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

A system of N classical particles in a two-dimensional periodic cell intera cting via a long-range attractive potential is studied numerically and theo retically. For low energy density U a collapsed phase is identified, while in the high energy limit the particles are homogeneously distributed. A pha se transition from the collapsed to the homogeneous state occurs at critica l energy U-c. A theoretical analysis within the canonical ensemble identifi es such a transition as first order. But microcanonical simulations reveal a negative specific heat regime near U-c. This suggests that the transition belongs to the universality class previously identified by Hertel and Thir ring [AM. Phys (N.Y.) 63, 520 (1970)] for gravitational lattice gas models. The dynamical behavior of the system is strongly affected by this transiti on: below U, anomalous diffusion is observed, while for U>U-c the motion of the particles is almost ballistic. In the collapsed phase, finite N effect s act like a "deterministic" noise sourer of variance O(1/N), that restores normal diffusion on a time scale that diverges with N. As a consequence, t he asymptotic diffusion coefficient will also diverge algebraically with N and superdiffusion will be observable at any time in the limit N-->infinity . A Lyapunov analysis reveals that for U>U-c the maximal exponent lambda de creases proportionally to N-1/3 and vanishes in the mean-field limit. For s ufficiently small energy, in spite of a clear nonergodicity of the system, a common scaling law lambda proportional to U-1/2 is observed for various d ifferent initial conditions. Ln the intermediate energy range, where anomol ous diffusion is observed, a strong intermittency is found. This intermitte nt behavior is related to two different dynamical mechanisms of chaotizatio n. [S1063-651X(99)10303-9].