Ballistic deposition with power-law noise: A variant of the Zhang model.

Abstract

We study a variant of the Zhang model [Y.-C. Zhang, J. Phys. (Paris) 51, 2113 (1990)], ballistic deposition of rods with the length l of the rods being chosen from a power-law distribution P(l)\ensuremath{\sim}${\mathit{l}}^{\mathrm{\ensuremath{-}}1\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\mu}}}$. Unlike in the Zhang model, the site at which each rod is dropped is chosen randomly. We confirm that the growth of the rms surface width w with length scale L and time t is described by the scaling relation w(L,t)=${\mathit{L}}^{\mathrm{\ensuremath{\alpha}}}$w(t/${\mathit{L}}^{\mathrm{\ensuremath{\alpha}}/\mathrm{\ensuremath{\beta}}}$), and we calculate the values of the surface-roughening exponents \ensuremath{\alpha} and \ensuremath{\beta}. We find evidence supporting the possibility of a critical value ${\mathrm{\ensuremath{\mu}}}_{\mathit{c}}$\ensuremath{\approxeq}5 for d=1 with \ensuremath{\alpha}=1/2 and \ensuremath{\beta}=1/3 for \ensuremath{\mu}>${\mathrm{\ensuremath{\mu}}}_{\mathit{c}}$, while for \ensuremath{\mu}${\mathrm{\ensuremath{\mu}}}_{\mathit{c}}$, \ensuremath{\alpha} and \ensuremath{\beta} vary smoothly, attaining the values \ensuremath{\alpha}=\ensuremath{\beta}=1 for \ensuremath{\mu}=2.