An extended critical region in swarms

Research summary

One paper, summarised for reference. Power laws of natural swarms are fingerprints of an extended critical region — R. González-Albaladejo & L. L. Bonilla, arXiv:2309.05064 (2024). arXiv:2309.05064.

The puzzle

Natural midge swarms show power laws (correlation length ∝ flock size, susceptibility, correlation time) and strong scale-free correlations — signatures of near-criticality — yet three facts sit awkwardly with the usual explanation (an ordering transition, à la Cavagna): swarms live in the disordered phase (low polarization), they form around markers on the ground (so they are not translation-invariant), and their dynamic scaling is imperfect — the normalized dynamic correlation function (NDCCF) collapses onto one curve only for short rescaled times ($0 < t/\xi^z < 4$). The ordering-transition RG gets the dynamic exponent right ($z\approx1.35$) but the static exponents wrong ($\nu\approx0.75,\gamma\approx1.17$ vs. observed $\nu\approx0.35,\gamma\approx0.9$), predicts a full NDCCF collapse, and can’t produce the marker/shape phenomenology. This paper offers a different mechanism.

The model: a scale-free-chaos transition

They use the harmonically confined Vicsek model (HCVM) — 3D Vicsek alignment plus a linear spring $\beta$ that binds the swarm (standing in for the marker) — with alignment noise $\eta$ and confinement $\beta$ as control parameters. The HCVM has a scale-free-chaos phase transition: chaotic dynamics (positive Lyapunov exponent) with scale-free correlations ($\xi \sim N^{1/3}$). Crucially, for finite $N$ this is not a point but an extended critical region bounded by three scale-free lines on the $(\eta,\beta)$ plane — the onset-of-chaos line $\beta_0$ (zero largest Lyapunov exponent), the single-to-multicluster line $\beta_c$, and the maximal-Lyapunov line $\beta_i$ — all collapsing onto the $\beta=0$ axis at the same rate as $N\to\infty$.

Power laws, and a bridge to the observable

As $\eta\to0$ the critical lines obey

$$\beta_j(N;\eta) = C_j\, N^{-1/(3\nu)}\,\eta^{m_j}, \qquad j=0,c,$$

with $\xi\sim\beta^{-\nu}$, $\chi\sim\beta^{-\gamma}$, $\tau\sim\xi^z$. Because the exponents drop out of the noise dependence, $\nu$ and $\gamma$ can be estimated at a fixed $N$ (you cannot grow the insect count at will) — giving $\nu\approx0.43$, $\gamma\approx0.92$. And the unmeasurable confinement $\beta$ is found to be proportional to the perception range (the mean nearest-neighbour distance), the same observable control parameter used in the Attanasi–Cavagna finite-size analysis — so the laws can be written in measurable terms.

The key move: mixtures of data

Real swarm data are pooled across different times, atmospheric conditions, species, and $N$ — so they do not sit at a single control-parameter value. The authors model this as a mixture of simulation data over the extended critical region (varying $N$, $\eta$, $\beta$ along $\beta_0$ and $\beta_c$). Applying the same fitting procedures used on real swarms, the mixture reproduces:

• the static exponents ($\nu\approx0.43$, $\gamma\approx0.92$ vs. observed $0.35$, $0.9$);
• the dynamic exponent (mixture: $z_{\text{LS}}\approx1.15$, $z_{\text{RMA}}\approx1.33$ vs. observed $1.16$, $1.37$ — noting LS underestimates $z$; on a single line $z\approx1$);
• the imperfect NDCCF collapse over only a finite time interval — a signature that several timescales are in play near the scale-free-chaos transition (an ordering transition has one timescale → full collapse);
• the observed swarm shape (a condensed core in an “insect vapour”).

The upshot

The ordering transition between homogeneous, translation-invariant phases does not belong to the universality class of insect swarms; the scale-free-chaos transition — with confinement/markers, chaos, an extended critical region, and multiple timescales — matches both the exponents and the qualitative features. The broader twist for the criticality debate: when a system’s control parameter is unmeasurable or varies across observations, apparent power laws may reflect a mixture over an extended critical region rather than fine-tuning to a single critical point (a cousin of Griffiths-phase / generic scale-invariance ideas).

References

González-Albaladejo & Bonilla (2024), arXiv:2309.05064 · González-Albaladejo, Carpio & Bonilla (2023), Phys. Rev. E 107, 014209 · González-Albaladejo & Bonilla (2023), Phys. Rev. E 107, L062601 · Attanasi et al. (2014), Phys. Rev. Lett. 113, 238102 · Cavagna et al. (2017), Nat. Phys. 13, 914 · Cavagna et al. (2023), Nat. Phys. 19, 1043 · Vicsek et al. (1995), Phys. Rev. Lett. 75, 1226 · Cavagna, Giardina & Grigera (2018), Phys. Rep. 728, 1 · Mora & Bialek (2011), J. Stat. Phys. 144, 268.