# Territory covered by N diffusing particles

@article{Larralde1992TerritoryCB, 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}, journal={Nature}, year={1992}, volume={355}, pages={423-426} }

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

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#### References

SHOWING 1-10 OF 14 REFERENCES

The Number of Distinct Sites Visited in a Random Walk on a Lattice

- Physics
- 1963

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 n… Expand

Random dispersal in theoretical populations.

- Mathematics
- 1951

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 growing… Expand

Random Walks on Lattices. II

- Physics
- 1965

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 boundary… Expand

Variance of the range of a random walk

- Mathematics
- 1986

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′), and… Expand

On the number of distinct sites visited in 2D lattices

- Chemistry
- 1982

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 and… Expand

Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications

- Physics
- 1990

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 may… Expand

Diffusion in regular and disordered lattices

- Physics
- 1987

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 to… Expand

On the range of random walk

- Mathematics
- 1968

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 integer… Expand

Random walks in biology

- Biology, Physics
- 1983

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 basis… Expand

Diffusion-Controlled Reactions

- Chemistry
- 1983

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 rate… Expand