OVERVIEW

Processi dinamici e stocastici in Fisica (codice 118258) is worth 6 CFU (52 hours) and is held during the second semester of the third LT year. Lectures are delivered in Italian.

Study material is available, to all student enrolled to the course, on Aulaweb.
 

AIMS AND CONTENT

LEARNING OUTCOMES

The purpose of the class is to introduce the fundamental models and techniques to study dynamical processes in classical and quantum systems. Effects due to nonlinearities, interaction with stochastic forces and the influence of the coupling with external environments will be discussed, illustrating these topics with several examples of relevande in Physics.
 

AIMS AND LEARNING OUTCOMES

The student must reach an adequate comprehension of the theoretical models and the necessary techniques to study dynamical phenomena in classical and quantum contexts. He will also need to be able to describe, by means of these models and techniques, the applications discussed in class both in the classical (e.g. critical slowing down, brownian motion) and in the quantum (e.g. time evolution of wave packets, the Aharanov-Bohm effect, the damped quantum harmonic oscillator) settings.
 

PREREQUISITES

Prerequisites are the contents of the General Physics and of the first semester of the Quantum Mechanics classes.
 

TEACHING METHODS

Traditional: oral lectures at the blackboard with worked out  examples and applications held by the Lecturers.

SYLLABUS/CONTENT

A) Classical nonlinear dynamical systems

A.1) Dynamics in phase space:
Nonlinear autonomous dynamical systems, fixed points and their classification. Complex structures: limit cycles.
Applications: The Duffing oscillator, the Van der Pol oscillator, the Volterra-Lotka model.

A.2) Bifurcation theory:
Parametric dependence of the dynamics.
Outstanding local bifurcations: saddle-node, transcritical, pitchfork, Hopf type.
Elementary introduction to the concepts of global and dynamical bifurcation.
Applications: critical slowing down and hysteresis near a bifurcation point.

A.3) Sensitive dependence on initial conditions and chaos:
Stability of the solutions w.r.t. a variation of the initial conditions.
Lyapunov exponents.
The Poincaré-Bendixson theorem, routes to chaos.
Applications: chaos in a forced Duffing oscillator and in the Lorentz system.

B) Classical stochastic dynamics

B.1) Fundamentals:
Essentials on the axiomatic formulation of probability.
Shannon entropy.
The concept of typicality.

B.2) Equations of the classical stochastic dynamics:
The Chapman-Kolmogorov equation. 
The Fokker-Planck equation. 
The classical Langevin equation.

B.3) The Brownian motion.
Theory of the diffusion coefficient.
Fluctuation-dissipation theorem.
Applications: classical brownian motion and motion of a particle in a double-well potential in the presence of external stochastical forces.

C) Elements of quantum dynamics

C.1) Quantum propagation phenomena:
General concepts on the time evolution in quantum mechanics.
Wavepacket evolution (in position and momentum representation).
The free wavepacket.
Applications: free Gaussian wavepacket, interaction with an infinite wall, confinement in a harmonic potential.

C.2) Quantum propagator and path integrals:
Generalities on the concept of propagator in quantum mechanics.
Applications: the free particle propagator, the quantum harmonic oscillator propagator.
From the propagator to the Feynman path integrals.
Applications: interpretation of the classical limit of quantum mechanics through path integrals, the Aharonov-Bohm effect.

C.3) Open quantum systems:
Introduction to open quantum systems and to the concept of dissipation in quantum mechanics.
The quantum Langevin equation.
Applications: the quantum Brownian motion, the damped quantum harmonic oscillator.
 

RECOMMENDED READING/BIBLIOGRAPHY

S. Strogatz, "Nonlinear Dynamics and Chaos" (CRC Press)
H. -P. Breuer and F. Petruccione, "The theory of Open Quantum Systems" (Oxford)
K. Konishi and G. Paffuti, "Quantum Mechanics - a new introduction"

Further reading material will be suggested by the Lecturers along the course.
 

TEACHERS AND EXAM BOARD

LESSONS

LESSONS START

According to the academic calendar approved by the Consiglio di Corso di Studi.

Class schedule

The timetable for this course is available here: Portale EasyAcademy

EXAMS

EXAM DESCRIPTION

Oral exam.

ASSESSMENT METHODS

The oral examination will allow to assess the knowledge of the physical models and techniquest discussed curing the lectures and the ability to identify their role in the explanation of the several different applications discussed in the course.