# General form of DMPK equation

## TL;DR

This paper derives a generalized form of the DMPK equation for quasi-one-dimensional systems, incorporating tensor diffusion and multiple approximation variants, extending the traditional model's scope.

## Contribution

It introduces a comprehensive generalization of the DMPK equation with tensor diffusion and explores three diagonal approximation forms, including reproductions of existing models.

## Key findings

- Derived a tensor-based diffusion form of the DMPK equation.
- Presented three diagonal approximation variants, including the classic and generalized forms.
- Identified additional terms in alternative variants that extend the traditional equation.

## Abstract

The Dorokhov-Mello-Pereyra-Kumar (DMPK) equation, using in the analysis of quasi-one-dimensional 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 entering them parameters. They contain additional terms, which are absent in the first variant.