Lyapunov Instability and Finite Size Effects in a System with Long-Range Forces

We study the largest Lyapunov exponent λ and the finite size effects of a system of N fully coupled classical particles, which shows a second order phase transition. Slightly below the critical energy density Uc, λ shows a peak which persists for very large N values (N0ex0ex=0ex0ex20000). We show, both numerically and analytically, that chaoticity is strongly related to kinetic energy fluctuations. In the limit of small energy, λ goes to zero with an N-independent power law: λ~√U. In the continuum limit the system is integrable in the whole high temperature phase. More precisely, the behavior λ~N^-1/3 is found numerically for U>Uc and justified on the basis of a random matrix approximation.