# The Reality of Casimir Friction

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Friction between Two Plates

_{1}of the lower plate is different from the particle density ρ

_{2}in the upper plate. This flexibility in formalism makes it possible to deal with the single-particle half-space problem simply by letting ρ

_{2}approach zero; we will turn to this problem later.

_{0}), he derived the following expressions in two limiting cases for the force per unit area A [28]:

## 3. Methodologies

_{B}T), where k

_{B}is Boltzmann's constant, and T is temperature. In this picture, Casimir forces can be interpreted as induced interactions due to quantized fluctuating dipole moments that interact via the radiating (time-dependent) dipole-dipole interaction. Within the “polymer” picture, arguments and methods of classical statistical mechanics can be applied to obtain Fourier transforms of time-dependent response functions given by quantum mechanical commutators. They follow from the corresponding correlation functions of the classical problem in imaginary time. It turns out that the viewpoint with fluctuating dipole moments is equivalent to the more traditional one where medium-induced changes in the ground state energies of the quantized electromagnetic field induce the Casimir forces.

## 4. Interaction between an Atom and a Plate

_{0}is the distance between the atom and the surface. This result agrees within a factor of 3/8 with that found in Reference [81]. The salient dependence is upon the cube of the velocity and the inverse tenth power of the distance. This friction can become appreciable only if the atom is extremely close to the surface. It is noteworthy that the same result (within a further factor of 5) was obtained by a perturbative method, related to that of Barton [71] in a recent paper [40], which argues that the linear velocity dependence found by Barton is an artifact of the particular velocity profile assumed by him. Scheel and Buhmann [87] like Barton had also obtained such a linear dependence. See also Reference [88]. (Some of the linear effects found reflect interactions with excited states, rather than with ground-state atoms, which is our focus here.) Another example of congruence of results appears in a recent paper [89] where a formula (Formula (41)) is given for the frictional force experienced by a dielectric particle moving in a thermal field, the problem considered by Einstein and Hopf [82,83]. This formula is exactly the same (apart from notation) as that derived in Equations (A4) and (A15) in Reference [27] and is given by:

^{2}= e

^{2}/(m α), for an oscillating charge e with mass m, and $\beta =1/{k}_{B}T$.

^{3}respectively. The former is influenced by the resistivity of the metal plane. Thus the corresponding friction force (3) contains two factors of σ.

## 5. Friction between Parallel Plates

_{x}is restricted to q

_{x}> 0 while integral (48) in Reference [52] does not have this restriction. (The factor of 6 between Pendry’s [28] and Volokitin and Persson’s result seems to be simply a miscalculation by Pendry.) We believe that essential convergence of results has thus been achieved.

## 6. Temperature Dependence

## 7. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

SI | Système international d’unités |

TM | Transverse magnetic |

C | Casimir |

CP | Casimir-Polder |

vdW | van der Waals |

## References

- Mohr, P.J.; Taylor, B.N.; Newell, D.B. CODATA Recommended Values of the Fundamental Physical Constants: 2006. Rev. Mod. Phys.
**2008**, 80, 633–730. [Google Scholar] [CrossRef] - Hanneke, D.; Fogwell, S.; Gabrielse, G. New Measurement of the Electron Magnetic Moment and the Fine Structure Constant. Phys. Rev. Lett.
**2008**, 100, 120801. [Google Scholar] [CrossRef] [PubMed] - Casimir, H.B.G.; Polder, D. The Influence of retardation on the London-van der Waals forces. Phys. Rev.
**1948**, 73, 360. [Google Scholar] [CrossRef] - Casimir, H.B.G. On the Attraction Between Two Perfectly Conducting Plates. Kon. Ned. Akad. Wetensch. Proc.
**1948**, 51, 793–795. [Google Scholar] - Milton, K.A.; Abalo, E.K.; Parashar, P.; Pourtolami, N.; Brevik, I.; Ellingsen, S.Å. Repulsive Casimir and Casimir-Polder Forces. J. Phys. A
**2012**, 45, 374006. [Google Scholar] [CrossRef] - Wilson, C.M.; Johansson, G.; Pourkabirian, A.; Simoen, M.; Johansson, J.R.; Duty, T.; Nori, F.; Delsing, P. Observation of the dynamical Casimir effect in a superconducting circuit. Nature
**2011**, 479, 376–379. [Google Scholar] [CrossRef] [PubMed] - Lähteenmäkia, P.; Paraoanua, G.S.; Hasselb, J.M.; Hakonena, P.J. Dynamical Casimir effect in a Josephson metamaterial. PNAS
**2013**, 110, 4234–4238. [Google Scholar] [CrossRef] - Moore, G.T. Quantum theory of the electromagnetic field in a variable-length one-dimensional cavity. J. Math. Phys.
**1970**, 11, 2679. [Google Scholar] [CrossRef] - Fullling, S.A. Nonuniqueness of Canonical Field Quantization in Riemannian Space-Time. Phys. Rev. D
**1973**, 7, 2850. [Google Scholar] [CrossRef] - Davies, P.C.W. Scalar production in Schwarzschild and Rindler metrics. J. Phys. A
**1975**, 8, 609. [Google Scholar] [CrossRef] - Unruh, W.G. Notes on black-hole evaporation. Phys. Rev. D
**1976**, 14, 870. [Google Scholar] [CrossRef] - Fulling, S.A.; Matsas, G.E.A. Unruh Effect. Scholarpedia
**2014**, 9, 31789. [Google Scholar] [CrossRef] - Chen, F.; Mohideen, U.; Klimchitskaya, G.L.; Mostepanenko, V.M. Experimental and theoretical investigation of the lateral Casimir force between corrugated surfaces. Phys. Rev. A
**2002**, 66, 032113. [Google Scholar] [CrossRef] - Munday, J.M.; Iannuzzi, D.; Barash, Y.; Capasso, F. Torque on birefringent plates induced by quantum fluctuations. Phys. Rev. A
**2005**, 71, 042102. [Google Scholar] [CrossRef] - Guérout, R.; Genet, C.; Lambrecht, A.; Reynaud, S. Casimir Torque between Nanostructured Plate. EPL
**2015**, 111, 44001. [Google Scholar] [CrossRef] - Milton, K.A. The Casimir Effect: Physical Manifestations of Zero-Point Energy; World Scientific: Singapore, 2001. [Google Scholar]
- Bordag, M.; Klimchitskaya, G.I.; Mohideen, U.; Mostepanenko, V.M. Advances in the Casimir Effect; Oxford University Press: Oxford, UK, 2009. [Google Scholar]
- Dalvit, D.A.R.; Milonni, P.; Roberts, D.; da Rosa, F. (Eds.) Casimir Physics; Springer: Berlin, Germany, 2011.
- Simpson, W.M.R.; Leonhardt, U. Force of the Quantum Vacuum: An Introduction to Casimir Physics; World Scientific: Singapore, 2015. [Google Scholar]
- Mate, C.M.; McClelland, G.M.; Erlandsson, R.; Chiang, S. Atomic-Scale Friction of a Tungston Tip on a Graphite Surface. Phys. Rev. Lett.
**1987**, 59, 1942. [Google Scholar] [CrossRef] [PubMed] - Berman, D.; Erdemir, A.; Zinovev, A.V.; Sumant, A.V. Nanoscale friction properties of graphene and graphene oxide. Diam. Rel. Mater.
**2015**, 54, 91–96. [Google Scholar] [CrossRef] - Levchenko, A.; Kamenev, A. Coulomb Drag at Zero Temperature. Phys. Rev. Lett.
**2008**, 100, 026805. [Google Scholar] [CrossRef] [PubMed] - Persson, B.N.J.; Zhang, Z. Theory of friction: Coulomb drag between two closely spaced solids. Phys. Rev. B
**1998**, 57, 7327. [Google Scholar] [CrossRef] - Teodorovich, E.V. Contribution of macroscopic van der Waals interactions to frictional force. Proc. R. Soc. Lond. A
**1978**, 362, 71–77. [Google Scholar] [CrossRef] - Levitov, L.S. Van der Waals friction. Europhys. Lett.
**1989**, 8, 499–504. [Google Scholar] [CrossRef] - Høye, J.S.; Brevik, I. Friction force between moving harmonic oscillators. Physica A
**1992**, 181, 413–426. [Google Scholar] [CrossRef] - Høye, J.S.; Brevik, I. Friction force with non-instantaneous interaction between moving harmonic oscillators. Physica A
**1993**, 196, 241–254. [Google Scholar] [CrossRef] - Pendry, J.B. Shearing the vacuum—Quantum friction. J. Phys. Condens. Matter
**1997**, 9, 10301–10320. [Google Scholar] [CrossRef] - Pendry, J.B. Can sheared surfaces emit light? J. Mod. Opt.
**1998**, 45, 2389–2408. [Google Scholar] [CrossRef] - Pendry, J.B. Quantum friction—Fact or fiction? New J. Phys.
**2010**, 12, 033028. [Google Scholar] [CrossRef] - Pendry, J.B. Reply to comment on “Quantum friction—Fact or friction?”. New J. Phys.
**2010**, 12, 068002. [Google Scholar] [CrossRef] - Landau, L.D.; Lifshitz, E.M. Statistical Physics, Part 2; Pergamon Press: Oxford, UK, 1980. [Google Scholar]
- Lambrecht, A.; Reynaud, S. Casimir force between metallic mirrors. Eur. Phys. J. D
**2000**, 8, 309–318. [Google Scholar] [CrossRef] - Volokitin, A.I.; Persson, B.N.J. Near-field radiative heat transfer and noncontact friction. Rev. Mod. Phys.
**2007**, 79, 1291–1329. [Google Scholar] [CrossRef] - Landau, L.D.; Lifshitz, E.M. Electrodynamics of Continuous Media; Pergamon Press: Oxford, UK, 1984. [Google Scholar]
- Rytov, S.M. Theory of Electrical Fluctuations and Thermal Radiation; Academy of Science USSR Publishing House: Moscow, Russia, 1953. [Google Scholar]
- Milonni, P.W. The Quantum Vacuum: An Introduction to Quantum Electrodynamics; Academic Press: Cambridge, MA, USA, 1994. [Google Scholar]
- Ginzburg, V.L. Applications of Electrodynamics in Theoretical Physics and Astrophysics; Gordon and Breach Science Publisher: Philadelphia, PA, USA, 1989. [Google Scholar]
- Ginzburg, V.L. Radiation by uniformly moving sources (Vavilov-Cherenkov effect, transition radiation, and other phenomena). Phys. Uspekhi
**1996**, 39, 973–982. [Google Scholar] [CrossRef] - Intravaia, F.; Mkrtchian, V.E.; Buhmann, S.; Schell, S.; Dalvit, D.A.R.; Henkel, C. Friction forces on atoms after acceleration. J. Phys. Condens. Matter
**2015**, 27, 214020. [Google Scholar] [CrossRef] [PubMed] - Pieplow, G.; Henkel, C. Cherenkov friction on a neutral particle moving parallel to a dielectric. J. Phys. Condens. Matter
**2015**, 27, 214001. [Google Scholar] [CrossRef] [PubMed] - Brevik, I.; Kolbenstvedt, H. Quantum point detector moving through a dielectric medium. II. Constant acceleration. Il Nuovo Cimento B
**1989**, 103, 45–62. [Google Scholar] [CrossRef] - Brevik, I.; Lautrup, B. Quantum Electrodynamics in Material Media; Munksgaard: Copenhagen, Denmark, 1970; Volume 38, pp. 1–37. [Google Scholar]
- Barnett, S.M. Resolution of the Abraham-Minkowski Dilemma. Phys. Rev. Lett.
**2010**, 104, 070401. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Wang, S.; Ng, J.; Xiao, M.; Chan, C.T. Electromagnetic stress at the boundary: Photon pressure or tension? Sci. Adv.
**2016**, 2, e1501485. [Google Scholar] [CrossRef] [PubMed] - Høye, J.S.; Brevik, I. Casimir friction force and energy dissipation for moving harmonic oscillators. EPL
**2010**, 91, 60003. [Google Scholar] [CrossRef] - Høye, J.S.; Brevik, I. Casimir friction force and energy dissipation for moving harmonic oscillators. II. Eur. Phys. J. D
**2011**, 61, 335–339. [Google Scholar] [CrossRef] - Høye, J.S.; Brevik, I. Casimir friction in terms of moving harmonic oscillators: Equivalence between two different formulations. Eur. Phys. J. D
**2011**, 64, 1–3. [Google Scholar] [CrossRef] - Høye, J.S.; Brevik, I. Casimir friction force between polarizable media. Eur. Phys. J. D
**2012**, 66, 149. [Google Scholar] [CrossRef] - Høye, J.S.; Brevik, I. Casimir friction force for moving harmonic oscillators. Int. J. Mod. Phys. A
**2012**, 27, 1260011. [Google Scholar] [CrossRef] - Høye, J.S.; Brevik, I. Casimir friction between dense polarizable media. Entropy
**2013**, 15, 3045–3064. [Google Scholar] [CrossRef] [Green Version] - Høye, J.S.; Brevik, I. Casimir friction at zero and finite temperatures. Eur. Phys. J. D
**2014**, 68, 61. [Google Scholar] [CrossRef] - Høye, J.S.; Brevik, I. Casimir friction: Relative motion more generally. J. Phys. Condens. Matter
**2015**, 27, 214008. [Google Scholar] [CrossRef] [PubMed] - Kubo, R. Lectures in Theoretical Physics; Brittin, W.E., Dunham, L.G., Eds.; Interscience: New York, NY, USA, 1959; Volume 1. [Google Scholar]
- Høye, J.S.; Stell, G. Quantum statistical mechanical model for polarizable fluids. J. Chem. Phys.
**1981**, 75, 5133. [Google Scholar] [CrossRef] - Thompson, M.J.; Schweizer, K.S.; Chandler, D. Quantum theory of polarization in liquids: Exact solution of the mean spherical and related approximations. J. Chem. Phys.
**1982**, 76, 1128–1135. [Google Scholar] [CrossRef] - Volokitin, A.I.; Persson, B.N.J. Theory of friction: The contribution from a fluctuating electromagnetic field. J. Phys. Condens. Matter
**1999**, 11, 345–359. [Google Scholar] [CrossRef] - Volokitin, A.I.; Persson, B.N.J. Noncontact friction between nanostructures. Phys. Rev. B
**2003**, 68, 155420. [Google Scholar] [CrossRef] - Volokitin, A.I.; Persson, B.N.J. Theory of the interaction forces and the radiative heat transfer between moving bodies. Phys. Rev. B
**2008**, 78, 155437. [Google Scholar] [CrossRef] - Volokitin, A.I.; Persson, B.N.J. Quantum friction. Phys. Rev. Lett.
**2011**, 106, 094502. [Google Scholar] [CrossRef] [PubMed] - Volokitin, A.I.; Persson, B.N.J. Comment on “Fully covariant radiation force on a polarizable particle”. New J. Phys.
**2014**, 16, 118001. [Google Scholar] [CrossRef] - Dedkov, G.V.; Kyasov, A.A. Vacuum attraction, friction and heating of nanoparticles moving nearby a heated surface. J. Phys. Condens. Matter
**2008**, 20, 354006. [Google Scholar] [CrossRef] - Dedkov, G.V.; Kyasov, A.A. Conservative-dissipative forces and heating mediated by fluctuating electromagnetic field: Two plates in relative nonrelativistic motion. Surf. Sci.
**2010**, 604, 562–567. [Google Scholar] [CrossRef] - Dedkov, G.V.; Kyasov, A.A. Dynamical van der Waals atom-surface interaction. Surf. Sci.
**2011**, 605, 1077–1081. [Google Scholar] [CrossRef] - Dedkov, G.V.; Kyasov, A.A. Dynamical Casimir-Polder atom-surface interaction. Surf. Sci.
**2012**, 606, 46–52. [Google Scholar] [CrossRef] - Dedkov, G.V.; Kyasov, A.A. A uniformly moving and rotating polarizable particle in thermal radiation field: Frictional force and torque, radiation and heating. 2015; arXiv:1504.01588. [Google Scholar]
- Polevoi, V.G. Tangential molecular forces between moving bodies by a fluctuating electromagnetic field. Sov. Phys. JETP
**1990**, 71, 1119. [Google Scholar] - Mkrtchian, V.E. Interaction between moving macroscopic bodies: Viscosity of the electromagnetic vacuum. Phys. Lett. A
**1995**, 207, 299–302. [Google Scholar] [CrossRef] - Rytov, S.M.; Kravtsov, Y.A. Elements of Random Fields. In Principles of Statistical Radiophysics; Springer: Berlin, Germany, 1989; Volume 3. [Google Scholar]
- Barton, G. On van der Waals friction: I. Between two atoms. New J. Phys.
**2010**, 12, 113044. [Google Scholar] [CrossRef] - Barton, G. On van der Waals friction. II: Between atom and half-space. New J. Phys.
**2010**, 12, 113045. [Google Scholar] [CrossRef] - Barton, G. On van der Waals friction between two atoms at nonzero temperature. New J. Phys.
**2011**, 13, 043023. [Google Scholar] [CrossRef] - Barton, G. On van der Waals friction between half-spaces at low temperature. J. Phys. Condens. Matter
**2011**, 23, 335004. [Google Scholar] [CrossRef] [PubMed] - Barton, G. Van der Waals friction: A Hamiltonian test bed. Int. J. Mod. Phys. A
**2012**, 27, 1260002. [Google Scholar] [CrossRef] - Philbin, T.G.; Leonhardt, U. No quantum friction between uniformly moving plates. New J. Phys.
**2009**, 11, 033035. [Google Scholar] [CrossRef] - Pieplow, G.; Henkel, C. Fully covariant radiation force on a polarizable particle. New J. Phys.
**2013**, 15, 023027. [Google Scholar] [CrossRef] - Maghrebi, M.F.; Golestanian, R.; Kardar, M. Quantum Cherenkov radiation and noncontact friction. Phys. Rev. A
**2013**, 88, 042509. [Google Scholar] [CrossRef] - Silveirinha, M.G. Theory of quantum friction. New J. Phys.
**2014**, 16, 063011. [Google Scholar] [CrossRef] - Fröhlich, J.; Gang, Z. Emission of Cherenkov Radiation as a Mechanism for Hamiltonian Friction. Adv. Math.
**2014**, 264, 183–235. [Google Scholar] [CrossRef] - Nesterenko, V.V.; Nesterenko, A.V. Macroscopic approach to the Casimir friction force. JETP Lett.
**2014**, 99, 581–584. [Google Scholar] [CrossRef] - Intravaia, F.; Behunun, R.O.; Dalvit, D.A.R. Quantum friction and fluctuation theorems. Phys. Rev. A
**2014**, 89, 050101(R). [Google Scholar] [CrossRef] - Einstein, A.; Hopf, L. Statistische Untersuchung der Bewegung eines Resonators in einem Strahlungsfeld. Ann. Phys.
**1910**, 338, 1105–1115. (In German) [Google Scholar] [CrossRef] - Einstein, A. Zur Quantentheorie der Strahlung. Phys. Zeitsch.
**1917**, 18, 121–128. (In German) [Google Scholar] - Mkrtchian, V.; Parsegian, V.A.; Podgornik, R.; Saslow, W.N. Universal Thermal Radiation Drag on Neutral Objects. Phys. Rev. Lett.
**2003**, 91, 220801. [Google Scholar] [CrossRef] [PubMed] - Høye, J.S.; Brevik, I.; Milton, K.A. Casimir friction between polarizable particle and half-space with radiation damping at zero temperature. J. Phys. A
**2015**, 48, 365004. [Google Scholar] [CrossRef] - Maghrebi, M.F.; Golestanian, R.; Kardar, M. Scattering approach to the dynamical Casimir effect. Phys. Rev. D
**2013**, 87, 025016. [Google Scholar] [CrossRef] - Scheel, S.; Buhmann, S.Y. Casimir-Polder Forces on Moving Atoms. Phys. Rev. A
**2009**, 80, 042902. [Google Scholar] [CrossRef] - Donaire, M.; Lambrecht, A. Velocity-dependent dipole forces on an excited atom. Phys. Rev. A
**2016**, 93, 022701. [Google Scholar] [CrossRef] - Volokitin, A.I. Blackbody friction force on a relativistic small neutral particle. Phys. Rev. A
**2015**, 91, 032505. [Google Scholar] [CrossRef] - Dedkov, G.V.; Kyasov, A.A. Attraction Force, Frictional Torque, and Heating of a Spherical Particle Rotating in the Evanescent Electromagnetic Field of a Heated Surface. Tech. Phys. Lett.
**2013**, 39, 609–611. [Google Scholar] [CrossRef] - Zhao, R.; Manjavacas, A.; Javier García de Abajo, F.; Pendry, J.B. Rotational Quantum Friction. Phys. Rev. Lett.
**2012**, 109, 123604. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**Standard configuration: Upper plate, having density ρ

_{2}, moving with velocity v; lower plate, with density ρ

_{1}, at rest. Gap width is d.

**Figure 2.**Atom moving above a dielectric or metallic surface. Although the medium and the atom may be taken to be isotropic, the electromagnetic interaction induces anisotropy. See, for example Reference [85].

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Milton, K.A.; Høye, J.S.; Brevik, I.
The Reality of Casimir Friction. *Symmetry* **2016**, *8*, 29.
https://doi.org/10.3390/sym8050029

**AMA Style**

Milton KA, Høye JS, Brevik I.
The Reality of Casimir Friction. *Symmetry*. 2016; 8(5):29.
https://doi.org/10.3390/sym8050029

**Chicago/Turabian Style**

Milton, Kimball A., Johan S. Høye, and Iver Brevik.
2016. "The Reality of Casimir Friction" *Symmetry* 8, no. 5: 29.
https://doi.org/10.3390/sym8050029