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Elephants can always remember: exact long-range memory effects in a non-Markovian random walk
Gunter M Schütz1, Steffen Trimper
1Institut für Festkörperforschung, Forschungszentrum Jülich, D-52425 Jülich, Germany. g.schuetz@fz-juelich.de
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|December 17, 2004
Summary
This study analyzes a discrete-time random walk with memory. It reveals critical transitions in walker behavior, from localization to superdiffusion, governed by a non-Markovian equation and an effective harmonic potential.
Area of Science:
- Statistical Physics
- Complex Systems
Background:
- Random walks are fundamental models in statistical physics.
- Understanding memory effects in stochastic processes is crucial for modeling complex systems.
Purpose of the Study:
- To investigate the behavior of a discrete-time random walk with history-dependent increments.
- To identify critical transitions and governing dynamics of such memory-influenced walks.
Main Methods:
- Exact calculation of mean and variance for the random walk position.
- Analysis of the probability distribution using a non-Markovian Fokker-Planck equation.
- Derivation of an effective harmonic oscillator potential for large-scale behavior.
Main Results:
- Identified a critical memory parameter (p=1/2) leading to a transition from localization to escape.
- Discovered a second critical value inducing superdiffusive behavior.
- The walker's dynamics are described by a Gaussian distribution within an effective time-dependent harmonic potential.
Conclusions:
- Memory effects significantly alter random walk dynamics, leading to distinct regimes and transitions.
- The non-Markovian nature and history dependence are captured by an effective harmonic oscillator model.
- The study provides insights into complex stochastic processes with long-range correlations.