STATISTICAL MECHANICS

Entropy, Information, and Quantum Systems

Graduate Program in Physics — UERJ

Prof. Marcelo Santos Guimarães

2nd semester 2026 — Mondays and Wednesdays, 10:40–12:20 — 1h40 per class

Course Overview

This is a graduate course in equilibrium statistical mechanics. It develops the standard machinery — ensembles, classical and quantum gases, phase transitions, the Ising model, and the renormalization group — and then uses information theory and quantum information to clarify what that machinery means.

The material is organized into 29 lectures across seven modules, interleaved with three in-person written assessments, running from 10 August to 9 December 2026.

Lectures include short concept questions that the whole class answers and then discusses in pairs. These are not graded, but they are where the vocabulary used throughout the course — deciding whether a given claim is exact, a thermodynamic-limit statement, a controlled approximation, or a heuristic — is actually practised.

Learning Outcomes

By the end of the course, a student should be able to:

• derive the microcanonical, canonical, and grand-canonical ensembles and move between their thermodynamic potentials;
• compute fluctuations and response functions, and state when ensembles become equivalent;
• distinguish thermodynamic, Boltzmann, Gibbs, Shannon, von Neumann, entanglement, and Rényi entropies;
• use classical and quantum relative entropy, mutual information, and maximum-entropy inference in statistical-mechanical problems;
• explain why microscopic reversibility does not by itself yield macroscopic entropy increase;
• solve standard classical and quantum-gas problems, including degenerate Fermi gases and Bose–Einstein condensation;
• analyze simple phase transitions through Landau theory and the Ising model;
• explain coarse graining, fixed points, relevance, and universality in the renormalization group;
• state clearly whether a result is exact, a thermodynamic-limit statement, a controlled approximation, a model-specific example, or a heuristic argument.

Main References

• [Kardar-P] M. Kardar, Statistical Physics of Particles, Cambridge University Press, 2007.
• [PB] R. K. Pathria & P. D. Beale, Statistical Mechanics, 3rd ed., Elsevier, 2011.
• [Reichl] L. E. Reichl, A Modern Course in Statistical Physics, 4th ed., Wiley, 2016.
• [Kardar-F] M. Kardar, Statistical Physics of Fields, Cambridge University Press, 2007 — for phase transitions and renormalization.

Additional references:

• [CT] T. M. Cover & J. A. Thomas, Elements of Information Theory, 2nd ed., Wiley, 2006.
• [NC] M. A. Nielsen & I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2000.
• [Wilde] M. M. Wilde, Quantum Information Theory, 2nd ed., Cambridge University Press, 2017.
• [Witten] E. Witten, “A Mini-Introduction to Information Theory,” Rivista del Nuovo Cimento 43 (2020), arXiv:1805.11965.
• [Tong] D. Tong, Lectures on Statistical Physics — a concise standard route through ensembles, gases, and phase transitions.
• [Touchette] H. Touchette, “The large deviation approach to statistical mechanics,” Physics Reports 478 (2009) — large deviations and ensemble equivalence.
• [Seifert] U. Seifert, “Stochastic thermodynamics, fluctuation theorems and molecular machines,” Rep. Prog. Phys. 75 (2012) — entropy production and nonequilibrium work.

Assessment System

Grades come only from individual, in-person written assessments. There are three, weighted 30% / 30% / 40%.

Structure of each assessment

Every assessment has the same two parts.

Part A — Referee exercise (40 points). You receive a short draft derivation, competently written and mostly correct, containing a small number of substantive defects: statements that are wrong, or that claim a stronger status than they possess. For each defect you must

1. locate it — quote the sentence or equation;
2. diagnose it — say what is wrong, and name the assumption, limit, or approximation that has been misused or hidden;
3. repair it — give the corrected statement together with its status: exact, thermodynamic-limit, controlled approximation, mean field, model-specific exact result, or heuristic.

Not everything unusual is wrong: flagging a correct statement costs points, so justify what you flag.

Part B — Calculations (60 points). Two problems of the same kind as the exercises in the lecture notes.

Conditions. One 1h40 meeting per assessment; individual work, in person; one handwritten double-sided A4 formula sheet; no books, internet, or electronic devices.

Practice

The lecture notes contain exercises for self-study. They are not submitted and not graded: the point of the assessments is not to reproduce a solution but to diagnose an argument and state precisely what it does and does not establish. Selected problems can be discussed in optional office hours.

Module Summary

Reading map

The lecture notes are organized into twelve chapters. Modules and chapters do not correspond one to one — Module IV in particular draws on three different chapters.

Each chapter opens with a box naming the lectures it covers, and markers in the margin of the text show where one lecture ends and the next begins.

Detailed Calendar

The teaching period runs from 10 August to 12 December, on Mondays and Wednesdays. Three meetings fall on official holidays (7 September, 12 October, 2 November); 28 October (Public Servants’ Day) is kept as a reserve date until the Institute confirms whether classes occur.

Closeout: classes officially end 12/12; a replacement or final exam may be held on 14/12; corrections and grade review 15–16/12; final grades released by 17/12.

Calendar Summary