hadronic transport coefficients from a microscopic transport model nasser demir, steffen a. bass...
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Hadronic Transport Coefficients from a Microscopic Transport Model
Nasser Demir, Steffen A. Bass
Duke University
April 22, 2007
Overview
• Motivation: “Low Viscosity Matter” at RHIC & Consequences
• Theory: Kubo Formalism for Transport Coefficients
• Analysis/Results: Equilibriation, Results for Viscosity
• Summary/Outlook: Time-dependence of Transport Coefficients!
Low Viscosity Matter at RHIC
initial state
pre-equilibrium
QGP andhydrodynamic expansion
hadronic phase
freeze-out
QGP-like phase at RHIC observed to behave verymuch like ideal fluid: ideal hydro treatment of QGP phase works well – but what about hadronic phase?
low viscosity
large viscosity
Why study hadronic phase?1) Need to know hadronicviscosity to constrain QGP viscosity.2) Viscosity changes as function of time in a heavy ion collision!
Two Questions re: “low viscosity”
1) How low? (AdS/CFT: η/s≥1/4π? KSS bound)
2) If there is a minimum, where is it? Near Tc?
PRL 94. 111601 (2005) Kovtun, Son, Starinets
Csernai, Kapusta, McLerran:nucl-th/0604032 PRL 97. 152303 (2006)
Pert. Theory N/A here.
What do we know thus far?
• Determining hadronic viscosity necessary to constrain viscosity of QGP.
• 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 near-equilibrium correlations!
Linear Transport Coefficients & Green-Kubo Relations
Green Kubo Relations: Near-Equilibrium
Stat. Mech
Green Kubo tells us we can compute linear transport coefficients by examining near-equilibrium fluctuations.
< … > indicate ensemble averaging once equilibrium has been reached.
OK, how to model the hadronic medium?
Suggests technique of molecular dynamics (MD) simulations.
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.-Interaction takes place only if:
(dmin is distance of closest approach between centers of two hadrons)
- 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!
What about Kinetic Equilibrium?
ε= 0.5 GeV/fm3
ρB =ρ0
ε= 0.5 GeV/fm3
ρB =ρ0
T=168.4 MeV
Calculating Correlation Functions
NOTE: correlation function found to empirically obey exponential decay.
Ansatz also used in Muronga, PRC 69:044901,2004
Entropy ConsiderationsMethod I: Gibbs formula for entropy:(extract μB for our system from SHAREv2,P and ε known from UrQMD.) Denote assGibbs.
Method II: Weight over specific entropies of particles, where s/n is a function of m/T & μB/T! Denote as sspecific
SHARE v2: Torrieri et.al.,nucl-th/0603026 -Tune particles/resonances to those in UrQMD.
Entropy Scaling
For system with fixed volume in equilibrium:
Summarizing our technology
• Use UrQMD in box mode to describe infinite equilibriated hadronic matter.
• Apply Green-Kubo formalism to extract transport coefficients.
• Calculate entropy by counting specific entropies of particles.
Perform analysis of η, η/s as a function of T and baryon # density for a hadron gas IN EQUILIBRIUM.
• Viscosity increases with Temperature.• Viscosity decreases with finite baryon number density.
Preliminary Results for η and η/s
- η/s decreases w. finite μB.- Minimum hadronic η/s ≈ 1.7/(4π)- Is minimum η/s near Tc? Need μ=0 results for T<100 MeV to answer this question with certainty. (IN PROGRESS)
Where is the minimum viscosity?
η increasing as function of T: Think specific binary collisions!
η ~p/σ: p increases w. T, and mean total CM energyshifts further to right of resonance peak.
T increases
σ decreases
E/V =0.3 GeV/cubic fm E/V =1.0 GeV/cubic fm
η decreasing w. finite μB: Think specific binary collisions!
η ~p/σ: Resonant πN crosssxns larger than ππ.Increasing μB!
ε=0.2 GeV/fm3ε=0.5 GeV/fm3
Summary/Outlook• Can apply Green-Kubo formalism to hadronic matter in equilibrium:
– Use UrQMD to model hadronic matter.– Use box mode to ensure equilibrium. Calculated entropy via 2 different methods (microscopic and macroscopic pictures self-consistent).
• Preliminary results:– Hadronic η /s satisfies viscosity bound from AdS/CFT (at least 1.7
times above bound).– η notably reduced at finite μB.In progress:
Analyzing μ=0 mesonic matter for T<100 MeV.• Outlook:
- Describe time-evolution of transport coefficient in relativistic heavy-ion reaction.
Full 3-d Hydrodynamics
QGP evolution
UrQMD
t fm/c
hadronic rescattering
Hadronization
TC TSW
Backup Slides
String theory to the rescue? A nice conjecture on viscosity.
Kovtun, Son, Starinets: hep-th/0405231
PRL 94. 111601 (2005)
Csernai, Kapusta, McLerran:nucl-th/0604032 PRL 97. 152303 (2006)
Strong coupling limit for η/s in QCD can’t be calculated!
Duality Idea: For a class of string theories, a black hole solution to a string theory (AdS5) equivalent to finite temperature solution for its dual field theory (N=4 SUSY YM).
Kovtun, Son, Starinets: hep-th/0405231
PRL 94. 111601 (2005)
New η/s measurement for ultra-cold atoms
cond-mat.other/arXiv:0707.2574v1
UrQMD EoS comparison with Statistical Model
Another computation of η/s from a cascade
Muroya, Sasaki ; Prog. Theor. Phys. 113, 2 (2005)“A Calculation of the Viscosity to Entropy Ratio of a Hadronic Gas”
Note: Muroya et. al have factor of 2 coefficient in viscosity formula, whereas we don’t.
Preliminary Results for Baryon Diffusion
(Units in fm)
A previous study of diffusionSasaki, Nonaka, et al. Europhys. Lett., 54 (1) (2004)
Idea : Compute Time-Evolution of Viscosity of System Losing Equilibrium
Full 3-d Hydrodynamics
QGP evolution
Cooper-Fryeformula
UrQMD
t fm/c
hadronic rescattering
Monte Carlo
Hadronization
TC TSW
PREMISE TO BE ESTABLISHED: Timescale over which η is extracted << timescale over which system alters macroscopic properties.
<π xy(0) π xy (t)>
< (πxy (0) )2 > yielding η(t + kΔt) =
< πxy(0) πxy(Δt )> corres. to η(t + (k-1)Δt) .
Recursion Relation: