Territory covered by N diffusing particles

  title={Territory covered by N diffusing particles},
  author={Hern{\'a}n Larralde and Paul A. Trunfio and Shlomo Havlin and Harry Eugene Stanley and G. H. Weiss},
THE number of distinct sites visited by a random walker after t steps is of great interest1–21, as it provides a direct measure of the territory covered by a diffusing particle. Thus, this quantity appears in the description of many phenomena of interest in ecology13–16, metallurgy5–7, chemistry17,18 and physics19–22. Previous analyses have been limited to the number of distinct sites visited by a single random walker19–22, but the (nontrivial) generalization to the number of distinct sites… Expand
Diffusion of a set of random walkers in Euclidean media. First passage times
When a large numberN of independent random walkers diffuse on ad-dimensional Euclidean substrate, what is the expectation valueht1;Ni of the time spent by the first random walker to cross a givenExpand
Order statistics of Rosenstock's trapping problem in disordered media.
  • S. B. Yuste, L. Acedo
  • Mathematics, Medicine
  • Physical review. E, Statistical, nonlinear, and soft matter physics
  • 2003
Simulation results for the two-dimensional incipient percolation aggregate confirm the predictions of the approach and show that the ratio between the nth cumulant and the n fourth moment of S(N)(t) is, for large N, very large in comparison with the same ratio in Euclidean media, and almost constant. Expand
Multiparticle trapping problem in the half-line
A variation of Rosenstock's trapping model in which N independent random walkers are all initially placed upon a site of a one-dimensional lattice in the presence of a one-sided random distributionExpand
Number of distinct sites visited by a random walker in the presence of a trap
The authors study the number of distinct sites visited by a random walker in d=1 after t steps, S(t), in the presence of a trap. They calculate the distribution q(S, t) of S(t) in the limit of largeExpand
Coloring of a one-dimensional lattice by two independent random walkers
A new type of question in random walk theory is formulated and solved for the particular case of a periodic one-dimensional lattice. A “red” and a “blue” random walker perform simultaneousExpand
Survival probability and order statistics of diffusion on disordered media.
  • L. Acedo, S. B. Yuste
  • Mathematics, Medicine
  • Physical review. E, Statistical, nonlinear, and soft matter physics
  • 2002
We investigate the first passage time t(j,N) to a given chemical or Euclidean distance of the first j of a set of N>>1 independent random walkers all initially placed on a site of a disorderedExpand
Number of common sites visited by N random walkers.
  • S. Majumdar, M. Tamm
  • Mathematics, Physics
  • Physical review. E, Statistical, nonlinear, and soft matter physics
  • 2012
The mean number of common sites, W(N)(t), visited by N independent random walkers each of length t and all starting at the origin at t = 0 in d dimensions is computed analytically. Expand
Exact distributions of the number of distinct and common sites visited by N independent random walkers.
It is shown that these two random variables can be mapped onto extreme value quantities associated with N independent random walkers and compute exactly their probability distributions for any value of N in the limit of large time t, where the randomWalkers can be described by Brownian motions. Expand
Note: About the Random Walk from Many Injection Points
The random walk on a discrete lattice has been analysed in completely different fields such as chemistry [1, 2], ecology [3, 4], and general physics [5, 6]. The general idea has been to insert aExpand
Distinct nodes visited by random walkers on scale-free networks
This work shows that the fraction of nodes of scale-free network not visited by random walkers in time $t$ has a stretched exponential form independent of the details of the network and number of walkers. Expand


The Number of Distinct Sites Visited in a Random Walk on a Lattice
A general formalism is developed from which the average number of distinct sites visited in n steps by a random walker on a lattice can be calculated. The asymptotic value of this number for large nExpand
Random dispersal in theoretical populations.
The random-walk problem is adopted as a starting point for the analytical study of dispersal in living organisms. The solution is used as a basis for the study of the expanson of a growingExpand
Random Walks on Lattices. II
Formulas are obtained for the mean first passage times (as well as their dispersion) in random walks from the origin to an arbitrary lattice point on a periodic space lattice with periodic boundaryExpand
Variance of the range of a random walk
Tn, the expectation of the square of the number of distinct sites occupied by a random walk in steps 1 throughn, is obtained from its relation to the dual first occupancy probabilityFij(x, x′), andExpand
On the number of distinct sites visited in 2D lattices
We present analytic results for the asymptotic behavior of Sn, the number of distinct sites visited in an n‐step random walk on two‐dimensional lattices using a combination of contour integration andExpand
Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications
Abstract The subject of this paper is the evolution of Brownian particles in disordered environments. The “Ariadne's clew” we follow is understanding of the general statistical mechanisms which mayExpand
Diffusion in regular and disordered lattices
Abstract Classical diffusion of single particles on lattices with frozen-in disorder is surveyed. The methods of continuous-time random walk theory are pedagogically developed and applications toExpand
On the range of random walk
AbstractLet {Sn, n=0, 1, 2, …} be a random walk (Sn being thenth partial sum of a sequence of independent, identically distributed, random variables) with values inEd, thed-dimensional integerExpand
Random walks in biology
This book is a lucid, straightforward introduction to the concepts and techniques of statistical physics that students of biology, biochemistry, and biophysics must know. It provides a sound basisExpand
Diffusion-Controlled Reactions
It is apparent that the kinetic rate kD influences the effective rate of reaction and that in certain circumstances (kr � kr) it becomes the rate­ limiting step [keff = kD]. Accordingly, the rateExpand