Physical Mathematics

Fractional and High Order Asymptotic Results of the MFPT

Abstract

Bar-Haim A

This work discusses the Mean First Passage Time (MFPT) and Mean Residence Time (MRT) of continuous-time nearest-neighbor random walks in a finite one-dimensional system with a trap at the origin and a reflecting barrier at the other end. The asymptotic results of the MFPT for random walks that start, for example, at the reflecting point, have a variety of dependencies with respect to its size N. For example, for the case of birth and death processes the MFPT~N; for the case of symmetric random walks the MFPT~N² and for the case of biased random walks the MFPT~α^N where α is a constant that depends on the system’s rates. In this work a transition matrix is derived in such a way that the MRT of the system is equal to (m+1)^d where m is the site number, and d is any arbitrary number satisfying the condition d>0. Since the MFPT is the sum of MRTs then the corresponding MFPT for such a transition matrix is MFPT~N^(1+d). Thus, one can determine the asymptotic result of the MFPT to be N^1+d for any arbitrary d>0, and based on it, obtain the corresponding transition matrix. Several examples of fractional and high order asymptotic results of the MFPT such as N^3.5, N⁵, N⁶, are presented.