Diffusive corrections to asymptotics of a strong-field quantum transport equation

The asymptotic analysis of a linear high-field Wigner-BGK equation is developed by a modified Chapman–Enskog procedure. By an expansion of the unknown Wigner function in powers of the Knudsen number ϵ , evolution equations are derived for the terms of zeroth and first order in ϵ . In particular, a quantum drift-diffusion equation for the position density of electrons, with an ϵ -order correction on the field terms, is obtained. Well-posedness and regularity of the approximate problems are established, and a rigorous proof that the difference between exact and asymptotic solutions is of order ϵ 2 , uniformly in time and for arbitrary initial data is given.