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Salt particles that have dissolved in water never spontaneously come back together in solution to reform solid particles. Moreover, a gas that has expanded in a vacuum remains dispersed and never spontaneously reassembles. The unidirectional nature of these phenomena is the result of a thermodynamic state function called entropy (S). Entropy is the measure of the extent to which the energy is dispersed throughout a system, or in other words, it is proportional to the degree of disorder of a...
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Microstate and Omega Complexity Analyses of the Resting-state Electroencephalography
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Structural Complexity as a Directional Signature of System Evolution: Beyond Entropy.

Donglu Shi1

  • 1Materials Science and Engineering, College of Engineering and Applied Science, University of Cincinnati, Cincinnati, OH 45221, USA.

Entropy (Basel, Switzerland)
|September 27, 2025
PubMed
Summary
This summary is machine-generated.

We introduce a universal framework using Kolmogorov Complexity (KC) and Fractal Dimension (FD) to measure system evolution. This approach quantifies increasing structural order in open systems, unlike entropy in closed systems.

Keywords:
Kolmogorov complexitycomplexity sciencedirectionalityentropy beyond equilibriumfractal dimensionnon-equilibrium thermodynamicsopen systemsself-organized criticalitystructural complexitysystem evolution

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Area of Science:

  • Complexity Science
  • Thermodynamics
  • Information Theory

Background:

  • Traditional entropy measures are limited to closed, equilibrium systems.
  • Understanding system evolution in open, non-equilibrium systems requires new metrics.
  • Schrödinger's concept of biological growth locally reducing entropy while increasing order provides a basis.

Purpose of the Study:

  • To propose a universal framework for system evolution based on structural complexity.
  • To introduce quantifiable metrics for organized complexity in open systems.
  • To formalize a new Universal Law for system evolution.

Main Methods:

  • Utilizing Kolmogorov Complexity (KC) to measure algorithmic complexity.
  • Employing Fractal Dimension (FD) to quantify geometric complexity.
  • Developing a non-decreasing function Ω(t) = α·KC(t) + β·FD(t) to model system evolution.

Main Results:

  • KC and FD provide scalable metrics for organized complexity in open, non-equilibrium systems.
  • The proposed Universal Law, Ω(t), parallels the Second Law of Thermodynamics but tracks algorithmic and geometric complexity.
  • The framework applies across physical, chemical, and biological domains.

Conclusions:

  • The proposed framework offers a directional signature for system evolution applicable across diverse scientific domains.
  • This approach integrates complexity science principles for a mathematically grounded understanding of system development.
  • The framework provides a novel lens for describing directional evolution from simple structures to complex cognition.