Motivated by a desire to understand quantum fluctuation energy densities and stress within a spatially varying dielectric medium, we examine the vacuum expectation value for the stress tensor of a scalar field with arbitrary conformal parameter, in the background of a given potential that depends on only one spatial coordinate. We regulate the expressions by incorporating a temporalspatial cutoff in the (imaginary) time and transverse-spatial directions. The divergences are captured by the zeroth-and second-order WKB approximations. Then the stress tensor is "renormalized" by omitting the terms that depend on the cutoff. The ambiguities that inevitably arise in this procedure are both duly noted and restricted by imposing certain physical conditions; one result is that the renormalized stress tensor exhibits the expected trace anomaly. The renormalized stress tensor exhibits no pressure anomaly, in that the principle of virtual work is satisfied for motions in a transverse direction. We then consider a potential that defines a wall, a one-dimensional potential that vanishes for z < 0 and rises like z α , α > 0, for z > 0. Previously, the stress tensor had been computed outside of the wall, whereas now we compute all components of the stress tensor in the interior of the wall. The full finite stress tensor is computed numerically for the two cases where explicit solutions to the differential equation are available, α = 1 and 2. The energy density exhibits an inverse linear divergence as the boundary is approached from the inside for a linear potential, and a logarithmic divergence for a quadratic potential. Finally, the interaction between two such walls is computed, and it is shown that the attractive Casimir pressure between the two walls also satisfies the principle of virtual work (i.e., the pressure equals the negative derivative of the energy with respect to the distance between the walls).

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...

Casimir entropies due to quantum fluctuations in the interaction between electrical bodies can often be negative, either caused by dissipation or by geometry. Although generally such entropies vanish at zero temperature, consistent with the third law of thermodynamics (the Nernst heat theorem), there is a region in the space of temperature and separation between the bodies where negative entropy occurs, while positive interaction entropies arise for large distances or temperatures. Systematic studies on this phenomenon in the Casimir-Polder interaction between a polarizable nanoparticle or atom and a conducting plate in the dipole approximation have been given recently. Since the total entropy should be positive according to the second law of thermodynamics, we expect that the self-entropy of the bodies would be sufficiently positive as to overwhelm the negative interaction entropy. This expectation, however, has not been explicitly verified. Here we compute the self-entropy of an electromagnetic δ-function plate, which corresponds to a perfectly conducting sheet in the strong coupling limit. The transverse electric contribution to the self-entropy is negative, while the transverse magnetic contribution is larger and positive, so the total self-entropy is positive. However, this self-entropy vanishes in the strong-coupling limit. In that case, it is the self-entropy of the nanoparticle that is just sufficient to result in a nonnegative total entropy.

Recently the Casimir self-entropy of an electromagnetic δ-function shell was considered by two different groups, with apparently discordant conclusions, although both had concluded that a region of negative entropy existed for sufficiently weak coupling. We had found that the entropy contained an infrared divergence, which we argued should be discarded on physical grounds. On the contrary, Bordag and Kirsten recently found a completely finite self-entropy, although they, in fact, have to remove an infrared divergence. Apart from this, the high-and low-temperature results for finite coupling agree precisely for the transverse electric mode, but there are significant discrepancies in the transverse magnetic mode. We resolve those discrepancies here. In particular, it is shown that coupling-independent terms do not occur in a consistent regulated calculation, they likely being an artefact of the omission of pole terms. The results of our previous analysis, especially, the existence of a negative entropy region for sufficiently weak coupling, are therefore confirmed. Finally, we offer some analogous remarks concerning the Casimir entropy of a thin electromagnetic sheet, where the total entropy is always positive. In that case, the origin of the analogous discrepancy can be explicitly isolated. * kmilton@ou.edu †

It has been increasingly becoming clear that Casimirand Casimir-Polder entropies may be negative in certain regions of temperature and separation. In fact, the occurrence of negative entropy seems to be a nearly ubiquitous phenomenon. This is most highlighted in the quantum vacuum interaction of a nanoparticle with a conducting plate or between two nanoparticles. It has been argued that this phenomenon does not violate physical intuition, since the total entropy, including the self-entropies of the plate and the nanoparticle, should be positive. New calculations, in fact, seem to bear this out at least in certain cases. The thermal Casimir puzzleFor 15 years there has been a controversy surrounding entropy in the Casimir effect. This is most famously centered around the issue of how to describe a real metal, in particular, the permittivity ε(ω) at zero frequency [1]. The latter determines the low-temperature and high temperature corrections to the free energy, and hence to the entropy. The issue involves how ω 2 ε(ω) behaves as ω → 0.Dissipation or finite conductivity implies this vanishes; this leads to a linear temperature dependence at low temperature, and a reduction of the high-temperature force. Most experiments [2-4], but not all [5], favor the nondissipative plasma model! For an overview of the status of both theory and experiment, see Ref.[6].The Drude model, and general thermodynamic and electrodynamic arguments, suggest that the transverse electric (TE) reflection coefficient at zero frequency for a good, but imperfect, metal plate should vanish. Careful calculations for lossy parallel plates show that at very low temperature the free energy approaches a constant quadratically in the temperature, thus forcing the entropy to vanish at zero temperature [7]. Thus, there is no violation of the third law of thermodynamics. However, there would persist a region at low temperature in which the entropy would be negative. This was not thought to be a problem, since the interaction Casimir free energy does not describe the entire system of the Casimir apparatus, whose total entropy must necessarily be positive. The physical basis for the negative entropy region remains somewhat mysterious. Here we address negative entropy arising from geometry. The interplay of geometry and material properties is further explored in Ref. [8].For some time it has been known that negative entropy regions can emerge geometrically. For example, when a perfectly conducting sphere is near a perfectly conducting plate, the entropy at room temperature can turn negative, with enhancement of the effect occurring for smaller spheres [9]. The occurrence of negative interaction entropies for a small sphere in front of a plane or another sphere is illustrated in Fig. 1 within the dipole and single-scattering approximation. Since the negative entropy region is enhanced by making the sphere small, this suggests that the phenomena be explored by considering the dipole approximation only. That is, we will examine point particles, atoms or nanopa...