FIAS Scientific Report 2011 - Frankfurt Institute for Advanced Studies ...

Aspects of transport coefficients in relativistic hydrodynamics
Collaborators: X. G. Huang 1,2 , P. Huovinen 1,2 , K. Kodama 3 , T. Koide 1,3 , D. H. Rischke 1,2 , A. Sedrakian 2
1 Frankfurt Institute for Advanced Studies, 2 Institute for Theoretical Physics, Frankfurt, Germany 3 Instituto de Fisica,
Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brasil
Relativistic hydrodynamics is one of the most often used approaches in studying the collective phenomena in
heavy-ion collisions, supernova explosions, and neutron stars. However, including viscosities in the relativistic
hydrodynamics is not easy. A naive relativistic generalization of Navier-Stokes theory does not work, because
it violates causality and is usually unstable, and hence does not permit numerical simulations. A solution to
this problem has been known since works of Mueller, and Israel and Stewart more than 30 years ago, and the
resulting theory is usually called second-order hydrodynamics. Different from Navier-Stokes theory where the
viscous stress tensors Π and π µν are completely fixed by the distribution of velocity fields, in second-order
hydrodynamics, Π and π µν are treated as independent dynamical variables which should be solved simultaneously
with the velocity fields. As was proven, second-order hydrodynamics is causal and stable, but the
cost is the appearance of new transport coefficients. The microscopic formulas for the shear viscosity η, the
bulk viscosity ζ and corresponding relaxation times τπ and τΠ of causal dissipative relativistic fluid-dynamics
are obtained at finite temperature and chemical potential by using the projection operator method. The nontriviality
of the finite chemical potential calculation is attributed to the arbitrariness of the operator definition
of the bulk viscous pressure. We show that, when the operator definition for the bulk viscous pressure Π is
appropriately chosen, the leading order result of the ratio of ζ and τΠ coincides with the same ratio obtained
at vanishing chemical potential. In applying these formulae to the pionic fluid, we find that the renormalizable
energy-momentum tensor should be employed to obtain consistent results.
Quarks produced in the early stage of non-central heavy-ion collisions could develop a global spin polarization
along the opposite direction of the reaction plane due to the spin-orbital coupling via parton interaction in a
medium that has finite longitudinal flow shear along the direction of the impact parameter. We study how such
polarization evolves via multiple scattering in a viscous quark-gluon plasma with an initial laminar flow. The
final polarization is found to be sensitive to the viscosity and the initial shear of local longitudinal flow.
Relativistic magnetohydrodynamics of strongly magnetized relativistic fluids is derived in the ideal and dissipative
cases, taking into account the breaking of spatial symmetries by a quantizing magnetic field. A complete set
of transport coefficients, consistent with the Curie and Onsager principles, is derived for thermal conduction,
as well as shear and bulk viscosities. It is shown that in the most general case the dissipative function contains
five shear viscosities, two bulk viscosities, and three thermal conductivity coefficients. We use Zubarev’s
non-equilibrium statistical operator method to relate these transport coefficients to correlation functions of equilibrium
theory. The desired relations emerge at linear order in the expansion of the non-equilibrium statistical
operator with respect to the gradients of relevant statistical parameters (temperature, chemical potential, and
velocity.) The transport coefficients are cast in a form that can be conveniently computed using equilibrium
(imaginary-time) infrared Green’s functions defined with respect to the equilibrium statistical operator.
Related publications in 2011:
1) X. -G. Huang, P. Huovinen and X. -N. Wang, Quark polarization in a viscous quark-gluon plasma, Phys.
Rev. C84, 054910 (2011)
2) X. -G. Huang, A. Sedrakian and D. H. Rischke, Kubo formulas for relativistic fluids in strong magnetic fields,
Annals Phys. 326, 3075 (2011)
3) X. -G. Huang and T. Koide, Shear viscosity, bulk viscosity and relaxation times of causal dissipative relativistic
fluid-dynamics at finite temperature and chemical potential, arXiv:1105.2483 [hep-th]
4) X. -G. Huang, T. Kodama, T. Koide and D. H. Rischke, Bulk viscosity and relaxation time of causal dissipative
relativistic fluid dynamics, Phys. Rev. C83, 024906 (2011)
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