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Summary
This summary is machine-generated.

This study quantifies complexity in finite-velocity diffusion using Fisher's information and Shannon's entropy. Results reveal relationships between non-local and local information measures in hyperbolic diffusion.

Keywords:
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Area of Science:

  • Physics
  • Information Theory
  • Mathematical Modeling

Background:

  • Diffusion processes are fundamental in various scientific fields.
  • Understanding the complexity of these processes is crucial for accurate modeling.
  • The telegrapher's equation describes a class of diffusion processes with finite velocity.

Purpose of the Study:

  • To calculate complexity measures for space-time distributions in finite-velocity diffusion.
  • To analyze Fisher's information, Shannon's entropy, and the Cramér-Rao inequality for the telegrapher's equation.
  • To explore the relationship between non-local and local information measures within hyperbolic diffusion.

Main Methods:

  • Numerical calculation of Fisher's information, Shannon's entropy, and Cramér-Rao inequality.
  • Analysis of a positively normalized solution to the telegrapher's equation.
  • Development of a perturbation theory for long-time entropy calculation.
  • Construction of a toy model for short-time ballistic regime analysis.

Main Results:

  • Quantification of complexity measures for the telegrapher's equation.
  • Demonstration of a relationship between non-local Fisher's information (x-parameter) and local Fisher's information (t-parameter).
  • Application of perturbation theory to determine long-time Shannon entropy.
  • Characterization of the system as an attenuated wave in the ballistic regime.

Conclusions:

  • The study provides a comprehensive analysis of complexity in finite-velocity diffusion.
  • Novel insights into information-theoretic measures within hyperbolic diffusion frameworks.
  • The findings contribute to a deeper understanding of diffusion processes and their mathematical descriptions.