Maurizio Bottaccio

Topological approach to neural complexity.

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.

Clustering in N-body gravitating systems

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.

Clustering in gravitating N-body systems

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.

The Holtsmark distribution of forces and its role in gravitational clustering

The evolution and the statistical properties of an infinite gravitating system represent an interesting and widely investigated subject of research. In cosmology, the standard approach is based on equations of hydrodynamics. In this paper, we analyse the problem from a different perspective, which is usually neglected. We focus our attention on the fact that at small scale the distribution is point-like, or granular, and not fluid-like. The basic result is that the discrete nature of the system is a fundamental ingredient in understanding its evolution. The initial configuration is a Poisson distribution in which the distribution of forces is governed by the Holtsmark function. Computer simulations show that the structure formation corresponds to the shift of the granularity from small to large scales. We also present a simple cellular automaton model that reproduces this phenomenon.

Gravitational clustering in N-body simulations

In this talk we discuss some of the main theoretical problems in the understanding of the statistical properties of gravity. By means of N-body simulations we approach the problem of understanding the role of gravity in the clustering of a finite set of N-interacting particles which samples a portion of an infinite system. Through the use of the conditional average density, we study the evolution of the clustering for the system putting in evidence some interesting and not yet understood features of the process.

Clustering in galaxy distribution: comparison between redshift surveys

The characterization of the galaxy clustering properties in the universe is one of the most interesting and controversial issues in modern cosmology. Here we present the results of the analysis of the recently available three-dimensional catalogues of galaxies, performed with the tools of modern statistical analysis. We focus on the problem of determination of crossover scale to homogeneity in the distribution of luminous matter.