General form of DMPK Equation

The Dorokhov–Mello–Pereyra–Kumar (DMPK) equation, using in the analysis of quasi-onedimensional systems and describing evolution of diagonal elements of the many-channel transfer-matrix, is derived under minimal assumptions on the properties of channels. The general equation is of the diffusion type with a tensor character of the diffusion coefficient and finite values of non-diagonal components. We suggest three different forms of the diagonal approximation, one of which reproduces the usual DMPK equation and its generalization suggested by Muttalib and co-workers. Two other variants lead to equations of the same structure, but with different definitions of their parameters. They contain additional terms, which are absent in the first variant. The coefficients of additional terms are shown to be finite beyond the metallic phase by calculation of the Lyapunov exponents and their comparison with numerical experiments. Consequences of the obtained equations for the problem of the conductance distribution and the status of the nonlinear sigma-models are discussed.