
Course unit
MODELS OF THEORETICAL PHYSICS
SCP8083597, A.A. 2019/20
Information concerning the students who enrolled in A.Y. 2019/20
ECTS: details
Type 
ScientificDisciplinary Sector 
Credits allocated 
Core courses 
FIS/02 
Theoretical Physics, Mathematical Models and Methods 
6.0 
Course unit organization
Period 
First semester 
Year 
1st Year 
Teaching method 
frontal 
Type of hours 
Credits 
Teaching hours 
Hours of Individual study 
Shifts 
Lecture 
6.0 
48 
102.0 
No turn 
Prerequisites:

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

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 nonhomogenous/disordered systems, which find applications even outside
the physical context in which they arose. 
Examination methods:

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

Critical knowledge of the course topics. Ability to present the studied material.
Discussion of the student project. 
Course unit contents:

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 (BerlinKac) model and 1/n
expansion.
Perturbative expansion. Introduction to renormalization group techniques and universality.
Generalized diffusion and stochastic differential equations. The FeynmanKac formula: diffusion with sinks and sources
Feynman path integrals and the quantum version of the FeynmanKac formula.
Quantum mechanics (solvable model: free particle, harmonic oscillator)
Quantum vs stochastic phenomena: quantum tunneling and stochastic tunneling
Stochastic amplification and stochastic resonance
Nonperturbative 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 machinelearning problem
Random energy model, replica trick
Cavity method, Random Field Ising Model 
Planned learning activities and teaching methods:

Lecture supported by tutorial, assignment, analytical and numerical problems 
Textbooks (and optional supplementary readings) 

Innovative teaching methods: Teaching and learning strategies
 Lecturing
 Problem solving
Innovative teaching methods: Software or applications used
 Moodle (files, quizzes, workshops, ...)

