Quantum Hydrodynamic Model by Moment Closure of Wigner Equation

TL;DR

This paper develops quantum hydrodynamic models from the Wigner equation using a Grad-type moment closure, ensuring well-posedness and regularization similar to classical Boltzmann models.

Contribution

It introduces a moment closure approach for the Wigner equation that leverages regularization techniques from Boltzmann theory, ensuring local well-posedness of the quantum model.

Findings

The moment expansion converts the Wigner potential into a linear source term.

Regularization achieves global hyperbolicity of the moment system.

The resulting quantum hydrodynamic model is locally well-posed.

Abstract

In this paper, we derive the quantum hydrodynamics models based on the moment closure of the Wigner equation. The moment expansion adopted is of the Grad type firstly proposed in \cite{Grad}. The Grad's moment method was originally developed for the Boltzmann equation. In \cite{Fan_new}, a regularization method for the Grad's moment system of the Boltzmann equation was proposed to achieve the globally hyperbolicity so that the local well-posedness of the moment system is attained. With the moment expansion of the Wigner function, the drift term in the Wigner equation has exactly the same moment representation as in the Boltzmann equation, thus the regularization in \cite{Fan_new} applies. The moment expansion of the nonlocal Wigner potential term in the Wigner equation is turned to be a linear source term, which can only induce very mild growth of the solution. As the result, the local well-posedness of the regularized moment system for the Wigner equation remains as for the Boltzmann equation.