  We present large-scale molecular dynamics simulations for a coarse-grained
model of polymer melts in equilibrium. From detailed Rouse mode analysis we
show that the time-dependent relaxation of the autocorrelation function (ACF)
of modes $p$ can be well described by the effective stretched exponential
function due to the crossover from Rouse to reptation regime. The ACF is
independent of chain sizes $N$ for $N/p<N_e$ ($N_e$ is the entanglement
length), and there exists a minimum of the stretching exponent as $N/p
\rightarrow N_e$. As $N/p$ increases, we verify the crossover scaling behavior
of the effective relaxation time $\tau_{{\rm eff},p}$ from the Rouse regime to
the reptation regime. We have also provided evidence that the incoherent
dynamic scattering function follows the same crossover scaling behavior of the
mean square displacement of monomers at the corresponding characteristic time
scales. The decay of the coherent dynamic scattering function is slowed down
and a plateau develops as chain sizes increase at the intermediate time and
wave length scales. The tube diameter extracted from the coherent dynamic
scattering function is equivalent to the previous estimate from the mean square
displacement of monomers.
