D.C. electrical resistivity as a functional of generalized dynamical susceptibilities—Comparison of different methods

The d.c. electrical resistivity is considered for a general solid state model characterized by local interactions of the charge carriers with a system of scatterers which may be e.g. quasiparticles, spins or impurities. A basic relationship - valid over the full temperature range - between the electrical resistivity of the charge carrier system (without magnetic field) and the generalized dynamical susceptibility of the scattering system is derived in the weak scattering limit. The equivalence of the following methods is shown: Boltzmann equation approach (variation procedure), Mori formalism, force-force correlation function method (for the total forces on the charge carriers), and the method of the transformed Liouville equation; herewith the equivalence with the Kubo or Green function method, the Zubarev formalism, the information-theoretical method and the Zwanzig formalism is also given. As a specific example, the s-d exchange model for the scattering of conduction electrons by localized spins is discussed including the corresponding electron-magnon system as a low-temperature approximation.