Uncertainty Relations in Hydrodynamics

TL;DR

This paper explores the uncertainty relations in hydrodynamics using the stochastic variational method, revealing differences between liquids and gases and proposing a new criterion for classifying fluid states.

Contribution

It extends quantum uncertainty principles to hydrodynamics and demonstrates their applicability to fluid classification through numerical analysis.

Findings

Uncertainty relations in hydrodynamics are derived from non-differentiable fluid trajectories.

Numerical results show qualitative differences in uncertainty behavior between liquids and gases.

The uncertainty relations can serve as a criterion to distinguish fluid states.

Abstract

The uncertainty relations in hydrodynamics are numerically studied. We first give a review for the formulation of the generalized uncertainty relations in the stochastic variational method (SVM), following the paper by two of the present authors [Phys. Lett. A382, 1472 (2018)]. In this approach, the origin of the finite minimum value of uncertainty is attributed to the non-differentiable (virtual) trajectory of a quantum particle and then both of the Kennard and Robertson-Schr\"{o}dinger inequalities in quantum mechanics are reproduced. The same non-differentiable trajectory is applied to the motion of fluid elements in hydrodynamics. By introducing the standard deviations of position and momentum for fluid elements, the uncertainty relations in hydrodynamics are derived. These are applicable even to the Gross-Pitaevskii equation and then the field-theoretical uncertainty relation is reproduced. We further investigate numerically the derived relations and find that the behaviors of the uncertainty relations for liquid and gas are qualitatively different. This suggests that the uncertainty relations in hydrodynamics are used as a criterion to classify liquid and gas in fluid.