Mobility and Linear Response Theory on Percolation Lattices

A real-space renormalization-group method is used to investigate particle motion on percolation lattices in the presence of an external force $\overrightarrow{\mathrm{K}}$. At the percolation threshold $p_c$ the particle position, $〈R〉$, varies as $〈R〉\propto t^{\chi }$, where $\chi <1\mathrm{}$ and $t^{\chi }$ gives the mean square displacement. Near $p_c$, the linear response theory is found to hold only asymptotically under the condition $\gamma \equiv \frac{\mathrm{Kl}}{k_{\mathrm{B}}T}\lesssim {\xi }^{-1\mathrm{}}$ or $\gamma \lesssim {\mathrm{}\left(\mathrm{wt}\right)\mathrm{}}^{-\frac{\chi }{2}}$, where $l$ is a lattice constant, $\xi$ is a coherence length, and $w$ is a transition rate.