Transport Coefficients of Transport Coefficients of Interacting Hadrons Interacting Hadrons

Anton WiranataAnton Wiranata&&

Madappa Prakash (Advisor)Madappa Prakash (Advisor)

Department of Physics and AstronomyOhio University, Athens, OH

Motivation Shear & bulk viscosities in the

Chapman-Enskog approximation Non relativistic limit for shear and bulk

viscosities Bulk viscosity and the speed of sound Inelastic collisions & transport coefficients Shear & bulk viscosities of mixtures

Transport coefficients (bulk & shear viscosities) are important inputs to viscous hydrodynamic simulations of relativistic heavy-ion collisions.

Collective motion with viscosity influences (reduces) the magnitude of elliptic flow relative to ideal (non-viscous) hydrodynamic motion.

“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 (e.g., bulk & shear viscosities) are then calculable

from relativistic kinetic theory.”

Deviation function

Equilibrium distribution function

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

the solution (deviation function) has the general structurethe solution (deviation function) has the general structure

The collision integral

Bulk viscosity

Heat conductivity Shear Viscosity

Reduced enthalpyRelativistic Omega Integrals

Relativity parameter

Ratio of specific heats

Transport cross sectionThermal weightRelative momentum dependent

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

Thermal variable

Contains collision cross sections

Shear viscosity

Non-relativistic case : z=m/kT >> 1

Non-relativistic omega integral

g : Dimensionless relative velocity

Note the g7 – dependence in the kernel, which favors high relative velocity particles in the heat bath (energy density & pressure carry a g4 – dependence) .

Note also the importance of the relative velocity and angle dependences of the cross section.

The quantities ci,j contain omega integrals

Hard sphere cross section –

Hard-sphere cross section –

It is desirable to reproduce these results in alternative approaches, such as variational & Green-Kubo calculations (in progress with collaborators from Minneapolis & Duke).

Bulk viscosity

Non relativistic case : z = m/kT >> 1

Non-relativistic omega integral

g : Dimensionless relative velocity

The quantities ai,j contain omega integrals

Hard sphere cross section –

Hard sphere cross section –

Illustration Continued : Interacting Pions Illustration Continued : Interacting Pions (Experimental Cross Sections)(Experimental Cross Sections)

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

1. The convergence of successive approximations to shear viscosity is significantly better than that for the bulk viscosity.

2. However, bulk viscosity is about 10-3 x shear viscosity, so its influence on the collision dynamics would be minimal, except possibly near the transition temperature.

Chapman-Enskog 1Chapman-Enskog 1stst approximation approximation

Adiabatic speed of sound

Features thermodynamic variables

The omega integral contains transport cross-section

Utilizing

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

Massive particles such that z >> 1 :Massive particles such that z >> 1 :

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

2nd order bulk viscosity

Feature thermodynamic variables

3rd order bulk viscosity

Terms that feature speed of sound dependences

Inelastic collisions can induce transitions to excited stated or result in new species

of particles. For the general formalism, see e.g., Kapusta(2008). In the

nonrelativistic case (applicable to heavy resonances) of

i + j k + l (Wang et al., 1964),

Energy of particle i Integration variable Energy difference

Inelastic terms

In the limit of the small Delta epsilon, inelastic collisions do not affect shear viscosity

Internal excitation and creation of new species of particles contribute to bulk viscosity

cint & cv are the internal heat capacity & heat capacity at fixed V per molecules

Inelastic term

In non-relativistic limit (z = m/kT >>1) , inelastic part of the cross section Contributes the most for bulk viscosity.

Thermodynamics termsThe same kind of particle collision

Different kind of particle collision

Relative momentumweight Thermal weight

Transport cross section

Relativistic Omega Integrals

Bulk viscosity of two type of spherical particles

Result 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

Coefficients to be determined

Calculation of ηs & ηs /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( In collaboration with Duke Univeristy).

Comparison of the Chapman-Enskog results with those of the Green-Kubo approach( In collaboration with Duke Univeristy).

Inclusion of decay processes (In collaboration with Minessota University

All the above for bulk viscosity ηv and ηv /s