Finite-size scaling in natural swarms

Research summary

One paper, summarised for reference. Finite-size scaling as a way to probe near-criticality in natural swarms — A. Attanasi, A. Cavagna, L. Del Castello, I. Giardina, S. Melillo, L. Parisi, O. Pohl, B. Rossaro, E. Shen, E. Silvestri & M. Viale, Phys. Rev. Lett. (2014) 113, 238102 (arXiv:1412.6975). doi:10.1103/PhysRevLett.113.238102.

The gap it fills

Earlier claims that biological groups sit near criticality had two weaknesses: the control parameter was always inferred through a model (prone to undersampling), and off-equilibrium conservation laws can produce long-range correlations generically (“generic scale invariance”) without any tuning. Worse, the critical point is sharp only in the thermodynamic limit — at finite $N$ the sole remnant is a susceptibility peak whose position depends on $N$. So a single bulk critical value is not critical across sizes (a tiny Ising model at bulk $T_c$ is deeply magnetised). The two missing pieces this paper supplies: (i) a direct measurement of the control parameter, and (ii) evidence that across sizes the control parameter varies with $N$ to stay near the susceptibility maximum. The lesson: it is the pair $(x, N)$ that must sit in the scaling region, not merely proximity to $x_c$.

System and observables

Wild midge swarms (Diptera) reconstructed in 3D, $N \approx 100$–$600$, in the disordered phase (low polarization $\Phi \approx 0.2$) yet strongly correlated. From velocity fluctuations $\delta\vec\varphi$ (with instantaneous global translation, rotation and dilatation subtracted — a non-equilibrium system has no usable time average), they build the connected correlation $C(r)$, take the correlation length $\xi \sim r_0$ (its first zero, far larger than the nearest-neighbour distance), and the susceptibility $\chi$ = integrated correlation up to $r_0$ (∝ number of correlated individuals). Natural swarms reach $\chi$ up to ~5.6 vs. ~0.1 for a non-interacting system (50× more correlated). The control parameter is the nearest-neighbour distance rescaled by the interaction range, $x = r_1/\lambda \approx r_1/l$ (the acoustic interaction range scales with body length $l$).

The Vicsek yardstick and finite-size scaling

The 3D Vicsek model has a density-driven order/disorder transition at $x_c$. Its bulk transition is first-order, but only above a huge crossover size ($N^\star\sim10^6$); below it — the swarm-relevant regime — a pseudo-second-order phenomenology holds and standard finite-size scaling applies:

$$x_{\max}(N) = x_c + N^{-1/3\nu}, \qquad y = (x - x_c)\,N^{1/3\nu},$$ $$\chi \sim N^{\gamma/3\nu}\,f(y), \qquad \xi \sim L\,g(y).$$

Two paths matter (map in the paper’s Fig. 2a): at fixed $x$, growing $N$ makes $\chi$ and $\xi$ saturate (you fall away from the peak); along constant $y$, $\chi \sim N^{\gamma/3\nu}$ and $\xi \sim L$ stay scale-free (you track the peak). 3D-Vicsek fit: $\nu\approx0.75$, $\gamma\approx1.6$, $x_c\approx0.42$ (near 3D Heisenberg).

The result

Natural swarms show no saturation of $\chi$ or $\xi$ with size — $\chi$ scales with $N$ and $\xi$ with $L$ up to the largest swarms. That places them on the constant-$y$ path: as $N$ grows, $x$ decreases (following the susceptibility peak while staying on the disordered side), evidenced by a direct $x$–$N$ correlation. So $(x,N)$ lives in the near-critical region, with $\chi \sim (x-x_c)^{-\gamma}$. The new content over prior work: the control parameter is measured, not inferred, and what is seen is not a generic vicinity but a mutual adjustment of $x$ and $N$ that keeps correlations scale-free — which rules out generic scale invariance. (The fitted swarm exponents are not conclusive given the limited size span, but the scaling result is fit-independent.)

Two readings of near-criticality

• Adjust $x$ given $N$ — tune the control parameter toward $x_{\max}(N)$; requires individuals to assess global correlation via a local proxy and readjust as $N$ changes.
• Grow $N$ given $x$ — a swarm aggregates up to a maximum sustainable size $N_{\max}(x)$: below it the group stays scale-free-correlated ($\xi/L$ constant via the Goldstone mode of the ordered phase), beyond it correlation deteriorates. A mating-driven aggregation mechanism plausibly enforces such a ceiling — and may explain why swarms stay disordered.

A caveat with reach: the second (aggregation) reading relies on continuous symmetry breaking (Goldstone → divergent bulk $\xi$). For discrete symmetry (Ising-like, as in many neural models) it fails, and an adaptive-$x$ mechanism seems required instead.

References

Attanasi, Cavagna et al. (2014), Phys. Rev. Lett. 113, 238102 · Cavagna et al. (2010), PNAS 107, 11865 · Vicsek et al. (1995), Phys. Rev. Lett. 75, 1226 · Mora & Bialek (2011), J. Stat. Phys. 144, 268 · Bialek et al. (2014), PNAS 111, 7212 · Chaté et al. (2008), Phys. Rev. E 77, 046113 · Toner & Tu (1998), Phys. Rev. E 58, 4828 · Grinstein, Lee & Sachdev (1990), Phys. Rev. Lett. 64, 1927.