A generalized entropy characterization of N -dimensional fractal control systems

TL;DR

This paper introduces a universal generalized entropy framework for N-dimensional fractal control systems, revealing self-similar properties and optimal coarse-graining scales that characterize their hierarchical structure.

Contribution

It develops a universal shape for the generalized entropy of hypercubes in N-dimensional fractal systems, independent of dimensionality, and links it to stationary fractal density profiles.

Findings

Generalized entropy shape is universal across N-dimensional systems.

Optimal coarse-graining scale enhances fractal dimension diversity.

Entropy shape relates to stationary fractal density distribution.

Abstract

It is presented the general properties of N-dimensional multi-component or many-particle systems exhibiting self-similar hierarchical structure. Assuming there exists an optimal coarse-graining scale at which the quality and diversity of the (box-counting) fractal dimensions exhibited by a given system are optimized, it is computed the generalized entropy of each hypercube of the partitioned system and shown that its shape is universal, as it also exhibits self-similarity and hence does not depend on the dimensionality N . For certain systems this shape may also be associated with the large time stationary profile of the fractal density distribution in the absence of external fields (or control).