The generalized transport equation for an electron : A solution in a simple case

The master equation to general order in the coupling responsible for the dissipative behaviour, derived in earlier papers by one of us (L.V.H.), is applied to the simple case of an electron in a system of randomly distributed, static, elastic scatters. Replacing Wu'(k, k’) by a constant, the equation can be solved approximately for general strength of the coupling. The solution, describing the approach to equilibrium of the electron momentum distribution, has an oscillatory character. The slightly more complicated case of an electron interacting with a vibrating harmonic lattice is also considered. It is shown how one can derive from the generalized master equation an equation describing the evolution of the electron alone, the phonons only entering through their coarse grained distribution in wave vector. Neglecting the phonon energies, one finds an equation of the same form as in the case of static, elastic scatterers