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 ϵϵ, evolution equations are derived for the terms of zeroth and first order in ϵϵ. In particular, a quantum drift-diffusion equation for the position density of electrons, with an ϵϵ-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 ϵ2ϵ2, uniformly in time and for arbitrary initial data is given.