Viscous control of minimum uncertainty state in hydrodynamics

TL;DR

This paper derives a minimum uncertainty state in viscous hydrodynamics using the Navier-Stokes-Korteweg equation, linking fluid uncertainty to viscosity and quantum analogies, with implications for high-energy physics.

Contribution

It introduces a generalized minimum uncertainty state in viscous fluids, connecting classical hydrodynamics with quantum uncertainty principles through the NSK framework.

Findings

Uncertainty depends on shear viscosity and can be smaller than inviscid limits.

The state generalizes coherent states with viscosity-dependent uncertainty.

Implications for fluctuating microscopic degrees of freedom in fluids.

Abstract

A minimum uncertainty state for position and momentum of a fluid element is obtained. We consider a general fluid described by the Navier-Stokes-Korteweg (NSK) equation, which reproduces the behaviors of a standard viscous fluid, a fluid with the capillary action and a quantum fluid, with the proper choice of parameters. When the parameters of the NSK equation is adjusted to reproduce Madelung's hydrodynamic representation of the Schreodinger equation, the uncertainty relation of a fluid element reproduces the Kennard and the Robertson-Schreodinger inequalities in quantum mechanics. The derived minimum uncertainty state is the generalization of the coherent state and its uncertainty is given by a function of the shear viscosity. The viscous uncertainty can be smaller than the inviscid minimum value when the shear viscosity is smaller than a critical value which is similar in magnitude to the Kovtun-Son-Starinets (KSS) bound. This uncertainty reflects the information of the fluctuating microscopic degrees of freedom in the fluid and will modify the standard hydrodynamic scenario, for example, in heavy-ion collisions.