  We discuss the relation between the micro-canonical and the canonical
ensemble for black holes, and highlight some problems associated with extreme
black holes already at the classical level. Then we discuss the contribution of
quantum fields and demonstrate that the partition functions for scalar and
Dirac (Majorana) fields in static space-time backgrounds, can be expressed as
functional integrals in the corresponding optical space, and point out that the
difference between this and the functional integrals in the original metric is
a Liouville-type action. The optical method gives both the correction to the
black hole entropy and the bulk contribution to the entropy due to the
radiation, while (if the Liouville term is ignored) the conical singularity
method just gives the divergent contribution to the black hole entropy. A
simple derivation of a general formula for the free energy in the
high-temperature approximation is given and applied to various cases. We
conclude with a discussion of the second law.
