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 de scribe 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-(1/3) to the leading order in N.