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

TL;DR

This paper develops a rigorous asymptotic analysis of a high-field quantum transport equation, deriving a quantum drift-diffusion model with corrections, and establishing the accuracy and well-posedness of the approximation.

Contribution

It introduces a modified Chapman-Enskog procedure to derive and justify a quantum drift-diffusion equation from the high-field Wigner-BGK model, including correction terms and error estimates.

Findings

Derived a quantum drift-diffusion equation with field-dependent corrections.

Proved the difference between exact and approximate solutions is of order ε².

Established well-posedness and regularity of the asymptotic models.

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