The harmonically confined Vicsek model (HCVM) — Vicsek's aligning self-propelled particles plus a harmonic trap mimicking the effective central spring observed in insect swarms — exhibits a scale-free-chaos phase transition: for finite particle number N, a critical curve in the confinement-noise plane separates single-cluster from multicluster chaotic attractors and carries scale-free correlations, collapsing to zero confinement as N goes to infinity. Its mean-field theory, a closed map for the swarm's center of mass, yields Landau-like static exponents (ν = b = 1/2, γ = 1), the dynamical exponent z = 1, a new exponent φ = 1/2 for the largest Lyapunov exponent, and distinctive measurable predictions: power laws for the winding number of the center-of-mass rotation and for the frequency of its maximal spectral amplitude. Both open cores survive in the literature as of August 2026: no beyond-mean-field (fluctuation or renormalization-group) theory of the transition exists — the authors' February 2026 review calls this 'a wide open problem' — and the predicted center-of-mass power laws have never been extracted from natural swarm data, although finite-N simulations reproduce the measured correlation-function exponents and a measurable proxy for the confinement (the perception range) is known. The problem asks for a controlled theory of the transition in finite dimensions, deriving the N → ∞ collapse of the critical curves and the dimension-dependent exponents, together with a quantitative experimental test of the predicted center-of-mass power laws in natural swarms.

Background

Self-propelled particle models describe collective motion: each particle moves at constant speed and aligns its velocity with the average velocity of nearby particles. The original such model, due to Vicsek and coworkers (Phys. Rev. Lett. 75, 1226 (1995)), exhibits an ordering transition between disordered motion and collective flocking. Natural mating swarms of midges, however, show no global order: they are cohesive but unpolarized, and experiments on laboratory swarms indicate that each insect behaves as if bound by an effective harmonic (spring-like) potential centered on the swarm. González-Albaladejo, Carpio and Bonilla therefore proposed the harmonically confined Vicsek model (HCVM): Vicsek aligning self-propelled particles updated in discrete time, with an additional restoring force pulling each particle toward the origin, whose strength β is called the confinement (Phys. Rev. E 107, 014209 (2023)).

For finitely many particles the HCVM has periodic, quasiperiodic and chaotic attractors; an attractor is chaotic when its largest Lyapunov exponent — the asymptotic exponential rate of separation of infinitesimally close trajectories — is positive. Numerical simulations revealed a 'scale-free-chaos phase transition': for each particle number N, a critical curve in the confinement-noise plane separates single-cluster from multicluster chaotic swarms, and on this curve the velocity-correlation length is proportional to the swarm size; the curve collapses to zero confinement as N → ∞. Unlike the Vicsek ordering transition, the 'ordered' phase here is a particular kind of chaos, and the transition reproduces several observed swarm traits: a condensed core surrounded by dilute vapor, and the collapse of the dynamic correlation function only for short scaled times, as observed in field data.

Field studies of wild midge swarms show scale-free correlations and power laws reminiscent of criticality: a correlation length growing with swarm size (ξ ~ N^(1/3)) and power-law susceptibility and correlation time, with static critical exponents ν ≈ 0.35 ± 0.10 and γ ≈ 0.9 ± 0.2 (Attanasi et al., Phys. Rev. Lett. 113, 238102 (2014)) and dynamical exponent z ≈ 1.2 (Cavagna et al., Nat. Phys. 13, 914 (2017)). A competing renormalization-group theory based on the ordering transition of active matter predicts z = 1.35, ν = 0.748 and γ = 1.171, matching the dynamical but not the static exponents (Cavagna et al., Nat. Phys. 19, 1043 (2023)).

The primary source, 'Mean field theory of chaotic insect swarms' by González-Albaladejo and Bonilla (Phys. Rev. E 107, L062601 (2023)), approximates the N → ∞ HCVM by a closed map for the swarm center of mass X(t): X(t+1) = X(t) + V(t+1), V(t+1) = v0 R_η[Θ(V(t) − β̃X(t))], where Θ(w) = w/|w|, R_η applies a random rotation of maximum amplitude η (the alignment noise), and β̃ is the confinement rescaled by the average number of neighbors within the interaction radius. This mean-field map reproduces the topology of the phase diagram and, at the scale-free-chaos transition (β = η = 0), yields analytically: the winding number w — the mean rotation rate of the center of mass about the origin, in turns per time step, which replaces the polarization as order parameter — scaling as w ~ β^(1/2); correlation length ξ ~ β^(−1/2); susceptibility χ ~ β^(−1); the dynamical exponent z = 1, relating the frequency Ω of the maximal spectral amplitude of the center-of-mass motion to the correlation length by Ω ~ ξ^(−1); and a new critical exponent φ = 1/2 for the largest Lyapunov exponent, λ1 ~ β^φ, together with the scaling relation φ = zν. These are the Landau (mean-field) values of equilibrium critical phenomena supplemented by two dynamical exponents. Because laboratory swarms driven at 1 Hz respond with coherent elliptic center-of-mass motion (Ni et al., Phys. Rev. Lett. 115, 118104 (2015)), such center-of-mass observables are measurable in principle, and the authors proposed extracting the predicted winding-number and spectral-peak power laws from experimental data as a test of the theory.

What is known beyond mean field comes from finite-N simulations: the transition exists in two and three dimensions, with static exponents ν = 0.30 ± 0.02 and γ = 0.78 ± 0.05 in 2D, ν = 0.436 ± 0.009 and γ = 0.92 ± 0.05 in 3D, and z ≈ 1 throughout — dimension-dependent deviations from the mean-field values. Later work showed that the unmeasurable confinement maps linearly onto the perception range (the time-averaged nearest-neighbor distance) that is measured in natural swarms, making the model's power laws empirically accessible (Phys. Rev. E 109, 014611 (2024)). No theory incorporating fluctuations — no renormalization-group treatment or controlled expansion around the mean-field map — exists for this transition, and the predicted center-of-mass power laws have not been tested against natural swarm data.

Problem Statement

Determine whether the scale-free-chaos phase transition of the harmonically confined Vicsek model is a genuine critical phenomenon of the fluctuating finite-dimensional system, and whether it quantitatively describes natural insect swarms. The problem comprises two linked results that jointly decide this question.

(1) A theory beyond mean field. Construct a controlled treatment of the HCVM scale-free-chaos phase transition — a renormalization-group calculation or an equivalent systematic inclusion of fluctuations — valid in finite spatial dimension d ≥ 2 and in the limit N → ∞ at vanishing confinement β → 0 and noise η → 0. It should derive (a) the collapse of the critical curves — the chaos boundary β0(N; η), the single-to-multicluster chaos boundary βc(N; η), and the curve of maximal largest Lyapunov exponent βi(N; η) — toward zero confinement as N → ∞, including their scaling rates; and (b) the full set of critical exponents: the correlation-length exponent ν, the susceptibility exponent γ, the order-parameter (winding-number) exponent b, the dynamical exponent z, and the Lyapunov exponent φ, thereby settling whether the mean-field values ν = b = φ = 1/2, γ = 1, z = 1 are exact or corrected by fluctuations, and accounting for the dimension dependence observed numerically (ν ≈ 0.30 in d = 2, ν ≈ 0.436 in d = 3).

(2) An experimental test. From measured center-of-mass trajectories of natural insect swarms (or laboratory swarms under controlled perturbation) spanning a range of swarm sizes and conditions, extract the winding number of the center-of-mass rotation and the frequency of the maximal spectral amplitude, determine their scaling with the measurable control parameter — the perception range, the established experimental proxy for the confinement — and test quantitatively whether these observables follow the predicted power laws with the model's exponents, including whether the center-of-mass dynamics is consistent with a noisy chaotic attractor whose largest Lyapunov exponent obeys λ1 ~ |x − xc|^φ with φ = zν, where x is the perception range and xc its critical value.

Result (1) establishes the transition as a controlled critical phenomenon; result (2) tests its distinctive fingerprints in real swarms; together they decide whether the criticality observed in natural swarms is the fingerprint of the scale-free-chaos transition. A complete resolution of the problem as posed requires both; a definitive derivation under (1), or a decisive confirmation or refutation under (2), is the corresponding partial resolution.

References

  1. R. González-Albaladejo and L. L. Bonilla, 'Mean field theory of chaotic insect swarms', Physical Review E 107, L062601 (2023). DOI: 10.1103/PhysRevE.107.L062601, https://doi.org/10.1103/PhysRevE.107.L062601 (arXiv:2305.14085, https://arxiv.org/abs/2305.14085)
  2. R. González-Albaladejo, A. Carpio and L. L. Bonilla, 'Scale-free chaos in the confined Vicsek flocking model', Physical Review E 107, 014209 (2023). DOI: 10.1103/PhysRevE.107.014209, https://doi.org/10.1103/PhysRevE.107.014209
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