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arxiv: 2604.02914 · v1 · submitted 2026-04-03 · ❄️ cond-mat.stat-mech

Hamiltonian flocks: Time-Reversal Symmetry and its consequences

Mathias Casiulis , Leticia F. Cugliandolo This is my paper

Pith reviewed 2026-05-13 18:20 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech
keywords hamiltonian flockstime-reversal symmetryfluctuation-dissipation theoremonsager-casimir relationspolar liquidscollective motionentropy production
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The pith

Hamiltonian flocks obey a generalized time-reversal symmetry that produces a mixed fluctuation-dissipation theorem.

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

The paper studies Hamiltonian flocks, conservative non-Galilean models of polar liquids that display collective motion without the usual driving activity. It demonstrates that these models remain invariant under a generalized time-reversal operation in which polarity reverses sign while positions and velocities follow standard reversal. This symmetry directly yields a fluctuation-dissipation theorem that couples position and polarity variables, together with Onsager-Casimir reciprocity relations arising from the odd parity of polarity. The same symmetry produces non-trivial long-time diffusion for spin orientations and shows that the ordinary time-reversal operation generates a spurious entropy-production signal.

Core claim

Hamiltonian flocks are invariant under a generalized time-reversal symmetry that treats polarity as odd. This invariance produces a fluctuation-dissipation theorem relating position fluctuations to polarity dissipation and vice versa. Because polarity changes sign under the operation, the linear-response relations satisfy Onsager-Casimir rather than standard Onsager reciprocity. The symmetry also governs spin-orientation dynamics, including a non-constant diffusion coefficient at long times. Applying the naive time-reversal operation instead yields a non-zero entropy-production rate that does not reflect any genuine departure from equilibrium.

What carries the argument

The generalized time-reversal symmetry that reverses time, flips velocities, and sends polarity to its negative while leaving the Hamiltonian invariant.

If this is right

  • Position and polarity fluctuations are linked by a single fluctuation-dissipation relation.
  • Linear response obeys Onsager-Casimir reciprocity rather than ordinary Onsager relations.
  • Spin orientations exhibit a non-trivial long-time diffusion constant fixed by the symmetry.
  • Naive application of ordinary time reversal produces a spurious positive entropy-production rate.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar generalized time-reversal symmetries may exist in other conservative active-looking models and could be used to derive new fluctuation-dissipation relations.
  • The exact equilibrium distribution implied by the symmetry may allow closed-form calculations of collective observables without simulation.
  • The non-trivial spin diffusion could be tested in microfluidic realizations of polar rods or in colloidal flocks with conservative interactions.

Load-bearing premise

The specific Hamiltonian for the flocks is unchanged when polarity is reversed together with the standard time-reversal operation on positions and velocities.

What would settle it

A direct numerical check that the predicted cross-correlations between position fluctuations and polarity dissipation exactly satisfy the mixed fluctuation-dissipation relation derived from the generalized symmetry.

Figures

Figures reproduced from arXiv: 2604.02914 by Leticia F. Cugliandolo, Mathias Casiulis.

Figure 1
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 o… view at source ↗
Figure 2
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 gra… view at source ↗
Figure 3
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… view at source ↗
Figure 4
Figure 4. Figure 4: Einstein-Smoluchowski-Sutherland Relations. [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
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] view at source ↗
Figure 6
Figure 6. Figure 6: Diagonal Fluctuation-Dissipation Equations. [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Confined angular dynamics as a Kramers problem. [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗
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 . In the insets, we show the relative error on the responses being opposite. Across this figure, β 1/2v0 = 5. verifies χxy = χyx. Because th… view at source ↗
read the original abstract

The fluctuation-dissipation theorem is a hallmark of equilibrium system that stem 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The paper claims that Hamiltonian flocks—conservative, non-Galilean models of polar liquids—obey a generalized time-reversal symmetry (flipping polarity while preserving the Hamiltonian structure). This symmetry produces a mixed fluctuation-dissipation theorem coupling position and polarity degrees of freedom. Because polarity (spin) is odd under the operation, the system obeys Onsager-Casimir reciprocity rather than standard Onsager relations. The work further derives non-trivial long-time spin diffusion and shows that the naive time-reversal operation yields a spurious entropy-production rate that could be misinterpreted as a distance from equilibrium.

Significance. If the derivation holds, the result supplies a concrete example of how a hidden generalized time-reversal symmetry can restore equilibrium-like relations in an apparently active system. The mixed FDT and Onsager-Casimir relations constitute a falsifiable prediction that could be tested in simulations or experiments on polar liquids. The caution against naive time reversal is a useful methodological point for the broader active-matter community.

major comments (2)
  1. [§3] §3 (definition of generalized TR): the claim that the operation leaves the Hamiltonian invariant is load-bearing for the entire FDT derivation; the explicit transformation of the interaction terms and the non-Galilean kinetic term must be written out to confirm that no additional assumptions are required.
  2. [§4] §4, Eq. (mixed FDT): the relation between the position-polarity cross-correlation and the corresponding response function is stated but the prefactors arising from the odd parity of spin are not derived explicitly; a short calculation showing how the sign flip enters the response function is needed to make the result quantitative.
minor comments (3)
  1. [Abstract] Abstract: 'stem from their time-reversal symmetry' should read 'stems from its time-reversal symmetry' for subject-verb agreement.
  2. [§2] Notation: the symbol for the generalized time-reversal operator is introduced late; defining it once in §2 and using it consistently thereafter would improve readability.
  3. [Figure 2] Figure 2 caption: the long-time diffusion constant is plotted but the fitting procedure (linear regression window) is not stated; adding this detail would allow readers to reproduce the reported value.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their positive assessment and constructive suggestions. We address each major comment below and will revise the manuscript accordingly to improve clarity and rigor.

read point-by-point responses

  1. Referee: [§3] §3 (definition of generalized TR): the claim that the operation leaves the Hamiltonian invariant is load-bearing for the entire FDT derivation; the explicit transformation of the interaction terms and the non-Galilean kinetic term must be written out to confirm that no additional assumptions are required.

    Authors: We agree that an explicit verification strengthens the presentation. In the revised manuscript we will expand §3 with a step-by-step calculation showing how the pairwise interaction terms and the non-Galilean kinetic term transform under the generalized time-reversal operation (polarity reversal with position and momentum unchanged). This will confirm invariance of the full Hamiltonian without extra assumptions. revision: yes

  2. Referee: [§4] §4, Eq. (mixed FDT): the relation between the position-polarity cross-correlation and the corresponding response function is stated but the prefactors arising from the odd parity of spin are not derived explicitly; a short calculation showing how the sign flip enters the response function is needed to make the result quantitative.

    Authors: We thank the referee for this observation. In the revised §4 we will insert a short explicit derivation that starts from the definition of the response function, incorporates the odd parity of the polarity variable under the generalized time reversal, and shows how the resulting minus sign appears in the prefactor relating the position-polarity cross-correlation to the response. This will make the mixed FDT fully quantitative. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation follows from Hamiltonian symmetry

full rationale

The paper's central results on generalized time-reversal symmetry, the mixed position-polarity fluctuation-dissipation theorem, and Onsager-Casimir reciprocity are derived directly from the invariance properties of the stated Hamiltonian under the authors' defined operation (flipping polarity while preserving the conservative non-Galilean structure). No step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation chain; the odd parity of spin is an input property of the model, and the consequences follow logically without renaming known results or smuggling ansatze. The derivation is self-contained against the model's equations.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that the specific Hamiltonian for flocks admits the generalized time-reversal symmetry defined in the paper; no free parameters or new entities are introduced beyond the standard structure of Hamiltonian mechanics for polar liquids.

axioms (1)
  • domain assumption The Hamiltonian flocks model is conservative and invariant under the generalized time-reversal operation that mixes position and polarity while reversing spin oddly.
    This invariance is the load-bearing premise that directly yields the mixed fluctuation-dissipation theorem and Onsager-Casimir relations.

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