  It has been recognized for some time that even for perfect conductors, the
interaction Casimir entropy, due to quantum/thermal fluctuations, can be
negative. This result was not considered problematic because it was thought
that the self-entropies of the bodies would cancel this negative interaction
entropy, yielding a total entropy that was positive. In fact, this cancellation
seems not to occur. The positive self-entropy of a perfectly conducting sphere
does indeed just cancel the negative interaction entropy of a system consisting
of a perfectly conducting sphere and plate, but a model with weaker coupling in
general possesses a regime where negative self-entropy appears. The physical
meaning of this surprising result remains obscure. In this paper we re-examine
these issues, using improved physical and mathematical techniques, partly based
on the Abel-Plana formula, and present numerical results for arbitrary
temperatures and couplings, which exhibit the same remarkable features.
