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Published on: September 5, 2019
Stochastic processes crossing from ballistic to fractional diffusion with memory: exact results
Valery Ilyin1, Itamar Procaccia, Anatoly Zagorodny
1Department of Chemical Physics, The Weizmann Institute of Science, Rehovot 76100, Israel.
Summary
This study models diffusion processes transitioning from ballistic to fractional behavior. We developed a method using a non-Markovian diffusion equation to accurately capture both short-time ballistic and long-time fractional diffusion dynamics.
Area of Science:
- Physics
- Physical Chemistry
- Statistical Mechanics
Background:
- Diffusion processes exhibit complex behaviors over time.
- Classical diffusion models often fail to capture transitions between ballistic and fractional regimes.
- Understanding these crossovers is crucial for various scientific applications.
Purpose of the Study:
- To develop a unified model for diffusion processes exhibiting ballistic and fractional behaviors.
- To accurately describe the crossover dynamics between short-time ballistic and long-time fractional diffusion.
- To provide a more accurate probability distribution function (pdf) for evolving diffusion systems.
Main Methods:
- Utilized the standard non-Markovian diffusion equation.
- Introduced a specific memory kernel to reconcile different asymptotic behaviors.
- Solved for the probability distribution function (pdf) within a ballistically expanding domain.
Main Results:
- Successfully demonstrated a method to choose a memory kernel for accurate asymptotic behavior.
- Obtained a continuous pdf solution that evolves within a ballistically expanding domain.
- The derived solution aligns with continuous random-walk models at long times.
Conclusions:
- The proposed model accurately captures the crossover from ballistic to fractional diffusion.
- The solution offers superior accuracy at shorter times compared to existing methods.
- This approach provides a more comprehensive understanding of complex diffusion phenomena.
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