Evaluation of causal shear viscosity and relaxation time in QCD

where the first (second) equation represents energymomentum (the number) conservation, and dissipative effects such as viscosities are completely neglected (the above set of equations is exact, but for the ideal hydrodynamics all the higher expansion parameters, transport coefficients, in T μν and N are set to zero). Even though the ideal hydrodynamic simulations nicely coincides with experimental data, concluding that RHIC provided the most perfect ideal fluid we have ever got, there still remain a lot of ambiguities: hydro simulations depend much on the initial condition, equation of state used in the hydro simulation, effects from dissipative parts, and so forth. In this work we rivet our eyes on the dissipation effects which are characterized by phenomenological transport coefficients in hydro equations, e.g., shear and bulk viscosities, and the particle number diffusion constant. The first step for the dissipative hydrodynamics should be the introduction of such transport coefficients, and add them to the ideal hydro equation. Since hydro equations describe a long range dynamics of the system in space time, one can expand the equation in terms of space-time derivatives of the flow velocity u(x, t). The expansion begins with the lowest power of the derivatives, and the transport coefficients are these expansion coefficients, which are in principle derived from an underlying microscopic theory. The introduction of only the first coefficients cause some problems in the relativistic case: acausality and instability. The former is related to instantaneous propagation of informations (e.g. sound velocity exceeds the speed of light), and the latter to time reversal in the equation. Both of them are due to the lack of relaxation time effects of the modes we are considering, and mathematically due to structure of the equation: difference in power of time and space derivatives, and the number of these terms. The relaxation time effect, for instance, for shear stress tensor T xy which is responsible for diffusion of the shear flow created in the system, is categorized into the second order coefficient, compared to shear viscosity which is the first order one in derivatives. Thus, we need to incorporate the relaxation time effect in a consistent way. The time evolution of spatial off-diagonal components of energy-momentum tensor T (k, t) is described, after the Markov approximation, by