## Entropy sightings

Entropy and its many avatars. (English summary)

*J. Math. Soc. Japan*67 (2015), no. 4, 1845–1857.

94A17 (37A35 60-02 60K35 82B05)

The author, a chief architect of the theory of large deviations, chronicles several manifestations of entropy. It made appearances in the realms indicated by these section headings:

- Entropy and information theory
- Entropy and dynamical systems
- Relative entropy and large deviations
- Entropy and duality
- Log Sobolev inequality
- Gibbs states
- Interacting particle systems

The topics are connected whenever a concept introduced in one section is treated in more depth in a later section. In this way, relative entropy is seen to play a key role in large deviations, Gibbs states, and systems of interacting particles.

Less explicit connections are left to the reader’s enjoyment and education. For example, the relation between Boltzmann entropy and Shannon entropy in the information theory section is a special case both of Sanov’s theorem, presented in the section on large deviations, and of the relation of free energy and relative entropy, in the section on Gibbs states.

The paper ends with a tribute to Professor Kiyosi Itô.

Reviewed by David R. Bickel

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