PATH-INTEGRAL APPROACH TO DIFFUSION IN RANDOM-MEDIA

Using the path-integral method, we derive the analytical solution for the following one-dimensional diffusion in random media: partial-P(x,t)/partial-t = D[partial-2P(x,t)/partial-x2] + lambda-V(x)P(x,t), where V is a white-noise Gaussian potential. A quantity tau = (16D/9-lambda-4)1/3 is introduced for the time scale. When the diffusion time t << tau, the behavior of the average < P(x,t) > is essentially diffusive. When t >> tau, the random potential plays a dominant role, and the average < P(x,t) > tends to [lambda-4t5/2/8(pi-D3)1/2]exp[(lambda-4t3/48D)(1 - x2/2Dt)].