Fractal geometry, information growth and nonextensive thermodynamics

TL;DR

This paper explores how fractal geometry influences information growth and thermodynamics in complex systems, linking nonadditive entropy to fractal phase space and deriving power law distributions.

Contribution

It introduces a geometrical approach to information evolution in fractal phase spaces using incomplete normalization and connects it to nonadditive entropies and thermodynamics.

Findings

Information growth is nonadditive and proportional to a trace-form involving $p_i$ and $p_i^q$

Power law distributions emerge from extremizing the information growth

The thermodynamics based on Tsallis entropy can preserve the Stefan-Boltzmann law

Abstract

This is a study of the information evolution of complex systems by geometrical consideration. We look at chaotic systems evolving in fractal phase space. The entropy change in time due to the fractal geometry is assimilated to the information growth through the scale refinement. Due to the incompleteness of the state number counting at any scale on fractal support, the incomplete normalization $\sum_ip_i^q=1$ is applied throughout the paper, where $q$ is the fractal dimension divided by the dimension of the smooth Euclidean space in which the fractal structure of the phase space is embedded. It is shown that the information growth is nonadditive and is proportional to the trace-form $\sum_ip_i-\sum_ip_i^q$ which can be connected to several nonadditive entropies. This information growth can be extremized to give power law distributions for these non-equilibrium systems. It can also be used for the study of the thermodynamics derived from Tsallis entropy for nonadditive systems which contain subsystems each having its own $q$. It is argued that, within this thermodynamics, the Stefan-Boltzmann law of blackbody radiation can be preserved.