Lévy walk dynamics in an external harmonic potential

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.