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Lévy walk dynamics in an external harmonic potential.

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Lévy walks (LWs) equilibrate exponentially under confinement, exhibiting a bimodal stationary distribution. This research clarifies LW behavior near boundaries, resolving long-standing questions in anomalous diffusion.

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Area of Science:

  • Statistical Physics
  • Complex Systems
  • Anomalous Diffusion

Background:

  • Lévy walks (LWs) are spatiotemporally coupled random processes modeling superdiffusion.
  • They are observed in diverse phenomena, including heat conduction, light propagation, and biological/robotic motion.
  • Understanding LW behavior under external potentials is crucial for their theoretical and applied relevance.

Purpose of the Study:

  • To investigate the response of Lévy walks to an external harmonic potential.
  • To characterize the equilibrium properties and stationary distributions of confined LWs.
  • To generalize LWs to scenarios involving confining forces and boundaries.

Main Methods:

  • Theoretical analysis of Lévy walk dynamics in a harmonic potential.
  • Derivation of the stationary distribution for confined LWs.
  • Investigation of boundary effects, specifically near a reflecting boundary at the origin.

Main Results:

  • Demonstrated exponential equilibration for Lévy walks in a harmonic potential.
  • Identified the possibility of a bimodal stationary distribution.
  • Showed a horizontal slope of the stationary distribution near a reflecting boundary, distinguishing LWs from other superdiffusive processes.

Conclusions:

  • Lévy walks exhibit predictable equilibration and stationary distributions under confinement.
  • The findings generalize LWs to confining forces and resolve existing theoretical puzzles.
  • The unique boundary behavior provides a new characteristic for identifying LWs in experimental systems.