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arxiv: 2604.18125 · v1 · submitted 2026-04-20 · ❄️ cond-mat.stat-mech

Stability and breakdown of chiral motion in non-reciprocal flocking

Aditya Kumar Dutta , Swarnajit Chatterjee , Matthieu Mangeat , Raja Paul This is my paper

Pith reviewed 2026-05-10 03:41 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech
keywords Vicsek modelnon-reciprocal interactionschiral motiontwo-species flockingcollective motionalignment and anti-alignmentphase transitionsfinite-size effects
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The pith

Chiral collective motion in a two-species non-reciprocal Vicsek model exists only within a narrow window of high density, low speed, and small system size.

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

The paper examines a discrete-time two-species Vicsek model in which particles of one species align with those of the other while the second species anti-aligns, with both species also aligning internally. It asks whether globally coherent chiral motion can arise as a stable large-scale state and finds that, for equal populations and motilities, this state occurs only inside a restricted window of high density, very low self-propulsion speed, and system size small compared with the interaction range. Inside the window, alignment must dominate anti-alignment; stronger anti-alignment instead produces segregation. Population or motility imbalances replace chirality with parallel or anti-parallel flocking, asynchronous oscillations, or clustered segregation of faster particles. The results show that non-reciprocal couplings do not generically produce chiral order but require these specific conditions.

Core claim

For equal populations and motilities, a globally coherent chiral motion state exists only within a restricted window characterized by high density, very low self-propulsion speed, and small system size relative to the interaction range. Within this window, chirality appears primarily when aligning interactions dominate over anti-alignment, whereas stronger anti-alignment leads to species segregation and suppresses chirality. Introducing species asymmetry through population imbalance drives transitions from chiral states to porous parallel-flocking or anti-parallel-flocking liquids; motility imbalance induces asynchronous oscillations and, in extreme cases, leads to segregation into moving 0.

What carries the argument

the discrete-time metric two-species Vicsek model with intra-species alignment and asymmetric inter-species couplings (one species aligns while the other anti-aligns)

If this is right

  • Stronger anti-alignment than alignment produces species segregation and eliminates chirality.
  • Population imbalance replaces the chiral state with porous parallel or anti-parallel flocking liquids.
  • Motility imbalance produces asynchronous oscillations or segregation of faster particles into moving clusters.
  • The chiral state remains stable only inside the window of high density, low speed, and small system size.

Where Pith is reading between the lines

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

  • The explicit dependence on small system size relative to interaction range implies the chiral state may disappear in the thermodynamic limit of infinite system size.
  • Non-reciprocal interactions alone do not guarantee chiral order; the additional requirements on density and speed indicate that chirality is a special rather than generic outcome.
  • The transitions seen under population or motility imbalance suggest that balanced parameters are necessary to prevent segregation from overtaking collective rotation.

Load-bearing premise

The discrete-time metric Vicsek rules with the chosen interaction range and periodic boundaries faithfully capture the large-scale hydrodynamic behavior without finite-size artifacts dominating the observed chiral window.

What would settle it

A simulation at the same high density and low speed but with system size much larger than the interaction range that fails to sustain a stable chiral state would show the reported window is an artifact of small-system effects.

Figures

Figures reproduced from arXiv: 2604.18125 by Aditya Kumar Dutta, Matthieu Mangeat, Raja Paul, Swarnajit Chatterjee.

Figure 1
Figure 1. Figure 1: (a) shows that the steady-state time evolution of the global mean orientations of the two species (cos ¯θs) [Eq. (5)] exhibits clear temporal oscillations for a dense system (ρ = 72) at low motility (v0 = 6 × 10−3 ), low noise (η = 0.02), and sufficiently strong inter-species cou￾pling (µ = 1). Moreover, the two species remain ap￾proximately in quadrature, ¯θA − ¯θB ≃ π/2, which is the hallmark of the chir… view at source ↗
Figure 2
Figure 2. Figure 2: (a-c). At fixed L and v0, increasing the density ρ drives the system into a strongly phase-locked state with Ψ, χ ≃ 1 [ [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: (color online) Stability phase diagrams of the chiral state. (a-c) Phase diagrams indicating the parameter regimes where the chiral state is stable, governed by the combined effects of v0, ρ, and L. (a) v0–ρ diagram at fixed system size L = 16. (b) ρ–L diagram at fixed particle speed v0 = 0.0015. (c) L–v0 diagram at fixed particle density ρ = 128. Solid symbols denote selected points. (d) Mean local neighb… view at source ↗
Figure 4
Figure 4. Figure 4: (color online) Stability diagram in the coupling space (JAB, JBA). Typical configurations of A (red) and B (blue) particles within different regions of the phase diagram are shown on the right. The diagonal line corresponds to JAB = −JBA. Parameters: Jself = 1, ρ = 72, η = 0.02, v0 = 0.006, and L = 10. duce inter-species segregation, hence suppressing the sus￾tained localized interactions required for chir… view at source ↗
Figure 5
Figure 5. Figure 5: (color online) Consequence of population imbalance in NRTSVM. (a–e) Representative snapshots of the observed states along with the corresponding phase diagram. In panels (a–d), particles are color-coded by species type in the top panel and by orientation in the bottom panel. In the orientation snapshot of (c), the white box marks a locally chiral region. Movies (movie2–5) corresponding to these states are … view at source ↗
Figure 6
Figure 6. Figure 6: (color online) Probability distribution of angu￾lar velocity. (a) High evader majority (m0 = −0.8), (b) low pursuer majority (m0 = 0.2), (c) intermediate pursuer ma￾jority (m0 = 0.6) and (d) high pursuer majority (m0 = 0.8). Parameters: ρ = 72, η = 0.02, v0 = 0.0015, L = 16, and µ = 1. promotes majority-species flocking, so the non-reciprocal frustration is no longer able to maintain a system-wide chiral m… view at source ↗
Figure 7
Figure 7. Figure 7: (color online) NRTSVM with fast and slow agents. (a–d) Snapshots of the observed states along with the corresponding phase diagram. In (a–c), particles are color-coded by species type in the top panel and by orientation in the bottom panel. Movies (movie6–8) corresponding to these states are available in Ref. [60]. (d) vA–vB phase diagram for intermediate inter-species interaction strength. The diagonal wh… view at source ↗
Figure 8
Figure 8. Figure 8: (color online) Chirality decay under motility asymmetry. (a, b) Time evolutions of cos ¯θA − ¯θB  for (a) passive pursuer, active evader system (vA = 0, vB = 0.003) [triangle] and (b) active pursuer, passive evader system (vA = 0.003, vB = 0) [square]. Movies (movie9–10) of snapshots corresponding to the time evolutions can be found at Ref. [60]. (c) Largest cluster size time evolution for active species.… view at source ↗
Figure 10
Figure 10. Figure 10: (color online) Chirality versus noise. Steady￾state time evolution of cos ¯θA − ¯θB  on varying noise strength η. The mean value deviates marginally from zero due to finite-size effects. Parameters: ρ = 72, v0 = 0.006, L = 12, and µ = 1. APPENDIX B: CROSSOVER FROM TRANSLATIONAL TO ROTATIONAL DYNAMICS [PITH_FULL_IMAGE:figures/full_fig_p012_10.png] view at source ↗
Figure 12
Figure 12. Figure 12: (color online) Non-reciprocity induced den￾sity fluctuations. ∆n 2 = ⟨n 2 ⟩ − ⟨n⟩ 2 versus average parti￾cle number ⟨n⟩ in a 200 × 200 simulation box for several µ. Parameters: ρ = 6, η = 0.4, v0 = 0.5, and L = 200. APPENDIX E: SUPPRESSION OF GIANT DENSITY FLUCTUATIONS OUTSIDE THE CHIRAL REGIME The Vicsek model and its reciprocal two-species ex￾tension are both known to exhibit giant density fluctua￾tions… view at source ↗
Figure 11
Figure 11. Figure 11: (color online) Representative steady states outside the global chiral regime. (a) Initial state with uniform spatial and orientational distribution of both species; parameters correspond to (d). (b–d) Breakdown of global chi￾rality at steady state under (b) low particle density ρ, (c) high propulsion speed v0, and (d) large system size L. Particles are color-coded by species and by orientation in the inse… view at source ↗
read the original abstract

We study a two-species Vicsek model with intra-species alignment and asymmetric inter-species couplings, where one species aligns with the other while the latter anti-aligns. Motivated by recent results showing that globally coherent chiral motion is not a generic large-scale state of finite-range non-reciprocal flocking, we ask whether a chiral state can nevertheless be stabilized in the discrete-time, metric, non-reciprocal two-species Vicsek model, and if so, under what conditions. For equal populations and motilities, we show that such a state exists only within a restricted window characterized by high density, very low self-propulsion speed, and small system size relative to the interaction range. Within this window, we also find that chirality appears primarily when aligning interactions dominate over anti-alignment, whereas stronger anti-alignment leads to species segregation and suppresses chirality. Conversely, introducing species asymmetry through population imbalance drives transitions from chiral states to porous parallel-flocking or anti-parallel-flocking liquids; motility imbalance induces asynchronous oscillations and, in extreme cases, leads to segregation into moving clusters of the faster species within a more dispersed background of slower particles. Overall, these results indicate that chirality in the non-reciprocal two-species Vicsek model arises within a restricted regime set by density, motility, inter-species coupling, and system size, rather than being a generic outcome of non-reciprocal interactions.

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 / 1 minor

Summary. The manuscript studies a two-species metric Vicsek model with intra-species alignment and asymmetric inter-species couplings (one species aligns, the other anti-aligns). Using discrete-time simulations with periodic boundaries, it reports that globally coherent chiral motion for equal populations and motilities exists only inside a narrow window of high density, very low self-propulsion speed, and small system size relative to the interaction range, with chirality favored when alignment dominates anti-alignment. Population or motility imbalance instead produces segregation, porous parallel/anti-parallel flocks, or asynchronous oscillations.

Significance. If the restricted chiral window survives finite-size extrapolation and statistical validation, the work would usefully demonstrate that non-reciprocal flocking does not generically produce large-scale chirality, supplying concrete parameter boundaries that can guide hydrodynamic modeling and experiments. The numerical mapping of density, motility, and coupling effects is a concrete contribution, though the purely observational character and absence of analytic derivations limit its reach beyond the specific discrete model.

major comments (2)
  1. [Abstract and Results] Abstract and main results: the claim that chiral motion exists only for small system size relative to interaction range (L/ξ) is load-bearing for the restricted-window conclusion, yet no finite-size scaling study, L→∞ extrapolation, or comparison with continuum hydrodynamic limits is presented; the observed chirality could therefore be an artifact of the chosen discrete-time metric rules and periodic boundaries rather than an intrinsic feature.
  2. [Methods and Results] Simulation protocol: the reported phase boundaries and chiral states lack error bars, details on the number of independent runs, time-averaging windows, or convergence tests with respect to integration timestep and noise realization, making it impossible to assess the statistical robustness of the narrow window.
minor comments (1)
  1. [Abstract] The abstract would be clearer if it briefly stated the precise form of the inter-species coupling terms and the metric interaction rule used.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We agree that additional statistical details and finite-size analysis would strengthen the manuscript and will revise accordingly to address the major points below.

read point-by-point responses

  1. Referee: [Abstract and Results] Abstract and main results: the claim that chiral motion exists only for small system size relative to interaction range (L/ξ) is load-bearing for the restricted-window conclusion, yet no finite-size scaling study, L→∞ extrapolation, or comparison with continuum hydrodynamic limits is presented; the observed chirality could therefore be an artifact of the chosen discrete-time metric rules and periodic boundaries rather than an intrinsic feature.

    Authors: We acknowledge that the manuscript does not include a systematic finite-size scaling analysis or L→∞ extrapolation, nor a direct comparison to continuum hydrodynamics. Our simulations for equal populations and motilities show that global chirality is stable only when L/ξ remains small (typically L/ξ ≲ 10 for the parameters studied), with larger systems exhibiting segregation or parallel flocking instead; this trend is visible in the data we have, but we did not quantify the scaling of the chiral window boundary with L. We will add a dedicated subsection on finite-size effects, including new simulation runs at larger L/ξ to illustrate the breakdown, and will explicitly note the absence of thermodynamic-limit extrapolation as a limitation of the present discrete-time study. This revision will make the restricted-window claim more precise without overstating its generality. revision: yes

  2. Referee: [Methods and Results] Simulation protocol: the reported phase boundaries and chiral states lack error bars, details on the number of independent runs, time-averaging windows, or convergence tests with respect to integration timestep and noise realization, making it impossible to assess the statistical robustness of the narrow window.

    Authors: We agree that the current manuscript omits these statistical details. In the revised version we will add error bars on all phase-boundary and order-parameter plots, specify the number of independent runs (typically 10–20 per parameter set), describe the time-averaging windows used after equilibration, and report convergence checks with respect to timestep and noise realizations. These additions will allow readers to evaluate the robustness of the narrow chiral window. revision: yes

Circularity Check

0 steps flagged

No circularity: results are direct numerical observations from model simulations

full rationale

The paper presents findings exclusively from discrete-time metric Vicsek simulations with specified interaction rules, periodic boundaries, and parameter sweeps. No analytical derivation, fitted functional form, or self-referential equation chain is used to obtain the restricted chiral window; the claims follow directly from running the model and observing outcomes under varying density, speed, system size, and coupling strengths. Self-citations appear only as motivation for the question and do not bear the load of the central results. The derivation chain is therefore self-contained empirical observation rather than any reduction to inputs by construction.

Axiom & Free-Parameter Ledger

4 free parameters · 2 axioms · 0 invented entities

The central claim rests on numerical exploration of the discrete-time metric two-species Vicsek model; no new physical entities are introduced and the only free parameters are the tunable simulation controls.

free parameters (4)
  • particle density
    Tuned to high values to observe the chiral window
  • self-propulsion speed
    Set to very low values for stability of chirality
  • system size relative to interaction range
    Kept small to satisfy the restricted regime
  • inter-species coupling asymmetry
    Varied between aligning and anti-aligning strengths
axioms (2)
  • domain assumption Intra-species alignment follows the standard Vicsek rule
    Core assumption of the flocking model
  • domain assumption Inter-species couplings are strictly asymmetric (align vs anti-align)
    Definition of the non-reciprocal two-species model

pith-pipeline@v0.9.0 · 5567 in / 1491 out tokens · 33042 ms · 2026-05-10T03:41:51.034526+00:00 · methodology

discussion (0)

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