Fluid moment hierarchy equations derived from gauge invariant quantum kinetic theory

F Haas,G Brodin,J Zamanian,M Marklund

doi:10.1088/1367-2630/12/7/073027

Abstract

The gauge invariant electromagnetic Wigner equation is taken as the basis of a fluid-like system describing quantum plasmas, derived from the moments of the gauge invariant Wigner function. The use of the standard, gauge-dependent Wigner function is shown to produce inconsistencies if a direct correspondence principle is applied. The propagation of linear transverse waves is considered and it is shown to be in agreement with the kinetic theory in the long-wavelength approximation, provided that an adequate closure is chosen for the macroscopic equations. A general recipe to solve the closure problem is suggested.

Highlights

The Wigner function is the quantum equivalent of the classical particle distribution function and can be used to calculate average values of physical observables [1]
In this situation the gauge invariance of the Wigner function should be assured from the very beginning in order to avoid inconsistencies, a point somewhat neglected in previous studies. It is the purpose of this work to stress the relevance and properties of the gauge invariant Wigner function (GIWF) [3, 4, 5] in connection with quantum plasmas problems
The moment hierarchy equations derived from the GIWF electromagnetic evolution equation is obtained

Summary

Introduction

The Wigner function is the quantum equivalent of the classical particle distribution function and can be used to calculate average values of physical observables [1]. The time evolution of the Wigner function is evaluated considering only scalar potentials, without the inclusion of magnetic fields. One reason for this is the considerable analytic complexity of the electromagnetic Wigner equation. The emergence of new areas like spintronics [2] where magnetic effects are crucial makes it desirable to have quantum kinetic models allowing for nonzero vector potentials In this situation the gauge invariance of the Wigner function should be assured from the very beginning in order to avoid inconsistencies, a point somewhat neglected in previous studies. It is included the Appendix A where the closure of the fluid-like system is discussed

Basic properties of the gauge invariant Wigner function

Fluid moments hierarchy

D Qi jk Dt

Transverse dispersion relation

Conclusion