Organizzazione della Didattica


Models of theoretical physics


Physics of matter


Frontali Esercizi Laboratorio Studio Individuale
ORE: 48 0 0 102


I anno1 semestre







Calendario Attività Didattiche



caratterizzanteTeorico e dei fondamenti della fisicaFIS/026

Responsabile Insegnamento

Prof. MARITAN AMOSFIS/03Dipartimento di Fisica e Astronomia "Galileo Galilei" - DFA

Altri Docenti

Prof. BAIESI MARCOMutuazioneFIS/02Dipartimento di Fisica e Astronomia "Galileo Galilei" - DFA

Attività di Supporto alla Didattica

Non previste


Good knowledge of mathematical analysis, calculus, elementary quantum mechanics and basic physics.

The purpose of the course is to provide the student with a wide vision on how theoretical physics can contribute to understand phenomena in a variety of fields ranging from “classical” subjects like difusionn quantum mechanics and more in general to the physics of complex systems. Particular emphasis will be placed on the relationships between different topics allowing for a unified mathematical approach where the concept of universality will play an important role. The course will deal with a series of paradigmatic physical systems that have marked the evolution of theoretical physics in the last century including the most recent challenges posed by disordered systems with applications to machine learning and neural networks. Each physical problem the modeling and the solution thereof will be described in detail using powerful mathematical techniques. The frst part of the course will provide the basic mathematical tools necessary to deal with most of the subjects of our interest. The second part of the course will be concerned with the key concepts of universality stochastic processes and emergent phenomena which justify the use of field theoretical models of interacting systems and tools like the renormalization group techniques. In the third part it will be shown how solutions of quantum systems can be mapped in solutions of difusion problems and vice versa using common mathematical techniques. The last part will deal with the most advanced theoretical challenges related to non-homogenous/disordered systems, which find applications even outside the physical context in which they arose.

Lecture supported by tutorial, assignment, analytical and numerical problems

Introduction; "The Unreasonable Effectiveness of Mathematics in the Natural Sciences (Wigner 1959)"; Gaussian integrals Wick theorem Perturbation theory connected contributions Steepest descent Legendre transformation Characteristic/Generating functions of general probability distributions/measures The Wiener integral geometric characteristics of Brownian paths and Hausdorff/fractal dimension Brownian paths and polymer physics biopolymer elasticity. The random walk generating function, the Gaussian field theory and coupled quantum harmonic oscillators Levy walks violation of universality Field theories as models of interacting systems O(n) symmetric Phi^4– theory. The large n limit: Spherical (Berlin-Kac) model and 1/n expansion. Perturbative expansion. Introduction to renormalization group techniques and universality. Generalized diffusion and stochastic differential equations. The Feynman-Kac formula: diffusion with sinks and sources Feynman path integrals and the quantum version of the Feynman-Kac formula. Quantum mechanics (solvable model: free particle, harmonic oscillator) Quantum vs stochastic phenomena: quantum tunneling and stochastic tunneling Stochastic amplification and stochastic resonance Non-perturbative methods: instantons Diffusion in random media and anomalous diffusion Quantum Mechanics in a random potential localization and random matrices Statistical physics of random spin systems and the machine-learning problem Random energy model, replica trick Cavity method, Random Field Ising Model

Final examination based on: Written and oral examination and weekly exercises proposed during the course

Critical knowledge of the course topics. Ability to present the studied material. Discussion of the student project.