Exact Fundamental Function for the One-Dimensional Random Walk with a Perfect Mirror under the External Field

• Published in: Bulletin of the Korean Chemical Society Vol. 37; no. 10; pp. 1638 - 1643
• Main Authors: Yoon, Joung-Hahn, Kim, Hyojoon
• Format: Journal Article
• Language: English
• Published: Weinheim Wiley-VCH Verlag GmbH & Co. KGaA 01.10.2016; Wiley‐VCH Verlag GmbH & Co. KGaA; 대한화학회
• Subjects: Diffusion-influenced reactions; Exact Green function; External field effects; One dimension; Random walk; 화학
• ISSN: 1229-5949; 0253-2964; 1229-5949
• DOI: 10.1002/bkcs.10921

The exact analytical probability function is presented for the random walk with a perfect mirror under the constant external field in one dimension. Unlike the field‐free solution, the symmetry around the boundary is broken by the external field and the fundamental function is not given by a simple form. We prove the solution by mathematical induction method and numerical simulations. Monte Carlo simulations can be replaced by the function without statistical noise. Based on this function, we also obtain the solution for the continuum diffusion‐influenced reaction, which is shown to be superior to the known solution especially for the system with a strong external field.