A detailed quantitative comparison between the results of
shear viscosities from the Chapman-Enskog and Relaxation Time methods is performed for the following test cases with specified elastic differential cross sections between interacting hadrons:
1. The non-relativistic, relativistic and ultra-relativistic hard sphere gas with angle and energy independent differential cross section sigma = a^2/4, where $a$ is the hard sphere radius,
2. The Maxwell gas with sigma (g,Theta)= mGamma(Theta)/2g, where $m$ is the mass of the heat bath particles, Gamma(Theta) is an arbitrary function of Theta, and $g$ is the relative velocity,
3. Chiral pions for which the $t-$averaged cross section sigma =
s/(64pi^2 f_pi^4) × (1+1/3 × cos^2 Theta), where $s$ and $t$ are the usual Mandelstam variables and $f_pi$ is the pion-decay constant, and
4. Massive pions for which the differential elastic cross section
is taken from experiments.
Quantitative results of the comparative study conducted revealed that
* the extent of agreement (or disagreement) depends very sensitively on the energy dependence of the differential cross sections employed, stressing the need to combine all available experimental knowledge concerning differential cross sections for low mass hadrons and to supplement it with theoretical guidance for the as yet unknown cross sections so that the temperature dependent shear viscosity to entropy ratio can be established for use in viscous hydordynamics.
* The result found for the ultra-relativistic hard sphere gas
for which the shear viscosity eta_s = 1.2676~k_BT~c^{-1} /(pi a^2) offers the opportunity to validate ultra-relativistic quantum molecular dynamical (URQMD) codes that employ Green-Kubo techniques.
* shear viscosity receives only small contributions from number changing inelastic processes.
The dependence of the bulk viscosity on the adiabatic speed of sound is studied in depth highlighting why only hadrons in the intermediate relativistic regime contribute the most to the bulk viscosity when only elastic collisions are considered. However, number changing inelastic processes dominate contributions to
the bulk viscosity as shown recently by other workers. These combined findings call for developing techniques to include number changing inelastic processes to reliably estimate the magnitude of bulk viscosity in a mixture for temperatures in the range 100 - 200 MeV in which hadrons likely exist during the space-time evolution of a heavy-ion collision.
Illustrative calculations in a binary mixture are performed by varying the mass ratio of the constituents paving the way for future calculations in a multi-component system of hadrons.
An important outcome of this study is that collaborative research to pursue comparative studies with Green-Kubo calculations of viscosities has been initiated with researchers from Duke University.