ELEMENTI DI FISICA STATISTICA E TEORIA DELL'INFORMAZIONE

The course introduces concepts of statistical mechanics, as well as the necessary theoretical technical background. The Information Theoretical Approach is followed, allowing an unified treatment of classical and quantum statistics. The course also provides the background for the understanding of concepts in quantum information theory, which is a timely topic both from the fundamental point of view and for applications, and which students will encounter in their subsequent studies.

• Knowledge and understanding – Knowledge of the main ideas and theoretical techniques in statitical mechanics for the representation of complex quantum system and the study of their dynamics. Knowledge of some numerical techniques with the software Wolfram-Mathematica.
• Applying knowledge and understanding – Ability in the application of basic theoretical techniques and approximate schemes for the analysis and the simulation of systems of interest for statistical mechanics.
• Communication skills – Ability in communication in the field of Statistical Mechanics and Information Theory.
• Learning skills – Acquiring skills allowing the continuous upgrade of the knowledge in the field, by accessing a research environment and specialized literature.

The course is structured in three main parts: (1) Foundational ideas in Statistical Mechanics and Information Theory; (2) Equilibrium statistical mechanics; [3] Nonequilibrium Statistical Mechanics.

1. Preliminary concepts
   Statistical physics: handling incomplete knowledge. Elements of probability Theory. Information: definition, information associated to a discrete/continuous probability. Density Matrix.
2. Statistical Mechanics
   Classical Mechanics, canonical transformations, calssical statistics and information. Quantum Mechanics, quantum statistics and information. Maximum missing information principle.
3. Equilibrium
   Past, future and irreversibility. Integrals of motion and thermal equilibrium. Derivation of the equilibrium state, as an optimization problem. Temperature. Adiabatic theorem, work and heat, ideal thermal machines.
4. Identical Particles
   Grand-canonical approach in classical statistical mechanics. Ideal quantum gas. Second quantization and grand-canonical approach.
5. Systems out of equilibrium
   Small deviations from equilibrium: Onsager relations, Einstin relation, fluctuation-dissipation theorem. Nonequilibrium: equations of motion, Boltzmann equation, phenomenological increase of entropy (H-theorem). Exact relations: Jarzynski relation and Crooks fluctuation theorem.

[1] Amnon Katz, Principles of Statistical, Mechanics. The Information Theory Approach, Freeman, San Francisco, 1967
[2] Carlo Di castro e Roberto Raimondi, Statistical Mechanics and Applications in Condensed Matter, Cambridge University Press, 2015.
[3] Stephen Wolfram, An Elementary Introduction to the Wolfram Language, Cambridge University Press, 2015.