Is galaxy distribution non-extensive and non-Gaussian?

Self-gravitating systems 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 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 statistical mechanics, (2) Fractal statistical mechanics, (3) Rényi-entropy-based (REB) statistical mechanics, and (4) Tsallis statistical mechanics, and use Akaike information criteria (AIC) for the fair comparison. We found that both Rényi-entropy-based statistical model and Tsallis statistical model are far better than the other two models. Therefore, the long-tail in the distribution function turns out to be essential.