Rafael González Albaladejo (LPTHE)

Salle 523, couloir 12-13, 5è étage

Swarming Theory: Scale-Free Chaos, Extended Criticality and Exact Solutions

Collective biological systems display power laws for macroscopic quantities and are fertile grounds for statistical physics. Besides power laws, natural insect swarms present strong scale-free correlations, suggesting closeness to phase transitions. Swarms exhibit imperfect dynamic scaling: their dynamical correlation functions collapse into single curves when written as functions of the scaled time \(t\xi^{-z}\) (\(\xi\): correlation length, z: dynamic exponent), but only for short times. Triggered by markers, natural swarms are not invariant under space translations. Measured critical exponents differ from those of equilibrium and many nonequilibrium phase transitions.

We consider the discrete-time Vicsek model with particles confined by a harmonic potential and calibrated by experimental data. The model exhibits periodic, quasiperiodic, and chaotic attractors, with scale-free lines among chaotic attractors as N increases.

In the parameter space of confinement strength and alignment noise, lines separating chaotic single-cluster and multicluster swarms, and chaotic from non-chaotic attractors, are scale-free and coalesce at the same rate as N>>1 to the zero-confinement line.

They characterize a scale-free-chaos phase transition. For finite N, these lines belong to an extended critical region. Finite-size scaling arguments allow calculating critical exponents for correlation length, time, and susceptibility. These power laws imply that confinement strength is proportional to the insect perception range. Observations under varying conditions are mimicked by mixing data on different critical lines and N. Our simulations reproduce key features of swarms and yield critical exponents matching observations.

We will also present exact solutions of the harmonically confined Vicsek model that capture all this discrete-time dynamics.

References:

R. González-Albaladejo, A. Carpio, and L. L. Bonilla, Scale free chaos in the confined Vicsek flocking model, Phys. Rev. E 107, 014209 (2023).

R. González-Albaladejo and L. L. Bonilla, Power laws of natural swarms as fingerprints of an extended critical region, Phys. Rev. E 109, 014611 (2024).

L. L. Bonilla and R. González-Albaladejo, Exact solutions of the harmonically confined Vicsek model, Chaos, Solitons & Fractals 191, 115826 (2025).