Measuring information growth in fractal phase space

TL;DR

This paper investigates how chaos in fractal phase spaces leads to nonadditive information growth, connecting it to nonextensive entropies and deriving power-law distributions for non-equilibrium stationary states.

Contribution

It introduces a framework for measuring information growth in fractal phase spaces using incomplete normalization, linking it to nonadditive entropies and non-equilibrium power-law distributions.

Findings

Information growth is nonadditive and proportional to a trace-form involving q-entropy.

The framework connects fractal geometry with nonextensive entropy concepts.

Power law distributions emerge as extremum of the information growth for non-equilibrium systems.

Abstract

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, at any scale, of the information calculation in fractal support, the incomplete normalization $\sum_ip_i^q=1$ is applied throughout the paper. It is shown that the information growth is nonadditive and is proportional to the trace-form $\sum_ip_i-\sum_ip_i^q$ so that it can be connected to several nonadditive entropies. This information growth can be extremized to give, for non-equilibrium systems, power law distributions of evolving stationary state which may be called ``maximum entropic evolution''.