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Territory covered by N random walkers.

S B Yuste1, L Acedo

  • 1Departamento de Física, Universidad de Extremadura, 06071 Badajoz, Spain. santos@unex.es

Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics
|April 24, 2002
PubMed
Summary

We analyze the coverage of distinct sites by N random walkers over time. The study reveals the asymptotic behavior and key corrective terms for large N, crucial for understanding diffusion on lattices.

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Area of Science:

  • Statistical Physics
  • Mathematical Physics
  • Condensed Matter Physics

Background:

  • Understanding the spatial extent of random processes is fundamental in various scientific fields.
  • Evaluating the number of distinct sites visited by random walkers is a classic problem in statistical mechanics.

Purpose of the Study:

  • To revisit and provide a detailed asymptotic analysis of the number of distinct sites covered by N random walkers up to time t.
  • To calculate the main term and the first two corrective terms for the coverage in a nontrivial time regime.
  • To investigate the behavior for a large number of walkers (N>>1).

Main Methods:

  • Asymptotic analysis of the coverage function S(N)(t).
  • Calculation of main and corrective terms in the asymptotic expansion.
  • Application to d-dimensional (d=1,2,3) simple cubic lattices.

Main Results:

  • The asymptotic behavior of S(N)(t) for N>>1 is determined.
  • The main term is identified as the volume of a hypersphere with a specific radius dependent on N, D, and t.
  • The first two corrective terms are calculated, showing a mild decay of 1/ln(m) N for the mth term.

Conclusions:

  • The study provides precise quantitative results for the coverage of random walkers on lattices.
  • Corrective terms reveal the impact of surface "roughening" on the visited sites.
  • The findings are applicable to understanding diffusion phenomena in physical systems.

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