  We study the multi-dimensional properties of neutrino transfer inside
supernova cores by solving the Boltzmann equations for neutrino distribution
functions in genuinely six dimensional (6D) phase space. Adopting
representative snapshots of the post-bounce core from other supernova
simulations in three dimensions, we solve the temporal evolutions to stationary
states of neutrino distribution functions by our Boltzmann solver. Taking
advantage of the multi-angle and multi-energy feature realized by the S$_n$
method in our code, we reveal the genuine characteristics of spatially three
dimensional (3D) neutrino transfer such as non-radial fluxes and non-diagonal
Eddington tensors. In addition, we assess the ray-by-ray approximation, turning
off the lateral-transport terms in our code. We demonstrate that the ray-by-ray
approximation tends to propagate fluctuations in thermodynamical states around
the neutrino-sphere along each radial ray and overestimate the variations
between the neutrino distributions on different radial rays. We find that the
difference in the densities and fluxes of neutrinos between the ray-by-ray
approximation and the full Boltzmann transport becomes ~20%, which is also the
case for the local heating rate, whereas the volume-integrated heating rate in
the Boltzmann transport is found to be only slightly larger (~2%) than the
counterpart in the ray-by-ray approximation due to cancellation among different
rays. These results suggest that we had better assess carefully the possible
influences of various approximations in the neutrino transfer employed in the
current simulations on supernova dynamics. Detailed information on the angle
and energy moments of neutrino distribution functions will be profitable for
the future development of numerical methods in neutrino-radiation
hydrodynamics.
