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Shear viscosity and out of equilibrium dynamics

Using Grad's method, we calculate the entropy production and derive a formula for the second-order shear viscosity coefficient in a one-dimensionally expanding particle system, which can also be considered out of chemical equilibrium. For a one-dimensional expansion of gluon matter with Bjorken boost invariance, the shear tensor and the shear viscosity to entropy density ratio $η/s$ are numerically calculated by an iterative and self-consistent prescription within the second-order Israel-Stewart hydrodynamics and by a microscopic parton cascade transport theory. Compared with $η/s$ obtained using the Navier-Stokes approximation, the present result is about 20% larger at a QCD coupling $α_s \sim 0.3$(with $η/s\approx 0.18$) and is a factor of 2-3 larger at a small coupling $α_s \sim 0.01$. We demonstrate an agreement between the viscous hydrodynamic calculations and the microscopic transport results on $η/s$, except when employing a small $α_s$. On the other hand, we demonstrate that for such small $α_s$, the gluon system is far from kinetic and chemical equilibrium, which indicates the break down of second-order hydrodynamics because of the strong noneqilibrium evolution. In addition, for large $α_s$ ($0.3-0.6$), the Israel-Stewart hydrodynamics formally breaks down at large momentum $p_T\gtrsim 3$ GeV but is still a reasonably good approximation.