Transport of excitations in disordered systems: Self-consistent density resummation

New self-consistent equations for the transport of excitations in disordered systems are derived based on a resummation procedure of the density expansion. Both local-time [partial-time-ordering prescription (POP)] and convolution-type [chronological-time-ordering prescription] equations are explored. Application is made to a system consisting of a random mixture of donors and traps with a general type of transfer rate $W\left(r\right)\mathrm{}$. We calculate the probability of remaining in the original site $G_0\mathrm{}\left(t\right)\mathrm{}$ as well as the second moment of the distribution of excitations $〈r^2\mathrm{}\left(t\right)\mathrm{}〉$. The POP equations reduce to the exact Forster solution for $G_0\mathrm{}\left(t\right)\mathrm{}$ in the case of traps only. We analyze the long-time behavior of $G_0\mathrm{}\left(t\right)\mathrm{}$ and its dependence on $W\left(r\right)\mathrm{}$. We find a percolation-type transition when $W\left(r\right)\mathrm{}$ has a cutoff [i.e., $W\left(r\right)=0\mathrm{}$ for $r>\mathrm{}{r}_0$] but in all other cases the system is diffusive at long times [$G_0\mathrm{}\left(t\right)\mathrm{}$ vanishes and $〈r^2\mathrm{}\left(t\right)\mathrm{}〉$ diverges linearly in time].