Shear Viscosity of a strongly interacting system: Green-Kubo vs. Chapman-Enskog and Relaxation Time Approximation

TL;DR

This paper compares numerical and analytical methods for calculating shear viscosity in a strongly interacting partonic system, highlighting the accuracy of Chapman-Enskog over Relaxation Time Approximation in heavy-ion collision conditions.

Contribution

It demonstrates that Chapman-Enskog approximation aligns well with Green-Kubo results, providing an analytical formula for kinetic transport modeling of shear viscosity in quark-gluon plasma.

Findings

RTA underestimates shear viscosity by 2-3 times.

CE approximation agrees with Green-Kubo results at first order.

Analytical formula enables fixed shear viscosity to entropy density ratio modeling.

Abstract

The shear viscosity $\eta$ has been calculated by using the Green-Kubo relation in the framework of a partonic transport approach solved at cascade level. We compare the numerical results for $\eta$ obtained from the Green-Kubo correlator with the analytical formulas in both the Relaxation Time Approximation (RTA) and the Chapman-Enskog approximation (CE). We investigate and emphasize the differences between the isotropic and anisotropic cross sections and between the massless and massive particles. We show that in the range of temperature explored in a Heavy Ion collision and for pQCD-like cross section the RTA significantly underestimates the viscosity by about a factor of 2-3, while a good agreement is found between the CE approximation and Gree-Kubo relation already at first order of approximation. The agreement with the CE approximation supplies an analytical formula that allows to develop kinetic transport theory at fixed shear viscosity to entropy density ratio, $\eta/s$. This open the possibility to explore dissipative non-equilibrium evolution of the distribution functions vs T-dependent $\eta/s$ and particle momenta in the dynamics of the Quark-Gluon Plasma created in ultra-relativistic heavy-ion collisions.