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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
Summary
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.
- 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.