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Free Energy and Entropy for Semi-classical Black Holes in the Canonical Ensemble

We consider the thermodynamics of a black hole coupled to thermal radiation in a spatially finite (spherical) region. Thermodynamic state functions are derived in the canonical ensemble, defined by elements of radius $r_o$ and boundary temperature $T(r_o)$. Using recent solutions of the semi-classical back reaction problem, we compute the $O(\hbar)$ corrections to the mass of the black hole, thermal energy, the entropy and free energy due to the presence of hot conformal scalars, massless spinors and U(1) gauge quantum fields in the vicinity of the hole. The free energy is particularly important for assessing under what conditions the nucleation of black holes from hot flat space is likely to occur.