Signatures of directed and spontaneous flocking

Martino Brambati, Giuseppe Fava, and Francesco Ginelli

Phys. Rev. E 106, 024608 – Published 22 August 2022

Abstract

Collective motion—or flocking—is an emergent phenomena that underlies many biological processes of relevance, from cellular migrations to animal group movements. In this work, we derive scaling relations for the fluctuations of the mean direction of motion and for the static density structure factor (which encodes static density fluctuations) in the presence of a homogeneous, small external field. This allows us to formulate two different and complementary criteria capable of detecting instances of directed motion exclusively from easily measurable dynamical and static signatures of the collective dynamics, without the need to detect correlations with environmental cues. The static one is informative in large enough systems, while the dynamical one requires large observation times to be effective. We believe these criteria may prove useful to detect or confirm the directed nature of collective motion in in vivo experimental observations, which are typically conducted in complex and not fully controlled environments.

Received 16 May 2022
Accepted 28 July 2022

DOI:https://doi.org/10.1103/PhysRevE.106.024608

Physics Subject Headings (PhySH)

Collective behavior Flocking

Collective dynamics Dry active matter

Theories of collective dynamics & active matter

Polymers & Soft MatterStatistical Physics & Thermodynamics

Authors & Affiliations

Martino Brambati¹, Giuseppe Fava ^2,3, and Francesco Ginelli ^2,3

¹Dipartimento di Scienza e Alta Tecnologia, Universitá degli Studi dell'Insubria, Como 22100, Italy
²Dipartimento di Scienza e Alta Tecnologia and Center for Nonlinear and Complex Systems, Universitá degli Studi dell'Insubria, Como 22100, Italy
³INFN sezione di Milano, Milano 20133, Italy

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Vol. 106, Iss. 2 — August 2022

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Images

Figure 1
Spontaneous (full black line) vs directed ( $h = 0.01$ , full red line) mean direction squared fluctuations as a function of time. The dashed blue line is the best fit of the spontaneous symmetry-breaking data (see text). We have chosen noise amplitude $η_{0} = 0.18$ and system size $L = 256$ , so that $N = ρ_{0} L^{2} \approx 10^{5}$ . In the inset: Zoom of the first 500 time steps.
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Figure 2
Scaling of the diffusion constant in the zero-field case. (a) Diffusion vs noise amplitude for system size $L = 256$ . The dashed red line marks quadratic growth, $D_{0} \sim η_{0}^{2}$ . (b) Diffusion vs total particle number for noise amplitude $η_{0} = 0.18$ and constant density. The dashed red line marks $1 / N$ decay. Error bars are given by two standard errors and plot are in a double-logarithmic scale.
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Figure 3
(a) Mean squared fluctuations of the flocking average direction as a function of time $t$ for different external field values. From bottom to top $h = 0.2$ , 0.1, 0.05, 0.02, 0.01, 0.005, 0.002, 0.001, 0. Data have been obtained with noise amplitude $η_{0} = 0.18$ and system size $L = 256$ . (b) Same data with both axes rescaled by $h$ (the $h = 0$ case has been removed). The dashed red line marks the rescaled crossover time $h τ_{c}$ . Both graphs are in a doubly logarithmic scale.
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Figure 4
(a) Asymptotic value of $〈 δ ψ^{2} 〉$ as a function of $h$ . Red diamonds report numerical estimates, while blue dots are the mean-field estimate of Eq. (21) (see text). The dashed black line marks the $\approx 1 / h$ scaling. Here $η_{0} = 0.18$ and $L = 256$ and the plot is in double-logarithmic scale. In the inset: Linear plot of the mean number of interacting active particles as a function of $h$ . Error bars are given by two standard errors.
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Figure 5
Isotropically averaged density structure factor $S (q, h)$ for the spontaneous (black dots) and directed ( $h = 0.01$ , red squares) case for systems of size $L = 1024$ . Data have been obtained by averaging the structure factor of ten different configurations sampled every 50 time steps (see text). The dashed blue line marks the power-law divergence $\approx q^{- ζ}$ , with $z = 1.33$ . Other parameters are $ρ = 2.0, v_{0} = 0.5, η = 0.18$ . Inset: Same parameters as in the main panel, but with system size $L = 256$ . All axes are in a double-logarithmic scale.
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Figure 6
(a) Isotropically averaged structure factor $S (q, h)$ for different external field amplitudes $h$ . From top to bottom $h = 0$ , 0.002, 0.005, 0.01, 0.02, 0.05, 0.1, 0.2. The dashed The dashed orange line marks the power-law divergence $\approx q^{- ζ}$ , with $z = 1.33$ . (b) Same as panel (a), but with the data rescaled by $h^{- 1 / z}$ (horizontal axis) and $h$ (vertical axis) in order to collapse the structure factor curves. The $h = 0$ data have been omitted. The vertical dashed red line marks the location of the (rescaled) crossover wave number (see text). All axes are in a double logarithmic scale, and model parameters are $ρ = 2.0, v_{0} = 0.5, η = 0.18$ , and $L = 512$ .
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