Pith. sign in

REVIEW 2 major objections 4 minor 2 cited by

Hamiltonian description of nonreciprocal interactions

T0 review · 2 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Nonreciprocal pairwise forces, however asymmetric, can be generated as the constrained dynamics of a reciprocal Hamiltonian.

desk verdict A clean Hamiltonian-embedding construction for a restricted class of nonreciprocal forces, but the advertised generality is not backed by the proofs. read the letter →

arxiv 2505.05246 v5 pith:KQ7QS2VZ submitted 2025-05-08 cond-mat.stat-mech cond-mat.othercond-mat.quant-gascond-mat.softquant-ph

Yu-Bo Shi , Roderich Moessner , Ricard Alert , Marin Bukov This is my paper · ORCID
classification cond-mat.stat-mechcond-mat.othercond-mat.quant-gascond-mat.softquant-ph
keywords nonreciprocalinteractionsHamiltonianembeddingauxiliarydegreesoffreedommirrorconstraintLangevindynamicsMonteCarlosimulationsFloquetengineeringXYspinswithvisioncones
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

This paper claims that any equation of motion whose nonreciprocal forces are pairwise can be rewritten as the constrained dynamics of a fully reciprocal Hamiltonian. The trick is to add one auxiliary degree of freedom per original one, couple the two layers symmetrically, and impose a 'mirror' constraint that locks each pair together. When the constraint holds at the initial time, it is preserved by Hamilton's equations, and each original degree of freedom evolves exactly as in the nonreciprocal system. If true, this gives nonreciprocal systems access to Monte Carlo sampling, canonical transformations, and periodic-drive engineering that were previously reserved for systems with an energy function.

What carries the argument

The central object is the Hamiltonian embedding: a doubled system in which each original degree of freedom $\theta_i$ is paired with an auxiliary 'mirror' degree of freedom $\varphi_i$ that is its canonical conjugate, with a symmetrized interaction built from the one-way potentials $U_{i\to j}$ and $U_{j\to i}$ plus auxiliary couplings that cancel one direction of each bond. The mirror constraint $\theta_i-\varphi_i=\pi$ (or $\mathbf r_i=\mathbf x_i$ with opposite momenta in the inertial case) selects the physical submanifold; the construction works because the constraint surface is invariant under the dynamics, so the nonreciprocal system is the constrained projection of a reciprocal Hamiltonian flow.

What would settle it

Take a pair of particles whose nonreciprocal force has a nonzero curl in the relative coordinate, so no one-way potential of the assumed form exists, and run the proposed constrained Hamiltonian dynamics. If the trajectories differ from the target nonreciprocal equation of motion, or if the mirror constraint is not preserved for all time, the central claim is false.

Watch

Extended reading notes

Core claim

Starting from pairwise nonreciprocal forces $F_{i\to j}(\mathbf r_i-\mathbf r_j)=-\partial_{\mathbf r_i}U_{i\to j}(\mathbf r_i-\mathbf r_j)$, the paper constructs a reciprocal Hamiltonian on a doubled configuration space: the original degrees of freedom plus a mirrored copy, coupled so that the unwanted half of each symmetric interaction is cancelled by the auxiliary layer. Hamilton's equations with the constraint $\theta_i-\varphi_i=\pi$ (for spins, with the general form $\mathbf r_i=\mathbf x_i$ and opposite momenta in the inertial case) reproduce the original nonreciprocal equations of motion exactly, and the constraint is dynamically preserved. The paper further shows that single-spin-flip Monte Carlo updates computed from the constrained Hamiltonian yield the same Fokker-Planck equation as the original Langevin dynamics, so stationary and time-dependent nonequilibrium states coincide, and that the symplectic structure supports a high-frequency expansion for periodic drives.

Load-bearing premise

The construction assumes each nonreciprocal pairwise force can be written as the gradient of a one-way potential $U_{i\to j}(\mathbf r_i-\mathbf r_j)$; forces that are not of this gradient form, such as those with a curl component, are outside its scope.

Editorial extensions

If this is right

  • Constrained single-spin-flip Monte Carlo dynamics built from the embedding reproduces both steady and nonstationary states of the original Langevin dynamics, so Monte Carlo methods can be used on nonreciprocal systems.
  • For vision-cone XY spins, the constrained Monte Carlo and Langevin simulations give the same transition temperature $T_c/J=0.79\pm0.04$, whereas the selfish-energy approach gives a different value.
  • The symplectic structure allows periodic-drive engineering; a drive whose amplitude tunes a Bessel-function zero can suppress interactions in one lattice direction, turning a square lattice into a set of one-dimensional chains.
  • The embedding applies to inertial as well as dissipative pairwise nonreciprocal dynamics, covering systems as varied as Janus colloids, walker robots, robotic metamaterials, feedback-controlled particles, and sedimenting particles.
  • Because the constraint manifold is closed and of measure zero in the embedding phase space, the nonreciprocal dynamics is fully contained inside a Hamiltonian flow and does not affect expectation values defined on the full phase space.

Reading between the lines

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

  • Inference: any invariant of the reciprocal embedding that restricts to the constraint manifold is automatically an invariant of the nonreciprocal dynamics, making the construction a potential source of conserved quantities for nonreciprocal systems.
  • Inference: if the embedding is exact for pairwise forces, the nonreciprocal phase space can be viewed as a constrained submanifold of a Hamiltonian phase space, which suggests that quantizing the embedding might produce a well-defined quantum theory for at least some nonreciprocal models.
  • Inference: applying the same constrained single-spin-flip rule to other nonreciprocal models, such as the soft vision-cone interaction studied in previous work, should reproduce the Langevin steady state for all measured observables; any deviation would pinpoint where the small-proposal-width expansion breaks down.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. This paper proposes a Hamiltonian embedding for nonreciprocal pairwise dynamics. An auxiliary copy of each degree of freedom is introduced, coupled reciprocally to the original variables, and a mirror constraint (e.g., θ_i−φ_i=π or r_i+x_i=0) is imposed. On the constraint manifold, Hamilton's equations are claimed to reproduce the original nonreciprocal equations of motion. The construction is worked out explicitly for vision-cone XY spins and chase-and-run XY spins; constrained Glauber dynamics based on the embedding is shown analytically and numerically to produce the same Fokker-Planck dynamics as the original Langevin equation, including nonstationary oscillatory states. The symplectic structure is then used to derive a Floquet-Magnus effective Hamiltonian for periodically driven nonreciprocal spins, predicting a Bessel-function-controlled dimensional crossover that is checked against simulations. The Supplemental Material contains the general embedding proof for velocity-independent one-way-potential forces, a velocity-dependent extension, and candidate embeddings for five experimental systems.

Significance. Should the construction hold at the advertised level of generality, it would give a principled way to transfer tools of equilibrium and Hamiltonian mechanics—Monte Carlo sampling, canonical transformations, and Floquet engineering—to nonreciprocal systems, provided the constraint submanifold remains invariant. The paper is transparent and detailed: constraint preservation is proved for the spin embeddings in Appendix C, the Glauber-Langevin equivalence is derived from independent Fokker-Planck equations in the Methods and Appendix D, and the numerical checks (critical temperature Tc/J=0.79±0.04 for both Langevin and constrained Glauber dynamics, and matching oscillatory order parameters for chase-and-run dynamics) are direct and convincing. The paper also honestly states the pairwise restriction and the one-way-potential assumption at the end. The main weakness is that the general proof is conditional on a one-way-potential assumption that several of the paper's own examples do not satisfy; this needs to be fixed or the claims need to be restricted.

major comments (2)
  1. [Sec. III and Appendix A 1, Eq. (A4)] The generality claim in Sec. III (a "general proof for arbitrary nonreciprocal interactions") is stronger than what the proof establishes. Appendix A 1 assumes that every velocity-independent nonreciprocal force is the gradient of a one-way potential, F_{i→j}=-∂_{r_i}U_{i→j}(r_i-r_j). The sedimenting-particle force in Eqs. (B32)-(B33) does not satisfy this assumption: for fixed gravity direction the Oseen-tensor force has a nonzero curl, so no scalar one-way potential exists in three dimensions. Yet Eq. (B34) presents a Hamiltonian for this system. Either the theorem should be stated with the one-way-potential condition as an explicit hypothesis, or a separate construction and constraint-preservation proof is needed for non-gradient pairwise forces such as the Oseen case.
  2. [Appendix B 2–B 6] For the particle-based examples the auxiliary equations of motion and constraint-preservation checks are not supplied, so these examples do not yet substantiate the "wide applicability" claim. For example, in the Janus-particle embedding, Eqs. (B6)-(B10), the constraints r_i+x_i=0 and n_i+m_i=0 are imposed after writing the equations for r_i and n_i, but no equations for dx_i/dt and dm_i/dt are given. With the usual choice dx_i/dt=-∂H/∂x_i, the position constraint is not preserved once x_i=-r_i; the residual is proportional to v0 n_i. If an opposite sign convention or a different bracket structure is intended, it must be stated and justified; otherwise the claimed Hamiltonian character of these examples is not established. The same omission affects the self-aligning and feedback-controlled examples in Eqs. (B13) and (B27).
minor comments (4)
  1. [Sec. IV A and Fig. 3] The text says the comparison is between "constrained Glauber dynamics (blue) vs. Langevin dynamics (green)", while the Fig. 3 legend identifies Langevin with red squares; the color assignments should be aligned.
  2. [Sec. V, Eq. (9)] For the periodically driven Hamiltonian H(t) and the effective Floquet Hamiltonian H_F^(0), the paper should state explicitly that the mirror constraint θ_i-φ_i=π is preserved under the time-dependent dynamics; this is a direct check analogous to Appendix C but is not written out.
  3. [Appendix A 2 c, Eq. (A43)] Equation (A43) writes the nonreciprocal force as F_{j→i}=-∂_{r_i}U^NR_{j→i}, but the preceding generalized-potential formalism, Eq. (A27), includes the additional term d/dt (∂_{\dot r_i}U^NR_{j→i}); please clarify whether (A43) is a definition, a special case, or a typographical omission.
  4. [Reference [2]] Reference [2] is cited as "arXiv:2602.11111 (2026)"; the arXiv identifier and date appear inconsistent and should be verified.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Hamiltonian embedding is a direct construction, the Glauber-Langevin equivalence is independently derived, and the only calibration is disclosed and non-load-bearing.

full rationale

The paper's central construction is a direct mathematical theorem, not an empirical prediction. In Sec. III and Appendix A, the Hamiltonian is explicitly written in terms of the given pairwise forces (e.g., Eq. A5), and the claim that the constrained Hamilton equations reduce to the original nonreciprocal equations of motion is verified by direct differentiation and substitution of the constraint. This is a calculation to be checked, not a fitted input dressed as a result. The Glauber-Langevin equivalence (Methods and Appendix D) is derived independently: both stochastic descriptions are shown to produce the same Fokker-Planck operator, with the time conversion t_Lan = (Delta^2)/(12T) obtained from the derivation. The numerical refinement alpha(Delta) ~ 2.37e-3/J used in the nonstationary comparison (Appendix G, Fig. 12) is explicitly disclosed as a period-matching calibration, and the same figure also demonstrates agreement using the analytical, parameter-free rescaling factor; it therefore does not constitute a hidden fit called a prediction. The Floquet claim is checked against direct integration of the driven Langevin equation, providing an external benchmark. No load-bearing self-citation or imported uniqueness theorem is used; the paper even states that the embedding 'does not introduce new physics content per se' (Sec. VI). The main caveat, namely that the general proof assumes one-way-potential forces (Eq. A4) and that the authors concede pairwise-only scope in Sec. VI, affects the breadth of applicability of the method but is not a circularity in the derivation itself.

Assumptions & free parameters 1 free parameters · 3 assumptions · 1 invented entities

The central construction itself introduces no free parameters: the embedding is determined by the given equations of motion. One calibration constant α appears only in the time-axis mapping for nonstationary simulations. The main input assumptions are the one-way-potential representability and the noise assignment on the doubled phase space.

free parameters (1)
  • α(Δ) time-rescaling factor = 2.37e-3 J (numerically fitted; theoretical prediction Δ²/(12T) ≈ 2.60e-3 J)
    Used in the chase-and-run comparison (Fig. 4 and Appendix G) to convert Monte Carlo time to Langevin time by matching the first oscillation period for L=200. This is a disclosed calibration of the time axis, not a parameter of the central construction.
assumptions (3)
  • domain assumption Nonreciprocal pairwise forces are gradients of one-way potentials U_{i→j}(r_i - r_j)
    Eq. (A4) in Appendix A; the entire embedding starts from this representation and the authors note in Sec. VI it may not apply to arbitrary non-potential systems.
  • domain assumption The mirror constraint is preserved when the same white-noise realization is applied to original and auxiliary degrees of freedom in the stochastic setting
    Used in Eq. (D5); the physical status of adding noise to the auxiliary DOF is assumed rather than derived from a bath.
  • standard math Floquet-Magnus expansion truncated at leading order is valid for J_ij/ω ≪ 1
    Invoked in Sec. V, Eq. (10); this is a standard asymptotic expansion.
invented entities (1)
  • Auxiliary degrees of freedom (φ_i spins, x_i positions, w_i displacements) with negative kinetic energy
    purpose: Symmetrize the nonreciprocal interaction and cancel one part of the symmetric coupling via the mirror constraint, recovering original dynamics
    These are mathematical constructs, not physical fields or particles; they are removed by the constraint and do not appear in observables. No falsifiable prediction is attached to them.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Hamiltonian description of nonreciprocal interactions." pith.science (2026). https://pith.science/paper/KQ7QS2VZ

@misc{pith2026250505246,
  author       = {Pith},
  title        = {Pith review of: Hamiltonian description of nonreciprocal interactions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KQ7QS2VZ}},
  note         = {Machine review of arXiv:2505.05246}
}
read the original abstract

In a vast class of systems, which includes members as diverse as sedimenting particles and bird flocks, interactions do not stem from a potential, and are in general nonreciprocal. Thus, it is not possible to define a conventional energy function, nor to use analytical or numerical tools that rely on it. Here, we overcome these limitations by constructing a Hamiltonian that includes auxiliary degrees of freedom; when subject to a constraint, this Hamiltonian yields the original nonreciprocal dynamics. We show that Glauber dynamics based on the constrained Hamiltonian reproduce both stationary and nonstationary states of the original Langevin dynamics, as we explicitly illustrate for dissipative XY spins with vision-cone interactions. Further, the symplectic structure inherent to our construction enables us to apply the well-developed notions of Hamiltonian engineering, which we demonstrate by varying the amplitude of a periodic drive to tune the spin interactions between those of a square and a chain lattice geometry. Overall, our framework for generic nonreciprocal pairwise interactions paves the way for bringing to bear the full conceptual and methodological power of conventional statistical mechanics and Hamiltonian dynamics to nonreciprocal systems.

Figures

Figures reproduced from arXiv: 2505.05246 by the authors.

Figure 1
Figure 1. FIG. 1. Classical systems can be classified as either Hamil [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. (a) Schematic illustration of XY spins with vision-cone interactions on a square lattice: each spin (blue arrow) described [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Nonequilibrium steady state in XY spins with vision [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (11 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Nonstationary state of XY spins with chase-and-run [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. The Hamiltonian embedding enables the use of Flo [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Schematic representation of various models exhibiting nonreciprocal equations of motion. (a) XY spins with vision-cone [PITH_FULL_IMAGE:figures/full_fig_p021_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Average magnetization [PITH_FULL_IMAGE:figures/full_fig_p041_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Average magnetization [PITH_FULL_IMAGE:figures/full_fig_p042_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Average magnetization [PITH_FULL_IMAGE:figures/full_fig_p042_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Snapshots of the spin configuration of nonreciprocal XY spins on a square lattice exhibiting chase-and-run dynamics, [PITH_FULL_IMAGE:figures/full_fig_p043_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. System size scaling of the oscillatory pattern obtained from Langevin dynamics. The real and imaginary parts of [PITH_FULL_IMAGE:figures/full_fig_p045_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12. System size scaling of the oscillatory pattern obtained from constrained Glauber dynamics. The real and imaginary [PITH_FULL_IMAGE:figures/full_fig_p046_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13. Comparison between physical Langevin time and rescaled Monte-Carlo time for different semi-widths ∆ [see legend]. [PITH_FULL_IMAGE:figures/full_fig_p047_13.png]
Figure 14
Figure 14. Figure 14: FIG. 14. Comparison of time evolution obtained from Langevin and constrained Glauber dynamics. The real and imaginary [PITH_FULL_IMAGE:figures/full_fig_p048_14.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Lattice-dependent orientational order in active crystals

    cond-mat.soft 2025-06 conditional novelty 7.0 of 10

    A distance-dependence parameter Ω controls how active particles on a lattice orient, yielding aligned or anti-aligned states, stripes, and frustration, via a mapping to an anisotropic spin model.

  2. Simulating Condensed Matter Physics on Quantum Hardware

    cond-mat.str-el 2026-06 unverdicted novelty 2.0 of 10

    A survey of quantum hardware platforms and methods for simulating condensed matter physics, covering ground states, topology, non-equilibrium dynamics, and the role of noisy devices as prototypes for fault-tolerant si...

Reference graph

Works this paper leans on

103 extracted references · 44 canonical work pages · cited by 2 Pith papers

  1. [1]

    ∂H ∂φi constr. ∆t+ √ 2T∆Wi(t) #

    Langevin steady-state distribution Recall that imposing the constraintθ i(t)−φ i(t) =π, and adding the same noise term for both the original and the auxiliary degrees of freedom results in the overdamped Langevin equations ˙θi ={θ i,H}+ √ 2Tηi(t) =∂ φiH|constr. + √ 2Tηi(t), ˙φi ={φ i,H}+ √ 2Tηi(t) =−∂ θiH|constr. + √ 2Tηi(t),(D5) with conjugate degrees of...

  2. [2]

    We will provide the explicit form of the Hamiltonian for the vision-cone XY spins, and explain how the dynamics naturally emerge from the constraint

    Undamped nonreciprocal systems Inspired by the approach in the previous subsection, we introduce a general procedure to construct the Hamiltonian embedding for undamped systems with nonreciprocal interactions. We will provide the explicit form of the Hamiltonian for the vision-cone XY spins, and explain how the dynamics naturally emerge from the constrain...

  3. [3]

    In particular, we focus on defining transition rates that are consistent with the constraint

    Glauber dynamics description We now present a Glauber algorithm that samples the steady state and the nonstationary distribution of the Langevin dynamics using the Hamiltonian embedding. In particular, we focus on defining transition rates that are consistent with the constraint. When the Hamiltonian is evaluated on the constraint, it vanishes identically...

  4. [4]

    D 2 that we use in the Main Text

    Constrained Glauber algorithm In this subsection, we introduce the details of the constrained Glauber algorithm from Sec. D 2 that we use in the Main Text. The Hamiltonian is introduced in Eq. (D1). We use Ψ (n) to represent the system configuration Ψ (n) = (θ(n) 1 ,θ (n) 2 ,...θ (n) i ,...θ (n) N ,φ (n) 1 ,φ (n) 2 ,...φ (n) i ,...φ (n) N ), wherendenotes...

  5. [5]

    Constrained Glauber algorithm after eliminating the auxiliary spins In Sec. D 2 c, we have shown that the auxiliary spins can be eliminated from the expression for the transition rate, and hence we can simulate the constrained Glauber dynamics using only the original spinsθ i. The system configuration is now Ψ (n) = (θ(n) 1 ,θ 2,...θ (n) N ). The algorith...

  6. [6]

    Finite-size effects In this subsection, we investigate the effects of system size as shown in Fig. 7. The average magnetization and specific heat from both Langevin dynamics and the constrained Glauber dynamics that we propose here exhibit similar behavior across ordered and disordered phases at different system sizes, differing significantly from the res...

  7. [7]

    Glauber dynamics without auxiliary spins To show the key role of the auxiliary spins in our Hamiltonian embedding construction, we also consider the Glauber dynamics for the Hamiltonian without auxiliary DOF HSS =− X ⟨ij⟩ [Jij(θi) +Jji(θj)] cos(θi−θj).(F5) 42 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 Temperature T (J) 0.0 0.2 0.4 0.6 0.8 1.0 Magnetization ⟨m⟩ (a) 0...

  8. [8]

    Soft-vision-cone interactions: Comparison between selfish energy and constrained Glauber dynamics To demonstrate that the agreement between the constrained Glauber dynamics and the Langevin dynamics is not a coincidence, we now adopt the vision-cone interaction proposed in Ref. [49] and apply it to the XY spins on the 43 0 100 200 x-direction 0 100 200y-d...

Show all 103 references
  1. [9]

    Langevin dynamics To enhance the numerical accuracy of the Langevin simulations, we adopt Heun’s method [93]. This method provides second-order accuracy in the time step, i.e.,O(δt 2), which significantly reduces numerical errors compared 45 0.0 0.5 1.0 Re⟨O⟩ L = 80 L = 120 L ...

  2. [10]

    conventional

    Constrained Glauber dynamics The Hamiltonian embedding for the nonreciprocal XY spins from Eq. (G1) reads as Htot =H SS +HSa−Haa, HSS =− X x,y (J→ +J←)[cos(θx,y−θx+1,y) + cos(θx,y−θx−1,y)] +J rec[cos(θx,y−θx,y+1) + cos(θx,y−θx,y−1)], HSa =− X x,y J← cos(θx,y−φx+1,y) +J→ cos(θx...

  3. [11]

    M. J. Bowick, N. Fakhri, M. C. Marchetti, and S. Ra- maswamy, Symmetry, thermodynamics, and topology in active matter, Phys. Rev. X12, 010501 (2022)

    2022

  4. [12]

    Fruchart and V

    M. Fruchart and V. Vitelli, Nonreciprocal many-body physics, arXiv:2602.11111 (2026)

    2026

  5. [13]

    Dzubiella, H

    J. Dzubiella, H. L¨ owen, and C. N. Likos, Depletion forces in nonequilibrium, Phys. Rev. Lett.91, 248301 (2003)

    2003

  6. [14]

    Golestanian, Collective behavior of thermally active colloids, Phys

    R. Golestanian, Collective behavior of thermally active colloids, Phys. Rev. Lett.108, 038303 (2012)

    2012

  7. [15]

    Soto and R

    R. Soto and R. Golestanian, Self-assembly of catalytically active colloidal molecules: Tailoring activity through sur- face chemistry, Phys. Rev. Lett.112, 068301 (2014)

    2014

  8. [16]

    Stark, Artificial chemotaxis of self-phoretic active col- loids: collective behavior, Acc

    H. Stark, Artificial chemotaxis of self-phoretic active col- loids: collective behavior, Acc. Chem. Res.51, 2681 (2018)

    2018

  9. [17]

    Liebchen and H

    B. Liebchen and H. L¨ owen, Which interactions dominate in active colloids?, J. Chem. Phys.150, 061102 (2019)

    2019

  10. [18]

    Agudo-Canalejo and R

    J. Agudo-Canalejo and R. Golestanian, Active phase sep- aration in mixtures of chemically interacting particles, Phys. Rev. Lett.123, 018101 (2019)

    2019

  11. [19]

    Romanczuk, I

    P. Romanczuk, I. D. Couzin, and L. Schimansky-Geier, Collective motion due to individual escape and pursuit response, Phys. Rev. Lett.102, 010602 (2009)

    2009

  12. [20]

    Barberis and F

    L. Barberis and F. Peruani, Large-scale patterns in a min- imal cognitive flocking model: incidental leaders, nematic patterns, and aggregates, Phys. Rev. Lett.117, 248001 (2016)

    2016

  13. [21]

    Durve, A

    M. Durve, A. Saha, and A. Sayeed, Active particle con- densation by non-reciprocal and time-delayed interac- tions, Eur. Phys. J. E41, 1 (2018)

    2018

  14. [22]

    Ramaswamy, Issues in the statistical mechanics of steady sedimentation, Advances in Physics50, 297 (2001)

    S. Ramaswamy, Issues in the statistical mechanics of steady sedimentation, Advances in Physics50, 297 (2001)

    2001

  15. [23]

    M. Smoluchowski, On the practical applicability of stokes’ law of resistance, and the modifications of it re- quired in certain cases, inProceedings of the Fifth Inter- national Congress of Mathematicians (Cambridge, 22–28 August 1912), Vol. 2 (Cambridge University Press, 1913...

    1912

  16. [24]

    M. Smoluchowski, ¨Uber die wechselwirkung von kugeln, die sich in einer z¨ ahen fl¨ ussigkeit bewegen, Bulletin Inter- national de l’Acad´ emie des Sciences de Cracovie, Classe des Sciences Math´ ematiques et Naturelles, S´ erie A , 28 (1911)

    1911

  17. [25]

    T. H. Tan, A. Mietke, J. Li, Y. Chen, H. Higinbotham, P. J. Foster, S. Gokhale, J. Dunkel, and N. Fakhri, Odd dynamics of living chiral crystals, Nature607, 287 (2022)

    2022

  18. [26]

    Schmidt, B

    F. Schmidt, B. Liebchen, H. L¨ owen, and G. Volpe, Light-controlled assembly of active colloidal molecules, J. Chem. Phys.150(2019)

    2019

  19. [27]

    V. Soni, E. S. Bililign, S. Magkiriadou, S. Sacanna, D. Bartolo, M. J. Shelley, and W. T. Irvine, The odd free surface flows of a colloidal chiral fluid, Nat. Phys. 15, 1188 (2019)

    2019

  20. [28]

    C. H. Meredith, P. G. Moerman, J. Groenewold, Y.-J. Chiu, W. K. Kegel, A. van Blaaderen, and L. D. Zarzar, Predator–prey interactions between droplets driven by non-reciprocal oil exchange, Nat. Chem.12, 1136 (2020)

    2020

  21. [29]

    Zhang, R

    J. Zhang, R. Alert, J. Yan, N. S. Wingreen, and S. Granick, Active phase separation by turning towards regions of higher density, Nat. Phys.17, 961 (2021)

    2021

  22. [30]

    C. Wu, J. Dai, X. Li, L. Gao, J. Wang, J. Liu, J. Zheng, X. Zhan, J. Chen, X. Cheng,et al., Ion-exchange enabled synthetic swarm, Nat. Nanotechnol.16, 288 (2021)

    2021

  23. [31]

    R. K. Gupta, R. Kant, H. Soni, A. Sood, and S. Ramaswamy, Active nonreciprocal attraction between motile particles in an elastic medium, Phys. Rev. E105, 064602 (2022)

    2022

  24. [32]

    E. S. Bililign, F. Balboa Usabiaga, Y. A. Ganan, A. Pon- cet, V. Soni, S. Magkiriadou, M. J. Shelley, D. Bartolo, and W. T. Irvine, Motile dislocations knead odd crystals 50 into whorls, Nat. Phys.18, 212 (2022)

    2022

  25. [33]

    Maity and A

    S. Maity and A. Morin, Spontaneous demixing of binary colloidal flocks, Phys. Rev. Lett.131, 178304 (2023)

    2023

  26. [34]

    S. Das, M. Ciarchi, Z. Zhou, J. Yan, J. Zhang, and R. Alert, Flocking by turning away, Phys. Rev. X14, 031008 (2024)

    2024

  27. [35]

    B¨ auerle, A

    T. B¨ auerle, A. Fischer, T. Speck, and C. Bechinger, Self- organization of active particles by quorum sensing rules, Nat. Commun.9, 3232 (2018)

    2018

  28. [36]

    F. A. Lavergne, H. Wendehenne, T. B¨ auerle, and C. Bechinger, Group formation and cohesion of active particles with visual perception–dependent motility, Sci- ence364, 70 (2019)

    2019

  29. [37]

    Brandenbourger, X

    M. Brandenbourger, X. Locsin, E. Lerner, and C. Coulais, Non-reciprocal robotic metamaterials, Nat. Commun.10, 4608 (2019)

    2019

  30. [38]

    Ghatak, M

    A. Ghatak, M. Brandenbourger, J. Van Wezel, and C. Coulais, Observation of non-hermitian topology and its bulk–edge correspondence in an active mechanical metamaterial, Proc. Natl. Acad. Sci.117, 29561 (2020)

    2020

  31. [39]

    Fruchart, R

    M. Fruchart, R. Hanai, P. B. Littlewood, and V. Vitelli, Non-reciprocal phase transitions, Nature592, 363 (2021)

    2021

  32. [40]

    A. V. Ivlev, J. Bartnick, M. Heinen, C.-R. Du, V. Nosenko, and H. L¨ owen, Statistical mechanics where newton’s third law is broken, Phys. Rev. X5, 011035 (2015)

    2015

  33. [41]

    N. P. Kryuchkov, A. V. Ivlev, and S. O. Yurchenko, Dis- sipative phase transitions in systems with nonreciprocal effective interactions, Soft Matter14, 9720 (2018)

    2018

  34. [42]

    Dinelli, J

    A. Dinelli, J. O’Byrne, A. Curatolo, Y. Zhao, P. Sollich, and J. Tailleur, Non-reciprocity across scales in active mixtures, Nat. Commun.14, 7035 (2023)

    2023

  35. [43]

    S. A. Loos, S. H. Klapp, and T. Martynec, Long-range order and directional defect propagation in the nonrecip- rocal xy model with vision cone interactions, Phys. Rev. Lett.130, 198301 (2023)

    2023

  36. [44]

    Y. Avni, M. Fruchart, D. Martin, D. Seara, and V. Vitelli, Nonreciprocal ising model, Phys. Rev. Lett. 134, 117103 (2025)

    2025

  37. [45]

    Huang, M

    Z.-F. Huang, M. t. Vrugt, R. Wittkowski, and H. L¨ owen, Active pattern formation emergent from single-species nonreciprocity, arXiv preprint arXiv:2404.10093 (2024)

    2024

  38. [46]

    V. I. Arnol’d,Mathematical methods of classical mechan- ics, Vol. 60 (Springer Science & Business Media, 2013)

    2013

  39. [47]

    Chan and M

    R. Chan and M. Gul´ acsi, The exact schrieffer–wolff trans- formation, Philos. Mag.84, 1265 (2004)

    2004

  40. [48]

    G. D. Birkhoff,Dynamical systems, Vol. 9 (American Mathematical Soc., 1927)

    1927

  41. [49]

    Gustavson, On constructing formal integrals of a hamiltonian system near an equilibrium point, Astron

    F. Gustavson, On constructing formal integrals of a hamiltonian system near an equilibrium point, Astron. J.71, 670–686 (1966)

    1966

  42. [50]

    J. R. Schrieffer and P. A. Wolff, Relation between the anderson and kondo hamiltonians, Phys. Rev.149, 491 (1966)

    1966

  43. [51]

    We define dissipation by lack of conservation of phase- space volume under the dynamics

  44. [52]

    Landau and K

    D. Landau and K. Binder,A guide to Monte Carlo simu- lations in statistical physics(Cambridge university press, 2021)

    2021

  45. [53]

    L. P. Dadhichi, J. Kethapelli, R. Chajwa, S. Ramaswamy, and A. Maitra, Nonmutual torques and the unimportance of motility for long-range order in two-dimensional flocks, Phys. Rev. E101, 052601 (2020)

    2020

  46. [54]

    Popli, A

    P. Popli, A. Maitra, and S. Ramaswamy, Ordering and defect cloaking in nonreciprocal lattice xy models, Phys. Rev. Lett.135, 088303 (2025)

    2025

  47. [55]

    Dopierala, H

    D. Dopierala, H. Chat´ e, X.-q. Shi, and A. Solon, In- escapable anisotropy of nonreciprocal xy models, Phys. Rev. Lett.135, 088302 (2025)

    2025

  48. [56]

    Bukov, L

    M. Bukov, L. D’Alessio, and A. Polkovnikov, Universal high-frequency behavior of periodically driven systems: from dynamical stabilization to floquet engineering, Adv. Phys.64, 139 (2015)

    2015

  49. [57]

    See Supplemental Material for the details

  50. [58]

    Baconnier, D

    P. Baconnier, D. Shohat, C. H. L´ opez, C. Coulais, V. D´ emery, G. D¨ uring, and O. Dauchot, Selective and collective actuation in active solids, Nat. Phys.18, 1234 (2022)

    2022

  51. [59]

    Rouzaire, D

    Y. Rouzaire, D. J. G. Pearce, I. Pagonabarraga, and D. Levis, Nonreciprocal interactions reshape topological defect annihilation, Phys. Rev. Lett.134, 167101 (2025)

    2025

  52. [60]

    Notice that, after applying the constraint, the vision cones forφ i andθ i point in the same directions, even thoughθ i−φ i =π

  53. [61]

    Baconnier, O

    P. Baconnier, O. Dauchot, V. D´ emery, G. D¨ uring, S. Henkes, C. Huepe, and A. Shee, Self-aligning polar active matter, Rev. Mod. Phys.97, 015007 (2025)

    2025

  54. [62]

    G. A. Weiderpass, M. Sharma, and S. Sethi, Solving the kinetic ising model with nonreciprocity, Phys. Rev. E 111, 024107 (2025)

    2025

  55. [63]

    A. K. Rajeev and A. Kumar, Ising model with non- reciprocal interactions, arXiv:2403.06875 (2024)

    2024 arXiv

  56. [64]

    D. S. Seara, A. Piya, and A. P. Tabatabai, Non-reciprocal interactions spatially propagate fluctuations in a 2d ising model, J. Stat. Mech.2023, 043209 (2023)

    2023

  57. [65]

    K. Blom, U. Thiele, and A. Godec, Local order controls the onset of oscillations in the nonreciprocal ising model, Phys. Rev. E111, 024207 (2025)

    2025

  58. [66]

    Osat and R

    S. Osat and R. Golestanian, Non-reciprocal multifarious self-organization, Nat. Nanotechnol.18, 79 (2023)

    2023

  59. [67]

    R. J. Glauber, Time-dependent statistics of the ising model, J. Math. Phys.4, 294 (1963)

    1963

  60. [68]

    C. R. Galley, Classical mechanics of nonconservative sys- tems, Phys. Rev. Lett.110, 174301 (2013)

    2013

  61. [69]

    Bandini, D

    G. Bandini, D. Venturelli, S. A. Loos, A. Jelic, and A. Gambassi, The xy model with vision cone: non- reciprocal vs. reciprocal interactions, arXiv preprint arXiv:2412.19297 (2024)

    2024 arXiv

  62. [70]

    Goldman and J

    N. Goldman and J. Dalibard, Periodically driven quan- tum systems: effective hamiltonians and engineered gauge fields, Phys. Rev. X4, 031027 (2014)

    2014

  63. [71]

    Eckardt, Colloquium: Atomic quantum gases in pe- riodically driven optical lattices, Rev

    A. Eckardt, Colloquium: Atomic quantum gases in pe- riodically driven optical lattices, Rev. Mod. Phys.89, 011004 (2017)

    2017

  64. [72]

    Newrock, C

    R. Newrock, C. Lobb, U. Geigenm¨ uller, and M. Octavio, The two-dimensional physics of josephson junction ar- rays, inSolid State Physics, Vol. 54 (Elsevier, 2000) pp. 263–512

    2000

  65. [73]

    D. H. Dunlap and V. M. Kenkre, Dynamic localization of a charged particle moving under the influence of an electric field, Phys. Rev. B34, 3625 (1986)

    1986

  66. [74]

    Lignier, C

    H. Lignier, C. Sias, D. Ciampini, Y. Singh, A. Zen- esini, O. Morsch, and E. Arimondo, Dynamical control of matter-wave tunneling in periodic potentials, Phys. Rev. Lett.99, 220403 (2007)

    2007

  67. [75]

    Arimondo, D

    E. Arimondo, D. Ciampini, A. Eckardt, M. Holthaus, and O. Morsch, Kilohertz-driven bose–einstein condensates in 51 optical lattices, inAdv. At. Mol. Opt. Phys., Vol. 61 (El- sevier, 2012) pp. 515–547

    2012

  68. [76]

    In the case of nonreciprocal interactions, the periodic drive leads to weakening and eventually com- plete decoupling along thex-bonds of the lattice

    In the literature on quantum systems [46, 60, 61], this phenomenon is known as dynamical localization, since an external drive is used to suppress the transfer of spin-state population between neighboring lattice sites, controlled by the corresponding spin-spin interaction str...

  69. [77]

    A. J. McRoberts, H. Zhao, R. Moessner, and M. Bukov, Prethermalization in periodically driven nonreciprocal many-body spin systems, Physical Review Research5, 043008 (2023)

    2023

  70. [78]

    Nutku, Hamiltonian structure of the lotka-volterra equations, Phys

    Y. Nutku, Hamiltonian structure of the lotka-volterra equations, Phys. Lett. A145, 27 (1990)

    1990

  71. [79]

    Bateman, On dissipative systems and related varia- tional principles, Phys

    H. Bateman, On dissipative systems and related varia- tional principles, Phys. Rev.38, 815 (1931)

    1931

  72. [80]

    Siegle, I

    P. Siegle, I. Goychuk, P. Talkner, and P. H¨ anggi, Marko- vian embedding of non-markovian superdiffusion, Phys. Rev. E81, 011136 (2010)

    2010

  73. [81]

    G. Jung, M. Hanke, and F. Schmid, Generalized langevin dynamics: construction and numerical integration of non-markovian particle-based models, Soft matter14, 9368 (2018)

    2018

  74. [82]

    T. J. Doerries, S. A. Loos, and S. H. Klapp, Correla- tion functions of non-markovian systems out of equilib- rium: Analytical expressions beyond single-exponential memory, Journal of Statistical Mechanics: Theory and Experiment2021, 033202 (2021)

    2021

  75. [83]

    S. H. Strogatz,Nonlinear dynamics and chaos: with ap- plications to physics, biology, chemistry, and engineering (CRC press, 2018)

    2018

  76. [84]

    Hanai, Nonreciprocal frustration: Time crystalline order-by-disorder phenomenon and a spin-glass-like state, Phys

    R. Hanai, Nonreciprocal frustration: Time crystalline order-by-disorder phenomenon and a spin-glass-like state, Phys. Rev. X14, 011029 (2024)

    2024

  77. [85]

    I. M. Georgescu, S. Ashhab, and F. Nori, Quantum sim- ulation, Rev. Mod. Phys.86, 153 (2014)

    2014

  78. [86]

    McDonald, T

    A. McDonald, T. Pereg-Barnea, and A. Clerk, Phase- dependent chiral transport and effective non-hermitian dynamics in a bosonic kitaev-majorana chain, Physical Review X8, 041031 (2018)

    2018

  79. [87]

    Wang and A

    Y.-X. Wang and A. Clerk, Non-hermitian dynamics with- out dissipation in quantum systems, Physical Review A 99, 063834 (2019)

    2019

  80. [88]

    Zhang and R

    Z. Zhang and R. Garcia-Millan, Entropy production of nonreciprocal interactions, Physical Review Research5, L022033 (2023)

    2023

  81. [89]

    S. A. Loos and S. H. Klapp, Irreversibility, heat and in- formation flows induced by non-reciprocal interactions, New Journal of Physics22, 123051 (2020)

    2020

  82. [90]

    Poncet and D

    A. Poncet and D. Bartolo, When soft crystals defy new- ton’s third law: Nonreciprocal mechanics and dislocation motility, Physical Review Letters128, 048002 (2022)

    2022

  83. [91]

    Guillet, A

    S. Guillet, A. Poncet, M. Le Blay, W. T. Irvine, V. Vitelli, and D. Bartolo, Melting of nonreciprocal solids: How dislocations propel and fission in flowing crystals, Pro- ceedings of the National Academy of Sciences122, e2412993122 (2025)

    2025

  84. [92]

    J. Yan, M. Han, J. Zhang, C. Xu, E. Luijten, and S. Granick, Reconfiguring active particles by electrostatic imbalance, Nature Mater15, 1095 (2016)

    2016

  85. [93]

    Szabo, G

    B. Szabo, G. Sz¨ oll¨ osi, B. G¨ onci, Z. Jur´ anyi, D. Selmeczi, and T. Vicsek, Phase transition in the collective migra- tion of tissue cells: experiment and model, Phys. Rev. E 74, 061908 (2006)

    2006

  86. [94]

    Deseigne, O

    J. Deseigne, O. Dauchot, and H. Chat´ e, Collective motion of vibrated polar disks, Phys. Rev. Lett.105, 098001 (2010)

    2010

  87. [95]

    Henkes, Y

    S. Henkes, Y. Fily, and M. C. Marchetti, Active jamming: Self-propelled soft particles at high density, Phys. Rev. E 84, 040301 (2011)

    2011

  88. [96]

    Okuma, K

    N. Okuma, K. Kawabata, K. Shiozaki, and M. Sato, Topological origin of non-hermitian skin effects, Phys. Rev. Lett.124, 086801 (2020)

    2020

  89. [97]

    Yao and Z

    S. Yao and Z. Wang, Edge states and topological in- variants of non-hermitian systems, Phys. Rev. Lett.121, 086803 (2018)

    2018

  90. [98]

    Tong, Lectures on the quantum hall effect, arXiv preprint arXiv:1606.06687 (2016)

    D. Tong, Lectures on the quantum hall effect, arXiv preprint arXiv:1606.06687 (2016)

    2016 arXiv

  91. [99]

    Berry and P

    M. Berry and P. Shukla, Hamiltonian curl forces, Proc. R. Soc. A471, 20150002 (2015)

    2015

  92. [100]

    Risken,The Fokker-Planck Equation: Methods of So- lution and Applications, second edition, 3rd printing ed., Springer Series in Synergetics No

    H. Risken,The Fokker-Planck Equation: Methods of So- lution and Applications, second edition, 3rd printing ed., Springer Series in Synergetics No. 18 (Springer, Berlin Heidelberg, 1996)

    1996

  93. [101]

    Sekimoto, Langevin equation and thermodynamics, Prog

    K. Sekimoto, Langevin equation and thermodynamics, Prog. Theor. Phys. Suppl.130, 17 (1998)

    1998

  94. [102]

    O. F. Sylju˚ asen and A. W. Sandvik, Quantum monte carlo with directed loops, Phys. Rev. E66, 046701 (2002)

    2002

  95. [103]

    R¨ uemelin, Numerical treatment of stochastic differ- ential equations, SIAM Journal on Numerical Analysis 19, 604 (1982)

    W. R¨ uemelin, Numerical treatment of stochastic differ- ential equations, SIAM Journal on Numerical Analysis 19, 604 (1982)

    1982

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.