  Not only are Casimir interaction entropies not guaranteed to be positive, but
also, more strikingly, Casimir self-entropies of bodies can be negative. Here,
we attempt to interpret the physical origin and meaning of these negative
self-entropies by investigating the Casimir self-entropy of a neutral spherical
nanoparticle. After extracting the polarizabilities of such a particle by
examining the asymptotic behavior of the scattering Green's function, we
compute the corresponding free energy and entropy. Two models for the
nanoparticle, namely a spherical plasma $\delta$-function shell and a
homogeneous dielectric/diamagnetic ball, are considered at low temperature,
because that is all that can be revealed from a nanoparticle perspective. The
second model includes a contribution to the entropy from the bulk free energy,
referring to the situation where the medium inside or outside the ball fills
all space, which must be subtracted on physical grounds in order to maintain
consistency with van der Waals interactions, corresponding to the self-entropy
of each bulk. (The van der Waals calculation is described in Appendix A.) The
entropies so calculated agree with known results in the low-temperature limit,
appropriate for a small particle, and are negative. But we suggest that the
negative self-entropy is simply an interaction entropy, the difference between
the total entropy and the blackbody entropy of the two bulks, outside or inside
of the nanosphere. The vacuum entropy is always positive and overwhelms the
interaction entropy. Thus the interaction entropy can be negative, without
contradicting the principles of statistical thermodynamics. Because the entropy
of blackbody radiation by itself plays an important role, it is also discussed,
including dispersive effects, in detail.
