Propulsion force and heat transfer for nonreciprocal nanoparticles

Kiryl Asheichyk; Laila Henkes; Matthias Kr\"uger

REVIEW 2 major objections 5 minor 64 references

Propulsion force and heat transfer for nonreciprocal nanoparticles

T0 review · 2 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Two nanoparticles at identical temperature can sustain a persistent heat current when one is optically nonreciprocal and the other anisotropic, a flow the paper derives and bounds by material passivity.

desk verdict Clean derivations, one genuinely new two-body persistent current, and one uncomputed environment balance that is probably fine but should be checked. read the letter →

arxiv 2412.03327 v2 pith:EARKRUEF submitted 2024-12-04 quant-ph

Laila Henkes , Kiryl Asheichyk , Matthias Krüger This is my paper · ORCID

classification quant-ph PACS 44.40.+a

keywords nonreciprocalnanoparticlespersistentheatcurrentradiativetransferCasimirforcespropulsionforcefluctuationalelectrodynamicsmagneto-opticalmaterialspoint-particlelimit

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 asks what nonreciprocal optical response — the kind a magneto-optical material develops in an external magnetic field — does to heat flow and Casimir forces between nanoparticles. Its central claim is a selection rule: split each body's response into a reciprocal '+' and a nonreciprocal '−' part, and the same-sign parts couple in self emission while the opposite-sign parts couple in heat transfer and lateral forces. That rule produces a persistent heat current between two passive particles held at the same temperature, possible only when one particle is nonreciprocal and the other anisotropic, and a propulsion force that can outweigh gravity. It also yields a passivity bound, showing the nonreciprocal contributions can never exceed the reciprocal ones, which caps the efficiency of any heat-engine use of the force. The results matter because they show energy and momentum can be moved between passive bodies with no temperature gradient, using only broken reciprocity.

What carries the argument

The engine of the paper is the trace-orthogonal decomposition $A = A^+ + A^-$ with $A^\pm = \tfrac{1}{2}(A \pm A^{\mathsf T})$, where the transposed operator includes swapping spatial arguments; because the trace of a symmetric times an antisymmetric operator vanishes, every physical quantity becomes a sum of same-sign and mixed-sign products with strict selection rules. In the point-particle limit the scattering operator collapses to $T = \frac{3}{R^3}\frac{\omega^2}{c^2}\,\boldsymbol\alpha$ with the polarizability tensor of Eq. (19), whose imaginary parts encode absorption of reciprocal, anisotropic, and nonreciprocal character; passivity, i.e., $\boldsymbol\alpha_I \ge 0$, then supplies the inequalities that bound the persistent current and propulsion force.

What would settle it

Measure the net heat flow between a magneto-optical nanoparticle and an anisotropic nanoparticle held at the same temperature, reversing the magnetic field at fixed separation: the current must change sign, follow the predicted $r^2 \cos(2\varphi)/d^5$ dependence, and vanish when the separation is parallel to the field or at $\varphi = \pi/4$. A null result in a geometry satisfying the point-particle condition would refute the central claim, as would a full finite-size calculation showing the current changes sign or disappears as the particle radius approaches the skin depth.

Watch

Extended reading notes

Core claim

Splitting every response operator into symmetric and antisymmetric parts reveals a strict coupling rule: self emission contains only same-sign products ($++$ and $--$), while heat transfer and lateral forces contain only mixed-sign products ($\pm\mp$). Applied to point particles, this yields a persistent heat current between two particles at equal temperature, $$H_{1\to 2}(T) = \frac{16 $r^{2}$ \cos(2\varphi)}{\pi $c^{3}$ $d^{5}$} \int_0^\infty d\omega\, \Theta(\omega,T)\, \$omega^{3}$ \left(\mathrm{Im}[\alpha_{1s}]\,\mathrm{Im}[\alpha_{2f}] - \mathrm{Im}[\alpha_{2s}]\,\mathrm{Im}[\alpha_{1f}]\right),$$ which is nonzero only if one particle is nonreciprocal ($\mathrm{Im}[\alpha_f] \neq 0$) and the other is anisotropic ($\mathrm{Im}[\alpha_s] \neq 0$), and which vanishes for identical particles. The current is linear in the magnetic field at small fields, reverses with the field direction, and vanishes when the separation is parallel to the field or at azimuths $\varphi = \pi/4, 3\pi/4$. The same machinery gives a lateral propulsion force on a nonreciprocal particle near a reciprocal plate that scales as $R^3/d^4$, agrees with earlier computations, and can exceed the gravitational force; a Fourier-based argument bounds all mixed-sign contributions by the same-sign ones, a direct consequence of passivity.

Load-bearing premise

The whole analysis assumes each particle is small compared with the thermal wavelength, the skin depth, and the distance to other objects, so its response is a point polarizability; if the particle is not that small, the explicit formulas for the persistent current and propulsion force need finite-size corrections that could change the results.

Editorial extensions

If this is right

A steady heat current flows between two passive bodies at equal temperature, with the environment absorbing and supplying equal power, driven purely by nonreciprocity plus anisotropy.
The propulsion force on a nonreciprocal particle scales as $R^3/d^4$ and can exceed gravity for materials such as n-doped InSb, making the force observable in a plate-particle setup.
The mixed-sign $\pm\mp$ terms — persistent current and lateral force — are bounded by the same-sign $\pm\pm$ emission terms through passivity, which limits the efficiency of this two-body system as a heat engine.
Self emission cannot reveal nonreciprocity when the surroundings are reciprocal: the nonreciprocal part of the particle contributes only at order $B^2$ and is even in the magnetic field.
The persistent current is linear in $B$ for small fields, reverses with the field direction, and its $\cos(2\varphi)$ angular dependence gives a geometric control over the direction of heat flow.

Reading between the lines

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

The $\cos(2\varphi)$ signature suggests a nanoscale thermal router: rotating the magnetic field around the pair should steer the direction of persistent heat flow continuously, a control knob the paper does not propose.
Because the current requires two dissimilar particles, a natural no-go generalization is that no single anisotropic nonreciprocal object can carry persistent current against a reciprocal environment; the paper proves only the two-particle case.
The same $\pm\mp$ selection rules should produce a persistent torque on an anisotropic–nonreciprocal pair, since antisymmetric response couples to angular momentum; the paper cites torques in other geometries but does not compute this one.
The equal-temperature current implies a circulating energy flow through the environment; in a confined geometry this should appear as directional heat flux in the substrate, a measurable signature that could be tested with scanning thermal microscopy.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Beth, here's my read on Henkes et al. The genuinely new thing is the two-body persistent heat current at equal temperatures—Eq. (33). Three-body persistent currents were known (Zhu-Fan), but for two point particles, one nonreciprocal and one anisotropic, the direct particle-to-particle transfer imbalance is new, and the paper gives an explicit formula with a nice angular structure. The self-emission decomposition into ++ and -- parts (Eqs. 8 and 21) is a clean structural contribution, and the force bound in Sec. VII ties to earlier work. The derivations are explicit and well-checked: Eq. (37) reproduces Milton et al. Eq. (8.2), and the reciprocal limits match Asheichyk et al. That is real evidence the machinery is right.

The soft spot is the one you flagged: the energy balance with the environment is asserted, not computed. The paper states that with all temperatures equal the net radiation from any particle vanishes and that the environment must supply compensating exchange equal in magnitude to Eq. (33). That is plausible—it follows from the general structure of fluctuational electrodynamics and the cited vanishing of net radiation—but they don't show the environment-mediated current explicitly, nor that a passive reciprocal background can provide the d^-5 scaling as d shrinks. I don't think this is a fatal flaw; it's a consistency condition that should hold by unitarity, and the direct current is a well-defined observable. But it is the least secure step, and an explicit check, even for one simple case, would close the gap.

Minor: the 'orders of magnitude above gravity' claim rests on a tuned numerical example with dielectric parameters chosen to overlap Im[alpha_f] and temperatures T1=300K, T2=10K, Tenv=0K. It is illustrative, but the abstract states it more flatly than the calculation supports.

The point-particle approximation is standard and acknowledged; the paper gives material parameters showing skin depth >400 nm, so R≈50 nm is consistent.

Overall: a solid theoretical contribution with a genuinely new result, one asserted-but-uncomputed environmental balance, and one slightly over-stated numerical illustration. The right call is to send it to peer review and ask for a check of the environment exchange in revision. I would cite the two-body current if I worked in the area.

Referee Report

2 major / 5 minor

Summary. The manuscript develops a fluctuational-electrodynamics description of heat transfer and lateral (propulsion) forces for nonreciprocal nanoparticles. By splitting polarizability, scattering, and Green's operators into symmetric ('+') and antisymmetric ('−') parts, the authors derive selection rules: self emission contains only ++ and −− terms, heat transfer contains ±∓ terms, and the persistent current at equal temperatures H_{1→2}(T) in Eq. (33) survives only when one particle is anisotropic and the other nonreciprocal. They further compute the lateral force on a nonreciprocal particle near a planar surface, recovering Eq. (8.2) of Ref. [44], and derive a passivity-based bound relating force and heat transfer (Sec. VII). Numerical examples for InSb particles illustrate the distance dependence of self emission and the size of the propulsion force relative to gravity.

Significance. The paper is valuable for its explicit symmetry decomposition, which gives a clear physical picture of how nonreciprocity enters heat transfer and Casimir forces. The derivations are self-contained and reproduce known benchmarks: Eq. (37) matches Ref. [44] and the reciprocal-particle limits match Ref. [52]; the passivity bound in Sec. VII is proven from V_iI≥0 and T_iI≥0. The predicted persistent two-body current, if confirmed by a full energy-balance calculation, is a novel and testable effect. The analytical selection rules are parameter-free and should be useful for designing experiments with magneto-optical nanoparticles.

major comments (2)

[Sec. IV B, Eq. (33) and following paragraph] The physical interpretation of Eq. (33) as a persistent heat current at equal temperature depends on the assertion that the imbalance H_{1→2}(T) is compensated by heat exchange with the environment. The authors state that the environment currents 'must equal in magnitude the result in Eq. (33)' but do not compute them. This is load-bearing, because without this balance Eq. (33) is only a direct pairwise transfer and not a demonstrated steady-state current. I request an explicit calculation of the environment contributions in the same point-particle expansion, or a self-contained proof from the fluctuation-dissipation theorem that the total heat current into each particle vanishes identically when all temperatures are equal. Note also that the required compensation scales as d^{-3} at fixed orientation (since r^2/d^5 = sin^2θ/d^3), so the consistency check should be performed at that order.
[Sec. VI and Fig. 4] The abstract's claim that the propulsion force can be 'orders of magnitude larger than gravitational forces' is based on a single numerical example with optimized plate permittivity parameters C1, ω1, γ1 and with Tenv=0 K. The paper does not state the particle radius used in Fig. 4; although the normalized ratio F/F_g is radius-independent in the near-field limit, the point-particle expansion of Appendix B requires R ≪ d. Please state R, verify that the point-particle conditions are satisfied at d=100 nm, and include a brief sensitivity check or at least a statement that the parameters were chosen to maximize the effect. Without this, the advertised magnitude is not fully substantiated.

minor comments (5)

[Sec. II C] The word 'antiymmetrical' should be 'antisymmetrical'.
[Sec. V] The word 'vanises' should be 'vanishes'.
[Appendix F] The phrase 'with with di being' contains a duplicated 'with' and should read 'with di being'.
[Fig. 2] The plot includes distances down to 0.01 µm while R1=10 nm, so d=R1 at the smallest plotted value; the point-particle condition d≫R is then violated. Please restrict the plotted range to d≫R or add an explicit statement that the asymptotic curves are shown beyond their quantitative domain of validity.
[Sec. IV B] After Eq. (33), the sentence 'Due to the factor r^2, the current also vanishes when d || B' is correct, but it may help to add that at fixed direction θ the prefactor r^2/d^5 scales as sin^2θ/d^3, so the distance scaling of the persistent current is d^{-3} rather than d^{-5}.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the persistent-current and force formulas are derived from general trace identities and checked against independent prior calculations; the uncomputed environment exchange is an acknowledged limitation, not a circular input.

full rationale

The paper's derivation chain is self-contained. Equation (33) follows algebraically from the general trace formula Eq. (16) by substituting the point-particle scattering operators of Sec. III and Appendix B; no parameter is fitted and renamed as a prediction. The reciprocal and plate limits are checked against Refs. [52] and [44] as benchmarks, not used as inputs. The passivity bound in Sec. VII is derived from the positivity of the Fourier-transformed Green's operator in Appendix E. The only self-citation is the statement "With all temperatures equal, net radiation from any particle must vanish [41]," used to argue that the unevaluated environment exchange balances Eq. (33); this is a consistency check rather than a premise of the calculation, and the paper explicitly concedes that this environment exchange is "not computed explicitly here." That acknowledged gap is a completeness or correctness limitation, not circular reasoning. The numerical example's material choice is illustrative and is not used to infer the central structural claims.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The structural derivations depend on standard fluctuational electrodynamics, passivity, and the point-particle approximation; the only hand-fitted inputs enter the illustrative numerical example, not the central claims.

free parameters (2)

Plate permittivity parameters C1, ω1, γ1 = C1=2, ω1=1.15e14 rad/s, γ1=7e10 rad/s
Chosen in Sec. VI A to optimize the overlap with Im[α2f] in the frequency integral, i.e., to maximize the computed propulsion force; these are not determined by the theory.
Illustrative thermal and geometric setup (d, T1, T2, Tenv, B, R1) = d=100 nm, T1=300 K, T2=10 K, Tenv=0 K, B variable (up to 10 T), R1=10 nm
Hand-picked for the figures; the quantitative claim of forces orders of magnitude above gravity is demonstrated for this setup, not for generic conditions.

assumptions (4)

domain assumption The scattering approach to fluctuational electrodynamics (trace formulas for heat transfer and forces) is valid.
The paper builds on Refs. [14-18, 49, 50] and uses Eqs. (2), (5), (35), (36) without re-deriving them.
domain assumption Objects are passive: V_iI >= 0, so H >= 0 and bounds hold.
Stated in Sec. II A and used for the positivity bound in Sec. VII and Appendix E.
domain assumption Point-particle limit with local permittivity and negligible magnetic response.
Appendix B approximates the scattering operator T by the electric polarizability α, assuming the particle is small and nonmagnetic.
domain assumption Permittivity tensor has the block form of Eq. (18) with parameters εp, εd, εs, εf.
This specific material model (magneto-optical or anisotropic, e.g., InSb) is used for all explicit calculations in Secs. IV and VI.

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Pith. "Pith review of Propulsion force and heat transfer for nonreciprocal nanoparticles." pith.science (2026). https://pith.science/paper/EARKRUEF

@misc{pith2026241203327,
  author       = {Pith},
  title        = {Pith review of: Propulsion force and heat transfer for nonreciprocal nanoparticles},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EARKRUEF}},
  note         = {Machine review of arXiv:2412.03327}
}

abstract

We analyze heat transfer and Casimir forces involving a nonreciprocal nanoparticle. By dissecting the resulting expressions into reciprocal and nonreciprocal contributions, we find that the particle's self emission contains $++$ and $--$ terms, i.e., the particle's reciprocal ($+$) and nonreciprocal ($-$) parts couple to the respective parts of its surrounding. In contrast, the heat transfer to the nanoparticle from the surrounding contains $-+$ and $+-$ contributions, which we find to persist at equal temperatures. For two nanoparticles, such persistent transfer is found to require one particle to be nonreciprocal and the other to be anisotropic. The propulsion force for the nanoparticle, for which our results agree with previous work, is dominated by $\pm\mp$ terms, making it distinct from forces found for reciprocal particles. The amplitude of the propulsion force can be orders of magnitude larger than gravitational forces. Despite being distinct, we find the $\pm\mp$ terms to be bound by $\pm\pm$ terms, a consequence of passivity of the objects. For the force, this bound limits the efficiency in a heat engine setup, as observed for parallel plates before.

Figures

Figures reproduced from arXiv: 2412.03327 by the authors.

**Figure 2.** FIG. 2. Two nonreciprocal particles, modelled by [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** FIG. 3. A reciprocal isotropic semi-infinite (occupying the [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: FIG. 4. Propulsion force acting on a small sphere modeled [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Reciprocal particles For reciprocal particles ( αif = 0, αid = αip ≡ αi), only the ++ term in Eq. (21) is finite. The self emission becomes H (2) 2 rec = H (2) 2,vac rec + 4 πd6 Z ∞ 0 dωΘ(ω, T2) Im[α2] × Im α1e2i ω c d 3 − 6i ω c d − 5 ω2 c2 d2 + 2i ω3 c3 d3 + ω4 c4 d4 , (23) where the vacuum part (i.e., without particle 1 present) is [52] H (2) 2,vac rec...

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Pith tools

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

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(21) is finite

Reciprocal particles For reciprocal particles ( αif = 0, αid = αip ≡ αi), only the ++ term in Eq. (21) is finite. The self emission becomes H (2) 2 rec = H (2) 2,vac rec + 4 πd6 Z ∞ 0 dωΘ(ω, T2) Im[α2] × Im α1e2i ω c d 3 − 6i ω c d − 5 ω2 c2 d2 + 2i ω3 c3 d3 + ω4 c4 d4 , (23) where the vacuum part (i.e., without particle 1 present) is [52] H (2) 2,vac rec...

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1), which gives rise to the polarizability tensor in Eq

Nonreciprocal particles: B and d parallel We assume nonreciprocity to occur because of an ex- ternal magnetic field B pointing along the x axis (see Fig. 1), which gives rise to the polarizability tensor in Eq. (19), with αf finite and αs = 0 for both particles. In this case, the self emission depends on the relative angle between B and d, as was also obs...

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We find in this case the same form as Eq

Nonreciprocal particles: B and d perpendicular For the perpendicular alignment, the terms ∼ B2 and ∼ (d × B) · (d × B) contribute. We find in this case the same form as Eq. (26), with the functions g∥ m replaced by g⊥ p (x) = e2ix 1 − 2ix − 3x2 + 2ix3 + x4 , (30a) g⊥ d (x) = e2ix 5 − 10ix − 7x2 + 2ix3 + x4 , (30b) g⊥ f (x) = 4e2ix −1 + 2ix + 2x2 − ix3 . (...

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persistent

Numerical example Figure 2 shows the self emission for parallel and per- pendicular cases using Eqs. (C2) to model the dielectric tensor, corresponding to n-doped InSb particles. As the reference situation for normalization, we choose particle 2 in isolation and with magnetic field strength B = 0, i.e., Eq. (24). Normalized in this way, the self emission ...

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