NONEQUILIBRIUM THERMODYNAMICS AND INFORMATION THEORY

In modern physics, the study of systems far from equilibrium and the role of information have become essential for understanding small-scale physical phenomena, where fluctuations dominate and necessitate an extension of classical thermodynamics.

This course introduces the mathematical methods and core concepts of stochastic thermodynamics and information theory. Topics range from the foundations of stochastic calculus to nonequilibrium fluctuation theorems (Jarzynski, Crooks, TURs), extending to the thermodynamic analysis of measurement and feedback (Sagawa-Ueda theorem), and the study of ensemble dynamics via information-theoretic metrics and geometric bounds (Thermodynamic Speed Limits).

Aimed at students and researchers with a theoretical or experimental-modeling focus, the course equips participants with the analytical tools to understand complex systems at the nanoscale and to rigorously evaluate the thermodynamic costs associated with control, precision, and information processing.

Students will be able to apply stochastic calculus and fluctuation theorems to nonequilibrium systems, rigorously quantifying the thermodynamic costs of information, control, and precision.

Basic knowledge of probability, statistical mechanics, and stochastic processes as provided in the Master's courses.

Introduction to stochastic calculus. Discrete and Continuous stochastic processes. Ito’s Lemma, and Ito vs. Stratonovich descriptions. Multidimensional Langevin equations and the Stochastic Heat Equation. Fundamental inequalities: Cauchy-Schwarz and Jensen.

Stochastic thermodynamics. Defining work and heat along individual paths. Fluctuation Theorems: Jarzynski, Crooks, Gallavotti-Cohen. Thermodynamic Uncertainty Relations (TURs) and the cost of precision in small systems. Cramer-Rao bound and TURs.

Information thermodynamics. Mutual information, Transfer Entropy, and the physics of measurement and feedback. Sagawa-Ueda fluctuation theorem, and the second law of information thermodynamics. Introduction to information-theoretic inequalities: data processing, and Stam’s isoperimetric inequality.

Nonequilibrium ensemble dynamics. Feynman–Kac theorem and the Fokker-Planck equation. The free energy and the Kullback-Leibler divergence. The Wasserstein-2 distance and the Benamou-Brenier fluid-dynamic formulation. Thermodynamic Speed Limits: Geometric bounds on the time-evolution of ensembles. The Otto-Villani HWI inequality.

Peliti, Luca, and Simone Pigolotti. Stochastic thermodynamics: an introduction. Princeton University Press, 2021.

The achievement of the course objectives is assessed through a small theoretical/computational project to be completed towards the end of the course.

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The instructor is responsible for ensuring the authenticity and originality of all examinations and coursework. In cases of suspected academic misconduct, an additional on-site assessment may be required during the exams, which may differ from the standard format.

A fully successful exam (27-30/30) requires demonstrating a solid and extensive mastery of the concepts discussed during the lectures. An average grade (22-26/30) will result from a fairly comprehensive understanding of individual topics, but with limited interconnections between the subjects. A passing grade (18-21/30) will correspond to a minimal knowledge of the individual concepts.

Lectures, tutorials, and coding sessions.