Dynamical theory of diffusion and localization in a random, static field

An approximation scheme is derived to calculate the velocity-autocorrelation function for the Lorentz model of overlapping hard discs and hard spheres. The theory describes a feedback between the particle-density correlations and the current relaxation rate, and is shown to give a percolation edge, a transition from the normal diffusion phase to nondiffusion phase characterized by a finite localization length $l_0$. Near the edge the low-frequency velocity spectrum for either phase is evaluated, thereby finding diffusivity to approach zero linearly with the separation parameter $\epsilon =\frac{(n-n_c)}{n_c}$, where $n_c$ is the critical density, while $l_0$ diverges like $\frac{1}{\sqrt{\epsilon }}$. A power-law long-time decay of the velocity-autocorrelation function is found for the diffusion phase. Upon approaching $n_c$ the hydrodynamic regime shrinks to zero, and a transition in the power-law exponent from its low-density value which is dependent on dimension to a value of $\frac{3}{2}$ for both dimensions is predicted.