BULK VISCOSITY OF INTERACTING HADRONS

Anton Wiranata and Madappa Prakash

Department of Physics and AstronomyOhio University, Athens, OH

Quark Matter 2009, March 30 – April 4, Knoxville, TN

Quark Matter 2009, March 30 – April 4, Knoxville, TN

Topics Overview of results Bulk viscosity in the Chapman–Enskog

approximation Bulk viscosity and speed of sound

Analysis in limiting situations Numerical results of ηv and ηv /s

Chiral pions Massive interacting pions

Multi-component mixturesBinary and N-species systems

Ongoing work and future plans

Bulk viscosity / entropy density vs T

Kapusta, arxiv:0809.3746v2

Quark Matter 2009, March 30 – April 4, Knoxville, TN

Wiranata & Prakash.

Chen and Wang, Arxiv:0711.4824v1

P.Arnold, et.al JHEP05, 051(2003) Kharzeev & Tuchin

JHEP09, 093(2008)

Chapman-Enskog Approximation “For small deviations from equilibrium, the distribution

function can be expressed in terms of hydrodynamic variables ( f(x,p) μ(x), u(x), T(x) ) and their gradients. Transport coefficients (viscosities, thermal conductivity, etc. ) are then calculable from relativistic kinetic theory.”

Equilibrium distribution function

Quark Matter 2009, March 30 – April 4, Knoxville, TN

μ(x) : Chemical potentialu(x) : Flow velocityT(x) : Temperature

φ(1) : Deviation function

Beginning with the relativistic Boltzman equation

and inserting f(0)

the solution (deviation function) has the general structure

Quark Matter 2009, March 30 – April 4, Knoxville, TN

Bulk ViscosityFirst approximation :

Reduced enthalpy : (ε + P) / nmc2

Relativistic Omega Integrals

Transport cross sectionThermal weightRelative momentum weight

Relative momentum : g = mc sinh ψTotal momentum : P = 2mc cosh ψ

Quark Matter 2009, March 30 – April 4, Knoxville, TN

Ratio of specific heats

Bulk Viscosity & the Speed of Sound

Chapman-Enskog 1st approximation

Quark Matter 2009, March 30 – April 4, Knoxville, TN

Utilizing

Features thermodynamic variables

The omega integral contains transport cross-section

Analysis for limiting cases

Nearly massless particles or very high temperatures (z << 1) :

Massive particles such that z >> 1 :

Lesson : For a given T, intermediate mass particles contribute significantly to ηv

Quark Matter 2009, March 30 – April 4, Knoxville, TN

Variational approximation Sean Gavin, Nucl. Phy. A 435, 826 (1985)

Gavin’s result can be rewritten as

Kapusta , arxiv:0809.3746v2

A similar analysis leads to the result that intermediate mass particles controlthe magnitude of ηv Quark Matter 2009, March 30 – April 4, Knoxville, TN

Numerical results (interacting pions)

Parametrization from Bertsch et al. , PR D37 (1988) 1202.

Quark Matter 2009, March 30 – April 4, Knoxville, TN

Entropy Density

15

Non-interacting part

Interacting part (up to 2nd virial expansion)

Entropy Density

Energy Density

Pressure

Fugacity

β =1/kT : Inverse Temperature

InteractionsThermal weightKinematic Factor

Quark Matter 2009, March 30 – April 4, Knoxville, TN

``Venugopalan & Prakash  Nucl. Phys. A546 (1992) 718''

Results (Interacting Pions)

16

ηv 3rd appro x 103

2nd

1st

ηs 1st appro

2nd

T[MeV]

[GeV

][fm

-2][c

-1]

Total

Non-in

terac

ting

Interacting

T[MeV]

[fm-3]

Entropy DensityBulk &Shear Viscosity

Prakash et al. ,Phys. Rep. 227 (1993) 321

Quark Matter 2009, March 30 – April 4, Knoxville, TN

Chen and Wang, Arxiv:0711.4824v1

Bulk viscosity / entropy density

Quark Matter 2009, March 30 – April 4, Knoxville, TN

QGP sum rule

LatticeMassless pions

For chiral pions ηv/s increases with T in contrast to ηv /s for massive interacting pions

Wiranata & Prakash

Multi-Component system

Transport cross sectionThermal weightRelative momentum weight

W.A. Leeuwen, et.al

Begin with binary mixture

Ratio of specific heats

Quark Matter 2009, March 30 – April 4, Knoxville, TN

Binary mixtureResults for constant cross-sections

Quark Matter 2009, March 30 – April 4, Knoxville, TN

Multi-component systemResult for N – species at pth order of approximation

Solubility conditions (assures 4-momentum conservation in collisions)

Coefficients to be determined Omega integral

Involves ratios of specific heats

Quark Matter 2009, March 30 – April 4, Knoxville, TN

1st order calculation

Coefficients to be determined

Quark Matter 2009, March 30 – April 4, Knoxville, TN

Ongoing work and future plans Calculation of ηv & ηv /s for a mixture of interacting

hadrons with masses up to 2 GeV. Development of an approach to calculate the needed

differentialcross-sections for hadron-hadron interactions including resonances up to 2 GeV.

Comparison of the Chapman-Enskog results with those of the Green-Kubo approach.

Inclusion of decay processes. All the above for shear viscosity ηv and ηv /s

Quark Matter 2009, March 30 – April 4, Knoxville, TN

Thank you

Quark Matter 2009, March 30 – April 4, Knoxville, TN