Lyapunov instability and finite size effects in a system with long-range forces

Abstract

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 $U_c$, $λ$ shows a peak which persists for very large N-values (N=20000). 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 a N-independent power law: $λ\sim \sqrt{U}$. In the continuum limit the system is integrable in the whole high temperature phase. More precisely, the behavior $λ\sim N^{-1/3}$ is found numerically for $U > U_c$ and justified on the basis of a random matrix approximation.