Multiparticle random walks

Details Multiparticle random walks Multiparticle random walks

2003-10-06

arxiv.org/abs/cond-mat/0310121v1

Recent Research Developments in Statistical Physics, Volume 2, Pages 83-106 (Transworld Research Network, Trivandrum, India, 2

Physics

Condensed Matter

Statistical Mechanics

14 pages, 13 figures, RevTex 4

Scientific paper

An overview is presented of recent work on some statistical problems on multiparticle random walks. We consider a Euclidean, deterministic fractal or disordered lattice and N >> 1 independent random walkers initially (t=0) placed onto the same site of the substrate. Three classes of problems are considered: (i) the evaluation of the average number of distinct sites visited (territory explored) up to time t by the N random walkers, (ii) the statistical description of the first passage time t_{j,N} to a given distance of the first j random walkers (order statistics of exit times), and (iii) the statistical description of the time \mathbf{t}_{j,N} elapsed until the first j random walkers are trapped when a Euclidean lattice is randomly occupied by a concentration c of traps (order statistics of the trapping problem). Although these problems are very different in nature, their solutions share the same form of a series in ln^{-n}(N) \ln^m \ln (N) (with n>=1 and 0<=m<=n) for N>>1. These corrective terms contribute substantially to the statistical quantities even for relatively large values of N.

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