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The Okinawa Lectures on Entropy — arXiv:2608.14523

Source: arXiv (en)
Original: https://arxiv.org/abs/2608.14523
License (as archived): CC0-1.0
Archived: 2026-10-19T04:13:30.294Z
SHA-256 of the archived text: 42bace3a3bd01826341dda4ce2d98e233abdef91c5b5249dc3eb7b1ac35ef834
Stable URL of this page: https://metraipolis.com/documento/dc_9f66242514fa

How to cite

The Okinawa Lectures on Entropy — arXiv:2608.14523 — arXiv (en), https://arxiv.org/abs/2608.14523 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_9f66242514fa (Metraipolis archive dc_9f66242514fa, sha256 42bace3a3bd01826341dda4ce2d98e233abdef91c5b5249dc3eb7b1ac35ef834)

BibTeX

@misc{dc9f66242514fa,
  title = {The Okinawa Lectures on Entropy — arXiv:2608.14523},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2608.14523},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_9f66242514fa; sha256 42bace3a3bd01826341dda4ce2d98e233abdef91c5b5249dc3eb7b1ac35ef834},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2608.14523. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

After a historical introduction, the most important classical and quantum entropies are introduced as constructions in classical and quantum probability theory. Classical entropies are studied from large deviation theory, including theorems of Sanov, Cramér, Gärtner-Ellis, and Varadhan, and are illustrated in some applications to both Boltzmannian and Gibbsian statistical physics. Quantum entropies superficially connect to classical entropies at the formula level, but more deeply do so via the crucial role of entropy in statistical hypothesis testing. The classical (relative) Kullback-Leibler entropy, its quantum counterpart introduced by Umegaki, as well as their deformations proposed by Renyi all fit naturally in this context. Quantum entropy faces the new problem of defining and computing the relative entropy of a pair of states on a subsystem, here formalized as a von Neumann algebra. This requires modular (aka Tomita-Takesaki) theory, which provides the framework for the relative quantum entropies introduced by Araki and Uhlmann (these encompass both the Kullback-Leibler and Umegaki entropies as special cases). To (re)define and compute these entropies in terms of density operators and traces, further constructions are needed, namely Haagerup's noncommutative L^p spaces. Von Neumann algebras also provide the setting for the Connes-Stormer-Narnhofer-Thirring entropy, which is a quantum version of the Kolmogorov-Sinai entropy in dynamical systems and ergodic theory, to which we also provide an introduction. This course was originally inspired by, and should be relevant to, black hole thermodynamics, although we discuss neither this application nor the second law. The course tries to be both mathematically rigorous and interesting to theoretical physicists. Prerequisites are undergraduate probability theory, functional analysis, and quantum theory.

Tema degli esploratori: fisica e astronomia (retro-catalogo arXiv: annunci del 2026-08-17 in gr-qc).
Categorie arXiv: Mathematical Physics (math-ph) ; General Relativity and Quantum Cosmology (gr-qc); High Energy Physics - Theory (hep-th).
Pagina del lavoro: https://arxiv.org/abs/2608.14523 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-19T04:09:10.986Z dalla pagina https://arxiv.org/catchup/gr-qc/2026-08-17?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).