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 knoledge 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 felds ranging from “classical” subjects like difusionn quantum mechanics andn more in generaln to the physics of complex systems. Particular emphasis will be placed on the relationships between diferent topics allowing for a unifed 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 problemn the modeling and the solution thereofn 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 universalityn stochastic processes and emergent phenomenan which justify the use of feld theoretical models of interacting systems. 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 systemsn which fnd applications even outside the physical context in which they arose.

Lecture supported by tutorial, assignment, analytical and numerical problems

Introduction; "The Unreasonable Efectiveness of Mathematics in the Natural Sciences (Wigner 1959)"; Gaussian integralsn Wick theorem Perturbation theoryn connected contributionsn Steepest descent Legendre transformationn Characteristic/Generating functions of general probability distributions/measures The Wiener integraln geometric characteristics of Brownian paths and Hausdorf/fractal dimension Brownian paths and polymer physicsn biopolymer elasticity. The random walk generating functionn the Gaussian feld theory and coupled quantum harmonic oscillators Levy walksn 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 expansionn Universalityn critical dimensions Generalized difusion and stochastic diferential equations Path integrals representation of stochastic processes with general difusion operator (Brownian motion in curved spaces) The Feynman-Kac formula: difusion with sinks and sources Quantum mechanics (solvable modeln harmonic oscillatorn free particle) Feynman path integrals and the quantum version of the Feynman-Kac formula. Quantum vs stochastic phenomena: quantum tunneling and stochastic tunneling Stochastic amplifcation and stochastic resonance Nonperturbative methods, instantons Difusion in random media and anomalous difusion Quantum Mechanics in a random potentialn 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.