Scale-free chaos in the confined Vicsek flocking model Articles uri icon

publication date

  • January 2023

start page

  • 1

end page

  • 24


  • 1


  • 107

International Standard Serial Number (ISSN)

  • 2470-0045

Electronic International Standard Serial Number (EISSN)

  • 2470-0053


  • The Vicsek model encompasses the paradigm of active dry matter. Motivated by collective behavior of insects in swarms, we have studied finite-size effects and criticality in the three-dimensional, harmonically confined Vicsek model. We have discovered a phase transition that exists for appropriate noise and small confinement strength. On the critical line of confinement versus noise, swarms are in a state of scale-free chaos characterized by minimal correlation time, correlation length proportional to swarm size and topological data analysis. The critical line separates dispersed single clusters from confined multicluster swarms. Scale-free chaotic swarms occupy a compact region of space and comprise a recognizable 'condensed'nucleus and particles leaving and entering it. Susceptibility, correlation length, dynamic correlation function, and largest Lyapunov exponent obey power laws. The critical line and a narrow criticality region close to it move simultaneously to zero confinement strength for infinitely many particles. At the end of the first chaotic window of confinement, there is another phase transition to infinitely dense clusters of finite size that may be termed flocking black holes.


  • Biology and Biomedicine
  • Chemistry
  • Materials science and engineering
  • Mathematics
  • Mechanical Engineering
  • Physics


  • chaos; collective behavior; collective dynamics; dynamical phase transitions; nonequilibrium statistical mechanics; scaling laws of complex systems; swarming