Generalized Mittag-Leffler relaxation: clustering-jump continuous-time random walk approach

Area of Science:

• Physics
• Statistical Mechanics
• Mathematical Modeling

Background:

• Continuous-time random walks (CTRWs) are fundamental models for anomalous diffusion and relaxation phenomena.
• Existing models often struggle to capture the complex dynamics observed in disordered systems.

Purpose of the Study:

• To propose a stochastic generalization of the renormalization-group transformation for CTRWs.
• To develop a novel framework for modeling relaxation processes with emergent power-law behaviors.

Main Methods:

• A renormalization technique is introduced, replacing clustered jump events with a single renormalized jump.
• The method involves transforming interjump time intervals corresponding to clustered jumps.
• This leads to a new class of coupled continuous-time random walks.

Main Results:

• The proposed method generates a new class of coupled CTRWs.
• Application to relaxation modeling results in general power-law properties.
• These properties align with empirical observations typically fitted by the Havriliak-Negami function.

Conclusions:

• The stochastic renormalization-group transformation offers a powerful tool for analyzing complex CTRW dynamics.
• This framework provides a theoretical basis for understanding the power-law relaxation observed in various physical systems.
• The approach bridges the gap between microscopic jump processes and macroscopic relaxation behavior.