# ELEMENTS OF STATISTICAL PHYSICS AND INFORMATION THEORY

**Academic Year 2022/2023**- Teacher:

**GIUSEPPE FALCI**

## Expected Learning Outcomes

The course introduces concepts of statistical mechanics and the necessary theoretical background. We adopt the Information Theoretical approach with a unified treatment of classical and quantum statistics. The course also provides the basis for the understanding of concepts in quantum information and quantum thermodynamics. Both are timely topics from the fundamental point of view and for applications, which students will encounter in their subsequent studies.

- Knowledge and understanding – Knowledge of the main ideas and theoretical techniques in statistical 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 fundamental theoretical techniques and approximate schemes for the analysis and the simulation of systems of interest for statistical mechanics.
- Communication skills – Ability in communicating 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.

## Course Structure

- Standard lectures, exercises and demonstrations with dedicated software (Mathematica). Seminars by experts will be organized.
- Should the circumstances require online or blended teaching, appropriate modifications to what is hereby stated may be introduced, in order to achieve the main objectives of the course.

## Required Prerequisites

Courses of classical mechanics, elementary quantum mechanics, condensed matter physics, lineart algebra, calculus, mathematical methods for physics.## Attendance of Lessons

Attending the course is compulsory, according to the Regolamento Didattico.## Detailed Course Content

- Preliminary concepts Goal of statistical mechanics. Handling incomplete information. Elements of kinetic theory and classical transport. Informazione: definition, information associated with a probability (discrete and continuous). Thermodynamics: from principles to thermodynamic potentials.
- Classical Statistical mechanics: equilibrium Canonical formalism. Past, future and irreversibility. Conserved quantities and thermal equilibrium. Principle of maximal (missing) information. Existence and unicity of the solution. Relation with thermodynamics: temperature, adiabatic theorem, work and heat, ideal thermal machines. Equipartition theorem in linear systems. Gibbs paradox. Discrete systems: paramagnets, impurities, mean-field theory, Ising model. Gran-canonical ensemble.
- Quantum Statistical mechanics: equilibrium density Matrix. Principle of maximal information. Distinguishable particles: spin systems and quantum computers. Identical particles, ideal quantum gas in second quantization (grand canonical). Fermi gas and metals. Bosons: phonons and specific heat, photons and Bose-Einstein condensation.
- Selected topic (only one!) -- Physical basis of the postulates: statistical ensembles, decoherence. Small deviations from equilibrium: Onsager relations, Einstein relation, fluctuation-dissipation theorem. Nonequilibrium: Boltzmann equations and H theorem. Jarzynski relation and Crooks fluctuation theorem. Landauer principle and Szilard engine.

## Textbook Information

[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] G. Falci, Lecture notes on Statistical Physics and Information Theory, 2020.

[4] D. Arovas, Lecture Notes on Thermodynamics and Statistical Mechanics (A Work in Progress), available on line, 2019.

[5] K. Huang, Introduction to Statistical Physics, Chapman & Hall, 2010.

[6] Stephen Wolfram, An Elementary Introduction to the Wolfram Language, Cambridge University Press, 2015.

[7] G. Baumann, Mathematica for Theoretical Physics, Springer, 2005.

## Course Planning

Subjects | Text References | |

1 | Preliminary topics | [1,2,4,5] |

2 | Classical statistical mechanics | [1,3,5] |

3 | Quantum statistical mechanics | [1,2,3] |

4 | Selected topics | [1,2,3] and specific material |

5 | Applications with Mathematica | [6,7] |