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A Continuous-Time Random Walk Extension of the Gillis Model
Gaia Pozzoli1,2, Mattia Radice1,2, Manuele Onofri1,2
1Center for Nonlinear and Complex Systems, Dipartimento di Scienza e Alta Tecnologia, Università degli Studi dell'Insubria, Via Valleggio 11, 22100 Como, Italy.
Entropy (Basel, Switzerland)
|December 23, 2020
Summary
We introduce a modified Gillis random walk with heavy-tailed waiting times. This process exhibits subdiffusion and ergodicity breaking, altering transport properties compared to normal diffusion.
Area of Science:
- Stochastic Processes
- Mathematical Physics
- Statistical Mechanics
Background:
- The Gillis random walk is a known mathematical model for non-homogeneous random walks.
- Standard random walks often assume finite waiting times, which limits their applicability to systems with anomalous diffusion.
Purpose of the Study:
- To generalize the Gillis random walk by incorporating heavy-tailed waiting-time distributions.
- To investigate the impact of these distributions on the process's properties, including diffusion and ergodicity.
Main Methods:
- Mathematical analysis of a continuous-time random walk with position-dependent drift and heavy-tailed waiting times.
- Derivation of exact results for various statistical quantities.
- Numerical simulations to validate theoretical predictions.
Main Results:
- Normal diffusion transitions to subdiffusion.
- Ergodicity is broken due to the heavy-tailed waiting times.
- Exact results obtained for hitting times, survival probabilities, occupation times, and record statistics.
Conclusions:
- The modified Gillis random walk provides a framework for studying anomalous transport phenomena.
- Heavy-tailed waiting times fundamentally alter the dynamics and statistical properties of random walks.
- The findings have implications for understanding complex systems exhibiting subdiffusion and non-ergodic behavior.
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