Quantum corrections at second order in derivatives to the dynamics of small non-relativistic fluids

TL;DR

This paper explores second-order quantum corrections to the dynamics of small, non-relativistic superfluid fluids, extending classical fluid models to include quantum effects relevant at mesoscopic scales.

Contribution

It identifies and analyzes second-order quantum terms in superfluid dynamics, extending the Gross-Pitaevskii and von Weizsäcker frameworks for mesoscopic fluids.

Findings

Quantum corrections significantly affect fluid expansion dynamics.

Second-order terms modify static and dynamic properties of superfluids.

Numerical simulations show observable differences in ultra-cold Fermi gas behavior.

Abstract

To capture the dynamics of macroscopic non-relativistic fluids consisting of very many atoms, it is typically sufficient to truncate the gradient expansion at order zero, leading to ideal fluid dynamics, or at order one, leading to the Navier-Stokes theory. For mesoscopic fluids consisting of a small number of atoms, second-order corrections can become significant. We investigate here specifically superfluids at vanishing temperature, and identify relevant second-order terms of quantum origin that contribute already in a static situation. The general form of these terms arises from an extension of the Gross-Pitaevskii theory. In the context of density functional theory, they are named after C. von Weizs\"acker. We assess the influence of these terms on numerical solutions of second-order fluid dynamic equations for the expansion of a mesoscopic ultra-cold Fermi gas released from an anisotropic harmonic trap in two spatial dimensions.