Applying the WIGNER-WEYL transformation to the LIOUVILLE equation gives the kinetic equation for the WIGNER function

where the kernel of the potential operator is given by

A practically used approximation to incorporate realistic scattering processes into the WIGNER equation is to utilize the BOLTZMANN scattering operator [220,219], or by an even simpler scheme such as the relaxation time approximation[221]. The inclusion of dissipation through the BOLTZMANN scattering operator, although intuitively appealing, raises some concerns about the validity of such procedure. The BOLTZMANN scattering operator is semi-classical by its nature, and represents a good approximation for sufficiently smooth device potentials. To account for scattering more rigorously, spectral information has to be included into the WIGNER function, resulting in an energy-dependence in addition to the momentum dependence [190].

The kinetic equation for the WIGNER function is similar to the semi-classical BOLTZMANN equation, except for a non-local potential term. In the case of a slowly varying potential this non-local term reduces to the local classical force term, and the semi-classical description given by the BOLTZMANN equation is obtained from the WIGNER equation. The BOLTZMANN equation is the basis for the standard models of electron transport in semiconductors in a semi-classical approximation. By far the most widely used technique for solving the BOLTZMANN equation has been the Monte Carlo method [222]. Transport models based on the BOLTZMANN transport equation can be derived using the method of moments [223,224,225] which yields the drift-diffusion model [226], the energy-transport and hydrodynamic models [227], or higher-order transport models [228]. Furthermore, an approximate solution can be obtained by expressing the distribution function as a series expansion which leads to the spherical harmonics approach [229,230].