Microscopic description of quantum transport by Langevin and Fokker-Planck equations

The present work focuses on the investigation of the stochastic properties of phase-coherent quantum transport in disordered systems by means of Langevin and Fokker-Planck equations. We discuss the case of cd-transport on the basis of the scattering properties of mesoscopic systems connected to perfectly ordered leads. The evolution of the scattering properties with the system length is a stochastic process which can be described by the multiplication of random transfer matrices T. Starting from a microscopic model of N coupled one-dimensional conductors the Langevin and Fokker-Planck equations for the probability distribution of TT are derived. We show that the method of Kree and Schmid for the description of the asymptotic distribution of TT of a one-dimensional conductor can be formally generalized to quasi-one-dimensional systems. This deepens the results of an earlier work by Dorokhov. The eigenvalues αj of ln TT/2L are expected to converge to the Lyapunov exponents of the transfer matrix with increasing system length. The generalization to quasi-one-dimensional systems shows that the asymptotic behaviour of the moments and the correlations of the αj is determined by the stationary probability distribution of the eigenvectors of TT. We propose a new method for the determination of this stationary distribution and apply it to simple models like the ''Equivalent Channel Model''. This model has equal probability of back-scattering from each channel into each other. It is shown that the orientations of the eigenvectors of this model are uniformly distributed. Finally we calculate exactly the mean value of TT for the special case that the conductors are only coupled by random hopping matrix elements. (orig.)