Rigorous drift-diffusion asymptotics of a high-field quantum transport equation

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 it is proved that the difference between exact and asymptotic solutions is of order $\epsilon ^2$, uniformly in time and for arbitrary initial data.