  In this paper we review a series of results obtained for 1D and 2D 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 microcanonical results can be derived
from numerical simulations. 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.
For both models, in the presence of a negative specific heat, a cluster of
collapsed particles coexists with a halo of higher energy particles which
perform long correlated flights, which lead to anomalous diffusion. The
dynamical origin of the "superdiffusion" is 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 $\lambda$ has a peak just in the region preceeding the phase
transition. Moreover, in the low energy limit $\lambda$ increases
proportionally to the square root of the internal energy, while in the high
energy region it vanishes as $N^{-1/3}$.
