# STATISTICAL PHYSICS II

## Learning outcomes of the course unit

Knowledge and understanding: At the end of the Statistical Physics course 2, the student will acquire advanced theoretical knowledge in the field of equilibrium and non-equilibrium statistical physics, with particular regard to phase transitions. She/he will know the main connections between the approaches and techniques of statistical physics and field theories and will understand its extensions to interdisciplinary research topics. She/he will also be able to understand the essential lines and content of research papers in the most recent literature in the field of statistical physics.

Applied knowledge and understanding: The student will acquire the ability to apply his / her knowledge to the modeling and analysis of statistical systems in condensed matter physics and in field theories. He will also be able to apply the approaches and techniques learned in the field of statistical physics to interdisciplinary problems, with particular regard to biological systems and computer science problems.

Making judgments: At the end of the course, the student will acquire the tools to understand how the theoretical approaches and the techniques studied can be used to study statistical systems, even in areas of original research. She/he will be able to identify the essential components of a systems, to develop a statistical model of a physical system and she/he will be able to evaluate the limits and potentials of the mathematical techniques described during the course.

Communication skills: The student will be able to effectively illustrate, also using computer support, the topics studied and will be able to explain the basic topics of the course in a clear way, also addressing non-specialists in the field.

Learning skills : The student will be ready to tackle advanced topics of statistical physics that are currently subject of scientific research and will have acquired the ability to undertake further studies in this field with a high degree of autonomy.

## Prerequisites

Recommended: Statistical Physics and Complex Systems

## Course contents summary

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 with how to cross scale and how to go from one scale to another, and how to deal with situations where this scale separation cannot work. It deals with systems composed of many interacting degrees of freedom.

Starting from microscopic models, it is able to describe new and unexpected behaviors 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 Statistical Physics course, we will address advanced topics on equilibrium and non equilibrium Statistical Physics. We will discuss phase transitions and universality in critical phenomena, the effects of disorder and its application to the statistical physics of learning, all discussed in an interdisciplinary perspective, as Statistical Physics is.

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

## Course contents

** Phase Transitions and Universality in Statistical Physics**

A summary of Statistical mechanics and probability theory. Thermodynamic limit, phase transitions and singularity of free energy. Order parameters and spontaneous symmetry breaking. Ergodicity breaking. Energy-entropy arguments. Low temperature expansions. The Peierls-Griffith proof for Ising. Lee-Yang's theorem and the zeros of the partition function. Discrete symmetry magnetic models: combinatorial solution and transfer matrix. Variational methods and the Bogoliubov Inequality. Mean field and Ising phase diagram in the mean field approximation. Curie Weiss Model. Systems at critical point, scaling hypotheses and universality. Field theories at the critical point: Landau Functional and the Landau Ginzburg's approach. The phi ^ 4 Model. Upper and lower Critical Dimensions. The renormalization group and the interpretation of universality: block transformations, renormalization flow and scaling . The renormalization group in k space. Relevant operators. Statistical Physics Models on Complex Networks and Phase Transitions on Graphs. First order phase transitions.

** Disorder **

Extensibility, additivity and self-averaging. Frustration and disorder. Quenched and annealed average. Disordered ferromagnets: dilution of sites. Harris Criterion for the relevance of disorder and Its Application. The Random Field Ising Model and the Criterion of Imry and Ma.

Supersymmetry and dimensional reduction. Introduction to spin glass. Overlaps between states and definition of P (q). Distance between states, hierarchical structure of spin glass states, and ultrametricity. Replica Method. Cavity Methods. Statistical Physics of Learning. Feed forward and recurrent Neural Networks. Hopfield's Model. Mean Field solutions for Finite Number of Patterns. Associative Memory in Neural Networks. Spin glasses and random optimization problems. K-Sat and phase transitions in classes of algorithmic complexity.

** Out of equilibrium **. Synchronization on networks and the Kuramoto model . Pattern formation and Turing's instability.

## Recommended readings

- Nigel Goldenfeld: Lectures on phase transitions and the Renormalization Group

Addison Wesley (cap 1,2,3,8,9)

- Le Bellac, Mortessagne and Batrouni - Equilibrium and non-equilibrium statistical thermodynamics,

Cambridge University Press (Cap. 4)

- Luca Peliti: Appunti di Meccanica Statistica, Bollati Boringhieri

(Cap. 5,6,10)

- R. Livi, P. Politi: Non equilibrium Statistical Physics: a Modern Perspective, Cambridge UP (cap 7)

Selected parts of textbooks and 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

Being an advanced course, learning will be verified through a presentation of one of the theoretical arguments discussed during the course, and typically applied to a particular physical system. This may require the study of recent research literature, which will be discussed with the teacher. This wil allow to verify the student's understanding of the arguments, the ability to apply the acquired knowledge and the ability to present them effectively and critically.