Dynamics and Statistics of Simple Models with Infinite-range Attractive Interaction

The treatment of long-range interacting systems (including Newtonian self-gravitating systems) remains a challenging issue in statistical mechanics. Due to the lack of extensivity, they present non-standard effects like negative specific heats, which shows the inequivalence of statistical ensembles (namely, microcanon-ical and canonical) even in the limit of infinite number of particles (N → ∞). In this paper we review a series of results obtained for one and two dimensional simple N-body dynamical models with infinite-range attractive interactions and without short distance singularities. The free energy of both models can be exactly obtained in the canonical ensemble, while information on the microcanonical ensemble and on the dynamical evolution can be derived from direct numerical simulations with simple O(N) codes, which make use of mean-field variables. Both models show a phase transition from a low energy clustered phase to a high energy gaseous state, in analogy with the models introduced in the early 70's by Thirring and Hertel. The phase transition is second order for the 1D model, first order for the 2D model. Negative specific heat appears in both models near the phase transition point, but while for the 2D model it is an equilibrium phenomenon, in the 1D case it is typical of transient metastable states, whose lifetime grows with N. For both models, in the presence of a negative specific heat, a cluster of collapsed particles coex-ists with a halo of higher energy particles which perform long correlated flights, which lead to anomalous diffusion: the mean square displacement grows faster than linear with time. The dynamical origin of this " superdiffusion " is however different in the two models, being related to particle trapping and untrapping in the cluster in 1D, while in 2D the channelling of particles in an egg-crate effective potential is responsible of the effect. Both models are Lyapunov unstable and the maximal Lyapunov exponent λ has a peak just in the region preceeding the phase transition. Moreover, in the low energy limit λ increases proportionally to the square root of the internal energy, while in the high energy region it vanishes as N −1/3. Since the 1D model is explicitely constructed considering the first modes of a Fourier expansion of a classical one dimensional gravity potential, the large scale properties of this model in the low-energy clustered phase resemble those of 1D gravity (mass-sheet models). The relation of both models with gravity remains to be …