Transport Coefficients of
Interacting Hadrons
Anton Wiranata
&
Madappa Prakash (Advisor)
Department of Physics and Astronomy
Ohio University, Athens, OH
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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
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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 potential
u(x) : Flow velocity
T(x) : Temperature
The collision integral
Bulk
viscosity
Heat conductivity
Shear Viscosity
the solution (deviation function) has the general structure
Relativity parameter
Reduced enthalpy
Relativistic Omega Integrals
Ratio of
specific heats
Relative momentum dependent
Thermal weight
Relative momentum : g = mc sinh ψ
Total momentum
: P = 2mc cosh ψ
Transport cross section
Thermal variable
Contains collision cross sections
Shear viscosity
The quantities ci,j contain omega integrals
Non-relativistic case : z=m/kT >> 1
g : Dimensionless relative velocity
Non-relativistic omega integral
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.
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
The quantities ai,j contain omega integrals
Non relativistic case : z = m/kT >> 1
g : Dimensionless relative velocity
Non-relativistic omega integral
Hard sphere cross section –
Hard sphere cross section –
Illustration Continued : Interacting Pions (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.
Adiabatic speed of sound
Chapman-Enskog 1st approximation
Features thermodynamic variables
The omega integral contains
transport cross-section
Utilizing
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
2nd order bulk viscosity
3rd order bulk viscosity
Feature thermodynamic variables
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 terms
The same kind of particle collision
Different kind of particle collision
Relativistic Omega Integrals
Relative momentum
weight
Thermal weight
Transport cross section
Bulk viscosity of two type
of spherical particles
Result for N – species at pth order of approximation
Coefficients to be determined
Omega integral
Solubility conditions (assures 4-momentum conservation in collisions)
Involves ratios of
specific heats
Coefficients to be determined
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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 differential
cross-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