A Fractional Entropy in Fractal Phase Space: Properties and Characterization

Abstract

A two-parameter generalization of Boltzmann-Gibbs-Shannon entropy based on natural logarithm is introduced. The generalization of the Shannon-Khinchin axioms corresponding to the two-parameter entropy is proposed and verified. We present the relative entropy, Jensen-Shannon divergence measure and check their properties. The Fisher information measure, the relative Fisher information, and the Jensen-Fisher information corresponding to this entropy are also derived. Also the Lesche stability and the thermodynamic stability conditions are verified. We propose a generalization of a complexity measure and apply it to a two-level system and a system obeying exponential distribution. Using different distance measures we define the statistical complexity and analyze it for two-level and five-level system.

Generalized Maxwell Relations in Thermodynamics with Metric Derivatives

“…Adding and subtracting Equation (16) to Equation (15) and considering that F q only depends on (V q , T q ), yields…”

“…The physical problems on a fractal (F ⊂ E 3 ) can be mapped into problems in the ν-dimensional space F ν ⊆ E 3 According to Reference [14], there is a relation between entropy and the fractal dimension; the latter can be considered a statistical index, measuring the complexity of a given pattern embedded in the topological space and characterizing a measure of the space-filling capacity. It is possible to define the statistical complexity by using different distance measures [15]. Thus, we claim that with the physical definition of the MD, the complexity can be inherently considered by the generalized thermodynamics characterized in entropic expressions.…”

Thermodynamic – Informational Paradigm in the Context of the Formation of a Mathematical Model of Transient Processes in an Open System

“…In the conditions of competition of information and physical processes, evolutionary development of complex systems begins to the determined by its information properties, including their relation to the external environment [1]- [4].…”

“…In the case, dissipative structures a rise, which are conserved only when there is pumping of energy from the surrounding medium [4].…”

“…Thus, when modeling self-organization processes in open systems that are in disequilibrium with the environment, it is necessary to take into account both the physical exchange processes between the open system and environment, and the exchange of information flow [4].…”