Asymptotically Exact Solution of the One-Dimensional Trapping Problem

Abstract

The survival probability of a particle that performs a random walk on a linear chain with randomly distributed traps is considered. An asymptotically exact solution was found, valid for large step numbers $n$ and all trap concentrations $c$. The survival probability depends upon the combination $x={[\ensuremath{-}\ensuremath{\pi}\mathrm{ln}(1\ensuremath{-}c)]}^{\frac{2}{3}}{n}^{\frac{1}{3}}$ of the variables $n$ and $c$ only and is in leading order equal to $(\frac{8}{\ensuremath{\pi}}){(\frac{2}{3}\ensuremath{\pi})}^{\frac{1}{2}}{x}^{\frac{3}{2}}\mathrm{exp}(\ensuremath{-}\frac{3x}{2})$. The analytical result includes correction terms and is confirmed by Monte Carlo simulations.