Number of distinct sites visited by N particles diffusing on a fractal.

Abstract

We study the mean number of distinct sites, ${\mathit{S}}_{\mathit{N}}$(t), visited up to time t by N\ensuremath{\gg}1 noninteracting random walkers all starting from the same origin on a fractal substrate of dimension ${\mathit{d}}_{\mathit{f}}$. Using analytic arguments and numerical simulations, we find ${\mathit{S}}_{\mathit{N}}$(t)\ensuremath{\sim}(lnN${)}_{\mathit{f}}^{\mathit{d}}$/\ensuremath{\delta}${\mathit{t}}_{\mathit{s}}^{\mathit{d}}$/2 for fractals with spectral dimension ${\mathit{d}}_{\mathit{s}}$==2${\mathit{d}}_{\mathit{f}}$/${\mathit{d}}_{\mathit{w}}$2, where \ensuremath{\delta}==${\mathit{d}}_{\mathit{w}}$/(${\mathit{d}}_{\mathit{w}}$-1) and ${\mathit{d}}_{\mathit{w}}$ is the fractal dimension of a random walk.