The Generalized DMPK equation revisited: A systematic derivation

The Generalized Dorokov-Mello-Pereyra-Kumar (DMPK) equation has recently been used to obtain a family of very broad and highly asymmetric conductance distributions for three dimensional disordered conductors. However, there are two major criticisms of the derivation of the Generalized DMPK equation: (1) certain eigenvector correlations were neglected based on qualitative arguments that can not be valid for all disorder, and (2) the repulsion between two closely spaced eigenvalues were not rigorously governed by symmetry considerations. In this work we show that it is possible to address both criticisms by including the eigenvalue and eigenvector correlations in a systematic and controlled way. It turns out that the added correlations determine the evolution of the Jacobian, without affecting the evaluation of the conductance distributions. They also guarantee the symmetry requirements. In addition, we obtain an exact relationship between the eigenvectors and the Lyapunov exponents leading to a sum rule for the latter at all disorder.