Statistical mechanics of SDSS galaxy distribution and cosmological N-body simulations

The self gravitating systems in the Universe are generally non-additive, and often show long-tails in various distribution functions. We quantitatively investigate these properties, by studying the four models of statistical mechanics: (1) Boltzmann, (2) Mono-Fractal-space, (3) Renyi, (4) Tsallis statistical mechanics. Firstly, we study the observational data of SDSS DR4 and found that Renyi and Tsallis models are consistent with observational galaxy distribution. Secondly, we investigate the data of cosmological N-body simulation of dark energy models. Although Boltzmann model well describes at the large redshift stage, we found that the best fit models evolve from Tsallis and Boltzmann to Tsallis and Renyi, as the Universe evolve into non-linear stage. Tsallis model universally describes the structure at all the stage. Thus in order to describe the formation of non-linear structure, the long-tail property is essential.