## Learning objectives

At the end of the course, the student will know the main techniques and advanced issues of equilibrium and non equilibrium statistical mechanics, with applications in problems that are current topics of research. He will know the fields of application of statistical mechanics, from condescend matter physics, to field theory, up to interdisciplinary applications in social networks, and biological systems. He will be able to apply this knowledge to selected problems in these fields.

## Prerequisites

Statistical Physics and Classical and Quantum Complex Systems

## Course unit content

"Natural phenomena that we observe occur at very different scales of length, time and energy, ranging from subatomic scale to that intergalactic. Surprisingly it is possible and in many cases

essential to discuss these levels independently."

Statistical Physics deals instead of how to cross scale and how to go from one scale to another.

Statistical Physics studies indeed systems composed of many interacting degrees of freedom. Starting from microscopic models, it is able to describe new and unexpected behavior on a large scale, in which collective effects are more than the sum of the behaviors of the individual constituents.

In this course, that is the natural extension of the course of Statistical Physics, we will address advanced topics on equilibrium and non equilibrium statistical mechanics. We will discuss phase transitions and properties of universality, out of equilibrium dynamics, the effects of disorder and complex networks.

Although the course is included in the curriculum of Theoretical Physics, the topics are inherently interdisciplinary with wide application outcomes, ranging from condensed matter physics, to quantum systems, up to recent applications to biological systems and social networks.

## Full programme

- Universality in Statistical Mechanics. Statistical mechanics and probability theory. Thermodynamic limit phase transitions and singularities of the free energy. Order parameters and spontaneous symmetry breaking. Energy-entropy arguments. Low temperature expansions. Peierls-Griffith proof. Lee-Yang theorem. Magnetic models with discrete symmetry: combinatorial and algebraic approach, transfer matrox. Variational methods and Bogoliubov inequalities. Mean Field methods. Systems at the critical point, scaling and universality hypothesis. Field theory at the critical point: Landau functional and Ginzburg Landau approach. Critical dimensions. The Wilson renormalization group and the interpretation of universality.

- Out of equilibrium.

Microscopic reversibility and macroscopic irreversibility. Linear response and the fluctuation dissipation theorem. Onsager relations. The stochastic dynamics. Brownian motion. Langevin equation and Fokker Planck equation. Stochastic differential equations and supersymmetric models. Anomalous transport and diffusion processes. Random walks and Lévy walks. Problems first passage and entrapment. Pattern formation and Turing instability.

- Complex networks.

The formalism of graph theory. Random graphs and deterministic networks. Growth patterns of complex networks. Geometric and topological characterization of complex networks: degree, betweeness, clustering, small world effect and scale free. Spectral properties of finite and infinite graphs. Models of statistical mechanics on complex networks. Dynamics on networks: oscillations, diffusion, propagation of information and models of infection. Synchronization of complex networks.

- The effects of disorder: how disorder changes the physical properties

How to describe disorder. Quenched and annealed averages. Harris criterion. Griffith singularities . Spin models on random field: supersymmetry and dimensional reduction. Models of spin glass and random optimization problems. Phase transitions in algorithmic complexity classes. Percolation.

## Bibliography

Lecture notes

## Teaching methods

General theoretical issues will be presented during the lectures, with reference to specific physical systems to which to apply the techniques studied. The course includes many examples applied to case studies in various areas of research.

## Assessment methods and criteria

This is an advanced course, so that learning will be assessed through a presentation by the students of one of the theoretical arguments discussed in class, applied to a particular physical system. This will check for the understanding of the topics and the ability to apply the knowledge gained.

## Other information

- - -