Journal article

Publication year: 1996

Pages: 161-168

Journal: Fractals: Complex Geometry, Patterns, and Scaling in Nature and Society

Volume number: 4

Issue number: 2

ISSN: 0218-348X

eISSN: 1793-6543

DOI Link: https://doi.org/10.1142/S0218348X96000212

Publisher: World Scientific Publishing

Abstract: 
The number of distinct sites visited by a random walker after t steps is of great interest, as it provides a direct measure of the territory covered by a diffusing particle. We review the analytical solution to the problem of calculating S-N(t), the mean number of distinct sites visited by N random walkers on a d-dimensional lattice, for d = 1, 2, 3 in the limit of large N. There are three distinct time regimes for S-N(t) A remarkable transition, for dimension greater than or equal to 2, in the geometry of the set of visited sites is found. This set initially grows as a disk with a relatively smooth surface until it reaches a certain size, after which the surface becomes increasingly rough. We also review the results for a model for migration and spreading of populations and diseases. The model is based on N diffusing species, where each species has a probability alpha(-) of dying (or recovery from a disease) and a probability alpha(+) to give birth (or to infect another species). It is found analytically that when alpha(+) approximate to alpha(-) not equal 0, after a crossover time t(x) similar to N/2 alpha(-), the territory covered by the population is localized around its center of mass while the center of mass diffuses regularly. When alpha(+) > alpha(-), the localization breaks down after a second crossover time and the species diffuse and spread around their center of mass. These results may explain the phenomena of migration and spreading of diseases and population appearing in nature.

Harvard Citation style: Havlin, S., Bunde, A., Larralde, H., Lereah, Y., Meyer, M., Trunfio, P., et al. (1996) Spreading of N diffusing species with death and birth features, Fractals: Complex Geometry, Patterns, and Scaling in Nature and Society, 4(2), pp. 161-168. https://doi.org/10.1142/S0218348X96000212

APA Citation style: Havlin, S., Bunde, A., Larralde, H., Lereah, Y., Meyer, M., Trunfio, P., & Stanley, H. (1996). Spreading of N diffusing species with death and birth features. Fractals: Complex Geometry, Patterns, and Scaling in Nature and Society. 4(2), 161-168. https://doi.org/10.1142/S0218348X96000212