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Group entropies, correlation laws, and zeta functions
1Departamento de Física Teórica II, Facultad de Físicas, Ciudad Universitaria, Universidad Complutense, E-28040 Madrid, Spain. p.tempesta@fis.ucm.es
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|September 21, 2011
Summary
A new concept of group entropy unifies existing entropy definitions like Boltzmann-Gibbs and Tsallis. This framework, based on group theory, reveals new connections in statistical mechanics and information theory.
Area of Science:
- Statistical Mechanics
- Information Theory
- Mathematical Physics
Background:
- Existing entropy definitions (Boltzmann-Gibbs, Tsallis, Abe, Kaniadakis) lack a unified framework.
- Understanding systems out of equilibrium requires advanced entropic functionals.
- Formal group theory offers a potential mathematical structure for correlations.
Purpose of the Study:
- To propose a unified and generalized concept of group entropy.
- To introduce new entropic functionals linked to correlation laws in non-equilibrium systems.
- To explore applications in information theory and establish connections with zeta functions.
Main Methods:
- Development of group entropy based on formal group theory.
- Introduction of entropic functionals associated with nontrivial correlation laws.
- Analysis of thermostatistics for systems with weak chaotic dynamics.
- Application to information theory, including generalizations of Kullback-Leibler divergence.
Main Results:
- Group entropy unifies diverse existing entropy definitions.
- New entropic functionals are derived for non-equilibrium systems.
- A connection between statistical mechanics and zeta functions is established.
- Tsallis entropy is explicitly related to the Riemann zeta function.
Conclusions:
- Group entropy provides a generalized mathematical framework for various entropic concepts.
- The theory offers insights into universality classes of systems out of equilibrium.
- This work bridges statistical mechanics, information theory, and number theory through zeta functions.
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