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Memory-induced anomalous dynamics in a minimal random walk model.

Upendra Harbola1, Niraj Kumar2, Katja Lindenberg3

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This study introduces a simplified non-Markovian random walker model. A single parameter variation can yield subdiffusion, normal diffusion, or superdiffusion, with perfect correlation needed for subdiffusion.

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Area of Science:

  • Physics
  • Statistical Mechanics
  • Complex Systems

Background:

  • Non-Markovian random walkers exhibit complex dynamics influenced by past events.
  • Previous models demonstrated parameter-driven diffusion behaviors (superdiffusion, normal diffusion, subdiffusion).

Purpose of the Study:

  • To propose and analyze an even simpler discrete-time model for non-Markovian random walkers.
  • To investigate the conditions under which this minimal model exhibits different diffusion regimes.

Main Methods:

  • Analysis of a discrete-time dynamics model with two states: move forward or stay at rest.
  • Variation of a single parameter to observe long-time dynamics.

Main Results:

  • The simplified model successfully reproduces asymptotic superdiffusion, normal diffusion, and subdiffusion.
  • Subdiffusive dynamics requires perfect correlation between memory of rest states and present dynamics.
  • Absence of this perfect correlation in unidirectional walks leads to only diffusive or superdiffusive behavior.

Conclusions:

  • A minimal non-Markovian random walker model can exhibit diverse diffusion behaviors.
  • Perfect correlation of memory is a critical condition for observing subdiffusion.
  • The findings offer insights into the fundamental mechanisms driving anomalous diffusion.