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

Abstract

We study the largest Lyapunov exponent $\ensuremath{\lambda}$ 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 ${U}_{c}$, $\ensuremath{\lambda}$ shows a peak which persists for very large $N$ values $(N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}20000)$. We show, both numerically and analytically, that chaoticity is strongly related to kinetic energy fluctuations. In the limit of small energy, $\ensuremath{\lambda}$ goes to zero with an $N$-independent power law: $\ensuremath{\lambda}\ensuremath{\sim}\sqrt{U}$. In the continuum limit the system is integrable in the whole high temperature phase. More precisely, the behavior $\ensuremath{\lambda}\ensuremath{\sim}{N}^{\ensuremath{-}1/3}$ is found numerically for $U>{U}_{c}$ and justified on the basis of a random matrix approximation.