Random Search Walks Inside Absorbing Annuli

Area of Science:

• Physics
• Applied Mathematics
• Statistical Mechanics

Background:

• Random search walks are fundamental in understanding diffusion and transport phenomena.
• The behavior of searchers in confined spaces with absorbing boundaries is crucial for various applications.
• Existing models often simplify step distributions, limiting their applicability to complex search scenarios.

Purpose of the Study:

• To investigate random search walks in two-dimensional (2D) space bounded by concentric absorbing annuli.
• To derive analytical and approximate expressions for search efficiency (η) using power-law step distributions.
• To analyze absorbing probabilities and Shannon entropy in relation to search dynamics.

Main Methods:

• Analytical derivation of search efficiency (η) in the ballistic limit.
• Approximation of η for searches starting far from boundaries.
• Numerical analysis of absorbing probabilities and Shannon entropy.
• Investigation of scaling behavior for small initial distances to the inner ring.

Main Results:

• Exact analytical result for search efficiency (η) in the ballistic limit.
• Approximate expression for η in the far-field regime.
• Numerical results show good agreement with theoretical findings.
• Power-law exponent for equiprobability and maximum entropy grows logarithmically with starting distance.

Conclusions:

• The study provides a comprehensive theoretical and numerical analysis of random search walks with power-law step distributions in 2D annular domains.
• The findings offer valuable insights into search efficiency and boundary interaction dynamics.
• This work serves as a mean-field approach to complex 2D search problems with significant real-world applications.