Docenti: prof. Giacomo Guarnieri, Università di Pavia

Numero ore: 14

Tipo di didattica: lezioni frontali

Programma:
This short course investigates the deep relationship between Thermodynamics, information theory, and computation, using Maxwell’s demon as a unifying theme.
Starting from elementary Thermodynamics and probability theory, the course develops the formalism of classical information theory (Shannon entropy, relative entropy, mutual information, data-processing inequalities) and shows how these purely mathematical quantities are actually extremely tied to our physical world, as they constrain energy interconversion, extractable work, and irreversibility. Maxwell’s demon and the Szilard engine are treated as concrete models in which
information about microscopic states is used to design feedback protocols for work extraction.
A central focus will be Landauer’s principle: the minimal thermodynamic cost of logically irreversible operations such as bit erasure. The course explores its derivation and its consequences for classical computation, including the theoretical limits of energy-efficient computing and the role of reversible computation. In the final part, the course introduces a few basic elements of quantum mechanics required to reframe the problem into the quantum realm, discussing how the information-thermodynamic bridge becomes even richer and deeper in this case.
The course is designed for 12–20 hours of lectures, and, although particular emphasis will be placed on mathematical rigour, it will be self-contained: no prior quantum mechanics is assumed, and the necessary probabilistic and linear-algebraic background is reviewed as needed.

Syllabus (list of topics)
The material below can be re-organized depending on allotted hours.

1. Foundations of thermodynamics and statistical mechanics
   ◦ Macroscopic thermodynamic quantities: internal energy, work, heat, entropy.
   ◦ First and second laws; entropy production and irreversibility.
   ◦ Microscopic description: phase space, Hamiltonian, canonical ensemble.
2. Probability theory and classical entropy
   ◦ Discrete probability distributions, expectation, conditional probability.
   ◦ Shannon entropy H(X), joint and conditional entropy.
   ◦ Relative entropy (Kullback–Leibler divergence) D(p∥q).
   ◦ Basic inequalities: non-negativity, convexity, chain rule.
3. Classical information theory and thermodynamic inequalities

◦ Mutual information I(X;Y) and conditional mutual information.
◦ Data-processing inequality and its implications.
◦ Relative entropy and elements of non-equilibrium thermodynamics.
◦ Entropic formulations of the second law; information as a resource.

4. Maxwell’s demon and the Szilard engine: classical analysis
   ◦ Formulation of Maxwell’s thought experiment.
   ◦ The Szilard engine as a single-molecule model.
   ◦ Detailed work and entropy balance over a demon-controlled cycle.
   ◦ Feedback control and the role of information in work extraction.
5. Landauer’s principle and classical computation
   ◦ Logical vs physical irreversibility; model of bit erasure.
   ◦ Derivation of Landauer’s bound W≥kB T ln 2 for bit erasure.
   ◦ Generalizations: erasure of biased bits and many-bit memories.
   ◦ Thermodynamic analysis of classical computation:
   ▪ Logical operations as transformations on probability distributions.
   ▪ Landauer cost of irreversible gates.
   ▪ Reversible computation (e.g. Toffoli gate) and its thermodynamic
   implications.

◦ Conceptual consequences: limits on energy-efficient computing.

6. Stochastic thermodynamics and fluctuation relations
   ◦ Trajectory-level description of work, heat, and entropy production.
   ◦ Jarzynski equality and Crooks fluctuation theorem (classical versions).
   ◦ Reformulation of the second law in terms of fluctuations.
   ◦ Feedback-controlled processes and information-modified second laws (conceptual level).
7. Introductory quantum perspective on information and thermodynamics
   ◦ Minimal quantum formalism: states as density matrices; measurements as projectors/
   POVMs.
   ◦ Von Neumann entropy and quantum relative entropy (definitions, basic properties).

◦ Quantum Gibbs states and quantum free energy.
◦ Quantum version of Landauer’s principle (statement and qualitative derivation).
◦ Comparison between classical and quantum demons:
▪ What changes (superposition, quantum friction, measurement disturbance).
▪ What remains structurally the same (information–entropy–work relations).
◦ Research perspectives.

Calendario:
Infrasettimanali serali 11 – 13 – 19 – 20 – 21 maggio (2 ore ciascuno, alle 20.30)
1 appuntamento di 4 ore Sabato 16 maggio (ore 10-12 e 14-16)
Aula B. Rossi