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1‐D random walk in presence of a totally reflecting barrier
Author(s) -
Orlowski Marius
Publication year - 2003
Publication title -
physica status solidi (b)
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.51
H-Index - 109
eISSN - 1521-3951
pISSN - 0370-1972
DOI - 10.1002/pssb.200301803
Subject(s) - random walk , statistical physics , diffusion , rectangular potential barrier , anomalous diffusion , continuous time random walk , mathematics , probability distribution , redistribution (election) , particle (ecology) , diffusion barrier , distribution (mathematics) , physics , materials science , mathematical analysis , condensed matter physics , statistics , computer science , nanotechnology , quantum mechanics , innovation diffusion , knowledge management , oceanography , layer (electronics) , politics , geology , political science , law
A rigorous analysis of counting paths for 1‐D random walk in the presence of a reflecting barrier is presented. This paper defines and distinguishes between partially and totally reflecting barriers. So far, in the literature only a special case of partially reflecting barrier has been dealt with. An exact combinatorial formula is proven which describes the probability distribution of a diffusing particle at a totally reflecting barrier, allowing computation of any random walk redistribution of a diffusing species near or at the totally reflecting barrier. The analysis shows that for a particle starting its random walk at the barrier, the probability of finding it at the interface is diminishing with the number of diffusion steps N as 1/( N /2 + 1) and that the peak of the probability distribution is moving away from the barrier with the increasing number of steps as . This analysis has implications on the treatment of diffusion of impurities and point defects in thin films and in subsurface layers.