Mesoscopic theory of flocking with alignment and anti-alignment copying

Chunming Zheng

arxiv: 2604.20251 · v1 · submitted 2026-04-22 · ❄️ cond-mat.stat-mech · cond-mat.soft· physics.bio-ph

Mesoscopic theory of flocking with alignment and anti-alignment copying

Chunming Zheng This is my paper

Pith reviewed 2026-05-09 23:50 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech cond-mat.softphysics.bio-ph

keywords flockingcollective motionalignmentanti-alignmentmesoscopic theorypolarizationfinite-size effectsstochastic dynamics

0 comments

The pith

Competing alignment and anti-alignment copying rules suppress long-range polar order in flocking models in the thermodynamic limit.

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

The paper examines a model where moving particles update their directions by copying from neighbors, with each interaction being either alignment or anti-alignment. Using a Fourier expansion of the master equation and a large particle number limit, it derives effective equations governing the overall polarization of the group. The central finding is that any mixture of the two interaction types prevents the emergence of stable, long-range order when the number of particles becomes very large. Finite groups, however, develop structured fluctuations whose details depend on the relative strength of alignment versus anti-alignment. This provides an analytically solvable description of how competing social rules affect collective motion.

Core claim

Starting from the microscopic stochastic process on the circle, where each particle selects a partner and adopts its orientation or the opposite with given probabilities, the analysis produces closed Fokker-Planck and stochastic differential equations for the polarization. In both annealed dynamics (interaction type chosen anew each time) and quenched dynamics (types fixed to particles), the competition between alignment and anti-alignment eliminates spontaneous long-range polar order in the infinite-size limit. Finite systems instead exhibit nontrivial, noise-driven polarization statistics controlled by the composition of the two interaction types.

What carries the argument

Fourier-mode expansion of the master equation combined with large-N truncation to obtain effective stochastic dynamics for the polarization under competing copying rules.

If this is right

Long-range polar order is suppressed in the thermodynamic limit for any positive fraction of anti-alignment interactions.
Finite systems show fluctuation-induced structure whose properties vary with the balance of aligning and anti-aligning agents.
The derived mesoscopic equations match direct simulations for both annealed and quenched interaction assignments.
Intrinsic noise plays a central role in generating observable patterns when global order is absent.

Where Pith is reading between the lines

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

Similar competing rules might explain the absence of large-scale coordination in systems with mixed cooperative and competitive behaviors, such as certain animal groups or social networks.
The approach could be extended to study other collective phenomena where individuals copy traits with opposing effects.
Experiments with controllable agent systems could test the predicted dependence of fluctuation statistics on system size and interaction composition.

Load-bearing premise

The Fourier-mode truncation and large-N expansion capture the essential dynamics even under strong competition between alignment and anti-alignment without missing important correlations.

What would settle it

If simulations with increasing particle numbers show persistent nonzero polarization even with mixed interactions, or if the predicted distributions from the effective equations deviate from microscopic simulations, the suppression claim would be falsified.

Figures

Figures reproduced from arXiv: 2604.20251 by Chunming Zheng.

**Figure 2.** Figure 2: FIG. 2. Distribution of polarization magnitudes and station [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Figure 3: , where the probability distributions P(m) for both dynamics remain nearly indistinguishable even for a moderate system size of N = 50. The excellent agreement between the microscopic histograms and the mesoscopic theory (both quenched and annealed) validates the perturbative argument that the frozen disorder only subtly modifies the noise statistics of the total polar mode. In contrast, the nematic mode… view at source ↗

read the original abstract

We study a stochastic model of collective motion in which individuals update their orientation through pairwise aligning or anti-aligning copying interactions. We analyze both annealed dynamics, where interaction types are chosen probabilistically at each update, and quenched dynamics, where individuals are permanently assigned to aligning or anti-aligning subpopulations. Starting from the microscopic master equation on the circle, we derive an exact mesoscopic description via a Fourier-mode expansion and a systematic large $N$ expansion, obtaining closed Fokker-Planck equations and effective stochastic differential equations for the polarization. We show that competing alignment and anti-alignment suppress long-range polar order in the thermodynamic limit in both cases, while finite systems display nontrivial fluctuation-induced structure controlled by the interaction composition. Our results, validated by Gillespie simulations, establish an analytically tractable framework for collective dynamics characterized by competing copying rules and intrinsic noise.

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.

Desk Editor's Note private letter to a colleague

This derives closed equations for polarization under competing alignment and anti-alignment copying and finds order suppression at large N, but the Fourier-large-N closure needs checking when the rules balance.

read the letter

The main point is that the paper starts from a microscopic master equation for orientations on the circle and uses a Fourier-mode expansion plus large-N limit to close the dynamics into Fokker-Planck and effective SDE equations for the polarization. It treats both annealed (probabilistic choice of rule at each step) and quenched (fixed subpopulations) cases, then shows that mixing alignment with anti-alignment kills long-range polar order in the thermodynamic limit while finite systems retain some fluctuation-driven structure set by the mix ratio. Gillespie simulations are used to check the analytic results. That combination of competing copying rules with an explicit closure is not standard in the flocking literature cited in the abstract, so the derivation itself is the concrete advance. The approach is systematic and gives analytic expressions rather than relying solely on numerics, which is useful for this class of models. The soft spot is the regime where alignment and anti-alignment compete strongly. There the steady-state distribution can spread or turn multimodal, so the truncation to the first Fourier mode and the implicit neglect of higher correlations may lose control. The abstract calls the equations exact after the expansion, yet supplies no explicit error bounds or convergence checks across competition strength. If the simulations stay in the mild-competition region, the reported suppression could be narrower than claimed. This is aimed at researchers who build mesoscopic theories for active matter or biological swarms and want closed equations that include intrinsic noise and mixed interactions. A reader who works with master-equation derivations or needs tractable models for collective motion with anti-alignment will find usable steps here. The work shows clear engagement with the starting stochastic process and the literature on flocking, so it deserves a serious referee. I would send it to peer review with the request that the authors add direct comparisons of analytic polarization variance to microscopic runs at increasing competition strength and state the range where the low-mode closure remains reliable.

Referee Report

2 major / 1 minor

Summary. The paper studies a stochastic model of collective motion on the circle where agents update orientations via pairwise aligning or anti-aligning copying. It considers annealed dynamics (probabilistic choice of interaction type) and quenched dynamics (fixed subpopulations). Starting from the microscopic master equation, the authors perform a Fourier-mode expansion followed by a systematic large-N expansion to obtain closed Fokker-Planck equations and effective SDEs for the polarization. They claim that competing alignment and anti-alignment suppress long-range polar order in the thermodynamic limit for both cases, while finite-N systems exhibit nontrivial fluctuation-induced structures controlled by the interaction composition; results are validated against Gillespie simulations.

Significance. If the central derivations and closure hold under competition, the work supplies an analytically tractable mesoscopic framework for collective dynamics with mixed copying rules, clarifying how intrinsic noise and interaction balance suppress order. The systematic expansion from the master equation and direct simulation validation are strengths that could enable extensions to other active-matter models with competing interactions.

major comments (2)

[Abstract] Abstract: the claim of an 'exact' mesoscopic description after the Fourier-mode and large-N expansions lacks an explicit error bound or convergence criterion. When alignment and anti-alignment compete strongly the steady-state orientation distribution can broaden or become multimodal, rendering the low-mode truncation and implicit neglect of higher-order correlations uncontrolled; this directly affects the central claim of order suppression in the N→∞ limit.
[Abstract] Validation (Gillespie simulations referenced in abstract): no quantitative comparisons are supplied between the analytic polarization variance (or order parameter) and microscopic runs as the competition strength is varied toward balance. Without such checks at increasing competition, it remains possible that the reported suppression is an artifact of the mesoscopic closure rather than a property of the underlying stochastic process.

minor comments (1)

The notation for the annealed versus quenched interaction probabilities could be introduced with a short table or explicit parameter definitions early in the model section to improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major comments point by point below.

read point-by-point responses

Referee: [Abstract] Abstract: the claim of an 'exact' mesoscopic description after the Fourier-mode and large-N expansions lacks an explicit error bound or convergence criterion. When alignment and anti-alignment compete strongly the steady-state orientation distribution can broaden or become multimodal, rendering the low-mode truncation and implicit neglect of higher-order correlations uncontrolled; this directly affects the central claim of order suppression in the N→∞ limit.

Authors: The Fourier-mode expansion of the master equation is formally exact, as the Fourier basis is complete for functions on the circle. The subsequent large-N expansion is systematic and yields the leading-order closed equations for the polarization. We agree that we do not supply explicit error bounds or a convergence criterion, and that strong competition can broaden the orientation distribution. However, in the thermodynamic limit the effective SDE for the polarization has a vanishing deterministic drift under competition, implying suppression of long-range order irrespective of higher-mode details. Near the disordered state the low-mode truncation is in fact better controlled. We will revise the abstract to replace 'exact' with 'leading-order large-N' and add a short discussion of the approximation's validity range. revision: partial
Referee: [Abstract] Validation (Gillespie simulations referenced in abstract): no quantitative comparisons are supplied between the analytic polarization variance (or order parameter) and microscopic runs as the competition strength is varied toward balance. Without such checks at increasing competition, it remains possible that the reported suppression is an artifact of the mesoscopic closure rather than a property of the underlying stochastic process.

Authors: The manuscript shows qualitative agreement with Gillespie simulations at selected parameter values, but we did not include a systematic quantitative comparison (e.g., error metrics or variance versus competition ratio) across the full range, including near balance. We have performed additional runs and will add a figure or table that directly compares the analytic steady-state polarization variance with simulation data as the aligning/anti-aligning fraction is varied. This will confirm that order suppression is a property of the microscopic dynamics. revision: yes

Circularity Check

0 steps flagged

Derivation from master equation via systematic expansions is self-contained

full rationale

The paper begins from the microscopic master equation for orientation copying on the circle and applies a Fourier-mode expansion followed by a systematic large-N expansion to obtain closed Fokker-Planck and SDE descriptions for the polarization. No parameters are fitted to data and then relabeled as predictions, no load-bearing steps rely on self-citations, and the suppression of long-range order emerges directly from the derived equations rather than being presupposed by construction. The procedure is a standard controlled approximation whose validity can be checked externally via simulation; it does not reduce the final results to the inputs by definition.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed from abstract only; full derivations, assumptions, and any fitted quantities are not visible.

axioms (1)

domain assumption Microscopic master equation on the circle governs the stochastic copying dynamics
Explicit starting point stated in the abstract for the Fourier-mode expansion

pith-pipeline@v0.9.0 · 5439 in / 1243 out tokens · 78134 ms · 2026-05-09T23:50:28.962768+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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J. Calvert, A. W. Richa, and D. Randall, How many more is different?, arXiv preprint arXiv:2510.06011 (2025)

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[1] [1]

Vicsek, A

T. Vicsek, A. Czir´ ok, E. Ben-Jacob, I. Cohen, and O. Shochet, Novel type of phase transition in a system of self-driven particles, Physical review letters75, 1226 (1995)

work page 1995

[2] [2]

Toner and Y

J. Toner and Y. Tu, Long-range order in a two- dimensional dynamical xy model: how birds fly together, Physical review letters75, 4326 (1995)

work page 1995

[3] [3]

I. D. Couzin, J. Krause, R. James, G. D. Ruxton, and N. R. Franks, Collective memory and spatial sorting in animal groups, Journal of theoretical biology218, 1 (2002)

work page 2002

[4] [4]

Bertin, M

E. Bertin, M. Droz, and G. Gr´ egoire, Boltzmann and hy- drodynamic description for self-propelled particles, Phys- ical Review E—Statistical, Nonlinear, and Soft Matter Physics74, 022101 (2006)

work page 2006

[5] [5]

C. Buhl, D. J. Sumpter, I. D. Couzin, J. J. Hale, E. Des- pland, E. R. Miller, and S. J. Simpson, From disorder to order in marching locusts, Science312, 1402 (2006)

work page 2006

[6] [6]

Ballerini, N

M. Ballerini, N. Cabibbo, R. Candelier, A. Cavagna, E. Cisbani, I. Giardina, V. Lecomte, A. Orlandi, G. Parisi, A. Procaccini,et al., Interaction ruling an- imal collective behavior depends on topological rather than metric distance: Evidence from a field study, Pro- ceedings of the national academy of sciences105, 1232 (2008)

work page 2008

[7] [7]

Cavagna, A

A. Cavagna, A. Cimarelli, I. Giardina, G. Parisi, R. San- tagati, F. Stefanini, and M. Viale, Scale-free correlations in starling flocks, Proceedings of the National Academy of Sciences107, 11865 (2010)

work page 2010

[8] [8]

A. P. Solon and J. Tailleur, Revisiting the flocking tran- sition using active spins, Physical review letters111, 078101 (2013)

work page 2013

[9] [9]

Feinerman, I

O. Feinerman, I. Pinkoviezky, A. Gelblum, E. Fonio, and N. S. Gov, The physics of cooperative transport in groups of ants, Nature Physics14, 683 (2018)

work page 2018

[10] [10]

Biancalani, L

T. Biancalani, L. Dyson, and A. J. McKane, Noise- induced bistable states and their mean switching time in foraging colonies, Physical review letters112, 038101 (2014)

work page 2014

[11] [11]

Dyson, C

L. Dyson, C. A. Yates, C. Buhl, and A. J. McKane, Onset of collective motion in locusts is captured by a minimal model, Physical Review E92, 052708 (2015)

work page 2015

[12] [12]

T. M. Liggett,Stochastic Interacting Systems: Contact, Voter and Exclusion Processes, Vol. 324 (Springer-Verlag Berlin Heidelberg, 1999)

work page 1999

[13] [13]

Castellano, S

C. Castellano, S. Fortunato, and V. Loreto, Statistical physics of social dynamics, Reviews of modern physics 81, 591 (2009)

work page 2009

[14] [14]

Jhawar, R

J. Jhawar, R. G. Morris, U. Amith-Kumar, M. Danny Raj, T. Rogers, H. Rajendran, and V. Guttal, Noise-induced schooling of fish, Nature Physics16, 488 (2020)

work page 2020

[15] [15]

Großmann, P

R. Großmann, P. Romanczuk, M. B¨ ar, and L. Schimansky-Geier, Vortex arrays and mesoscale turbulence of self-propelled particles, Physical review letters113, 258104 (2014)

work page 2014

[16] [16]

Zheng and R

C. Zheng and R. T¨ onjes, Noise-induced swarming of ac- tive particles, Physical Review E106, 064601 (2022)

work page 2022

[17] [17]

Chatterjee, M

S. Chatterjee, M. Mangeat, C.-U. Woo, H. Rieger, and J. D. Noh, Flocking of two unfriendly species: The two- species vicsek model, Physical Review E107, 024607 (2023)

work page 2023

[18] [18]

Escaff, Self-organization of anti-aligning active parti- cles: Waving pattern formation and chaos, Physical Re- view E110, 024603 (2024)

D. Escaff, Self-organization of anti-aligning active parti- cles: Waving pattern formation and chaos, Physical Re- view E110, 024603 (2024)

work page 2024

[19] [19]

K¨ ursten, J

R. K¨ ursten, J. Mihatsch, and T. Ihle, Emergent flocking in mixtures of antialigning self-propelled particles, Phys- ical Review E111, L023402 (2025)

work page 2025

[20] [20]

Lardet, L

E. Lardet, L. Chen, and T. Bertrand, Flocking beyond one species: Novel phase coexistence in a generalized two- species vicsek model, arXiv preprint arXiv:2503.17617 (2025)

work page arXiv 2025

[21] [21]

Hong and S

H. Hong and S. H. Strogatz, Kuramoto model of coupled oscillators with positive and negative coupling parame- ters: an example of conformist and contrarian oscillators, Physical Review Letters106, 054102 (2011)

work page 2011

[22] [22]

the hung elections scenario

S. Galam, Contrarian deterministic effects on opinion dy- namics:“the hung elections scenario”, Physica A: Statis- tical Mechanics and its Applications333, 453 (2004)

work page 2004

[23] [23]

Brunel, Dynamics of sparsely connected networks of excitatory and inhibitory spiking neurons, Journal of computational neuroscience8, 183 (2000)

N. Brunel, Dynamics of sparsely connected networks of excitatory and inhibitory spiking neurons, Journal of computational neuroscience8, 183 (2000)

work page 2000

[24] [24]

D. T. Gillespie, Exact stochastic simulation of coupled chemical reactions, The journal of physical chemistry81, 2340 (1977)

work page 1977

[25] [25]

N. G. Van Kampen and W. P. Reinhardt, Stochastic pro- cesses in physics and chemistry (1983). 9

work page 1983

[26] [26]

Minors, T

K. Minors, T. Rogers, and C. A. Yates, Noise-driven bias in the non-local voter model, Europhysics letters122, 10004 (2018)

work page 2018

[27] [27]

Fruchart, R

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

work page 2021

[28] [28]

P. W. Anderson, More is different: broken symmetry and the nature of the hierarchical structure of science., Sci- ence177, 393 (1972)

work page 1972

[29] [29]

Calvert, A

J. Calvert, A. W. Richa, and D. Randall, How many more is different?, arXiv preprint arXiv:2510.06011 (2025)

work page arXiv 2025