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Random walks near a disk exhibit unique behavior. Trajectories concentrate in a specific strip, revealing insights into diffusion processes.

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Area of Science:

  • Physics
  • Statistical Mechanics
  • Fluid Dynamics

Background:

  • Understanding particle behavior in confined spaces is crucial.
  • Random walks model diffusion processes in various systems.
  • Flow around obstacles influences particle distribution.

Purpose of the Study:

  • To analyze the stationary radial distribution of a random walk.
  • To investigate the effect of tangential velocity and an impenetrable disk on particle trajectories.
  • To determine the asymptotic behavior of the distribution for large disk radii.

Main Methods:

  • Mathematical modeling of a random walk with diffusion and tangential velocity.
  • Derivation of the stationary radial distribution function P(ρ).
  • Asymptotic analysis for the case where disk radius R is much larger than D/V.

Main Results:

  • The radial distribution converges to a function involving the Airy function.
  • Particle trajectories are localized within a specific circular strip [R, R+δR^(1/3)].
  • The localization width depends on diffusion coefficient D and velocity V, but not R.

Conclusions:

  • The presence of an impenetrable disk significantly alters random walk distributions.
  • Airy functions naturally emerge in describing diffusion near boundaries.
  • The localization phenomenon provides a key characteristic of these confined random walks.