Abstract
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.
Highlights
The expressions of fundamental laws of physics should be independent of the choice of coordinates.This requirement is naturally satisfied when dynamics is formulated in the variational principle.The variational approach is of wide application and describes the behaviors of particles and fields in classical and quantum systems [1]
These relations are applicable to the trapped Bose gas described by the Gross-Pitaevskii equation and the field-theoretical uncertainty relation is reproduced
We showed that the quantum-mechanical uncertainty relations can be reformulated in the stochastic variational method
Summary
The expressions of fundamental laws of physics should be independent of the choice of coordinates. Uncertainty relations are known to be an important property in quantum mechanics, which characterizes the correlation between fluctuations of, for example, position and momentum of a quantum particle Using this property appropriately [15,16,17,18,19], we can investigate a fundamental limitation for simultaneous measurements as was pointed out by Heisenberg [20]. If SVM is a natural framework of quantum mechanics, the same property should be obtained from the stochasticity of a quantum particle without introducing operators Such a study will enable us to define the uncertainty relations in hydrodynamics. By introducing the standard deviations of position and momentum for fluid elements, the Kennard-type and Robertson-Schrödinger-type relations are derived for the fluid described by the NSF equation These relations are applicable to the trapped Bose gas described by the Gross-Pitaevskii equation and the field-theoretical uncertainty relation is reproduced.
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