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A system of N classical particles in a 2D periodic cell interacting vialong-range attractive potential is studied. For low energy density $U$ acollapsed phase is identified, while in the high energy limit the particles arehomogeneously distributed. A phase transition from the collapsed to thehomogeneous state occurs at critical energy U_c. A theoretical analysis withinthe canonical ensemble identifies such a transition as first order. Butmicrocanonical simulations reveal a negative specific heat regime near $U_c$.The dynamical behaviour of the system is affected by this transition : belowU_c anomalous diffusion is observed, while for U > U_c the motion of theparticles is almost ballistic. In the collapsed phase, finite $N$-effects actlike a noise source of variance O(1/N), that restores normal diffusion on atime scale diverging with N. As a consequence, the asymptotic diffusioncoefficient will also diverge algebraically with N and superdiffusion will beobservable at any time in the limit N \to \infty. A Lyapunov analysis revealsthat for U > U_c the maximal exponent \lambda decreases proportionally toN^{-1/3} and vanishes in the mean-field limit. For sufficiently small energy,in spite of a clear non ergodicity of the system, a common scaling law \lambda\propto U^{1/2} is observed for any initial conditions.Comment: 17 pages, Revtex - 15 PS Figs - Subimitted to Physical Review E - Two  column version with included figures : less paper waste

Equilibrium and dynamical properties of two-dimensionalN-body systems with long-range attractive interactions