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Two-parameter deformations of logarithm, exponential, and entropy: a consistent framework for generalized statistical

G Kaniadakis1, M Lissia, A M Scarfone

  • 1Dipartimento di Fisica and Istituto Nazionale di Fisica della Materia (INFM), Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy. giorgio.kaniadakis@polito.it

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|May 21, 2005
PubMed
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This study introduces a generalized statistical mechanics framework using the maximum entropy principle. The new framework yields a class of entropies and power-law distributions relevant for anomalous systems.

Area of Science:

  • Statistical mechanics
  • Thermodynamics
  • Information theory

Background:

  • The maximum entropy principle is a fundamental tool in statistical mechanics.
  • Existing entropy measures may not fully capture the behavior of anomalous systems.

Purpose of the Study:

  • To develop a consistent generalization of statistical mechanics.
  • To derive a new class of entropies and associated power-law distributions.

Main Methods:

  • Applying the maximum entropy principle to a trace-form entropy.
  • Ensuring preservation of physically motivated mathematical properties.
  • Solving the resulting differential-functional equation.

Main Results:

  • A two-parameter class of generalized logarithms and entropies was derived.

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  • These generalized entropies exhibit desirable properties like positivity, concavity, and Lesche stability.
  • The derived power-law distributions are applicable to anomalous systems.
  • The Boltzmann-Shannon entropy is a special case within this framework.

    • Conclusions:

    • The generalized framework provides a consistent extension of statistical mechanics.
    • The new entropies and distributions offer a powerful tool for analyzing anomalous systems.
    • Further investigation into the deformed algebras associated with these entropies is warranted.