Caputo Fractional Derivative and Quantum-Like Coherence.

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.