Clustered continuous-time random walks: diffusion and relaxation consequences

Area of Science:

• Physics
• Mathematics
• Statistical Mechanics

Background:

• Continuous-time random walks (CTRWs) are fundamental models for anomalous diffusion.
• Existing CTRW models often assume independent jumps and waiting times.

Purpose of the Study:

• To introduce a novel CTRW model with clustered jumps and coupled waiting times.
• To analyze the scaling limits and fractional diffusion equations governing these clustered CTRWs.
• To demonstrate the model's ability to capture experimentally observed two power-law relaxation patterns.

Main Methods:

• Development of a new class of CTRWs with clustered jumps.
• Mathematical analysis of scaling limits, identifying them as time-changed processes.
• Derivation of fractional diffusion equations based on coupling dependencies.

Main Results:

• The scaling limits of the proposed CTRWs are shown to be time-changed processes.
• Two distinct fractional diffusion equations arise, depending on the coupling of waiting times to preceding or following jumps.
• These equations successfully model diverse two power-law relaxation patterns observed in experiments.

Conclusions:

• Clustered jumps and coupled waiting times in CTRWs provide a unified framework for understanding complex relaxation dynamics.
• The derived fractional diffusion equations offer powerful tools for analyzing experimental data with power-law behaviors.
• The model parameters directly relate to observable exponents and frequencies in relaxation phenomena.