  We propose a method which enables a QCD-based calculation of a relaxation
time for a dissipative current in the causal and dissipative hydrodynamic
equation derived by Israel and Stewart. We point out that the Israel-Stewart
equation is not unique as a causal and dissipative hydrodynamic equation, and
the form of the causal and dissipative hydrodynamic equation is determined by
the shape of a spectral function reflecting the properties of elementary
excitations in the system we consider. Our method utilizes a relaxation
function, which can be calculated from QCD using the linear response theory. We
show that the relaxation function can be derived from a spectral function for a
microscopic representation of the dissipative current. We also show that the
Israel-Stewart equation is acceptable only as long as the calculated relaxation
function is approximated well by a exponentially damping function, and the
relaxation time can be obtained as its damping time constant. Taking a
baryon-number dissipative current of a plasma consisting of charm quarks and
gluons as a simple example, we present the first calculation of the relaxation
function with use of the spectral function derived employing the quenched
lattice QCD together with the maximum entropy method. The calculated relaxation
function shows a strongly-oscillation damping behaviour due to the charmed
vector hadron $J/\Psi$ surviving above the deconfinement phase transition
temperature in QCD. This result suggests that the applicability of the
Israel-Stewart equation to the baryon-number dissipative current of the charm
quark-gluon plasma is quite doubtful. We present an idea for the improvement of
the Israel-Stewart equation by deriving the hydrodynamic equation consistent
with the strongly-oscillation damping relaxation function.
