Entropy measures and their applications: A comprehensive review

TL;DR

This survey assembles a broad panorama of entropy measures used to quantify uncertainty and complexity across disciplines, tracing the lineage from Boltzmann and Gibbs to Shannon and beyond. It catalogues about 60 entropy variants organized into seven theoretical families, detailing their definitions, key properties, and interrelationships, with explicit attention to continuous/discrete forms and limits to Shannon entropy. The article also maps practical applications across thermodynamics, communications, finance, categorical data, AI, signal processing, chemistry, and biology, and catalogs publicly available datasets (49 open datasets among 81 references) that underpin entropy-based studies. Finally, it discusses data resources, highlights current trends, and outlines promising directions such as causal inference, extropy, fractional-order constructs, and data-driven maximum-entropy frameworks for future research and applications.

Abstract

Entropy has emerged as a dynamic, interdisciplinary, and widely accepted quantitative measure of uncertainty across different disciplines. A unified understanding of entropy measures, supported by a detailed review of their theoretical foundations and practical applications, is crucial to advance research across disciplines. This review article provides motivation, fundamental properties, and constraints of various entropy measures. These measures are categorized with time evolution ranging from Shannon entropy generalizations, distribution function theory, fuzzy theory, fractional calculus to graph theory, all explained in a simplified and accessible manner. These entropy measures are selected on the basis of their usability, with descriptions arranged chronologically. We have further discussed the applicability of these measures across different domains, including thermodynamics, communication theory, financial engineering, categorical data, artificial intelligence, signal processing, and chemical and biological systems, highlighting their multifaceted roles. A number of examples are included to demonstrate the prominence of specific measures in terms of their applicability. The article also focuses on entropy-based applications in different disciplines, emphasizing openly accessible resources. Furthermore, this article emphasizes the applicability of various entropy measures in the field of finance. The article may provide a good insight to the researchers and experts working to quantify uncertainties, along with potential future directions.

Figures (14)

• Figure 1: Evolution of entropy measures over years
• Figure 2: I
• Figure 3: II
• Figure 4: Some applications of entropy measures discussed in this section
• Figure 5: Entropy Measures and their Applications in Thermodynamic Systems