Out of equilibrium physics is a fascinating and multi-disciplinary topic that finds applications in various contexts, where standard equilibrium approach cannot be applied. The course is meant to provide an introduction to some general theoretical methods and techniques to approach out of equilibrium problems. A particular focus will be devoted to quantum systems, with examples and a hands-on approach to solve problems in condensed matter physics and to discuss their experimental realization.

After this course the student is expected to learn some general theoretical methods and techniques to approach problems of out of equilibrium physics. Furthermore, the student will be able to apply these methods to some problems of condensed matter physics, and to explicitly compute quantities that are experimentally measurable.

Quantum Mechanics and Equilibrium Statistical Mechanics

Part 1: The Classical Boltzmann Equation -Short review of motion of a classical particle under a stochastic force, the phenomenological Drude model for transport. -Construction of the Boltzmann equation, hypothesis, the collision integral, and properties. -Example: The Boltzmann Equation in a random potential. The diffusion limit and Brownian motion Part 2: Towards out of equilibrium physics of quantum systems -Quantum ingredients in the Semi-Classical Boltzmann Equation -The relaxation-time approximation. -Linear response theory, Fluctuation-dissipation theorem. Onsager relations and its applications -Applications to Condensed Matter: derivation of Ohm’s law, scattering processes of electrons in solids, charge and heat transport Part 3: Quantum approaches to out of Equilibrium Quantum Systems A) The coherent limit. Introduction to quantum mesoscopic physics. Scattering matrix formalism, the Landauer-Büttiker approach to compute the electron current in mesoscopic systems. The quantum of resistance. Examples of quantum interference phenomena in nanosystems (Aharonov-Bohm effect, Weak localization). Optional: The noise of electrons. Josephson effect and mesoscopic effects with superconductors B) Effects of dissipation and decoherence. -Phenomenological approach: the T1-T2 model, applications to a two-level system exposed to a time-dependent perturbation Depending on student interest: -Microscopic approach 1: The Density matrix formalism, applications to a two-level system coupled to a thermal bath. Markov limit, reduced density matrix, the Quantum Optical Master Equation. -Microscopic approach 2: The Keldysh-Schwinger formalism and non-equilibrium Green functions. Applications to quantum transport in nanodevices: the Meir-Wingreen formula for conductance, the Coulomb blockade effect.

51 ore: Lezioni in Aula 9 ore: Esercitazioni in Aula

R. Zwanzig, Nonequilibrium Statistical Mechanics (Oxford Univ Press) L. E. Reichl, A modern course in Stitistical Physics (John Wiley & Sons) J. Rammer, Quantum Field Theory of nonequilibrium states (Cambridge Univ. Press) A. Kamenev, Field Theory of Nonequilibrium systems (Cambridge Univ. Press) H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics and Financial markets (World Scientific) Haug-Jauho, Quantum kinetics in Transport and optics of semiconductors (Springer)

Slides; Dispense;

Modalità di esame: Prova orale obbligatoria;

Exam: Compulsory oral exam;

... The exam is based on an oral test at the blackboard, with questions on 2-3 topics of the material discussed during the course. The first topic can be chosen by the student. The knowledge of the out of equilibrium physics methods is tested by asking the student to illustrate the key concepts, to derive proofs of the main results, and to apply the techniques to some examples and applications discussed during the course.

Gli studenti e le studentesse con disabilità o con Disturbi Specifici di Apprendimento (DSA), oltre alla segnalazione tramite procedura informatizzata, sono invitati a comunicare anche direttamente al/la docente titolare dell'insegnamento, con un preavviso non inferiore ad una settimana dall'avvio della sessione d'esame, gli strumenti compensativi concordati con l'Unità Special Needs, al fine di permettere al/la docente la declinazione più idonea in riferimento alla specifica tipologia di esame.