Emergence of vorticity and viscous stress in finite-scale quantum hydrodynamics

TL;DR

This paper develops a coarse-grained quantum hydrodynamics framework that allows for finite vorticity and viscous-like stresses, extending the irrotational Madelung description to include vortex dynamics.

Contribution

It introduces a finite-scale closure in quantum hydrodynamics enabling the emergence of vorticity and viscous stress terms, bridging quantum and classical fluid behaviors.

Findings

Finite vorticity can arise in quantum fluids through coarse-graining.

The vorticity dynamics follow an equation similar to classical hydrodynamics.

A novel stress term analogous to viscous stress appears in the quantum fluid equations.

Abstract

The Madelung equations offer a hydrodynamic description of quantum systems, from single particles to quantum fluids. In this formulation, the probability density is mapped onto the fluid density and the phase is treated as a scalar potential generating the velocity field. As examples of potential flows, quantum fluids described in this way are inherently irrotational, but quantum vortices may arise at discrete points where the phase is undefined. In this paper, starting from this irrotational description of a quantum fluid, a coarse-graining procedure is applied to arrive at a macroscopic description of the quantum fluid in terms of a hierarchy of moments in which the role of velocity is played by a Favre average of the microscopic velocity field. This hierarchy is truncated using an explicit closure derived from an expansion in a finite length scale. The resulting coarse-grained fields are shown to allow for finite vorticity at any point in the fluid. Furthermore, it is shown that this vorticity obeys a similar equation to the vorticity equation in classical hydrodynamics and includes a vortex-stretching term. The particular closure employed here also gives rise to a novel stress term in the fluid equations, which in the appropriate limit appears analogous to an artificial viscous stress from computational fluid dynamics.