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This study reveals that processes with long-range memory can exhibit slow diffusion, challenging the typical superdiffusion associated with Lévy flights. These findings offer new insights into anomalous motion and its potential applications.

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Area of Science:

  • Physics
  • Statistical Mechanics
  • Complex Systems

Background:

  • Markovian processes, like Lévy flights, are known for superdiffusion (faster-than-Brownian motion).
  • Understanding anomalous diffusion is crucial for modeling complex systems.

Purpose of the Study:

  • To investigate limit distributions of processes with long-range memory.
  • To explore anomalous motion beyond typical superdiffusion models.
  • To analyze the modification of the central limit theorem in such systems.

Main Methods:

  • Analysis of path-dependent processes with frequent relocations to visited sites.
  • Derivation of modified central limit theorem for analytic and nonanalytic characteristic functions.
  • Development of a fluctuation-dissipation relation.

Main Results:

  • Lévy laws and Gaussian distributions can emerge as limit distributions in processes exhibiting logarithmic, slow diffusion.
  • Anomalous motion arises from path dependence and site revisitation.
  • The central limit theorem is shown to be modified under these conditions.

Conclusions:

  • Processes with long-range memory can display sub-diffusive behavior, contrasting with Lévy flight superdiffusion.
  • The findings provide a new framework for understanding anomalous motion.
  • Potential applications include modeling animal and human displacement patterns.