The Kramers problem for quantum fermi-gases with constant collision frequency and specular - diffusive boundary conditions

TL;DR

This paper develops a new generalized method for solving the Kramers problem for quantum Fermi gases with boundary conditions, using integral equations and Neumann series to achieve arbitrary accuracy.

Contribution

It introduces a novel approach to boundary problems in kinetic theory, representing boundary conditions as sources and solving via Fredholm integral equations.

Findings

Method achieves any desired accuracy in solutions.

Reduces boundary problem to Fredholm integral equation.

Provides explicit solutions using Neumann series.

Abstract

The Kramers problem for quantum fermi-gases with specular - diffuse boundary conditions of the kinetic theory is considered. On an example of Kramers problem the new generalised method of a source of the decision of the boundary problems from the kinetic theory is developed. The method allows to receive the decision with any degree of accuracy. At the basis of a method lays the idea of representation of a boundary condition on distribution function in the form of a source in the kinetic equation. By means of integrals Fourier the kinetic equation with a source is reduced to the integral equation of Fredholm type of the second kind. The decision is received in the form of Neumann's series.