Viscous control of minimum uncertainty state in hydrodynamics

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 Schrödinger equation, the uncertainty relation of a fluid element reproduces the Kennard and the Robertson–Schrödinger 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 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.