Entropy, temperature and internal energy of trapped gravitons and corrections to the Black Hole entropy

In this paper we study the proposal present in \cite{1} concerning the statistical description of trapped gravitons and applied to derive the semi-classical black hole (BH) entropy $S_{BH}$. We study the possible configurations depending on physically reasonable expressions for the internal energy $U$. In particular, we show that expressions for $U\sim R^k, k\geq 1$, with $R$ the radius of the confining spherical box, can have a semi-classical description, while behaviors with $k<1$ derive from thermodynamic or quantum fluctuations. There, by taking a suitable physically motivated expression for $U(R)$, we obtain the well known logarithmic corrections to the BH entropy, with the usual behaviors present in the literature of BH entropy. Moreover, a phase transition emerges with a positive specific heat $C$ at Planckian lengths instead of the usual negative oneat non-Planckian scales, in agreement with results present in the literature. Finally, we show that evaporation stops at a radius $R$ of the order of the Planck length.