Fluid moment hierarchy equations derived from quantum kinetic theory

TL;DR

This paper derives quantum hydrodynamic equations from the Wigner kinetic equation, revealing quantum effects in transport phenomena and connecting quantum kinetic theory with fluid models, supported by analytical and numerical results.

Contribution

It introduces a set of quantum fluid equations derived directly from kinetic theory without assuming local equilibrium, highlighting quantum effects in transport equations.

Findings

Quantum diffraction effects appear explicitly in the heat flux transport equation.

A quantum-modified Bohm-Gross dispersion relation is derived.

Numerical solutions of nonlinear waves in one dimension are presented.

Abstract

A set of quantum hydrodynamic equations are derived from the moments of the electrostatic mean-field Wigner kinetic equation. No assumptions are made on the particular local equilibrium or on the statistical ensemble wave functions. Quantum diffraction effects appear explicitly only in the transport equation for the heat flux triad, which is the third-order moment of the Wigner pseudo-distribution. The general linear dispersion relation is derived, from which a quantum modified Bohm-Gross relation is recovered in the long wave-length limit. Nonlinear, traveling wave solutions are numerically found in the one-dimensional case. The results shed light on the relation between quantum kinetic theory, the Bohm-de Broglie-Madelung eikonal approach, and quantum fluid transport around given equilibrium distribution functions.