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REVIEW 3 major objections 5 minor 51 references

Spontaneous Torque on an Inhomogeneous Chiral Body out of Thermal Equilibrium

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read A chiral body out of thermal equilibrium with the vacuum should spontaneously rotate, even if made of ordinary reciprocal materials.

desk verdict A careful second-order torque derivation with explicit geometries, but the Drude-gold examples strain the perturbative expansion. read the letter →

arxiv 2412.03336 v2 pith:AX2ALXH5 submitted 2024-12-04 quant-ph hep-th

Kimball A. Milton , Nima Pourtolami , Gerard Kennedy This is my paper · ORCID
classification quant-phhep-th
keywords quantumvacuumtorqueCasimirthermalnonequilibriumchiralityelectricsusceptibilityfrictionangularmomentumfluxfluctuation-dissipationtheorem
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper argues that a body made of ordinary (reciprocal) materials will experience a spontaneous torque from the quantum vacuum if the body is chiral, inhomogeneous, and at a temperature different from its surroundings. In second order in the electric susceptibility, the torque is nonzero only when both chirality and inhomogeneity are present; a uniform reciprocal body gets no torque at this order. The authors derive an explicit formula, $\tau = (1/2\pi^2)\int d\omega/(2\pi)\, X_{AB}(\omega)\left[(e^{\beta\omega}-1)^{-1} - (e^{\beta'\omega}-1)^{-1}\right] J_{AB}(\omega)$, and evaluate it for a dual Allen wrench and a dual flag. For micrometer-scale examples the resulting terminal angular velocity is of order $10^{-3}$ to $10^{-2}$ radians per second, which they argue should be observable in the laboratory. The significance is a new mechanical effect of vacuum fluctuations: a self-rotation driven purely by thermal disequilibrium, with no external fields or moving parts.

What carries the argument

The load-bearing object is the second-order torque identity $\boldsymbol{\tau} = \frac{1}{2\pi^2}\int \frac{d\omega}{2\pi} X_{AB}(\omega)\left[\frac{1}{e^{\beta\omega}-1}-\frac{1}{e^{\beta'\omega}-1}\right]\mathbf{J}_{AB}(\omega)$, built from the antisymmetric susceptibility product $X_{AB}(\omega) = \operatorname{Im}\chi_A(\omega) \operatorname{Re}\chi_B(\omega) - \operatorname{Re}\chi_A(\omega) \operatorname{Im}\chi_B(\omega)$ and the geometric factor $\mathbf{J}_{AB}(\omega) = -\int_A d\mathbf{r}\int_B d\mathbf{r}'\, (\mathbf{r}\times\mathbf{r}')/|\mathbf{r}-\mathbf{r}'|^8\, \phi(\omega|\mathbf{r}-\mathbf{r}'|)$. The function $\phi(v)$, defined in Eq. (2.18), encodes the correlated vacuum Green's dyadic; its small- and large-distance behavior guarantees the integral converges and, together with the positivity of $\phi(v)+4v^8/9$, fixes the sign of the torque. The mechanism is back-reaction: a chiral body emits and absorbs thermal radiation asymmetrically, so an imbalance between body temperature $T'$ and environment temperature $T$ produces a net flux of angular momentum away from the body.

What would settle it

Suspend a dual Allen wrench with a gold shaft and dielectric tags in vacuum at 300 K, heat it to 600 K, and watch for rotation: the formula predicts a terminal angular velocity near $3\times 10^{-3}$ rad/s for a micrometer-scale version and around $10^{-4}$ rad/s for a larger version, so the absence of rotation at that scale would refute the prediction; conversely, measuring a spontaneous torque on a uniform chiral gold pinwheel in a regime where second-order perturbation theory should hold would refute the claim that inhomogeneity is necessary.

Watch

Extended reading notes

Core claim

In the paper's own terms, the discovery is that a spontaneous vacuum torque appears at second order in the electric susceptibility for a reciprocal body, provided the body is both chiral and inhomogeneous and is out of thermal equilibrium with blackbody radiation. For a two-part body with isotropic uniform susceptibilities $\chi_A$ and $\chi_B$, the torque is $\boldsymbol{\tau} = \frac{1}{2\pi^2}\int_0^\infty \frac{d\omega}{2\pi} X_{AB}(\omega)\left[\frac{1}{e^{\beta\omega}-1}-\frac{1}{e^{\beta'\omega}-1}\right]\mathbf{J}_{AB}(\omega)$, where $X_{AB} = \operatorname{Im}\chi_A \operatorname{Re}\chi_B - \operatorname{Re}\chi_A \operatorname{Im}\chi_B$ and $\mathbf{J}_{AB}$ is a purely geometric integral over the two volumes. The authors confirm the formula by an independent calculation of angular momentum flux in the radiation zone, and they show that the torque reverses sign when the body is colder rather than hotter than its environment. They conclude that inhomogeneity, not just chirality, is required for a reciprocal-body vacuum torque through second order.

Load-bearing premise

The torque formula is derived by truncating the expansion in powers of the electric susceptibility at second order, and the results could change if higher-order (or nonperturbative) terms contribute significantly for real materials such as gold.

Editorial extensions

If this is right

  • A uniform reciprocal body will not experience a vacuum torque through second order; both spatial inhomogeneity and chirality are required, so any observation of torque on a homogeneous chiral particle would signal higher-order or nonperturbative physics.
  • A chiral body released hotter or colder than the vacuum will spin up and then settle at a terminal angular velocity as it thermalizes; for the dual Allen wrench with micrometer dimensions the terminal rate is about $3\times 10^{-3}$ rad/s.
  • Using thin two-dimensional flags instead of wire tags enhances the torque by roughly the ratio of the flag length to its thickness, raising the terminal velocity to about $3\times 10^{-2}$ rad/s for a small object.
  • The sense of rotation is fixed by the temperature imbalance and material asymmetry: a body hotter than the vacuum rotates so that local forces point toward the metallic part, and a colder body rotates the opposite way.
  • The effect persists for large bodies: for objects larger than about 10 micrometers at room temperature the geometric factor grows linearly with size, so the torque does not vanish in the macroscopic limit.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the second-order torque is real, the same formula should produce torques for many material pairs with crossing susceptibilities, not just dielectric-on-metal; scanning material combinations could tune both magnitude and sign of the effect.
  • The requirement that the body stay out of equilibrium suggests a practical route: an optically trapped chiral particle held at elevated temperature by laser absorption should exhibit a measurable steady spin, which would constitute a clean test of the mechanism.
  • The sharpest discriminator between this perturbative picture and nonperturbative effects is the homogeneous-chiral-body case: a measurement of torque on a uniform gold pinwheel at conditions where second order should dominate would directly test whether inhomogeneity is truly necessary.
  • Because the torque is proportional to the difference of Bose-Einstein factors, it vanishes at equal temperatures; this offers a built-in null check for experimental searches.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper claims that an inhomogeneous, chiral body made of reciprocal (ordinary) materials experiences a spontaneous quantum vacuum torque when it is out of thermal equilibrium with its environment. The torque is derived in second order in the electric susceptibility, both from the classical torque formula and from the angular-momentum flux in the radiation zone, leading to the central formula Eq. (2.21) with an antisymmetric susceptibility product X_AB and a geometric integral J_AB. Two concrete configurations are analyzed, the dual Allen wrench (Sec. IV) and the dual flag (Sec. V), for which the paper computes torques, cooling times, and terminal angular velocities, concluding that the effect could be observable. The paper also emphasizes that in second order inhomogeneity is required for reciprocal bodies, in contrast to the nonperturbative result of Ref. [47].

Significance. If the central result holds, the paper extends the authors' previous work on spontaneous quantum-thermal forces to torques and provides a concrete, in-principle observable prediction. The two independent derivations of the central torque formula, the closed-form expressions for the geometric integrals, and the explicit treatment of thermalization and terminal angular velocity are valuable and make the paper more than a formal exercise. The significance is, however, conditional on the second-order perturbative truncation being controlled for the material parameters used in the examples; this is precisely where the manuscript is currently weakest.

major comments (3)
  1. [Sec. IV, Eqs. (4.5), (4.6), (4.8), (4.15)] The quantitative examples use Drude gold with ω_p=9 eV and ν=0.035 eV (Eq. (4.5)). At the dominant thermal frequency ω≈k_B T=0.026 eV, |χ|=ω_p^2/[ω(ω^2+ν^2)^{1/2}]≈7×10^4, so the susceptibility is very large. The paper does not identify a small parameter controlling the O(χ^2) truncation for these parameters. Footnote 9 concedes that forces and torques appear in third order, and Ref. [47] reports a nonperturbative torque on a homogeneous chiral gold body. Unless the next order is shown to be negligible for the Drude parameters used, the computed torques and terminal angular velocities in Secs. IV and V are not established for the actual materials considered.
  2. [Sec. II, Eqs. (2.17), (2.18)] The central geometric kernel ϕ(v) in Eq. (2.18), together with Δ(v) in Eq. (2.17), is taken without derivation from the authors' Ref. [48]. This function enters every subsequent numerical result, including the sign and magnitude of the torque and the positivity argument for J_AB. The manuscript should either derive these functions or explicitly state that the results depend on the correctness of the corresponding derivation in Ref. [48].
  3. [Sec. III and Conclusions] The structural conclusion that inhomogeneity is required for a reciprocal body ('Again, inhomogeneity is required for both torque and force, in second order') is stated, but the derivation of the EE-fluctuation contributions is only sketched: the text says they give the corresponding structure and leaves the verification to the reader. Because the abstract and conclusions present the inhomogeneity requirement without the 'second order' qualifier in the opening paragraph, the manuscript should either provide the full derivation or state prominently that the conclusion is a second-order perturbative statement, especially in light of footnote 9 and Ref. [47].
minor comments (5)
  1. [Eq. (2.7)] In Eq. (2.7), the measure 'dν/dπ' appears where 'dν/2π' is clearly intended; this should be corrected.
  2. [Fig. 1 and Fig. 9 captions] The phrase 'invariant under reflection in the origin' should read 'invariant under inversion through the origin' or 'reflection through the origin', since the symmetry is a central inversion, not a mirror reflection.
  3. [Ref. [47]] Reference [47] is cited as unpublished; the arXiv identifier arXiv:1708.01985 is available and should be included for reproducibility.
  4. [Eq. (4.16) and surrounding text] The prefactor estimate in Eq. (4.16) is labeled 'roughly' but it is unclear whether the numerical value 5×10^{-10} s^{-1} includes the factor 33/40 and all material parameters; a short worked evaluation would help the reader check this important order-of-magnitude claim.
  5. [Sec. IV A] The cooling model assumes that only the metal part A radiates, while the dielectric tags are taken to be lossless. The text acknowledges this in footnote 8, but the implication for the terminal angular velocity should be stated more explicitly in the main text, since a lossy dielectric would shorten the cooling time and reduce ω_T.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the torque formula is derived from the classical torque expression plus FDT, with an independent radiation-zone cross-check; self-citations supply rederivable algebraic identities and external cooling results, not the target prediction.

full rationale

The central result, Eq. (2.21), is not assumed. It follows from the classical torque formula (2.1), the FDT correlations (2.3), and an explicit second-order expansion of P and E; the algebra leading to Eqs. (2.19)-(2.21) is shown in the text. Section III provides an independent radiation-zone derivation of the same torque, which is a genuine cross-check rather than a circular restatement. The Green's dyadic product identity (2.16)-(2.18) is taken from the authors' own Ref. [48], but it is a parameter-free algebraic identity that does not contain the target torque, so citing it is not circular. Similarly, the cooling power P(T,T') in Eq. (4.12) is taken from Ref. [43], but it is an independent quantal-radiation result used only for the terminal-velocity estimate, not to construct the torque itself. The examples use standard Drude parameters from Eq. (4.5) and analytic integrals; no quantity is fitted to the predicted torque. The paper itself flags its main limitation: footnote 9 concedes that forces and torques also appear in third order, and Ref. [47] finds a nonperturbative torque on a homogeneous chiral gold body. That is a convergence/accuracy concern about the second-order truncation for the Drude examples, not a circularity. No step in the derivation chain reduces to its own input.

Assumptions & free parameters 4 free parameters · 9 assumptions · 0 invented entities

The central derivations rest on standard fluctuation electrodynamics: the classical torque expression, the fluctuation-dissipation theorem, and the vacuum Green's dyadic. The paper introduces no new entities. The quantitative examples depend on material parameters from the literature (Drude omega_p and nu for gold), an unspecified dielectric constant chi_B, and chosen dimensions. The main non-standard input is the identity for Delta(v) and phi(v) taken from the authors' own Ref [48], and the thermalization model from Ref [43]. These are self-citations but not circular, because they do not assume the torque result.

free parameters (4)
  • Drude plasma frequency of gold, omega_p = 9 eV
    Taken from Ref [51]; fitted to gold optical data and used in Eqs. (4.5), (4.6), (4.8), and (4.12).
  • Drude damping rate of gold, nu = 0.035 eV
    From Ref [51]; determines thermal factors, cooling power, and the frequency integrals.
  • Dielectric tag susceptibility, chi_B = not specified (real constant)
    Chosen as dispersionless; the torque and terminal velocity estimates scale linearly with this value, so the quantitative claims depend on an unspecified constant.
  • Geometric dimensions a, b, SA, SB, LB = e.g., a=b=1 micron or a=1 cm; wire radius 40-50 nm; LB=50 nm
    Chosen for the example estimates; these set the magnitude of J_AB and the moment of inertia.
assumptions (9)
  • standard math Classical torque formula for a dielectric body, Eq. (2.1), with internal and external torque terms.
    Taken from Milton and Schwinger (Ref [50]); used as the starting point for the perturbative expansion.
  • standard math Fluctuation-dissipation theorem correlating free polarization and electric field fluctuations, Eqs. (2.3a) and (2.3b).
    The coth factors encode body and environment temperatures; this is the standard FDT for thermal and quantum fluctuations.
  • domain assumption Electric susceptibility is local in space and the perturbative expansion in powers of susceptibility converges.
    Used throughout Sec II; if nonlocality or strong coupling invalidates the expansion, the second-order truncation fails.
  • domain assumption The body is at a uniform temperature T'.
    Stated in Sec VI; allows the body temperature to enter through a single coth factor; thermal conduction within the body is ignored.
  • domain assumption The Green's dyadic identities for Delta(v) and phi(v), Eqs. (2.16)-(2.18), are correct.
    Borrowed from the authors' previous Ref [48]; the compact torque formula (2.19) depends on it.
  • domain assumption Drude model describes the metal part (gold) and a dispersionless real susceptibility describes the dielectric tags.
    Used for numerical estimates in Secs IV and V; the torque values are model-dependent.
  • standard math The angular momentum conservation law and radiation-zone stress tensor, Eqs. (3.1)-(3.3), give the torque on the body.
    Used as the second independent derivation in Sec III.
  • domain assumption Thermalization is dominated by radiation from the Drude metal, using the power expression from Ref [43], and quantum frictional torque is negligible.
    Used for terminal angular velocity estimates in Sec IV A; the dielectric tags are treated as non-emitting.
  • domain assumption Specific heat obeys the Debye or Dulong-Petit model with Debye temperature about 170 K for gold.
    Sets the cooling time scale t0 in Eq. (4.14).

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Cite this review

Pith. "Pith review of Spontaneous Torque on an Inhomogeneous Chiral Body out of Thermal Equilibrium." pith.science (2026). https://pith.science/paper/AX2ALXH5

@misc{pith2026241203336,
  author       = {Pith},
  title        = {Pith review of: Spontaneous Torque on an Inhomogeneous Chiral Body out of Thermal Equilibrium},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AX2ALXH5}},
  note         = {Machine review of arXiv:2412.03336}
}
read the original abstract

In a previous paper we showed that an inhomogeneous body in vacuum will experience a spontaneous force if it is not in thermal equilibrium with its environment. This is due to the asymmetric asymptotic radiation pattern such an object emits. We demonstrated this self-propulsive force by considering an expansion in powers of the electric susceptibility: A torque arises in first order, but only if the material constituting the body is nonreciprocal. No force arises in first order. A force does occur for bodies made of ordinary (reciprocal) materials in second order. Here we extend these considerations to the torque. As one would expect, a spontaneous torque will also appear on an inhomogeneous chiral object if it is out of thermal equilibrium with its environment. Once a chiral body starts to rotate, it will experience a small quantum frictional torque, but much more important, unless a mechanism is provided to maintain the nonequilibrium state, is thermalization: The body will rapidly reach thermal equilibrium with the vacuum, and the angular acceleration will essentially become zero. For a small, or even a large, inhomogeneous chiral body, a terminal angular velocity will result, which seems to be in the realm of observability.

Figures

Figures reproduced from arXiv: 2412.03336 by the authors.

Figure 1
Figure 1. FIG. 1. A reflected L-shaped body (“dual Allen wrench”) composed of a thin wire [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The geometrical factor [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. The dependence the geometric factor [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: FIG. 4. The integral ˆτ [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. The torque on a small Allen wrench, ˆτ [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Time taken (scaled by the prefactor [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. The terminal angular velocity for the object shown in Fig. 1, apart from the prefactor, (4.16), as a function of the [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. The dimensionless integral ˆω [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. A two-dimensional chiral object, consisting of a central metallic wire [PITH_FULL_IMAGE:figures/full_fig_p014_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Reduced geometric integrals [PITH_FULL_IMAGE:figures/full_fig_p015_10.png]

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Works this paper leans on

51 extracted references · 37 canonical work pages

  1. [47]

    M. T. H. Reid, O. D. Miller, A. G. Polimeridis, A. W. Rodriguez, E. M. Tomlinson, and S. G. Johnson, Photon torpedoes and Rytov pinwheels: Integral-equation modeling of non-equilibrium fluctuation-induced forces and torques on nanoparticles, arXiv:1708.01985, unpublished

    arXiv

  2. [48]

    K. A. Milton, N. Pourtolami, and G. Kennedy, Quantum self-propulsion of an inhomogeneous object out of thermal equilibrium, Phys. Rev. A 110, 042814 (2024), [arXiv:2405.15061]

    arXiv 2024

  3. [1]

    E. V. Teodorovich, Contribution of macroscopic van der Waals interactions to frictional force. Proc. R. Soc. Lond. A 362, 71–77 (1978). 18

    work page 1978

  4. [2]

    L. S. Levitov, Van der Waals friction. Europhys. Lett. 8, 499–504 (1989)

    work page 1989

  5. [3]

    J. B. Pendry, Shearing the vacuum—quantum friction, J. Phys. Condens. Matter 9, 10301 (1997)

    work page 1997

  6. [4]

    A. I. Volokitin and B. N. J. Persson, Theory of friction: The contribution from a fluctuating electromagnetic field, J. Phys. Condens. Matter 11, 345 (1999)

    work page 1999

  7. [5]

    G. V. Dedkov and A. A. Kyasov, The relativistic theory of fluctuation electromagnetic interactions of moving neutral particles with a flat surface, Phys. Solid State 45, 1815 (2003)

    work page 2003

  8. [6]

    P. C. W. Davies, Quantum vacuum friction, J. Opt. B 7, S40 (2005)

    work page 2005

Show all 51 references
  1. [7]

    Barton, On van der Waals friction

    G. Barton, On van der Waals friction. II: Between atom and half-space, New J. Phys. 12, 113045 (2010)

    2010

  2. [8]

    R. Zhao, A. Manjavacas, F. J. G. de Abajo, and J. B. Pendry, Rotational quantum friction, Phys. Rev. Lett. 109, 123604 (2012)

    2012

  3. [9]

    M. G. Silveirinha, Theory of quantum friction, New J. Phys. 16, 063011 (2014)

    2014

  4. [10]

    Pieplow and C

    G. Pieplow and C. Henkel, Cherenkov friction on a neutral particle moving parallel to a dielectric, J. Phys. Condens. Matter 27, 214001 (2015)

    2015

  5. [11]

    Intravaia, V

    F. Intravaia, V. E. Mkrtchian, S. Y. Buhmann, S. Scheel, D. A. R. Dalvit, and C. Henkel, Friction forces on atoms after acceleration, J. Phys. Condens. Matter 27, 214020 (2015)

    2015

  6. [12]

    J. S. Høye, I. Brevik and K. A. Milton, The reality of Casimir friction [arXiv:1508.00626], Symmetry 8 (5), 29 (2016), doi:10.3390/sym8050029

    2016 arXiv

  7. [13]

    G. V. Dedkov and A. A. Kyasov, Nonlocal friction forces in the particle-plate and plate-plate configurations: Nonretarded approximation, Surface Sci. 700, 121681 (2020)

    2020

  8. [14]

    M. B. Far ´ ıas, F. C. Lombardo, A. Soba, P. L. Villar, and R. Decca, Towards detecting traces of non-contact quantum friction in the corrections of the accumulated geometric phase, npj Quantum Information 6, 25 (2020). https://doi.org/10.1038/s41534-020-0252-x

    2020 doi

  9. [15]

    Oelschl¨ ager, Fluctuation-induced phenomena in nanophotonic systems, Ph.D

    M. Oelschl¨ ager, Fluctuation-induced phenomena in nanophotonic systems, Ph.D. thesis, Institut f¨ ur Physik, Humboldt Universit¨ at zu Berlin, 2020

    2020

  10. [16]

    G. V. Dedkov, Nonequilibrium Casimir-Lifshitz friction force and anomalous radiation heating of a small particle, Appl. Phys. Lett. 121, 231603 (2022)

    2022

  11. [17]

    T.-B. Wang, Y. Zhou, H.-Q. Mu, K. Shehzad, D.-J. Zhang, W.-X. Liu, T.-B. Yu, and Q.-H. Liao, Enhancement of lateral Casimir force on a rotating body near a hyperbolic material, Nanotechnology 33, 245001 (2022)

    2022

  12. [18]

    Brevik, B

    I. Brevik, B. Shapiro, and M. G. Silveirinha, Fluctuational electrodynamics in and out of equilibrium, Int. J. Mod. Phys. A 37, 2241012 (2022)

    2022

  13. [19]

    Z. Xu, P. Ju, K. Shen, Y. Jin, Z. Jacob, and T. Li, Observation on non-contact Casimir friction, arXiv:2403.06051

    arXiv

  14. [20]

    Einstein and L

    A. Einstein and L. Hopf, Statistische Untersuchung der Bewegung eines Resonators in einem Strahlungsfeld, Ann. Phys. (Leipzig) 338, 1105 (1910)

    1910

  15. [21]

    Mkrtchian, V

    V. Mkrtchian, V. A. Parsegian, R. Podgornik, and W. M. Saslow, Universal thermal radiation drag on neutral objects, Phys. Rev. Lett. 91, 220801 (2003)

    2003

  16. [22]

    G. V. Dedkov and A. A. Kyasov, Tangential force and heating rate of a neutral relativistic particle mediated by equilibrium background radiation, Nucl. Instrum. Methods Phys. Res., Sect. B 268, 599 (2010)

    2010

  17. [23]

    G. Lach, M. DeKieviet, and U. D. Jentschura, Einstein-Hopf drag, Doppler shift of thermal radiation and blackbody drag: Three perspectives on quantum friction, Cent. Eur. J. Phys. 10, 763 (2012)

    2012

  18. [24]

    Pieplow and C

    G. Pieplow and C. Henkel, Fully covariant radiation force on a polarizable particle, New J. Phys. 15, 023027 (2013)

    2013

  19. [25]

    A. I. Volokitin, Friction force at the motion of a small relativistic neutral particle with respect to blackbody radiation, JETP Letters, 101, 427–433 (2015)

    2015

  20. [26]

    A. I. Volokitin and B. N. J. Persson, Electromagnetic Fluctuations at the Nanoscale(Springer, Berlin, 2017)

    2017

  21. [27]

    X. Guo, K. A. Milton, G. Kennedy, W. P. McNulty, N. Pourtolami, and Y. Li, Energetics of quantum vacuum friction: Field fluctuations, Phys. Rev. D 104, 116006 (2021), [arXiv:2108.01539]

    2021 arXiv

  22. [28]

    X. Guo, K. A. Milton, G. Kennedy, W. P. McNulty, N. Pourtolami, and Y. Li, Energetics of quantum vacuum friction. II: Dipole fluctuations and field fluctuations, Phys. Rev. D 106, 016008 (2022), [arXiv:2204.10886]

    2022 arXiv

  23. [29]

    Reiche, F

    D. Reiche, F. Intravaia, J.-T. Hsiang, K. Busch, and B.-L. Hu, Nonequilibrium thermodynamics of quantum friction, Phys. Rev. A 102, 050203(R) (2020)

    2020

  24. [30]

    Khandekar, S

    C. Khandekar, S. Buddhiraju, P. R. Wilkinson, J. K. Gimzewski, A. W. Rodriguez, C. Chase, S. Fan, Nonequilibrium lateral force and torque by thermally excited nonreciprocal surface electromagnetic waves, Phys. Rev. B 104, 245433 (2021)

    2021

  25. [31]

    Gelbwaser-Klimovsky, N

    D. Gelbwaser-Klimovsky, N. Graham, M. Kardar, and M. Kr¨ uger, Near field propulsion forces from nonreciprocal media, Phys. Rev. Lett. 126, 170401 (2021)

    2021

  26. [32]

    Kr¨ uger, T

    M. Kr¨ uger, T. Emig, and M. Kardar, Nonequilibrium electromagnetic fluctuations: Heat transfer and interactions, Phys. Rev. Lett. 106, 210404 (2011)

    2011

  27. [33]

    A. Ott, P. Ben-Abdallah, and S.-A. Biehs, Circular heat and momentum flux radiated by magneto-optical nanoparticles, Phys. Rev. B 97, 205414 (2018)

    2018

  28. [34]

    M. F. Maghrebi, A. V. Gorshkov, and J. D. Sou, Fluctuation-induced torque on a topological insulator out of thermal equilibrium, Phys. Rev. Lett. 123, 055901 (2019)

    2019

  29. [35]

    D. Pan, H. Xu, and F. J. Garc ´ ıa de Abajo, Magnetically activated rotational vacuum friction, Phys. Rev. A 99, 062509 (2019)

    2019

  30. [36]

    Khandekar and Z

    C. Khandekar and Z. Jacob, Thermal spin photonics in the near-field of nonreciprocal media, New J. Phys. 21, 103030 (2019) 19

    2019

  31. [37]

    H. C. Fogedby and A. Imparato, Autonomous quantum rotator, EPL 122, 10006 (2018)

    2018

  32. [38]

    Guo and S

    C. Guo and S. Fan, Theoretical constraints on reciprocal and non-reciprocal many-body radiative heat transfer, [arXiv:2007.13274] Phys. Rev. B 102, 085401 (2020)

    2020 arXiv

  33. [39]

    Guo and S

    Y. Guo and S. Fan, A single gyrotropic particle as a heat engine, [arXiv:2007.11234] ACS Photonics 8, 1623–1629 (2021)

    2021 arXiv

  34. [40]

    Strekha, S

    B. Strekha, S. Molesky, P. Chao, M. Kr¨ uger, and A. Rodriguez, Trace expressions and associated limits for non-equilibrium Casimir torque, [arXiv:2207.13646] Phys. Rev. A 106, 042222 (2022)

    2022 arXiv

  35. [41]

    Ge, Negative vacuum friction in terahertz gain systems, Phys

    L. Ge, Negative vacuum friction in terahertz gain systems, Phys. Rev. B 108, 045406 (2023)

    2023

  36. [42]

    V. S. Asadchy, M. S. Mirmoosa, A. Diaz-Rubio, S. Fan, and S. A. Tretyakov, Tutorial on electromagnetic nonreciprocity and its origins, Proc. IEEE, 108, (10) 1684–1727 (2020) [arXiv:2001.04848]

    2020 arXiv

  37. [43]

    K. A. Milton, X. Guo, G. Kennedy, N. Pourtolami, and D. M. DelCol, Vacuum torque, propulsive forces, and anomalous tangential forces: Effects of nonreciprocal media out of thermal equilibrium, [arXiv:2306.02197], Phys. Rev. A, 108, 022809 (2023), DOI:10.1103/PhysRevA.108.022809

    2023 arXiv

  38. [44]

    Kennedy, Quantum torque on a non-reciprocal body out of thermal equilibriium and induced by a magnetic field of arbitrary strength, Eur

    G. Kennedy, Quantum torque on a non-reciprocal body out of thermal equilibriium and induced by a magnetic field of arbitrary strength, Eur. Phys. J. Spec. Top. 232, 3197 (2023)

    2023

  39. [45]

    L. Zhu, Y. Guo, and S. Fan, Theory of many-body radiative heat transfer without the constraint of reciprocity, Phys. Rev. B 97, 094302 (2018)

    2018

  40. [46]

    M¨ uller and M

    B. M¨ uller and M. Kr¨ uger, Anisotropic particles near surfaces: Propulsive force and friction, Phys. Rev. A 93. 032511 (2016)

    2016

  41. [49]

    J. R. Deop-Ruano, F. J. Garc ´ ıa de Abajo, and A. Manjavacas, Thermal radiation forces on planar structures with asym- metric optical response, Nanophotonics, 2024, https://doi.org/10.1515/nanoph-2024-0121

    2024 doi

  42. [50]

    Milton and J

    K. Milton and J. Schwinger, Classical Electrodynamics, 2nd edition, (CRC Press [Taylor & Francis], Boca Raton, 2024), ISBN: 978-0-367-50207-2

    2024

  43. [51]

    Lambrecht and S

    A. Lambrecht and S. Reynaud, Casimir force between metallic mirrors, Eur. Phys. J. D 8, 309 (2000)

    2000

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.