PACS. 05.20.-y -Classical statistical mechanics. PACS. 45.50.Jf -Few-and many-body systems. PACS. 45.50.-j -Dynamics and kinematics of a particle and a system of particles.Abstract. -We study gravitational clustering of mass points in three dimensions with random initial positions and periodic boundary conditions (no expansion) by numerical simulations. Correlation properties are well defined in the system and a sort of thermodynamic limit can be defined for the transient regime of clustering. Structure formation proceeds along two paths: (i) fluid-like evolution of density perturbations at large scales and (ii) shift of the granular (non fluid) properties from small to large scales. The latter mechanism finally dominates at all scales and it is responsible for the self-similar characteristics of the clustering.One of the fundamental challenges of modern cosmology is the understanding of the formation of the structures in the Universe. These structures consist of clusters of galaxies and show complex properties extended to very large scales [1]. Usually the simulations and the models aimed at the understanding of these structures are based on three essential elements: (i) the dynamics under the effect of the gravitational forces; (ii) some particular type of initial conditions; (iii) a model for the Hubble expansion [2,3]. In addition simulations are usually run up to a time which is supposed to represent the present state.Here we would like to take inspiration from these studies and formulate the problem of clustering by gravity in the perspective of the statistical physics of dynamical systems. So we will single out the role of each individual effect at the expense of a loss in realism. We try therefore to identify simple fundamental mechanisms which can be studied in great detail and followed up to their asymptotic state. As a first problem we consider the simple, basic question: how does a random distribution of point masses evolve under gravity? The comparison with the expanding case may then allow us to identify the specific role of this effect. Simulations similar to ours, but in a cosmological context, were performed long ago, e.g., by Itoh et al. [4]. However, we are going to see that the general problematic we consider and the final interpretation will be substantially different.The main results are: (i) the existence of a well defined thermodynamic limit for the transient regime; (ii) the nature of the clustering process arising from the shift of the granular (non fluid-like) characteristics from small to large scales. (iii) the evolution of correlations shows self similar characteristics.c EDP Sciences

Self-gravitating systems have acquired growing interest in statistical mechanics, due to the peculiarities of the 1/r potential. Indeed, the usual approach of statistical mechanics cannot be applied to a system of many point particles interacting with the Newtonian potential, because of (i) the long range nature of the 1/r potential and of (ii) the divergence at the origin. We study numerically the evolutionary behavior of self-gravitating systems with periodical boundary conditions, starting from simple initial conditions. We do not consider in the simulations additional effects as the (cosmological) metric expansion and/or sophisticated initial conditions, since we are interested whether and how gravity by itself can produce clustered structures. We are able to identify well defined correlation properties during the evolution of the system, which seem to show a well defined thermodynamic limit, as opposed to the properties of the "equilibrium state". Gravity-induced clustering also shows interesting self-similar characteristics.

Considerable effort in modern statistical physics is devoted to the study of networked systems. One of the most important example of them is the brain, which creates and continuously develops complex networks of correlated dynamics. An important quantity which captures fundamental aspects of brain network organization is the neural complexity C(X) introduced by Tononi et al. [Proc. Natl. Acad. Sci. USA 91, 5033 (1994)]. This work addresses the dependence of this measure on the topological features of a network in the case of a Gaussian stationary process. Both analytical and numerical results show that the degree of complexity has a clear and simple meaning from a topological point of view. Moreover, the analytical result offers a straightforward and faster algorithm to compute the complexity of a graph than the standard one.

The study of the properties of cosmic structures in the universe is one of the most fascinating subject of the modern cosmology research. Far from being predicted, the large scale structure of the matter distribution is a very recent discovery, which continuosly exhibits new features and issues. We have faced such topic along two directions; from one side we have studied the correlation properties of the cosmic structures, that we have found substantially different from the commonly accepted ones. ¿From the other side, we have studied the statistical properties of the very simplified system, in the attempt to capture the essential ingredients of the formation of the observed strucures.