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Hadronic Transport Coefficients from a Microscopic Transport Model

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Hadronic Transport Coefficients from a Microscopic Transport Model ... Motivation: 'Low Viscosity Matter' ... Pyx. Vx= v1. Vx= v2. Shear Viscosity Coefficient: ... – PowerPoint PPT presentation

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Title: Hadronic Transport Coefficients from a Microscopic Transport Model


1
Hadronic Transport Coefficients from a
Microscopic Transport Model
Nasser Demir, Steffen A. Bass Duke
University April 22, 2007
2
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!

3
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!

4
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.
5
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?
6
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!
7
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?
8
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.

9
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!
10
What about Kinetic Equilibrium?
T168.4 MeV
e 0.5 GeV/fm3 ?B ?0
e 0.5 GeV/fm3 ?B ?0
11
Calculating Correlation Functions
NOTE correlation function found to empirically
obey exponential decay.
Ansatz also used in Muronga, PRC 69044901,2004
12
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
13
Entropy Scaling
For system with fixed volume in equilibrium
14
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.

15
Preliminary Results for ? and ?/s
  • Viscosity increases with Temperature.
  • Viscosity decreases with finite baryon number
    density.

16
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)

17
? 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
18
? 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
19
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.

20
  • Backup Slides

21
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)
22
Kovtun, Son, Starinets hep-th/0405231 PRL 94.
111601 (2005)
23
New ?/s measurement for ultra-cold atoms
cond-mat.other/arXiv0707.2574v1
24
UrQMD EoS comparison with Statistical Model
25
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.
26
Preliminary Results for Baryon Diffusion
(Units in fm)
27
A previous study of diffusion
Sasaki, Nonaka, et al. Europhys. Lett., 54 (1)
(2004)
28
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
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