Pith. sign in

REVIEW 5 minor 77 references

Hamiltonian flocks: Time-Reversal Symmetry and its consequences

T0 review · 0 major / 5 minor · reviewed 2026-07-13 · grok-4.5

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

Pith's one-line read Hamiltonian flocks obey a generalized time-reversal symmetry that yields a mixed position-polarity fluctuation-dissipation theorem and zero entropy production.

desk verdict Clean derivation of generalized TRS for Hamiltonian flocks: mixed FDT, Onsager-Casimir, and a concrete warning that naïve entropy production is spurious. read the letter →

arxiv 2604.02914 v2 pith:VLRFXGAF submitted 2026-04-03 cond-mat.stat-mech

Mathias Casiulis , Leticia F. Cugliandolo This is my paper · ORCID
classification cond-mat.stat-mech
keywords Hamiltonianflockstime-reversalsymmetryfluctuation-dissipationtheoremOnsager-Casimirreciprocityentropyproductionnon-Galileandynamicspolarliquidsactivematter
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

Hamiltonian flocks are conservative, non-Galilean models of polar liquids that can form moving clusters without any active driving. This paper shows that they still obey a generalized time-reversal symmetry: positions reverse, spins flip sign, and the bath velocity reverses. That symmetry produces a fluctuation-dissipation theorem that mixes position and polarity degrees of freedom, enforces Onsager-Casimir reciprocity rather than ordinary Onsager relations, and implies identically zero entropy production. Using the ordinary time-reversal operation that leaves spins unchanged generates a spurious nonzero entropy-production rate that would be misread as a distance from equilibrium. The same coupling also produces nontrivial long-time angular diffusion and Kramers-like barrier crossings when a global velocity is present. The result supplies a concrete warning for active-matter studies: apparent irreversibility can be an artifact of choosing the wrong reversal, and looking for hidden extensions of time-reversal symmetry may recover equilibrium structure where none was expected.

What carries the argument

The generalized time-reversal map T (r_t → r_−t, θ_t → θ_−t + π so s → −s, v_0 → −v_0) that leaves the Martin-Siggia-Rose-Janssen-De Dominicis action of the Langevin dynamics invariant and thereby generates the mixed FDT and zero entropy production.

What would settle it

Compute the entropy-production rate from the ratio of forward and reverse path probabilities under the spin-flipping map; if the rate is nonzero, or if the mixed position-spin FDT fails in the co-moving frame, the claimed generalized symmetry is false.

Watch

Extended reading notes

Core claim

The model is invariant under the generalized time reversal that sends positions to their time-reversed counterparts, spins to their negatives, and the imposed bath velocity to its negative. That invariance leaves the dynamical action unchanged, yields the connected FDT relating every pair of position and spin correlators and responses, produces Onsager-Casimir reciprocity because spin is odd, and forces the entropy production rate to vanish identically. The naïve reversal that keeps spins unchanged produces a nonzero spurious production rate.

Load-bearing premise

The friction and noise terms obtained from the linear bath-oscillator construction correctly realize the canonical ensemble that also fixes the center-of-mass velocity.

Editorial extensions

If this is right

  • Position mean-square displacements obey the ordinary Einstein relation once displacements are measured in the frame moving at the bath velocity.
  • Long-time angular diffusion is controlled by an effective friction that depends on the spin-velocity coupling K, not the bare rotational friction alone.
  • When a nonzero global velocity is present the spins feel an effective confining potential and exhibit Kramers barrier-crossing dynamics before diffusing.
  • Entropy-production measurements that ignore spin-oddness under time reversal will report a false distance from equilibrium.
  • Any polar system whose Galilean invariance is broken may admit an analogous generalized reversal that restores an equilibrium FDT.

Reading between the lines

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

  • The same construction may allow classes of active polar models to be rewritten as higher-dimensional non-Galilean Hamiltonian systems, giving access to equilibrium sampling tools.
  • Experimental estimates of extractable work in living or synthetic active matter could change once hidden polarity degrees of freedom are restored to the time-reversal map.
  • The overdamped limit with a tachostat still preserves the Boltzmann measure for the angles, so the mixed FDT survives even when inertia is dropped.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 5 minor

Summary. The paper studies Hamiltonian flocks: conservative, non-Galilean polar liquids with spin-velocity coupling K that can exhibit collective motion without activity. After deriving Langevin dynamics (with a tachostat fixing center-of-mass velocity v0) via a Zwanzig-Mori construction, the authors construct the MSRJD action and show it is invariant under a generalized time-reversal map that reverses positions, flips spins (theta to theta+pi), and reverses v0. This symmetry yields mixed fluctuation-dissipation relations Ra,b(tau)=-beta partial_tau C^c_a,b(tau) for a,b in {r,s}, Onsager-Casimir reciprocity (because spin is odd under T), vanishing entropy production under the correct reversal, and a spurious nonzero production rate under the naive spin-preserving reversal. Angular dynamics are shown to be rich, with an effective long-time rotational diffusion constant and Kramers-type confinement when Kv0 is nonzero. Analytic predictions for MSDs/MSADs and overdamped limits are checked against single-particle Langevin simulations.

Significance. If correct, the work supplies a clean equilibrium baseline for systems that look active because they break Galilean invariance. The mixed FDT, Onsager-Casimir relations, and the explicit demonstration that naive TRS produces a spurious entropy production rate are concrete, falsifiable diagnostics that active-matter studies can use when only partial observables are available. Strengths include the fully explicit path-integral construction (Appendices B.2-B.6), the dual Zwanzig-Mori derivations that both collapse to the same white-noise Langevin equations, closed-form MSAD expressions (including the self-consistent DR and the Kramers picture), and direct numerical checks of the FDT and angular diffusion. The cautionary message about mis-diagnosing distance to equilibrium is timely and well-supported by the calculation.

minor comments (5)
  1. Figures 1-3 and 6-7 would benefit from explicit parameter boxes (beta, K, v0, gamma) in every panel; some insets are hard to read at print size.
  2. Notation for the co-moving connected correlator C^c is introduced late; a short definition table or early equation would help readers who jump to the FDT summary.
  3. Appendix C's recursive expansion for DR is useful; a short numerical comparison of D^(0)_R versus the fully self-consistent root for a few kappa values would make the validity range more transparent.
  4. A few typos remain (e.g., 'time-reveralsymmetry', 'na\"ive' spacing, and occasional missing spaces around equations). A light copy-edit pass would clean them.
  5. The discussion of possible mappings of active systems onto higher-dimensional non-Galilean Hamiltonians is intriguing but brief; one or two concrete pointers to existing non-reciprocal-to-Hamiltonian constructions would strengthen the outlook.

Circularity Check

1 steps flagged · score 1.0 of 10

No significant circularity: FDT, Onsager-Casimir, and zero entropy production follow algebraically from an explicitly constructed generalized TRS of the MSRJD action; self-citations supply model context only.

  1. self citation load bearing [Introduction / Sec. 2 (model definition) and Discussion]
    "In Hamiltonian flocks, conservative yet non-Galilean models of polar liquids, previous work reported collective motion without the activity that usually underlies it. ... Hamiltonian flocks have blurred the line between equilibrium systems and active ones [38-41, 43, 44, 73]."

    The paper's premise that the model is equilibrium (canonical measure with tachostat) rests partly on the authors' own prior characterizations. This is ordinary self-citation for model context and is not load-bearing for the new TRS/FDT derivation, which is self-contained in Appendices A-B; it therefore contributes only a minor score increment.

full rationale

The central claims are derived, not assumed or fitted. Appendix A obtains the Langevin equations (3)-(4) from two independent Zwanzig-Mori linear couplings (angles and spins) that both collapse to the same white-noise form; the friction/noise are therefore fixed by the bath construction rather than tuned to force an FDT. Appendix B.2 then exhibits an explicit transformation T (r_t o r_{-t}, heta_t o heta_{-t}+ heta, v_0 o -v_0, with the corresponding response-field shifts) under which both the deterministic and dissipative pieces of the action are invariant and the initial-condition term recovers the tachostat-augmented canonical weight. The mixed FDT R_{a,b}( au)=-eta heta_ au C^c_{a,b}( au) for a,b heta {r,s}, the Onsager-Casimir sign flip for the crossed responses, and the identically vanishing entropy production (B.82) are then algebraic consequences of that invariance; the naïve (spin-preserving) reversal produces a nonzero spurious rate by construction of the incomplete map. Model parameters K, heta, v_0 enter as free inputs of the Hamiltonian, not as quantities adjusted to data. Self-citations ([38-44]) supply earlier equilibrium characterizations of the same non-Galilean model and do not close a logical loop that forces the present TRS or FDT results. The single minor self-citation load is therefore non-load-bearing, yielding score 1.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central claims rest on the Hamiltonian definition of the model (already in the literature), the standard construction of Langevin dynamics from a harmonic bath, the white-noise limit, and the definition of the generalized time-reversal map that flips spins. No new particles or forces are postulated; free parameters are ordinary model coefficients, not fitted constants that force the FDT.

free parameters (3)
  • spin-velocity coupling K
    Dimensionless parameter that breaks Galilean invariance; chosen by hand in simulations (e.g. √βK = 5) but not fitted to force any FDT relation.
  • translational and rotational frictions γt, γr
    Bath damping coefficients appearing in the Langevin equations; set equal for simplicity in most analytics and numerics.
  • imposed center-of-mass velocity v0
    Tachostat parameter that appears in the canonical measure and in the dissipative action; chosen by hand for trajectory illustrations.
assumptions (4)
  • domain assumption Zwanzig-Mori linear coupling of independent harmonic oscillators to particle positions/angles (or spins) yields, after integrating out the bath and taking the white-noise limit, the stated Langevin equations with additive friction and noise that satisfy the fluctuation-dissipation relation at the si
    Appendix A; standard but non-trivial for a non-Galilean system that also requires a tachostat.
  • ad hoc to paper The correct time-reversal map multiplies every spin by −1 (equivalently θ → θ + π) and reverses the bath velocity v0 → −v0.
    Introduced in Sec. 3 and Appendix B.2; justified a posteriori by invariance of the action, but chosen by the authors rather than derived from a more primitive principle.
  • standard math Path-integral (MSRJD / Onsager-Machlup) representation of the Langevin dynamics and the usual rules for extracting response and correlation functions from the action.
    Appendices B.1 and B.4; textbook technique.
  • domain assumption In the co-moving frame the single-time averages become stationary, so connected correlators may be used and time-translation invariance holds.
    Used throughout Appendix B.3 to convert two-time FDT statements into the familiar one-time-difference form.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Hamiltonian flocks: Time-Reversal Symmetry and its consequences." pith.science (2026). https://pith.science/paper/VLRFXGAF

@misc{pith2026260402914,
  author       = {Pith},
  title        = {Pith review of: Hamiltonian flocks: Time-Reversal Symmetry and its consequences},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VLRFXGAF}},
  note         = {Machine review of arXiv:2604.02914}
}
read the original abstract

The fluctuation-dissipation theorem is a hallmark of equilibrium systems that stems from their time-reversal symmetry. In many non-equilibrium systems, in particular active ones, extensions and explicit violations of this theorem are used to assess their ''distance'' to equilibrium. In Hamiltonian flocks, conservative yet non-Galilean models of polar liquids, previous work reported collective motion without the activity that usually underlies it. In this paper, we show that this model obeys a generalized time-reversal symmetry that yields a fluctuation-dissipation theorem that mixes position and polarity degrees of freedom. Due to the oddness of spin under time reversal, the system also obeys Onsager-Casimir reciprocity rather than standard Onsager relations. The coupling also induces rich spin orientation dynamics, including a non-trivial diffusion constant at long times. Finally, we show that considering the na\"ive time-reversal operation rather than the generalized one that leaves the system invariant leads to a spurious entropy production rate, that could be wrongly interpreted as a distance to equilibrium. Our findings suggest looking for possible extensions of time-reversal symmetry in active-looking systems, which may lead to yet unknown generalizations of the fluctuation-dissipation theorem.

Figures

Figures reproduced from arXiv: 2604.02914 by the authors.

Figure 1
Figure 1. Single-particle trajectories. Example trajectories for single particles following the dynamics defined by Eqs. (3) and (4) for γt = γr = γ. In the top row, v0 = 0 with (a) K = 0 (Brownian particle) and (b) √ βK = 5. In the bottom row, v0 = v0eˆx with √ βv0 = 1, and (c) √ βK = 0.5 or (d) √ βK = 10. Insets of the bottom row show the same trajectories in the frame moving at v0. Throughout panels, a gradient of colors r… view at source ↗
Figure 2
Figure 2. Overdamped trajectories. Example trajectories for single particles following the overdamped dynamics defined by Eqs. (9) and (10) for γt = γr = γ. In the top row, v0 = 0 with (a) K = 0 (Brownian particle) and (b) √ βK = 5. In the bottom row, v0 = v0eˆx with √ βv0 = 1, and (c) √ βK = 0.5 or (d) √ βK = 10. Insets of the bottom row show the same trajectories in the frame moving at v0. Throughout panels, a gradient of c… view at source ↗
Figure 3
Figure 3. Undamped and noiseless trajectories. Conservative single-particle trajectories obtained by integrating Eqs. (11) and (12). (a) Galilean particle (K = 0) with finite p and ω. (b) Circular trajectory for K > 0, p = 0, ω > 0. (c) Stable￾oscillation regime of the spin for K > 0, p > 0, and ω < ωc. (d) Winding regime of the spin for K > 0, p > 0, and ω > ωc. Throughout the figure, time flows from dark purple to light pin… view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Einstein-Smoluchowski-Sutherland Relations. [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]
Figure 5
Figure 5. Figure 5: Oscillatory behavior. MSAD against dimensionless time for v0 = 0, analogous to [PITH_FULL_IMAGE:figures/full_fig_p015_5.png]
Figure 6
Figure 6. Figure 6: Diagonal Fluctuation-Dissipation Equations. [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]
Figure 7
Figure 7. Figure 7: Confined angular dynamics as a Kramers problem. [PITH_FULL_IMAGE:figures/full_fig_p017_7.png]
Figure 8
Figure 8. Figure 8: Onsager-Casimir Reciprocity. (a) Integrated response of θ to a force along y, χθy and integrated response of y to a positive torque, χyθ, across a few values of K. (b) Response of sx = cos θ to a force along x, χsxx, and response of x to a magnetic field along x, χxsx …

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

77 extracted references · 1 linked inside Pith

  1. [1]

    swim pressure

    Introduction Play a movie of Brownian particles backwards, and you will not be able to tell the difference with the original movie from any observable. This is because their dynamics are invariant under time reversal – a property shared with all systems in thermodynamic equilibrium. This time-reversal symmetry (TRS) is associated to conservation laws obey...

  2. [2]

    tachostat

    Model We consider a Hamiltonian flock model. ForNparticles ind= 2dimensions, following the dimensionless rewriting of Refs. [38–40] to set masses and moments of inertia to unity, it is described by the Hamiltonian H= NX i=1 p2 i 2 + ω2 i 2 −Kp i·si + 1 2 X k̸=i [U(r ik)−J(r ik)s i·sk],(1) where each particle is characterized by its positionr i ∈R 2, its p...

  3. [3]

    self-interaction

    Discussion We have derived FDT equations that apply to particles following non-Galilean dynamics and thus obeying an extended time-reversal symmetry. We have shown that these FDT equations, in spite of the introduction of a tachostat in non-Galilean dynamics, are very similar to the standard ones, differing only by the need to work in the co- moving frame...

  4. [4]

    thesis The University of Tokyo

    Murashita Y 2021Fluctuation Theorems under Divergent Entropy Production and their Applications for Fundamental Problems in Statistical PhysicsPh.D. thesis The University of Tokyo

  5. [5]

    Fodor É, Nardini C, Cates M E, Tailleur J, Visco P and Van Wijland F 2016 Physical Review Letters117038103

    2016

  6. [6]

    Pigolotti S, Neri I, Roldán É and Jülicher F 2017Physical Review Letters119 140604

  7. [7]

    Mandal D, Klymko K and DeWeese M R 2017Physical Review Letters119258001

  8. [8]

    Nardini C, Fodor E, Tjhung E, van Wijland F, Tailleur J and Cates M E 2017 Physical Review X7021007

    2017

Show all 77 references
  1. [9]

    Shankar S and Marchetti C M 2018Physical Review E98020604(R)

  2. [10]

    Caprini L, Marconi U M B, Puglisi A and Vulpiani A 2019Journal of Statistical Mechanics: Theory and Experiment2019053203

  3. [11]

    Dabelow L, Bo S and Eichhorn R 2019Physical Review X921009

  4. [12]

    Borthne Ø L, Fodor É and Cates M E 2020New Journal of Physics22123012

  5. [13]

    Chiarantoni P, Cagnetta F, Corberi F, Gonnella G and Suma A 2020Journal of Physics A: Mathematical and Theoretical5336LT02

  6. [14]

    Caprini L 2021Journal of Statistical Mechanics: Theory and Experiment2021 063202

  7. [15]

    Caprini L, Puglisi A and Sarracino A 2021Symmetry1381

  8. [16]

    Cengio S D, Levis D and Pagonabarraga I 2021Journal of Statistical Mechanics: Theory and Experiment2021043201

  9. [17]

    Markovich T, Fodor É, Tjhung E and Cates M E 2021Physical Review X11021057

  10. [18]

    Skinner D J and Dunkel J 2021Physical Review Letters127198101

  11. [19]

    RoS,GuoB,ShihA,PhanTV,AustinRH,LevineD,ChaikinPMandMartiniani S 2022Physical Review Letters129220601

  12. [20]

    Caprini L, Marconi U M B, Puglisi A and Löwen H 2023Journal of Chemical Physics159041102

  13. [21]

    Semeraro M, Gonnella G, Suma A and Zamparo M 2023Physical Review Letters 131158302

  14. [22]

    Boffi N M and Vanden-Eijnden E 2024Proceedings of the National Academy of Sciences of the United States of America121e2318106121

  15. [23]

    Agranov T, Jack R L, Cates M E and Fodor É 2025New Journal of Physics27 104602

  16. [24]

    Ferretti F, Giardina I, Grigera T, Pisegna G and Veca M 2025Physical Review Research7L032006

  17. [25]

    Johnsrud M K and Golestanian R 2025Physical Review Research7L032053 REFERENCES64

  18. [26]

    Anand S, Ma X, Guo S, Martiniani S and Cheng X 2024Proceedings of the National Academy of Sciences121e2411608121

  19. [27]

    Knight J, Kaveh F and Pruessner G 2025Arxiv Preprintarxiv:2507.22199

    arXiv

  20. [28]

    Metzger J, Ro S and Tailleur J 2025Arxiv Preprint2503.11902

  21. [29]

    Bore S L, Schindler M, Lam K D N T, Bertin É and Dauchot O 2016Journal of Statistical Mechanics: Theory and Experiment2016033305

  22. [30]

    Casiulis M, Tarzia M, Cugliandolo L F and Dauchot O 2020Journal of Statistical Mechanics: Theory and Experiment2020013209

  23. [31]

    Casiulis M, Tarzia M, Cugliandolo L F and Dauchot O 2020Physical Review Letters 124198001

  24. [32]

    Casiulis M 2019Study of a non-Galilean Hamiltonian liquid: collective motion without activityPhd thesis Sorbonne Université

  25. [33]

    Tasaki H 2020Physical Review Letters125220601

  26. [34]

    Bhattacharya A, Horbach J and Karmakar S 2025Physical Review E111015429

  27. [35]

    Chen L and Davis L K 2026Arxiv Preprint2602.11104

  28. [36]

    Takatori S C, Yan W and Brady J F 2014Physical Review Letters113028103

  29. [37]

    Bissinger T and Fuchs M 2023Journal of Chemical Physics158044902

  30. [38]

    Diu B, Guthmann C, Lederer D and Roulet B 1996Physique Statistique(Hermann)

  31. [39]

    Caravelli F and Vuffray M 2026Europhysics Letters15321003

  32. [40]

    Zwanzig R W 1961 Statistical Mechanics of IrreversibilityLectures in Theoretical Physics, Volume IIIed Brittin W E, Downs B W, Downs J and Boulder Summer institute for theoretical Physics (New York; London: Interscience Publishers) pp 106–141

    1961

  33. [41]

    Mori H 1965Progress of Theoretical Physics33423

  34. [42]

    Martin P C, Siggia E D and Rose H A 1973Physical Review A8423

  35. [43]

    Janssen H K 1976Zeitschrift for Physik B23377

  36. [44]

    De Dominicis C 1976Le Journal de Physique Colloques37C1–247

  37. [45]

    De Dominicis C and Peliti L 1978Physical Review B18353

  38. [46]

    Arnoulx de Pirey T, Cugliandolo L F, Lecomte V and van Wijland F 2022Advances in Physics711

  39. [47]

    Aron C, Biroli G and Cugliandolo L F 2010Journal of Statistical Mechanics: Theory and Experiment2010P11018

  40. [48]

    Aron C, Barci D G, Cugliandolo L F, Arenas Z G and Lozano G S 2016Journal of Statistical Mechanics: Theory and Experiment2016053207

  41. [49]

    Gomez-Solano J R, Petrosyan A, Ciliberto S, Chetrite R and Gawedzki K 2009 Physical Review Letters103040601

    2009

  42. [50]

    Crisanti A, Puglisi A and Villamaina D 2012Physical Review E85061127

  43. [51]

    Onsager L 1931Physical Review37405 REFERENCES65

  44. [52]

    Onsager L 1931Physical Review382266

  45. [53]

    Casimir H B 1945Reviews of Modern Physics17343

  46. [54]

    Cugliandolo L, Kurchan J and Parisi G 1994Journal de Physique I41641

  47. [55]

    Pottier N 2003Physica A: Statistical Mechanics and its Applications317371

  48. [56]

    Fily Y and Marchetti M C 2012Physical Review Letters108235702

  49. [57]

    Festa R and D’Agliano E G 1978Physica A90A229

  50. [58]

    Risken H 1989The Fokker-Planck Equation(Berlin Heidelberg: Springer-Verlag)

  51. [59]

    Benenti G, Casati G, Saito K and Whitney R S 2017Physics Reports6941–124

  52. [60]

    Onsager L and Machlup S 1953Physical Review911505

  53. [61]

    Seifert U 2005Physical Review Letters95040602

  54. [62]

    5: Statistical Physics, Part I3rd ed (Pergamon Press)

    Landau L D and Lifshitz E M 1980Course of Theoretical Physics Vol. 5: Statistical Physics, Part I3rd ed (Pergamon Press)

  55. [63]

    Dieball C and Godec A 2025Journal of Chemical Physics162090901

  56. [64]

    Casiulis M, Tarzia M, Cugliandolo L F and Dauchot O 2019Journal of Chemical Physics150154501

  57. [65]

    Shi Y B, Moessner R, Alert R and Bukov M 2026Nature Physics

  58. [66]

    Ford G W, Kac M and Mazur P 1965Journal of Mathematical Physics6504

  59. [67]

    Bertotti G, Mayergoyz I D and Serpico C 2009Nonlinear Magnetization Dynamics in Nanosystems(Elsevier Science)

  60. [68]

    Cugliandolo L F, Déjardin P M M, Lozano G S and Van Wijland F 2015Physical Review E - Statistical, Nonlinear, and Soft Matter Physics91032139

  61. [69]

    Brown W F J 1963Physical Review51677

  62. [70]

    Aron C, Barci D G, Cugliandolo L F, Arenas Z G and Lozano G S 2014Journal of Statistical Mechanics: Theory and Experiment2014P09008

  63. [71]

    dos Santos G, Romá F, Tranchida J, Castedo S, Cugliandolo L F and Bringa E M 2023Physical Review B108134417

  64. [72]

    Gardiner C 1983Handbook of Stochastic Methods(Berlin: Springer)

  65. [73]

    Sylvester J 1883The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science16267

  66. [74]

    Liverpool T B and Marchetti M C 2003Physical Review Letters90138102

  67. [75]

    Kurzthaler C, Leitmann S and Franosch T 2016Scientific Reports636702

  68. [76]

    Zhang G and Yeomans J M 2023Physical Review Letters13038202

  69. [77]

    Kahn H and Marshall A W 1953Journal of Operations Research Society of America 1263–278

Pith tools

Reviewed July 13, 2026 · model on record in the stance chip above.