed balance!
Kubo Formalism: Calculating Correlation Functions
NOTE: correlation function found to empirically obey exponential decay.
Exponential ansatz used in Muronga, PRC 69:044901,2004
T=67.9 +/- 0.7 MeVμ = 0
Τπ = 136 +/- 11 fm/c
Entropy Scaling
For system with fixed volume in equilibrium:
Compute entropy using Gibbs formula:
Viscosity Results (μ~0)
- η/s decreases with increasing T in hadronic phase, but levels off at higher hadronic T.- (η/s)min(T~160 MeV)~0.9.
Fugacities in Hadronic Phase• Chemical Freezeout: Tchem~160 MeV.
• Kinetic Freezeout: Tkin~130 MeV.
• In hydro evol.: – introduce non-unit fugacities for species (pions, kaons). (Equiv.
to λi=exp(μi/T)>1)
NOTE: Hadronic phase acquires increasingly non-unit fugacities as system evolves out of chem. equil in HIC!
c.f. PCE scheme (T. Hirano & K. Tsuda: Nucl. Phys. A715, 821 (2003)P.F. Kolb & R. Rapp: Phys. Rev. C67, 044903 (2003))
• Initialize matter w/ equilibrium distributions,but off-equilibrium yields, corresponding to desired fugacities.
•Perform viscosity measurement before system relaxes into equilibrium.•Verify fugacities at time of measurement w/statistical model analysis.
Chemical Non-equilibrium in HIC w/ UrQMD:
- η/s reduced, and result can be understood classically.- (η/s)min(T~160 MeV) ~(0.4-0.5) (reduced by factor of 2!)- Suggests possible complicated structure of evolution of η/s in HIC.
Effect of Finite Baryochem Pot & chem. non-equil
Where do we go from here?
- Need precise parametrization of η/s as function of T,λi ’s for vRFD calculations.- Future progress to be expected from hybrid vRFD+micro models. - Trajectory of η/s in a HIC as a function of temperaturemay have complicated structure.
Summary/Outlook Use Green-Kubo formalism to calculate η/s of hadronic matter:– UrQMD to model hadronic matter.– box mode to ensure kinetic equilibrium, then calculate viscosity
both at unit and non-unit fugacities. Verified entropy calculation via scaling law.
Results:– Hadronic η /s satisfies viscosity bound from AdS/CFT.– η/s reduced at non-unit fugacities, and this suggests complicated
structure of η/s as it evolves from T≈Tc+ !
Outlook:- Map out η /s trajectory probed by collision in hadronic phase
- Complete calculations for hadronic bulk viscosity and possible effect of system evolving out of chemical equilibrium.
- Trajectory of viscositie(s) in a HIC crucial input to T-dep vRFD calculations and vRFD+micro models.
Backup Slides
Analyzing effect of chemical non-equilibrium in HIC w/ UrQMD.
• Initialize matter w/ equilibrium distributions,but off-equilibrium yields, corresponding to desired fugacities.
• Perform viscosity measurement before system relaxes into equilibrium.• Verify fugacities at time of measurement
w/statistical model analysis.
Entropy ConsiderationsMethod I: Gibbs formula for entropy:(extract μB for our system from SHAREv2,P and ε known from UrQMD.) Denote assGibbs.
SHARE v2: Torrieri et.al.,nucl-th/0603026 -Tune particles/resonances to those in UrQMD.
Method II: Weight over specific entropies of particles, where s/n is a function of m/T & μB/T! Denote as sspecific