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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

large viscosity

QGP and hydrodynamic expansion

initial state

freeze-out

low viscosity

pre-equilibrium

hadronic phase

QGP-like phase at RHIC observed to behave

very much like ideal fluid ideal hydro treatment

of QGP phase works well but what about

hadronic phase?

- Why study hadronic phase?
- Need to know hadronic
- viscosity to constrain QGP
- viscosity.
- Viscosity changes as function
- of time in a heavy ion collision!

Two Questions re low viscosity

- How low? (AdS/CFT ?/s1/4p? KSS bound)
- 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?

Linear Transport Coefficients Green-Kubo

Relations

Phenomenological Transport Equation

thermodynamic/mechanical flux linearly

proportional to applied field in small field

limit.

Examples of transport coefficients thermal

conductivity, diffusion, shear viscosity.

Shear Viscosity Coefficient

y

x

Vx v2

ya

Pyx

y0

Vx v1

Green-Kubo compute linear transport coefficients

by examining near-equilibrium correlations!

Green Kubo Relations Near-Equilibrium Stat. Mech

Green Kubo tells us we can compute linear

transport coefficients by examining

near-equilibrium fluctuations.

lt gt indicate ensemble averaging once

equilibrium has been reached.

Suggests technique of molecular dynamics (MD)

simulations.

OK, how to model the hadronic medium?

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 s of experimentally measurable

processes input from experimental data.

- 55 baryon- and 32 meson species, among those 25

N, ? 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

- Chemical equilibrium DISABLE multibody

decays/collisions. ? RESPECT detailed balance!

- Kinetic Equilibrium Compute TEMPERATURE by

fitting to Boltzmann distribution!

What about Kinetic Equilibrium?

T168.4 MeV

e 0.5 GeV/fm3 ?B ?0

e 0.5 GeV/fm3 ?B ?0

Calculating Correlation Functions

NOTE correlation function found to empirically

obey exponential decay.

Ansatz also used in Muronga, PRC 69044901,2004

Entropy Considerations

Method I Gibbs formula for entropy (extract µB

for our system from SHAREv2, P and e known from

UrQMD.) Denote as sGibbs.

- 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

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 ?, ?/s as a function of T

and baryon density for a hadron gas IN

EQUILIBRIUM.

Preliminary Results for ? and ?/s

- Viscosity increases with Temperature.
- Viscosity decreases with finite baryon number

density.

Where is the minimum viscosity?

- - ?/s decreases w. finite µB.
- Minimum hadronic ?/s 1.7/(4p)
- Is minimum ?/s near Tc? Need µ0 results for

Tlt100 MeV to answer this question with certainty.

(IN PROGRESS)

? increasing as function of T Think specific

binary collisions!

? p/s p increases w. T, and mean total CM

energy shifts further to right of resonance peak.

T increases

E/V 0.3 GeV/cubic fm

E/V 1.0 GeV/cubic fm

s decreases

? decreasing w. finite µB Think specific binary

collisions!

? p/s Resonant pN crosssxns larger than pp.

Increasing µB!

e0.5 GeV/fm3

e0.2 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 ? /s satisfies viscosity bound from

AdS/CFT (at least 1.7 times above bound). - ? notably reduced at finite µB.
- In progress
- Analyzing µ0 mesonic matter for Tlt100 MeV.
- Outlook
- - Describe time-evolution of transport

coefficient in relativistic heavy-ion reaction.

- Backup Slides

String theory to the rescue? A nice conjecture on

viscosity.

Strong coupling limit for ?/s in QCD cant 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 (N4 SUSY YM).

Csernai, Kapusta, McLerran nucl-th/0604032 PRL

97. 152303 (2006)

Kovtun, Son, Starinets hep-th/0405231 PRL 94.

111601 (2005)

Kovtun, Son, Starinets hep-th/0405231 PRL 94.

111601 (2005)

New ?/s measurement for ultra-cold atoms

cond-mat.other/arXiv0707.2574v1

UrQMD EoS comparison with Statistical Model

Another computation of ?/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 dont.

Preliminary Results for Baryon Diffusion

(Units in fm)

A previous study of diffusion

Sasaki, Nonaka, et al. Europhys. Lett., 54 (1)

(2004)

Idea Compute Time-Evolution of Viscosity of

System Losing Equilibrium

PREMISE TO BE ESTABLISHED Timescale over which ?

is extracted ltlt timescale over which system

alters macroscopic properties.

ltp xy(0) p xy (t)gt

lt (pxy (0) )2 gt yielding ?(t k?t) lt pxy(0)

pxy(?t )gt corres. to ?(t (k-1)?t) .

Recursion Relation