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

We study the largest Lyapunov exponent $\lambda$ 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 $U_c$, $\lambda$ shows a peak which persists for very large $N$ values $(N=20\mathrm{}000)$. We show, both numerically and analytically, that chaoticity is strongly related to kinetic energy fluctuations. In the limit of small energy, $\lambda$ goes to zero with an $N$-independent power law: $\lambda \sim \sqrt{U}$. In the continuum limit the system is integrable in the whole high temperature phase. More precisely, the behavior $\lambda \sim N^{-1/3\mathrm{}}$ is found numerically for $U>\mathrm{}{U}_c$ and justified on the basis of a random matrix approximation.