## Learning outcomes of the course unit

At the end of the course

The student will know different models of equilibrium and non-equilibrium statistical mechanics, learning both analytical and numerical techniques.

He will be able to understand how such models can be applied to different systems both in physical fields but also in interdisciplinary applications of biology, sociology, economics and informatics.

He will also be able to apply the numerical and analytical techniques for the analysis of the statistical physics model.

## Course contents summary

The course is devoted to the study of different kinds of systems showing typical complex behaviours due to the presence of a large number of degrees of freedom. We will discuss several theoretical models adopting both analytical and numerical techniques; our aim is to find the phenomenological laws describing the global behaviour of such systems. First, we will treat purely statistical models, then we will focus on stochastic dynamics, finally we will consider graphs and complex networks. 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. Application of dynamical mean field to quantum systems Gutzwiller equation and discrete nonlinear Schrodeinger equation for bosons on lattices. Synchronization and Kuramoto model. Neural networks dynamics.

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.

## 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 (15 pts). In the second consists of an oral exam focused on the key points of the course (15 pts).