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We propose a method which enables a QCD-based calculation of a relaxationtime for a dissipative current in the causal and dissipative hydrodynamicequation derived by Israel and Stewart. We point out that the Israel-Stewartequation is not unique as a causal and dissipative hydrodynamic equation, andthe form of the causal and dissipative hydrodynamic equation is determined bythe shape of a spectral function reflecting the properties of elementaryexcitations in the system we consider. Our method utilizes a relaxationfunction, which can be calculated from QCD using the linear response theory. Weshow that the relaxation function can be derived from a spectral function for amicroscopic representation of the dissipative current. We also show that theIsrael-Stewart equation is acceptable only as long as the calculated relaxationfunction is approximated well by a exponentially damping function, and therelaxation time can be obtained as its damping time constant. Taking abaryon-number dissipative current of a plasma consisting of charm quarks andgluons as a simple example, we present the first calculation of the relaxationfunction with use of the spectral function derived employing the quenchedlattice QCD together with the maximum entropy method. The calculated relaxationfunction shows a strongly-oscillation damping behaviour due to the charmedvector hadron $J/\Psi$ surviving above the deconfinement phase transitiontemperature in QCD. This result suggests that the applicability of theIsrael-Stewart equation to the baryon-number dissipative current of the charmquark-gluon plasma is quite doubtful. We present an idea for the improvement ofthe Israel-Stewart equation by deriving the hydrodynamic equation consistentwith the strongly-oscillation damping relaxation function.

Relaxation Time and Relaxation Function of Quark-Gluon Plasma with Lattice QCD