On entropy, entropy-like quantities, and applications

Abstract

This is a review on entropy in various fields of mathematics and science. Its scope is to convey a unified vision of the classical as well as some newer entropy 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 to 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.

arXiv comment35 pages, no figures. This is a corrected version of a paper published in 2015

References in corpus (19)

• Measuring Information Transfer
• Quantifying Causal Coupling Strength: A Lag-specific Measure For Multivariate Time Series Related To Transfer Entropy
• Topological entropy of nonautonomous dynamical systems
• Conditional entropy of ordinal patterns
• On uniqueness theorems for Tsallis entropy and Tsallis relative entropy
• Generalized entropies and logarithms and their duality relations
• Causation entropy from symbolic representations of dynamical systems
• Space-time directional Lyapunov exponents for cellular automata
• Forbidden ordinal patterns in higher dimensional dynamics
• Information directionality in coupled time series using transcripts
• Permutation complexity of interacting dynamical systems
• Tsallis entropy: How unique?
• Entropy determination based on the ordinal structure of a dynamical system
• Kolmogorov-Sinai entropy via separation properties of order-generated sigma-algebras
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• The world according to Renyi: Thermodynamics of multifractal systems
• A simplified algorithm for the topological entropy of multimodal maps
• Directional uniformities, periodic points, and entropy
• Directional complexity and entropy for lift mappings

Cited by in corpus (3)

• Permutation group entropy: a new route to complexity for real-valued processes
• Complexity-based permutation entropies: from deterministic time series to white noise
• Entropy monotonicity and superstable cycles for the quadratic family revisited