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Ultraslow diffusion in an exactly solvable non-Markovian random walk
M A A da Silva1, G M Viswanathan2, J C Cressoni1
1Departamento de Física e Química, FCFRP, Universidade de São Paulo, 14040-903 Ribeirão Preto, SP, Brazil.
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|October 30, 2014
Summary
This study introduces a novel random walk model with memory, revealing ultraslow diffusion and stationary localization regimes. The findings extend understanding of complex stochastic processes beyond continuous-time models.
Area of Science:
- Statistical Physics
- Complex Systems
- Stochastic Processes
Background:
- Non-Markovian random walks exhibit complex memory effects.
- Existing models like Scher-Montroll can be simplified using operational time.
- Continuous-time random walks may not capture all memory-dependent dynamics.
Purpose of the Study:
- To investigate a discrete-time non-Markovian random walk with strong memory and pauses.
- To analyze the impact of continuous stochastic dynamics on diffusion regimes.
- To analytically solve for moments and determine the Hurst exponent.
Main Methods:
- Analytical solution of equations of motion for the first two moments.
- Determination of the Hurst exponent.
- Construction of a complete phase diffusion diagram.
Main Results:
- Exact results demonstrating ultraslow diffusion and stationary diffusion (localization).
- Identification of anomalous diffusion regimes: superdiffusion, subdiffusion, ultraslow, and stationary.
- Analysis of persistence and statistics in key regions.
Conclusions:
- The discrete-time non-Markovian model exhibits unique ultraslow and stationary diffusion regimes.
- The inclusion of continuous stochastic dynamics, even in subdiffusion limits, significantly alters behavior.
- The study provides a comprehensive phase diagram for anomalous diffusion phenomena.
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