  For long black holes have been considered as endowed with a definite
temperature. Yet when the Schwarzschild black hole is treated as a canonical
ensemble three problems arise: incompatibility with the Hawking radiation,
divergence of the partition function, and a formally negative mean-square
fluctuation of the energy. We solve all three problems by considering the
Schwarzschild black hole as a grand canonical ensemble, with the Hamiltonian
(the ADM mass) and the horizon surface area, separately, as observable
parameters. The horizon area simulates the number of particles in statistical
mechanics since its spectrum is assumed to be discrete and equally spaced. We
obtain a logarithmic correction to the Bekenstein-Hawking entropy and a
Gaussian type distribution for the energy levels.
