Many-particle quantum hydrodynamics: exact equations and pressure tensors

TL;DR

This paper derives exact many-particle quantum hydrodynamics equations for systems with multiple particle types, analyzes two pressure tensors in depth, and discusses their interpretations and coordinate transformations.

Contribution

It introduces a comprehensive derivation of MPQHD equations for multi-species systems and compares two pressure tensors, highlighting gauge freedom and interpretability issues.

Findings

Derived exact MPQHD equations for multi-species systems.

Analyzed Wyatt and Kuzmenkov pressure tensors in detail.

Identified gauge freedom and interpretability differences between tensors.

Abstract

In the first part of this paper, the many-particle quantum hydrodynamics (MPQHD) equations for a system containing many particles of different sorts are derived exactly from the many-particle Schr\"odinger equation. It includes the derivation of the many-particle continuity equations (MPCE), many-particle Ehrenfest equations of motion (MPEEM), and many-particle quantum Cauchy equations (MPQCE) for any of the different particle sorts and for the total particle ensemble. The new point in our analysis is that we consider a set of arbitrary particles of different sorts in the system. In MPQCEs, there appears a quantity called pressure tensor. In the second part of this paper, we analyze two versions of this tensor in depth -- the Wyatt pressure tensor and the Kuzmenkov pressure tensor. There are different versions because there is a gauge freedom for the pressure tensor similar to that for potentials. We find that the interpretation of all quantities contributing to the Wyatt pressure tensor is understandable but for the Kuzmenkov tensor, it is difficult. Furthermore, the transformation from Cartesian coordinates to cylindrical coordinates for the Wyatt tensor can be done in a clear way, but for the Kuzmenkov tensor, it is rather cumbersome.