On entropy, entropy-like quantities, and applications
2015
This is a review on entropy in various fields of mathematics and science. Its
scope is to convey a unified vision of the classical entropies and some newer,
related notions to a broad audience with an intermediate background in dynamical
systems and ergodic theory. Due to the breadth and depth of the subject, we
have opted for a compact exposition whose contents are a compromise between
conceptual import and instrumental relevance. The intended technical level and
the space limitation born furthermore upon the final selection of the topics,
which cover the three items named in the title. Specifically, the first part
is devoted to the avatars of entropy in the traditional contexts: many
particle physics, information theory, and dynamical systems. This
chronological order helps present the materials in a didactic manner. The
axiomatic approach will be also considered at this stage to show that, quite remarkably,
the essence of entropy can be encapsulated in a few basic properties.
Inspired by the classical entropies, further akin quantities have been
proposed in the course of time, mostly aimed at specific needs. A common
denominator of those addressed in the second part of this review is their
major impact on research. The final part shows that, along with its profound
role in the theory, entropy has interesting practical applications beyond
information theory and communications technology. For this sake we preferred
examples from applied mathematics, although there are certainly nice
applications in, say, physics, computer science and even social sciences. This
review concludes with a representative list of references.
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