Termite and ant diffusions in the d-dimensional lattice trapping model

Abstract

The coherent medium approximation of Odagaki and Lax (1981) is generalised to the trapping model. The frequency-dependent diffusion constant in the d-dimensional hypercubic lattice is studied when the jump rate obeys a bimodal distribution. The coherent medium approximation gives the correct static diffusion constant. The imaginary part of the AC part of the diffusion constant vanishes linearly in frequency omega when d>2, as omega ln omega when d=2 and as omega d/2 when d 4, as omega 2 ln omega when d=4 and as omega d/2 when d<4. The termite limit is studied by taking the limit that one of the two jump rates (probability p) becomes infinite. In the termite diffusion the static diffusion constant is critical at p=1 and the critical exponents are the same as those for the termite diffusion in the hopping model. The ant (or ant lion) limit is defined by the limit of one jump rate being zero. The imaginary and real parts of the AC diffusion constant vanish linearly and quadratically in frequency, respectively. The critical exponent of the leading real part of the diffusion constant is one less than the corresponding exponent for the ant diffusion in the hopping model below the percolation threshold, while the leading imaginary part has the same critical exponent as those in the hopping model.