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

ABSTRACT The asymptotic analysis of a linear high-field Wigner-BGK equation is developped by a modified Chapman-Enskog procedure. By an expansion of the unknown Wigner function in powers of the Knudsen number $\epsilon$, evolution equations are derived for the terms of zeroth and first order in epsilon. In particular, it is obtained a quantum drift-diffusion equation for the position density, which is corrected by field-dependent terms of order epsilon. 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 epsilon^2, uniformly in time and for arbitrary initial data is given. Key words: Asymptotic analysis, quantum drift-diffusion model, Wigner equation, open quantum systems, singularly perturbed parabolic equations.