# Generalized entropies and logarithms and their duality relations

@article{Hanel2012GeneralizedEA, title={Generalized entropies and logarithms and their duality relations}, author={Rudolf Hanel and Stefan Thurner and Murray Gell-Mann}, journal={Proceedings of the National Academy of Sciences}, year={2012}, volume={109}, pages={19151 - 19154} }

For statistical systems that violate one of the four Shannon–Khinchin axioms, entropy takes a more general form than the Boltzmann–Gibbs entropy. The framework of superstatistics allows one to formulate a maximum entropy principle with these generalized entropies, making them useful for understanding distribution functions of non-Markovian or nonergodic complex systems. For such systems where the composability axiom is violated there exist only two ways to implement the maximum entropy…

## 25 Citations

Beyond the Shannon-Khinchin Formulation: The Composability Axiom and the Universal Group Entropy

- Computer Science
- 2014

Generalized information and entanglement entropy, gravitation and holography

- Physics
- 2015

A nonextensive statistical mechanics entropy that depends only on the probability distribution is proposed in the framework of superstatistics. It is based on a Γ(χ2) distribution that depends on β…

A theorem on the existence of trace-form generalized entropies

- Computer ScienceProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
- 2015

An analytic technique is proposed, which allows to generate many new examples of entropic functionals generalizing the standard Boltzmann–Gibbs entropy. Our approach is based on the existence of a…

Generalized Entanglement Entropy and Holography

- Physics
- 2015

A nonextensive statistical mechanics entropy that depends only on the probability distribution is proposed in the framework of superstatistics. It is based on a Γ(χ2) distribution that depends on β…

Generalized Fokker-Planck equations derived from nonextensive entropies asymptotically equivalent to Boltzmann-Gibbs.

- PhysicsPhysical review. E
- 2020

This work derives generalized Fokker-Planck equations (FPEs) based on two nonextensive entropy measures S_{±} that depend exclusively on the probability and finds that there are models regarding biological sciences, for the study of congregation and aggregation behavior, the structure of which coincides with the one of the models derived.

How multiplicity determines entropy and the derivation of the maximum entropy principle for complex systems

- Computer ScienceProceedings of the National Academy of Sciences
- 2014

This paper proves that a MEP indeed exists for complex systems and derive the generalized entropy, and finds that it belongs to the class of the recently proposed (c,d)-entropies, and shows that path-dependent random processes with memory naturally require specific generalized entropies.

Phase space volume scaling of generalized entropies and anomalous diffusion scaling governed by corresponding non-linear Fokker-Planck equations

- MathematicsScientific Reports
- 2018

This paper model stochastic processes by a family of generalized Fokker-Planck equations whose stationary solutions are equivalent to the maximum entropy distributions according to generalized entropies and shows that at asymptotically large times and volumes, the scaling exponent of the anomalous diffusion process and the phase space volume scaling exponent determine each other via a simple algebraic relation.

Information Geometric Duality of ϕ-Deformed Exponential Families

- Computer ScienceEntropy
- 2019

There exists another duality that arises in the context of information geometry, relating the Fisher information of ϕ-deformed exponential families that correspond to linear constraints to those that are based on escort constraints, and this information geometric duality is demonstrated for the case of (c,d)-entropy.

Generalization of the maximum entropy principle for curved statistical manifolds

- Computer SciencePhysical Review Research
- 2021

By establishing a bridge between non-Euclidean geometry and the MEP, this proposal sets a solid foundation for the numerous applications of the Rényi entropy, and enables a range of novel methods for complex systems analysis.

Generalized (c, d)-Entropy and Aging Random Walks

- Computer ScienceEntropy
- 2013

This work generalizes acceleration, path-dependent and aging random walks to a much wider class of stochastic systems that can be characterized as “aging” walks and shows that for particular aging walks, Sc,d is again the correct extensive entropy.

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