Hilbert entropy for measuring the complexity of high-dimensional systems

TL;DR

This paper introduces a new method using Hilbert curves and generalized entropy to measure the complexity of high-dimensional physical systems, effectively capturing intrinsic properties and phase transitions.

Contribution

A novel approach combining Hilbert space-filling curves with entropy measures to quantify high-dimensional system complexity while preserving contextual information.

Findings

Hilbert entropy accurately identifies phase transition points.

Linear relationship between Hilbert entropy and fractal dimension.

Method applicable to various high-dimensional physical systems.

Abstract

Measuring the complexity of high-dimensional data in physical systems becomes a critical factor in determining the information and quality of the systems. However, traditional metrics, such as Lyapunov exponent, fractal dimension, and information entropy, are limited in measuring contextual higher-dimensional data in that they do not elucidate the intrinsic nature of physical systems. Herein, we introduce a novel methodology for quantifying the complexity of high-dimensional data through dimension reduction yet retaining context using a space-filling curve such as the Hilbert curve along with generalized entropy measures. We validate this methodology in measuring critical phenomena, including phase transitions in spin and percolation models. Our findings demonstrate a high degree of concordance between the Hilbert entropy and theoretical phase transition points. Moreover, we further proceed to an exploration of the hidden relationship between the Hilbert entropy and the fractal dimension, such as a linear relationship between scaling exponent and the Euclidean dimension of scale-invariant 2D/3D geometries. The present methodology offers a promising new framework for understanding and analyzing complex systems in higher dimensions, with potential applications across various fields of physics.