Path-integral approach to diffusion in random media.

Abstract

Using the path-integral method, we derive the analytical solution for the following one-dimensional diffusion in random media: \ensuremath{\partial}P(x,t)/\ensuremath{\partial}t=D[${\mathrm{\ensuremath{\partial}}}^{2}$P(x,t)/\ensuremath{\partial}${\mathit{x}}^{2}$]+\ensuremath{\lambda}V(x)P(x,t) , where V is a white-noise Gaussian potential. A quantity \ensuremath{\tau}=(16D/9${\ensuremath{\lambda}}^{4}$${)}^{1/3}$ is introduced for the time scale. When the diffusion time t\ensuremath{\ll}\ensuremath{\tau}, the behavior of the average 〈P(x,t)〉 is essentially diffusive. When t\ensuremath{\gg}\ensuremath{\tau}, the random potential plays a dominant role, and the average 〈P(x,t)〉 tends to [${\ensuremath{\lambda}}^{4}$${\mathit{t}}^{5/2}$/8(\ensuremath{\pi}${\mathit{D}}^{3}$${)}^{1/2}$]exp[(${\ensuremath{\lambda}}^{4}$${\mathit{t}}^{3}$/48D) (1-${\mathit{x}}^{2}$/2Dt)]. t)].