Using the phase-space formulation of quantum mechanics, we derive a four-component Wigner equation for a system composed of spin- fermions (typically, electrons) including the Zeeman effect and the spin–orbit coupling. This Wigner equation is coupled to the appropriate Maxwell equations to form a self-consistent mean-field model. A set of semiclassical Vlasov equations with spin effects is obtained by expanding the full quantum model to first order in the Planck constant. The corresponding hydrodynamic equations are derived by taking velocity moments of the phase-space distribution function. A simple closure relation is proposed to obtain a closed set of hydrodynamic equations.
This article is part of the themed issue ‘Theoretical and computational studies of non-equilibrium and non-statistical dynamics in the gas phase, in the condensed phase and at interfaces’.
One contribution of 11 to a theme issue ‘Theoretical and computational studies of non-equilibrium and non-statistical dynamics in the gas phase, in the condensed phase and at interfaces’.
Electronic supplementary material is available online at https://dx.doi.org/10.6084/m9.figshare.c.3685624.
- Accepted October 26, 2016.
- © 2017 The Author(s)
Published by the Royal Society. All rights reserved.