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

A system of N classical particles in a two-dimensional periodic cell interacting via a long-range attractive potential is studied numerically and theoretically. For low energy density U a collapsed phase is identified, while in the high energy limit the particles are homogeneously distributed. A phase transition from the collapsed to the homogeneous state occurs at critical energy $U_c.$ A theoretical analysis within the canonical ensemble identifies 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 Thirring [Ann. Phys. (N.Y.) 63, 520 (1970)] for gravitational lattice gas models. The dynamical behavior of the system is strongly affected by this transition: below $U_c$ anomalous diffusion is observed, while for $U>\mathrm{}{U}_c$ the motion of the particles is almost ballistic. In the collapsed phase, finite N effects act like a “deterministic” noise source of variance $\mathcal{O}\mathrm{}(1/N),$ that restores normal diffusion on a time scale that diverges with N. As a consequence, the asymptotic diffusion coefficient will also diverge algebraically with N and superdiffusion will be observable at any time in the limit $\overrightarrow{N}\infty .$ A Lyapunov analysis reveals that for $U>\mathrm{}{U}_c$ the maximal exponent $\lambda$ decreases proportionally to $N^{-1/3}$ and vanishes in the mean-field limit. For sufficiently small energy, in spite of a clear nonergodicity of the system, a common scaling law $\lambda \propto U^{1/2}$ is observed for various different initial conditions. In the intermediate energy range, where anomalous diffusion is observed, a strong intermittency is found. This intermittent behavior is related to two different dynamical mechanisms of chaotization.