  Self gravitating systems (SGS) in the Universe are generally thought to be
non-extensive, and often show long-tails in various distribution functions. In
principle, these non-Boltzmann properties are naturally expected from the
peculiar property of gravity, long-range and unshielded. Therefore the ordinary
Boltzmann statistical mechanics would not be applicable for these self
gravitating systems (SGS) in its naive form. In order to step further, we
quantitatively investigate the above two properties, non-extensivity and
long-tails, by explicitly introducing various models of statistical mechanics.
We use the data of CfA II South redshift survey and apply the count-in-cell
method. We study four statistical mechanics, (1) Boltzmann, (2) Fractal, (3)
R\'enyi, and (4) Tsallis, and use Akaike information criteria (AIC) for the
fair comparison.
