Mean field theory of chaotic insect swarms

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 [Physical Review 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 $\varphi=0.5$ 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.