Accelerating and retarding anomalous diffusion: A Bernstein function approach

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.