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

Abstract

The parametric instability contribution to the largest Lyapunov exponent ${\ensuremath{\lambda}}_{1}$ 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 ${\ensuremath{\lambda}}_{1}$ 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 ${\ensuremath{\lambda}}_{1}$ scaling as ${N}^{\ensuremath{-}(1/3)}$ to the leading order in $N$.