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

We study the largest Lyapunov exponent A 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 = 20 000). 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 similar to root U. In the continuum limit the system is integrable in the whole high temperature phase. More precisely, the behavior lambda similar to N-1/3 is found numerically for U > U-c and justified on the basis of a random matrix approximation. [S0031-9007(97)05121-1].