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Caputo Fractional Derivative and Quantum-Like Coherence.

Garland Culbreth1, Mauro Bologna2, Bruce J West3

  • 1Center for Nonlinear Science, University of North Texas, P.O. Box 311427, Denton, TX 76201, USA.

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Summary
This summary is machine-generated.

This study explores anomalous diffusion using fractional derivatives and time-dependent coefficients. The research demonstrates their equivalence and proposes methods to analyze complex systems, linking diffusion to cognition and self-organization.

Keywords:
cognitioncomplexitycrucial eventsfractional derivatives

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Area of Science:

  • Physics
  • Mathematics
  • Complex Systems

Background:

  • Anomalous diffusion deviates from standard Brownian motion.
  • Fractional calculus and time-dependent coefficients offer new models for diffusion.
  • Understanding these deviations is crucial for complex systems analysis.

Purpose of the Study:

  • To investigate two forms of anomalous diffusion: Caputo fractional derivative and time-dependent diffusion coefficient.
  • To demonstrate the equivalence of these two modeling approaches.
  • To propose methods for analyzing systems exhibiting anomalous diffusion, particularly in relation to cognition and self-organization.

Main Methods:

  • Utilized the Caputo fractional derivative to model anomalous diffusion.
  • Incorporated a time-dependent diffusion coefficient.
  • Employed phenomenological and theoretical projection methods to derive and compare diffusion equations.
  • Designed a time series to validate the proposed anomalous diffusion equation.

Main Results:

  • Proved the mathematical equivalence between the Caputo fractional derivative model and the time-dependent diffusion coefficient model.
  • Developed a time series consistent with the derived anomalous diffusion equation.
  • Established a link between anomalous diffusion, cognition, and self-organization.
  • Proposed a criterion to distinguish self-organization from quantum coherence effects.

Conclusions:

  • The Caputo fractional derivative serves as an indicator of the link between cognition and self-organization.
  • Anomalous diffusion arises from distinct mechanisms, including quantum coherence.
  • Diffusion entropy analysis can aid in studying physiological processes with deviations from ordinary scaling.