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# Shear viscosity and out of equilibrium dynamics

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### Abstract

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 $$\eta/s$$ 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 $$\eta/s$$ obtained using the Navier-Stokes approximation, the present result is about 20% larger at a QCD coupling $$\alpha_s \sim 0.3$$(with $$\eta/s\approx 0.18$$) and is a factor of 2-3 larger at a small coupling $$\alpha_s \sim 0.01$$. We demonstrate an agreement between the viscous hydrodynamic calculations and the microscopic transport results on $$\eta/s$$, except when employing a small $$\alpha_s$$. On the other hand, we demonstrate that for such small $$\alpha_s$$, 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 $$\alpha_s$$ ($$0.3-0.6$$), the Israel-Stewart hydrodynamics formally breaks down at large momentum $$p_T\gtrsim 3$$ GeV but is still a reasonably good approximation.

### Most cited references7

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### Viscosity in Strongly Interacting Quantum Field Theories from Black Hole Physics

(2004)
The ratio of shear viscosity to volume density of entropy can be used to characterize how close a given fluid is to being perfect. Using string theory methods, we show that this ratio is equal to a universal value of $$\hbar/4\pi k_B$$ for a large class of strongly interacting quantum field theories whose dual description involves black holes in anti--de Sitter space. We provide evidence that this value may serve as a lower bound for a wide class of systems, thus suggesting that black hole horizons are dual to the most ideal fluids.
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### Transverse interactions and transport in relativistic quark-gluon and electromagnetic plasmas

(1990)
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### The Effect of Shear Viscosity on Spectra, Elliptic Flow, and HBT Radii

(2003)
I calculate the first correction to the thermal distribution function of an expanding gas due to shear viscosity. With this modified distribution function I estimate viscous corrections to spectra, elliptic flow, and HBT radii in hydrodynamic simulations of heavy ion collisions using the blast wave model. For reasonable values of the shear viscosity, viscous corrections become of order one when the transverse momentum of the particle is larger than 1.7 GeV. This places a bound on the $$p_{T}$$ range accessible to hydrodynamics for this observable. Shear corrections to elliptic flow cause $$v_{2}(p_{T})$$ to veer below the ideal results for $$p_{T} \approx 0.9$$ GeV. Shear corrections to the longitudinal HBT radius $$R^{2}_{L}$$ are large and negative. The reduction of $$R_{L}^2$$ can be traced to the reduction of the longitudinal pressure. Viscous corrections cause the longitudinal radius to deviate from the $$\frac{1}{\sqrt{m_T}}$$ scaling which is observed in the data and which is predicted by ideal hydrodynamics. The correction to the sideward radius $$R^{2}_{S}$$ is small. The correction to the outward radius $$R^{2}_{O}$$ is also negative and tends to make $$R_{O}/R_{S} \approx 1$$.
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### Author and article information

###### Journal
15 December 2008
2009-06-18
###### Article
10.1103/PhysRevC.79.044914
0812.2762