# COMPLEX SYSTEMS

## Learning outcomes of the course unit

The student will know: several model of statistical physics of interdisciplinary relevance, non-equilibrium statistical physics evidencing conceptual differences with respect the equilibrium case, network theory focusing on relevant themes in statistical physics.

The student will be able to apply numerical and analytical techniques for analyzing statistical physics models. Particular relevance is given to numerical simulations.

Making judgements: the student will understand the possible applications of the subjects of the course to several physical systems and to interdisciplinary fields such as biology, social and economic sciences and computer science.

The student will be able to illustrate effectively, using also computer facilities, the results of numerical simulations. Moreover, the student will be able to present clearly the basic topics of the course in an interdisciplinary way addressing not only to a specialized audience.

The student will be ready to face advanced subjects in statistical physics that are now subject of scientific research.

## Prerequisites

Basic concepts of statistical physics and phase transitions.

## Course contents summary

The course is devoted to the study of different kinds of systems showing typical behaviors of complex systems. We will discuss several theoretical models adopting both analytical and numerical techniques. First, we will treat purely statistical models, then we will focus on stochastic dynamics, finally we will consider graphs and complex networks. The role of stochasticity and the effects due to the dynamics will be topics of particular theoretical relevance. We will discuss applications in the fields of physics, biology, epidemics, informatics and economy. Due to the interdisciplinary nature of the subjects and to the different possible applications, the course is recommended to students in different fields.

## Course contents

1 Equilibrium statistical mechanics

Ensemble theory, mean field and phase transitions. Finite-size scaling and Binder cumulant. Focus on some statistical models relevant for their phenomenology and for their applications: interdisciplinary applications of the Ising model, Potts, p-spin, Hopfield model, XY model (Kosterlitz Thouless transition), polymeric chains, percolation.

2 Dynamics

Montecarlo method, detailed balance. Master equations and random walks. Brownian motion, Langevin and Fokker-Plank equations. Out of equilibrium systems. Arrhenius law. Time dependent linear response theory. Transport and Einstein equation. Entropy productions in time dependent dynamics. Subdiffusion in continuous time random walks and Superdiffusion in Lévy walks. Slow dynamics: coarsening of magnetic domains for the Ising model, dynamical scaling exponent.

Purely dynamical models: SIS and SIR models in epidemics. Contact process and directed percolation as a paradigm of dynamical phase transition. Dynamical mean field. Voter model. Sand-pile model and self-organized criticality. Synchronization and Kuramoto model. Neural networks dynamics.

Surface growth and KPZ model.

Application of dynamical mean field to quantum systems Gutzwiller equation and discrete nonlinear Schrodeinger equation for bosons on lattices.

3 Graphs and complex networks

Definition of graph: degree, radius, adjacency matrix. Linear models on graphs: harmonic oscillators, electric networks and random walks. Fractal dimension and spectral dimension. Anomalous diffusion on fractals. Complex networks, small world and scale free network (Watts-Strogatz e di preferential attachment). Study of some statistical models on complex networks: percolation and epidemic models.

4 Applications. In the different subjects we discuss applications in physics but also in interdisciplinary fields: biology, epidemics, informatics and sociology.

## Recommended readings

Lecture notes.

L. Peliti Appunti di Meccanica statistica.

B. Cowan Topics in Statistical Mechanics.

R. Livi e P. Politi Nonequilibrium statistical physics.

M. Henkel, H. Hinrichsen e S. Lübeck Non-Equilibrium Phase Transitions.

J. Klafter and I.M. Sokolov First Steps in Random Walks.

A. Barrat, M Berthélemy e A. Vespignani Dynamical processes in Complex Networks.

## Teaching methods

Class lectures. Moreover, the student will prepare a simple numerical simulation on a subject of the course, in order to get acquainted with the numerical and analytical techniques. Possible criticalities in the numerical simulations can be discussed with the teacher outside of the class lectures.

## Assessment methods and criteria

The examination consists of an oral proof divided into two parts. In the first the student will presents the results of the numerical simulations prepared during the course (20 pts). In the second consists of an oral exam focused on the key points of the course (10 pts).