# Negativity of the Casimir Self-Entropy in Spherical Geometries

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## Abstract

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## 1. Introduction

## 2. Transverse Magnetic Free Energy of Plasma-Shell Sphere

## 3. Weak Coupling

## 4. Low Temperature

#### 4.1. Euclidean Frequency Argument

#### 4.2. Abel–Plana Analysis

## 5. High Temperature

## 6. Numerical Analysis

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

TE | Transverse electric |

TM | Transverse magnetic |

## References

- Schrödinger, E. What Is Life—The Physical Aspect of the Living Cell; Cambridge University Press: Cambridge, UK, 1944. [Google Scholar]
- Cvetic, M.; Nojiri, S.; Odintsov, S.D. Black hole thermodynamics and negative entropy in de Sitter and anti-de Sitter Einstein-Gauss-Bonnet gravity. Nucl. Phys. B
**2002**, 628, 295–330. [Google Scholar] [CrossRef] [Green Version] - Nojiri, S.; Odintsov, S.D. The final state and thermodynamics of dark energy universe. Phys. Rev. D
**2004**, 70, 103522. [Google Scholar] [CrossRef] [Green Version] - Bordag, M. Entropy in some simple one-dimensional configurations. arXiv
**2018**, arXiv:1807.10354. [Google Scholar] - Bordag, M.; Muñoz-Castañeda, J.M.; Santamaría-Sanz, L. Free energy and entropy for finite temperature quantum field theory under the influence of periodic backgrounds. Eur. Phys. J. C
**2020**, 80, 221. [Google Scholar] [CrossRef] [Green Version] - Brevik, I.; Ellingsen, S.A.; Milton, K.A. Thermal corrections to the Casimir effect. New J. Phys.
**2006**, 8, 236. [Google Scholar] [CrossRef] - Bezerra, V.B.; Klimchitskaya, G.L.; Mostepanenko, V.M.; Romero, C. Lifshitz theory of atom-wall interaction with applications to quantum reflection. Phys. Rev. A
**2008**, 78, 042901. [Google Scholar] [CrossRef] [Green Version] - Canaguier-Durand, A.; Maia Neto, P.A.; Lambrecht, A.; Reynaud, S. Thermal Casimir effect in the plane-sphere geometry. Phys. Rev. Lett.
**2010**, 104, 040403. [Google Scholar] [CrossRef] [Green Version] - Canaguier-Durand, A.; Maia Neto, P.A.; Lambrecht, A.; Reynaud, S. Thermal Casimir effect for Drude metals in the plane-sphere geometry. Phys. Rev. A
**2010**, 82, 012511. [Google Scholar] [CrossRef] [Green Version] - Bordag, M.; Pirozhenko, I.G. Casimir entropy for a ball in front of a plane. Phys. Rev. D
**2010**, 82, 125016. [Google Scholar] [CrossRef] [Green Version] - Rodriguez-Lopez, P. Casimir energy and entropy in the sphere–sphere geometry. Phys. Rev. B
**2011**, 84, 075431. [Google Scholar] [CrossRef] [Green Version] - Rodriguez-Lopez, P. Casimir energy and entropy between perfect metal spheres. Int. J. Mod. Phys. Conf. Ser.
**2012**, 14, 475–484. [Google Scholar] [CrossRef] [Green Version] - Khusnutdinov, N.R. The thermal Casimir-Polder interaction of an atom with a spherical plasma shell. J. Phys. A Math. Theor.
**2012**, 45, 265301. [Google Scholar] [CrossRef] [Green Version] - Milton, K.A.; Guérout, R.; Ingold, G.-L.; Lambrecht, A.; Reynaud, S. Negative Casimir entropies in nanoparticle interactions. J. Phys. Condens. Matter
**2015**, 27, 214003. [Google Scholar] [CrossRef] [Green Version] - Ingold, G.-L.; Umrath, S.; Hartmann, M.; Guérout, R.; Lambrecht, A.; Reynaud, S.; Milton, K.A. Geometric origin of negative Casimir entropies: A scattering-channel analysis. Phys. Rev. E
**2015**, 91, 033203. [Google Scholar] [CrossRef] - Li, Y.; Milton, K.A.; Kalauni, P.; Parashar, P. Casimir self-entropy of an electromagnetic thin sheet. Phys. Rev. D
**2016**, 94, 085010. [Google Scholar] [CrossRef] [Green Version] - Milton, K.A.; Li, Y.; Kalauni, P.; Parashar, P.; Guérout, R.; Ingold, G.-L.; Lambrecht, A.; Reynaud, S. Negative entropies in Casimir and Casimir-Polder interactions. Fortschr. Phys.
**2017**, 65, 1600047. [Google Scholar] [CrossRef] [Green Version] - Balian, R.; Duplantier, B. Electromagnetic waves near perfect conductors. 2. Casimir effect. Ann. Phys.
**1978**, 112, 165–208. [Google Scholar] [CrossRef] - Milton, K.A.; Kalauni, P.; Parashar, P.; Li, Y. Casimir self-entropy of a spherical electromagnetic δ-function shell. Phys. Rev. D
**2017**, 96, 085007. [Google Scholar] [CrossRef] [Green Version] - Milton, K.A.; Kalauni, P.; Parashar, P.; Li, Y. Remarks on the Casimir self-entropy of a spherical electromagnetic δ-function shell. Phys. Rev. D
**2019**, 99, 045013. [Google Scholar] [CrossRef] [Green Version] - Bordag, M.; Kirsten, K. On the entropy of a spherical plasma shell. J. Phys. A
**2018**, 51, 455001. [Google Scholar] [CrossRef] [Green Version] - Bordag, M. Free energy and entropy for thin sheets. Phys. Rev. D
**2018**, 98, 085010. [Google Scholar] [CrossRef] [Green Version] - Parashar, P.; Milton, K.A.; Shajesh, K.V.; Brevik, I. Electromagnetic δ-function sphere. Phys. Rev. D
**2017**, 96, 085010. [Google Scholar] [CrossRef] [Green Version] - Graham, N.; Jaffe, R.L.; Khemani, V.; Quandt, M.; Schröder, O.; Weigel, H. The Dirichlet Casimir problem. Nucl. Phys. B
**2004**, 677, 379–404. [Google Scholar] [CrossRef] [Green Version] - Milton, K.A.; Parashar, P.; Brevik, I.; Kennedy, G. Self-stress on a dielectric ball and Casimir–Polder forces. Ann. Phys.
**2020**, 412, 168008. [Google Scholar] [CrossRef]

**Figure 1.**The TM free energy for low temperature in terms of $\xi =\alpha \sqrt{3/\left(2{\lambda}_{0}\right)}$. Shown are the coincident results for the Formula (22) and for Equation (25a) with two different values of $\alpha $, $\alpha =0.1$ and $\alpha =0.01$. Plotted is the free energy apart from a factor of ${(2{\lambda}_{0}/3)}^{2}/\left(\pi a\right)$. Although the slope is negative (positive entropy) for small $\xi $ (strong coupling), it is positive (negative entropy) for large enough $\xi $ (weak enough coupling).

**Figure 2.**The TM free energy (in units of 1/a) computed from the exact Formula (6) (left panel) or the low-temperature Formula (22) or (25b) (right panel) plotted as a function of aT for the same intermediate values of ${\lambda}_{0}$, ${\lambda}_{0}$ = 0.5, 1, and 2, in increasing order on the right side of each figure. Although the low-temperature formula would not seem to be applicable here, since the temperature is not particularly low, it gives results that are qualitatively identical to the exact free energy seen in Figure 2a, with significant deviations apparent only at higher T.

**Figure 3.**TM free energy relative to the strong-coupling low-temperature limit. The left panel shows the exact TM free energy (6) as a function of temperature T (in units of 1/a) relative to the strong-coupling low-temperature limit (21), for various values of the coupling ${\lambda}_{0}$. For a very low temperature, the free energy agrees with the limit (21). The nonmonotonicity is quite striking. The right panel shows the ratio R of Equation (22) to (21) as a function of aT. It is seen that the general low-temperature expression (22) captures most of the behavior shown in Figure 3a. The different curves in Figure 3b correspond to the same values of the coupling as in Figure 3a, namely, ${\lambda}_{0}$ = 0.5 (blue, solid), ${\lambda}_{0}$ = 1 (red, dotted), and ${\lambda}_{0}$ = 2 (black, dashed).

**Figure 4.**The behavior of the TM free energy for low temperatures (in units of 1/a), for even smaller values of the coupling, relative to the limiting value for low temperature and very small ${\lambda}_{0}$, Equation (23). The left panel shows the exact free energy (6), while the same ratio R is plotted in the right panel, except that the TM free energy is computed from the general low-temperature expression (22). The different curves are for the same values of ${\lambda}_{0}$ as in Figure 4a: ${\lambda}_{0}={10}^{-4}$ (blue, solid), ${\lambda}_{0}=2\times {10}^{-4}$ (red, dotted), and ${\lambda}_{0}=4\times {10}^{-4}$ (black, dashed). The fact that ${F}_{H}$ turns negative for very small temperatures reflects the limit (21).

**Figure 5.**The left panel shows the ratio of the exact TM free energy (6) to the strong-coupling, low-temperature limit (21) for relatively low temperatures, as a function of ${\lambda}_{0}$. The reversal of sign for low ${\lambda}_{0}$ reflects the transition from the regime where Equation (23) applies to the strong-coupling, low-temperature limit (21). The right panel shows the same ratio, except that, instead of the exact free energy, the general low-temperature expression (22) is used for the same values of temperature. The two graphs are nearly indistinguishable. In both panels, the different curves correspond to the temperatures ${a}_{T}=2.5\times {10}^{-2}$, $5\times {10}^{-2}$, $1\times {10}^{-1}$, from bottom to top on the right of each panel.

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**MDPI and ACS Style**

Li, Y.; Milton, K.A.; Parashar, P.; Hong, L.
Negativity of the Casimir Self-Entropy in Spherical Geometries. *Entropy* **2021**, *23*, 214.
https://doi.org/10.3390/e23020214

**AMA Style**

Li Y, Milton KA, Parashar P, Hong L.
Negativity of the Casimir Self-Entropy in Spherical Geometries. *Entropy*. 2021; 23(2):214.
https://doi.org/10.3390/e23020214

**Chicago/Turabian Style**

Li, Yang, Kimball A. Milton, Prachi Parashar, and Lujun Hong.
2021. "Negativity of the Casimir Self-Entropy in Spherical Geometries" *Entropy* 23, no. 2: 214.
https://doi.org/10.3390/e23020214