KABANOV, V.V. and ALEXANDROV, A.S., 2008. Electron relaxation in metals: theory and exact analytical solutions. Physical Review B, 78(17), 174514 pp. 1-8.
The nonequilibrium dynamics of electrons is of a great experimental and theoretical value, providing important
microscopic parameters of the Coulomb and electron-phonon interactions in metals and other cold
plasmas. Because of the mathematical complexity of collision integrals, theories of electron relaxation often
rely on the assumption that electrons are in a “quasiequilibrium” QE with a time-dependent temperature, or
on the numerical integration of the time-dependent Boltzmann equation. We transform the integral Boltzmann
equation to a partial differential Schrödinger-type equation with imaginary time in a one-dimensional “coordinate”
space reciprocal to energy which allows for exact analytical solutions in both cases of electron-electron
and electron-phonon relaxations. The exact relaxation rates are compared with the QE relaxation rates at high
and low temperatures.