Shear viscosity of a strongly interacting system: Green-Kubo correlator versus Chapman-Enskog and relaxation-time approximations

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 (CE) approximation. 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 perturbative-QCD-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 us to develop a kinetic transport theory at fixed shear-viscosity to entropy-density ratio, eta/s. This opens the possibility of exploring dissipative nonequilibrium evolution of the distribution functions versus T-dependent eta/s and particle momenta in the dynamics of the quark-gluon plasma created in ultrarelativistic heavy-ion collisions.