The Boltzmann transport equation describes the dynamics of electrons via
the time evolution of a 6-D scalar field. This semiclassical
description is valid in any device where the external field is
relatively constant over the decoherence length. Near equilibrium, the
electron distribution can be characterized by a local chemical potential
and the lattice temperature, leading to a simplified electron state
described by a 3-D scalar field. When external fields are large, this
approximation breaks down as the electrons are accelerated far away from
a lattice temperature thermal equilibrium. If the external field is
quasi-homogeneous, the local field or average kinetic energy can be used
to characterize the shape of the distribution function, leading to an
electron state described by one or two 3-D scalar fields. However, if
the external field is not quasi-homogeneous, there is currently no
significant simplification of the Boltzmann transport equation that is
widely accepted as being theoretically sound.