To the class of Hamiltonian models with long range forces can be ascribed a large number of physical systems (e.g. magnetic systems, self-gravitating systems, turbulent systems etc.) N-body hamiltonian systems with long range forces reveal several interesting phenomena, such as clustering and anomalous relaxation to the equilibrium. In some of these models the dynamics of the single particle depends self-consistently on the time-behaviour of "mean-field" quantities, obtained averaging over all the particles of the system (Antoni and Ruffo). Such self-consistent dynamics lead to propagation of "soliton-like" cluster, at least in one-dimension, and to clustering and de-clustering phenomena that can be related to a phase transition in the system.
In 2 dimensions we have shown that the clustering transition is first order and that it is related to anomalies in the thermodynamics of the system (e.g. a region of ``negative specific heat'' has been observed near the transition) as well as in its dynamical behaviour: diffusion of particles in these systems is not ruled by usual Brownian motion. But a competition of localized and ballistic behaviours instead leads to anomalous diffusion. Some preliminary indications suggest that these motions can be interpreted within the class of the Levy-walks. Moreover, the observed superdiffusive behaviour is clearly related to the existence of a clustered state. For the first time a transition from a clustered to a homogeneous state has been related to a dynamical transition from anomalous to ballistic motion [35,41].
A recent review concerning 1-d and 2-d self-gravitating system is reported in Ref. [44] of my pubblication list.
We are presently working on a generalization of the previous model in 2d, this new model depending on a parameter is able to exhibit both first and second order clustering transitions. In particular, we are observing that a negative specific heat regime is present only in connection with first order transitions [53,54]. We plan to investigate in more details these findings due to their relevance not only for the statistical mechanics community but also for the astrophysicists.
More recently we have examined for this 2d generalized model dynamical features connected to the first order transition, in particular we observed that the escaping time from the metastable states diverge exponentially with the number of particles and that the exponential rate is proportional to the entropy barrier between the metastable and the unstable state [62].
The cited references refer to the publication list