IKT approach for quantum hydrodynamic equations

TL;DR

This paper proposes a unique definition of quantum temperatures within an inverse kinetic theory framework for quantum hydrodynamics, based on a constant H-theorem, enhancing the classical-quantum fluid analogy.

Contribution

It introduces a criterion using the H-theorem to uniquely define quantum temperatures in an inverse kinetic theory approach for quantum hydrodynamics.

Findings

A new criterion for defining quantum temperatures is proposed.

The approach ensures a consistent dynamical evolution of quantum fluid fields.

It clarifies the non-uniqueness issue in previous definitions.

Abstract

A striking feature of standard quantum mechanics is its analogy with classical fluid dynamics. In particular it is well known the Schr\"{o}dinger equation can be viewed as describing a classical compressible and non-viscous fluid, described by two (quantum) fluid fields ${\rho ,% \mathbf{V}} $, to be identified with the quantum probability density and velocity field. This feature has suggested the construction of a phase-space hidden-variable description based on a suitable inverse kinetic theory (IKT; Tessarotto et al., 2007). The discovery of this approach has potentially important consequences since it permits to identify the classical dynamical system which advances in time the quantum fluid fields. This type of approach, however requires the identification of additional fluid fields. These can be generally identified with suitable directional fluid temperatures $T_{QM,i}$ (for $i=1,2,3$), to be related to the expectation values of momentum fluctuations appearing in the Heisenberg inequalities. Nevertheless the definition given previously for them (Tessarotto et al., 2007) is non-unique. In this paper we intend to propose a criterion, based on the validity of a constant H-theorem, which provides an unique definition for the quantum temperatures.