  We numerically solve the one dimensional Boltzmann equation of the neutrino
and anti-neutrino transport in accretion disks and obtain the fully energy
dependent and direction dependent neutrino and anti-neutrino emitting spectra,
under condition that the distribution of the mass density,temperature and
chemical components are given. Then, we apply the resulting neutrino and
anti-neutrino emitting spectra to calculate the corresponding annihilation rate
of neutrino pairs above the neutrino dominated accretion disk and find that the
released energy resulting from the annihilation of neutrino pairs can not
provide sufficient energy for the most energetic short gamma ray bursts whose
isotropic luminosity can be as high as $10^{52}$ ergs/s unless the high
temperature zone where the temperature is beyond 10 MeV can stretch over 200 km
in the disk. We also compare the resulting luminosity of neutrinos and
anti-neutrinos with the results from the two commonly used approximate
treatment of the neutrino and anti-neutrino luminosity: the Fermi-Dirac black
body limit and a simplified model of neutrino transport, i.e., the gray body
model, and find that both of them overestimate the neutrino/anti-neutrino
luminosity and their annihilation rate greatly. Additionally, as did in Sawyer
(2003), we also check the validity of the two stream approximation, and find
that it is a good approximation to high accuracy.
