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: