Although a lot of research work has been focused on explaining the physics of horizon, our knowledge about the horizon degrees of freedom is still far from perfect. The simplest choice (not realistic) for the quantum degrees of freedom on the horizon is a non-interacting boson gas. We consider a simple model for the quantum degrees of freedom on the horizon. We assume that the horizon degrees of freedom are in a condensed state whose particle number is equal to the number of space-time quantum bits. For a more realistic model of quantum degrees of freedom on the horizon, we should presumably consider interacting bosons (gravitons). An ideal gas with intermediate statistics could be considered as an effective theory for interacting bosons. This analysis shows that we may obtain a correct entropy just by a suitable choice of parameter in the intermediate statistics. So this model may explain some aspects of the quantum structure of space - time . According to Verlinde's idea, the space - time is built of holographic screens . A boson gas can be used as a probe to explore equipartition theorem and entropy of quantum bits of such screens. We calculate the dimension of quantum channel of radiation from a Schwarzschild, BTZ and Lovelock black holes . Our results indicate that for odd D dimensional space-time, the dimension of transmission from pure Lovelock black holes is equal to D and for even D dimensional space-times, the dimension of quantum channel becomes , where is cosmological constant. It is interesting that cosmological constant may put some constraint on dimension of quantum channel in even dimensional spacetimes. The thermodynamics of the inner horizon plays a crucial role in understanding the black hole entropy microscopically. We will show that for some black holes, the entropy product of the outer and inner horizons is independent of the mass of the black hole , and this is equivalent to the relation , where and being the Hawking temperatures and the entropies of the outer and inner horizons respectively. Here a generalized mass independent relation for Myers-Perry (MP) black holes in higher dimensions and Kerr-AdS black hole in is obtained . Then we will generalize to another relation of for Reier-Nordstrom-AdS black hole in 4 dimensions.