2022/01/18 by Samuel Hidalgo‐Caballero, Samuel Hidalgo-Caballero, Alvaro Cassinelli +7 · 1 citation
Biochemistry, Genetics and Molecular Biology · Mathematics · Physics and Astronomy · #Data Analysis #Diffusion and Search Dynamics #FOS: Physical sciences #Mathematical Physics (math-ph) #Statistics and Probability (physics.data-an) #math-ph #math.MP #physics.data-an
paper · pdf · doi:10.48550/arxiv.2201.07142
14 pages, 5 figures
arxiv created 2022/01/18 · openalex publication_date 2022/01/18 · arxiv updated 2022/01/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A remarkable result from integral geometry is Cauchy's formula, which relates the mean path length of ballistic trajectories randomly crossing a convex 2D domain, to the ratio between the region area and its perimeter. This theorem has been generalized for non-convex domains and extended to the case of Brownian motion to find many applications in various fields including biological locomotion and wave physics. Here, we generalize the theorem to arbitrary closed trajectories exploring arbitrary domains. We demonstrate that, regardless of the complexity of the trajectory, the mean arc length still satisfies Cauchy's formula provided that no trajectory is entirely contained in the domain. Below this threshold, the mean arc length decreases with the size of the trajectory. In this case, an approximate analytical formula can still be given for convex trajectories intersecting convex domains provided they are small in comparison. To validate our analysis, we performed numerical simulations of different types of trajectories exploring arbitrary 2D domains. Our results could be applied to retrieve geometric information of bounded domains from the mean first entrance-exit length.