Analytic estimation of the Lyapunov exponent in a mean-field model undergoing a phase transition

The parametric instability contribution to the largest Lyapunov exponent ${\lambda }_1$ is derived for a mean-field Hamiltonian model, with attractive long-range interactions. This uses a recent Riemannian approach to describe Hamiltonian chaos with a large number $N$ of degrees of freedom. Through microcanonical estimates of suitable geometrical observables, the mean-field behavior of ${\lambda }_1$ is analytically computed and related to the second-order phase transition undergone by the system. It predicts that chaoticity drops to zero at the critical temperature and remains vanishing above it, with ${\lambda }_1$ scaling as $N^{-\mathrm{}(1/3)\mathrm{}}$ to the leading order in $N$.