Mean field theory of chaotic insect swarms Articles uri icon

publication date

  • June 2023

start page

  • L062601-1

end page

  • L062601-6


  • 6


  • 107

International Standard Serial Number (ISSN)

  • 2470-0045

Electronic International Standard Serial Number (EISSN)

  • 2470-0053


  • The harmonically confined Vicsek model displays qualitative and quantitative features observed in natural insect swarms. It exhibits a scale-free transition between single and multicluster chaotic phases. Finite-size scaling indicates that this unusual phase transition occurs at zero confinement [Phys. Rev. E 107, 014209 (2023)]. While the evidence of the scale-free-chaos phase transition comes from numerical simulations, here we present its mean-field theory. Analytically determined critical exponents are those of the Landau theory of equilibrium phase transitions plus dynamical critical exponent z = 1 and a new critical exponent phi = 0.5 for the largest Lyapunov exponent. The phase transition occurs at zero confinement and noise in the mean-field theory. The noise line of zero largest Lyapunov exponents informs observed behavior: (i) the qualitative shape of the swarm (on average, the center of mass rotates slowly at the rate marked by the winding number and its trajectory fills compactly the space, similarly to the observed condensed nucleus surrounded by vapor) and (ii) the critical exponents resemble those observed in natural swarms. Our predictions include power laws for the frequency of the maximal spectral amplitude and the winding number.


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


  • chaos; collective behavior; dynamical phase transitions; nonequilibrium statistical mechanics; scaling laws of complex systems; swarming; active matter; collective dynamics; theories of collective dynamics & active matter; vicsek model