
Course unit
MODELS OF THEORETICAL PHYSICS
SCP8083597, A.A. 2018/19
Information concerning the students who enrolled in A.Y. 2018/19
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 
Start of activities 
01/10/2018 
End of activities 
18/01/2019 
Examination board
Examination board not defined
Prerequisites:

Good knoledge 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 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 nonhomogenous/disordered systemsn which fnd 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 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 (BerlinKac) 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 FeynmanKac formula: difusion with sinks and sources
Quantum mechanics (solvable modeln harmonic oscillatorn free particle)
Feynman path integrals and the quantum version of the FeynmanKac 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 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 based learning
 Problem solving

