Buffon-Laplace Needle Problem as a geometric probabilistic approach to filtration process

• Published in: arXiv.org 2024-11
• Main Authors: Yan-Jie, Min, De-Quan, Zhu, Jin-Hua, Zhao
• Format: Article
• Language: English
• Subjects: Exact solutions; Mathematical analysis; Monte Carlo simulation
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.2402.06670

Buffon-Laplace Needle Problem considers a needle of a length \(l\) randomly dropped on a large plane distributed with vertically parallel lines with distances \(a\) and \(b\) (\(a \geqslant b\)), respectively. As a classical problem in stochastic probability, it serves as a mathematical basis of various physical literature, such as the efficiency of a filter and the emergence of clogging in filtration process. Yet its potential application is limited by previous focus on its original form of the `short' needle case of \(l < b\) and its analytical difficulty in a general sense. Here, rather than a `short' needle embedded in two-dimensional space, we analytically solve problem versions with needles and spherocylinders of arbitrary length and radius embedded in two- and three-dimensional spaces dropped on a grid with any rectangular shape. We further confirm our analytical theory with Monte Carlo simulation. Our framework here helps to provide a geometric analytical perspective to filtration process, and also extend the analytical power of the needle problem into unexplored parameter regions for physical problems involving stochastic processes.