Order statistics of the trapping problem
Abstract
When a large number N of independent diffusing particles are placed upon a site of a d-dimensional Euclidean lattice randomly occupied by a concentration c of traps, what is the m-th moment <t^m_{j,N}> of the time t_{j,N} elapsed until the first j are trapped? An exact answer is given in terms of the probability Phi_M(t) that no particle of an initial set of M=N, N-1,..., N-j particles is trapped by time t. The Rosenstock approximation is used to evaluate Phi_M(t), and it is found that for a large range of trap concentracions the m-th moment of t_{j,N} goes as x^{-m} and its variance as x^{-2}, x being ln^{2/d} (1-c) ln N. A rigorous asymptotic expression (dominant and two corrective terms) is given for <t^m_{j,N}> for the one-dimensional lattice.
Cite
@article{arxiv.cond-mat/0110172,
title = {Order statistics of the trapping problem},
author = {Santos B. Yuste and Luis Acedo},
journal= {arXiv preprint arXiv:cond-mat/0110172},
year = {2009}
}
Comments
11 pages, 7 figures, to be published in Phys. Rev. E