Casimir self-entropy of a spherical electromagnetic δ -function shell

Abstract

In this paper we continue our program of computing Casimir self-entropies of idealized electrical bodies. Here we consider an electromagnetic δ-function sphere ("semitransparent sphere") whose electric susceptibility has a transverse polarization with arbitrary strength. Dispersion is incorporated by a plasma-like model. In the strong coupling limit, a perfectly conducting spherical shell is realized. We compute the entropy for both low and high temperatures. The TE self-entropy is negative as expected, but the TM self-entropy requires ultraviolet and infrared subtractions, and, surprisingly, is only positive for sufficiently strong coupling. Results are robust under different regularization schemes. I. INTRODUCTIONThe usual expectation, based on the notion that entropy is a measure of disorder, is that entropy should be positive. However, there are circumstances in which entropy can take on negative values. For example, negative entropy is often discussed in connection with biological systems [1]. More interesting physically is the occurrence of negative entropy in black-hole and cosmological physics [2,3].In Casimir physics, perhaps the first appearance of negative entropy occurred in connection with the description of the quantum vacuum interaction between parallel conducting plates. If dissipation is present, the entropy of the interaction is positive at large distances, aT ≫ 1, where a is the separation between the plates and T is the temperature, but turns negative for short distances. Considered as a function of temperature, the sign of the entropy changes as the temperature decreases, but does tend to zero as the temperature tends to zero, in accordance with the Nernst heat theorem [4]. Although perhaps surprising, this was not thought to be a problem because this phenomenon only referred to the interaction part of the free energy, and the total entropy of the system was expected to be positive. Somewhat later it was discovered that negative Casimir entropies also occurred purely geometrically, for example between a perfectly conducting sphere and a perfectly conducting plane without dissipation [5][6][7], or between two spheres [8,9]. When the distance times the temperature (in natural units) is of order unity, typically a negative entropy region was present. Since the effect was dominant in the dipole approximation, this led to a systematic study of the phenomenon of negative entropy arising between polarizable particles, characterized by electric and magnetic polarizabilities, or between such particles and a conducting plate. For appropriate choices of these polarizabilities, these nanoparticles behaved like small conducting spheres. We found that sometimes the entropy started off negatively for small aT , before eventually turning positive, and sometimes the entropy was first positive, turned negative for a while, and then turned positive again as aT increased [10,11]. The combined effects of both geometry and dissipation are considered in Refs. [12,13].The occurrence of negative entropy, geometricall...

Remarks on the Casimir self-entropy of a spherical electromagnetic δ -function shell

“…(6.22) of Ref. [3], while for large ξ the result coincides with the high-temperature limit of the exact result for the TM free energy in O(λ 0 ):…”

“…3 in Ref. [3]. This further shows that the TM entropy (the negative slope of the free energy with respect to T ) is positive for strong coupling (small ξ), and negative for weak coupling (large ξ), with the transition occurring near ξ 0 = 1.75 .…”

“…This is an alternative "closed-form" to that shown in Ref. [3], and gives the limits shown above in Eqs. (2.2), (2.3).…”

“…The entropy due to electromagnetic field fluctuations, or Casimir entropy, of a perfectly conducting spherical shell (of radius a) was computed many years ago by Balian and Duplantier [1], who found the following low and high temperature behaviors for the free energy, Here the subscript is a reminder that the conductivity of the sphere is considered infinite, and the ∆ means this is the correction to the zero-temperature Casimir energy of the sphere, first calculated by Boyer [2]. Only recently was this calculation generalized to a spherical shell with a finite electromagnetic coupling, a so-called electromagnetic δ-function shell, or a spherical plasma shell [3,4]. The former is described by the background permittivity ε(r) − 1 = λ(1 − rr)δ(r − a), (1.2) which describes a sphere of radius a centered at the origin.…”

Free energy and entropy for finite temperature quantum field theory under the influence of periodic backgrounds

“…Meanwhile, the entropy of Casimir effect related configurations was calculated for quite a large number of model systems. For three dimensional ones, a plasma plane or a plasma sphere, in [19,20] and [21,22] and for some simple one dimensional examples in [23]. In all these cases the single standing objects were considered and the complete entropy, except the black body * bordag@uni-leipzig.de † jose.munoz.castaneda@uva.es ‡ lucia.santamaria@uva.es part (contributions from the empty space), was computed.…”