Special solutions of the Fokker-Planck transport equation for arbitrary band structures; Mobility and diffusivity in a uniform field of force

A Fokker-Planck equation can be derived from a transition-type transport equation if the transition rates are nearly local in momentum space compared with the inhomogeneity length of the distribution. It is a second-order differential equation, whose coefficients depend on the band structureE(k), the viscosity tensor (k), and the temperatureT. Classical solutions of the Fokker-Planck equation deal with the parabolic band structure of free Brownian particles in a field of force. Mobility and diffusivity are then independent of the applied field. Here the explicit solution for the stationary state and the time-integrated conditional probability will be given in one dimension. This suffices to determine mobility and diffusivity. Assuming = 1, these quantities become independent of the field and the band structure, if the latter is nonperiodic, though the distribution still depends on it. This property even holds in three dimensions fork-independent viscosity tensors. Field-dependent mobility and diffusivity are obtained for ak-dependent viscosity or = 1 and periodic band structures. The latter is demonstrated for the caseE-cosk, which is also related to the noise problem in Josephson junctions.