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Group Entropies: From Phase Space Geometry to Entropy Functionals via Group Theory
Henrik Jeldtoft Jensen1,2, Piergiulio Tempesta3,4
1Centre for Complexity Science and Department of Mathematics, Imperial College London, South Kensington Campus, London SW7 2AZ, UK.
Entropy (Basel, Switzerland)
|December 3, 2020
Summary
Group theoretic entropies generalize Boltzmann-Gibbs entropy by replacing additivity with composability. These generalized entropies depend on degrees of freedom (N) and offer new information-theoretical perspectives.
Area of Science:
- Information Theory
- Statistical Mechanics
- Group Theory
Background:
- The standard Boltzmann-Gibbs entropy relies on four axioms, including additivity.
- Generalized entropies are crucial for understanding systems with varying degrees of freedom.
Purpose of the Study:
- To explore group theoretic entropies as a generalization of Boltzmann-Gibbs entropy.
- To investigate the role of the composability axiom and degrees of freedom (N).
Main Methods:
- Reviewing group theoretic entropies and their mathematical framework.
- Analyzing the composability axiom as a replacement for additivity.
- Examining the dependence of entropy on the number of degrees of freedom (N).
Main Results:
- Group theoretic entropies generalize standard entropy by incorporating a composability axiom.
- These entropies are shown to depend significantly on the number of degrees of freedom (N).
- The functional form of group entropies relates to the dependence of phase space volume (W) on N.
Conclusions:
- Group entropies offer a more general framework than Boltzmann-Gibbs entropy.
- The composability axiom and dependence on N are key features of group entropies.
- Group entropies hold particular relevance for information-theoretical applications.
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