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We discovered a duality relation for anomalous diffusion using Bernstein functions. This allows unified control over retarding and accelerating diffusion, explaining complex biological processes and relaxation dynamics.

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Area of Science:

  • Mathematical Physics
  • Complex Systems Dynamics
  • Biophysics

Background:

  • Anomalous diffusion describes non-standard transport phenomena observed in various complex systems.
  • Bernstein functions offer a powerful framework for analyzing stochastic processes, including diffusion.
  • Distinguishing between retarding (subdiffusion) and accelerating (superdiffusion) processes is crucial for understanding system dynamics.

Purpose of the Study:

  • To establish a duality relation between infinitely divisible subordinators for anomalous diffusion.
  • To demonstrate how conjugate Bernstein functions can naturally generate both retarding and accelerating anomalous diffusion.
  • To provide a unified approach for modeling transient anomalous diffusion in biological systems.

Main Methods:

  • Utilizing the special Bernstein function approach.
  • Analyzing Laplace exponents of conjugate Bernstein function pairs.
  • Applying the framework to understand relaxation dynamics and power laws.

Main Results:

  • A duality relation between infinitely divisible subordinators producing anomalous diffusion was discovered.
  • Conjugate Bernstein functions were shown to generate both retarding and accelerating anomalous diffusion.
  • The findings offer a unified method to control diffusion dynamics in complex biological processes.

Conclusions:

  • The Bernstein function approach provides a unified framework for understanding anomalous diffusion.
  • This duality relation enhances the explanation of relaxation diagrams, including the Havriliak-Negami law.
  • The findings have implications for interpreting single-particle tracking experiments and complex biological dynamics.