Magnetic Corrections to the Boltzmann Transport Equation

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Abstract
The usual derivations of the Boltzmann transport equation suffer from a number of weaknesses. Using a simple model, this paper attempts to treat on as rigorous a basis as possible the influence of a magnetic field on the transport equation. The approach presented here establishes the relationship between the classical Boltzmann equation and the corresponding quantum-mechanical formalism. It is shown how the exact guage-dependent Liouville equation, determining the density matrix, can be transformed into a completely gauge-independent equation satisfied by a new density matrix. This equation is solved for a model consisting of noninteracting free electrons being elastically scattered by randomly placed scattering centers. The new density matrix is developed in ascending powers of the strength of the scattering potential. In carrying out this development, the product of the cyclotron frequency and the collision relaxation time is assumed to be of order unity. In this case the familiar Boltzmann transport equation in the presence of a magnetic field represents an approximation valid in the limiting cases of very weak or very dilute scatterers. The corresponding velocity operator is shown to be the usual gauge-independent expression, just the ordinary free-particle momentum operator divided by the mass. Higher order corrections to the transport equation are found, some of which involve the magnetic field.

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