[Paper Review] Rigorous drift-diffusion asymptotics of a high-field quantum transport equation

M. Chiara Manzini, Giovanni Frosali|ArXiv.org|Dec 13, 2006

Gas Dynamics and Kinetic Theory33 references3 citations

TL;DR

This paper rigorously derives a quantum drift-diffusion (QDD) equation with explicit, field-dependent mobility and diffusion coefficients from a high-field Wigner-BGK equation using a modified Chapman-Enskog asymptotic expansion in the Knudsen number $\epsilon$. The analysis proves that the difference between the exact and asymptotic solutions is of order $\epsilon^2$, uniformly in time, and establishes well-posedness and regularity for the approximate problem.

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 $ε$, evolution equations are derived for the terms of zeroth and first order in $ε$. In particular, it is obtained a quantum drift-diffusion equation for the position density, which is corrected by field-dependent terms of order $ε$. 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 $ε^2$, uniformly in time and for arbitrary initial data.

Motivation & Objective

• To derive a quantum drift-diffusion model with explicit, field-dependent transport coefficients from a high-field quantum kinetic equation.
• To establish a rigorous asymptotic link between the linear high-field Wigner-BGK equation and a macroscopic QDD model using a modified Chapman-Enskog procedure.
• To prove the well-posedness and regularity of the resulting approximate QDD problem in Sobolev spaces.
• To quantify the error between the exact and asymptotic solutions, showing it is of order $\epsilon^2$ uniformly in time.
• To extend semi-classical asymptotic results to the quantum regime with explicit dependence on the Knudsen number $\epsilon$.

Proposed method

• Apply a modified Chapman-Enskog procedure to expand the Wigner function in powers of the Knudsen number $\epsilon$.
• Derive evolution equations for the zeroth- and first-order terms in $\epsilon$, leading to a quantum drift-diffusion equation with $\mathcal{O}(\epsilon)$ corrections.
• Include field-dependent transport coefficients arising from the high-field regime, consistent with semi-classical derivations.
• Use a singularly perturbed parabolic PDE framework to model the QDD equation with non-homogeneous, $\epsilon$-dependent coefficients.
• Decompose the error into bulk and boundary parts and estimate each using semigroup theory and operator bounds.
• Employ a time-localization technique via a $C^\infty$ cutoff function $\eta_\epsilon$ to handle non-uniform behavior of the inhomogeneous term near $t=0$.

Experimental results

Research questions

• RQ1 Can a quantum drift-diffusion model with explicit, field-dependent mobility and diffusion coefficients be rigorously derived from a high-field Wigner-BGK equation?
• RQ2 What is the asymptotic structure of the Wigner function in the high-field, small-$\epsilon$ limit?
• RQ3 How does the error between the exact and asymptotic solutions scale with $\epsilon$?
• RQ4 Is the resulting QDD problem well-posed and does it preserve regularity for arbitrary initial data?
• RQ5 Can the error estimate be uniform in time and independent of initial data regularity?

Key findings

• The asymptotic analysis yields a quantum drift-diffusion equation corrected by $\mathcal{O}(\epsilon)$ terms, including field-dependent mobility and diffusion coefficients derived from the Wigner-BGK model.
• The difference between the exact solution of the Wigner-BGK equation and the asymptotic QDD solution is bounded by $C\|w_0\|_{H^{4}_{k+1}}\epsilon^2$ for all $t \in [0,T]$.
• The $\mathcal{O}(\epsilon^2)$ error bound holds uniformly in time and is independent of the initial data's regularity, provided $w_0 \in H^{4}_{k+1}$.
• Well-posedness of the $\mathcal{O}(\epsilon^2)$-approximated problem is established in Sobolev spaces $X_k$, with exponential decay of the solution operator.
• The regularity estimates and well-posedness results complement prior work on singularly perturbed parabolic PDEs with constant coefficients.
• The method confirms the validity of the QDD model as a macroscopic approximation of high-field quantum transport with controlled error.