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Article

Uncertainty Relations in Hydrodynamics

1
Instituto de Física, Universidade Federal do Rio de Janeiro, C.P. 68528, Rio de Janeiro 21941-972, Brazil
2
Instituto de Física, Universidade Federal Fluminense, Niterói 24210-346, Brazil
*
Author to whom correspondence should be addressed.
Water 2020, 12(11), 3263; https://doi.org/10.3390/w12113263
Received: 29 September 2020 / Revised: 9 November 2020 / Accepted: 18 November 2020 / Published: 21 November 2020
(This article belongs to the Special Issue Stochastic Modeling in Fluid Dynamics)
The qualitative behaviors of uncertainty relations in hydrodynamics are numerically studied for fluids with low Reynolds numbers in 1+1 dimensional system. We first give a review for the formulation of the generalized uncertainty relations in the stochastic variational method (SVM), following the work by two of the present authors [Phys. Lett. A 382, 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-Schrödinger inequalities in quantum mechanics are reproduced. The same non-differentiable trajectory is applied to the motion of fluid elements in the Navier-Stokes-Fourier equation or the Navier-Stokes-Korteweg equation. 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. View Full-Text
Keywords: Navier-Stokes-Fourier equation; Navier-Stokes-Korteweg equation; uncertainty relations; stochastic calculus; variational principle Navier-Stokes-Fourier equation; Navier-Stokes-Korteweg equation; uncertainty relations; stochastic calculus; variational principle
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MDPI and ACS Style

Gonçalves de Matos, G.; Kodama, T.; Koide, T. Uncertainty Relations in Hydrodynamics. Water 2020, 12, 3263. https://doi.org/10.3390/w12113263

AMA Style

Gonçalves de Matos G, Kodama T, Koide T. Uncertainty Relations in Hydrodynamics. Water. 2020; 12(11):3263. https://doi.org/10.3390/w12113263

Chicago/Turabian Style

Gonçalves de Matos, Gyell, Takeshi Kodama, and Tomoi Koide. 2020. "Uncertainty Relations in Hydrodynamics" Water 12, no. 11: 3263. https://doi.org/10.3390/w12113263

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