Diffusion on fractals with singular waiting-time distribution

摘要: We study diffusion in lattices of arbitrary dimensions with a power-law distribution waiting times \ensuremath{\tau}, P(\ensuremath{\tau})\ensuremath{\sim}${\ensuremath{\tau}}^{\ensuremath{\alpha}\mathrm{\ensuremath{-}}2}$, \ensuremath{\alpha}1, \ensuremath{\tau}\ensuremath{\ge}1. Using general scaling arguments we find that the asymptotic behavior mean-square displacement random walker is given , where d${\ifmmode\bar\else\textasciimacron\fi{}}_{w}$=${d}_{w}$ for \ensuremath{\alpha}0 and d${\ifmmode\bar\else\textasciimacron\fi{}}_{w}$=${d}_{w}${1+${d}_{s}$\ensuremath{\alpha}/[2(1-\ensuremath{\alpha})]} 0\ensuremath{\le}\ensuremath{\alpha}1 ${d}_{s}$\ensuremath{\le}2. Here ${d}_{w}$ (conventional) exponent constant ${d}_{s}$ fracton dimension substrate. Our expression d${\ifmmode\bar\else\textasciimacron\fi{}}_{w}$ holds Euclidean as well deterministic fractals. have also investigated properties function P\ifmmode \tilde{}\else \~{}\fi{}(l,t) corresponding moments 〈${l}^{q}$〉, l chemical distance traveled time t. To test our theoretical expressions performed extensive computer simulations on incipient percolation cluster d=2, using exact enumeration method. The numerical results agree predictions.