Entropy

Last updated: 09/Sep/2010

Entropy is one of my current research topics

I gave a talk in Rio in 08/2010 about entropy of geometric structures (including Poisson structures), and am writing up a text about it. It generalizes the notion

of (geometric) entropy studied by Ghys, Langevin, Walczak, Bis, etc., and works also for singular distrbutions, and foliations, Lie algebroids, etc. I’ll post my article here when available.

I’m interested in understanding more about entropy, and in applying its concept and ideas to other situations (not only mathematics, but also in socio-economics).

Some references:

Physical References:

- Roger Balian, Entropy, a protean concept, Séminaire Poincaré 2(2003), 13–27. (Brief introduction to different definitions of entropy).

- O. Darrigol, The origins of the entropy concept , Séminaire Poincaré 2(2003), 1–12. (History of entropy, involving Carnot, Clausis, Maxwell, Boltzmann, Gibbs, Plank, Poincaré, Einstein, etc.)

- Elliott H. Lieb & Jakob Yngvason, Elliott H. Lieb & Jakob Yngvason ; A Guide to Entropy and the Second Law of Thermodynamics, Notices of the American Mathematical Society 45 (1998), 571-581. ArXiv : math-ph/9805005 and references therein (axiomatic approach to physical entropy, modern but similar to Caratheodory; interesting is the fact that the entropy function can be shown to be unique and does not depend on the way you define it)

- Wikipedia, entropy & entropie (in English and French: the pages in English and in French contain different complementary informations !)

- Calcul de l’entropie d’un corps pur (l’entropie varie comme le logarithme de la température) Temparature <–> number of states, and that’s why entropy is its log function ? (In the axiomatic approach, temperature is obtained as the derivation of entropy !)

Mathematical references:

- Shannon entropy

- Kolmogorov, Sinai (metric entropy)

- Adler et al., Bowen, Dinaburg, … (Topological entropy)

- Ghys et al. (Geometric entropy)

- Bis (Entropy of a regular distribution)

Notes/questions:

Renyi’s & Tsallis’ definitions entropy: are they of any use ?

How to connect mathematical entropy to physical entropy (physically: not just similar formulas, but interpret the mathematical definition as part of the real world entropy ??)

Additivity of entropy in mathematics ?

How to relate physical entropy to information volume ?

Can we define entropy of a few-body system ?

What is the analog of temperature in Shannon/Kolmogorov entropy ?