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Area of Science:

  • Information Theory
  • Statistical Mechanics
  • Group Theory

Background:

  • Generalized entropy measures are crucial in various scientific domains.
  • Existing frameworks for entropy composition lack a unified theoretical basis.
  • The Shannon-Khinchin axioms provide foundational rules for classical information entropy.

Purpose of the Study:

  • To establish a unifying group-theoretical framework for generalized entropy.
  • To develop a generalized information measure based on group structure.
  • To explore the connection between generalized entropy and physical likelihood functions.

Main Methods:

  • Employed a group-theoretical approach to define entropy composition.
  • Extended the Shannon-Khinchin axioms to a generalized context.
  • Derived a generalized information measure from group-theoretic principles.

Main Results:

  • A group structure was identified for well-defined entropy composition of independent systems.
  • A generalized information measure satisfying additivity was associated with a class of entropies.
  • Einstein's likelihood function emerged naturally from the informational interpretation of entropies.

Conclusions:

  • The proposed group-theoretical approach unifies generalized entropy and information theory.
  • Composable entropies are validated for applications in both physical and social sciences.
  • The framework provides a novel perspective on the relationship between information and physical systems.