  We study the thermodynamics of a quantum field in a spherical shell around a
static black hole. We implement brick wall regularization by imposing Dirichlet
boundary conditions on the field at the boundaries and analyze their effects on
the free energy and the entropy. We also consider the possibility of using
Neumann boundary conditions. We examine both bosonic and fermionic fields in
Schwarzschild, Reissner-Nordstr\"om (RN), extreme RN, and dilatonic
backgrounds. We show that the horizon divergences get contributions from the
boundary (brick wall) which at the Hawking temperature are comparable to the
bulk contributions. It is also shown that the leading divergence is the same
for all the backgrounds considered and that the subleading logarithmic
divergence is given by a specific function of surface gravity and horizon area.
We will also consider the near horizon geometry and show the existence of a
finite term gen- erated by the logarithmic divergence which involves the
logarithm of the horizon area. The thermodynamics of quantum fields is examined
in the ultrarelavisitc regime through the high temperature/small mass
expansion. We derive the high temperature expansion in the presence of chemical
potential by Mellin transform and heat kernel methods as applied to the
harmonic sum representation of the free energy.
