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: ∂P(x,t)/∂t=D[${\mathrm{\partial }}^2$P(x,t)/∂${\mathit{x}}^2$]+λV(x)P(x,t) , where V is a white-noise Gaussian potential. A quantity τ=(16D/9${\lambda }^4$ ${)}^{1/3}$ is introduced for the time scale. When the diffusion time t≪τ, the behavior of the average 〈P(x,t)〉 is essentially diffusive. When t≫τ, the random potential plays a dominant role, and the average 〈P(x,t)〉 tends to [${\lambda }^4$ ${\mathit{t}}^{5/2}$/8(π${\mathit{D}}^3$ ${)}^{1/2}$]exp[(${\lambda }^4$ ${\mathit{t}}^3$/48D) (1-${\mathit{x}}^2$/2Dt)]. t)].