2020/09/05 by G. Gonçalves de Matos, Gyell Gonçalves de Matos, T. Kodama +1
Mathematics · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Classical mechanics #Differentiable function #Mathematical analysis #Mathematics #Momentum (technical analysis) #Physics #Position (finance) #Quantum #Quantum Mechanics and Applications #Quantum hydrodynamics #Quantum mechanics #Statistical Mechanics and Entropy #Statistical physics #Trajectory #physics.flu-dyn
paper · pdf · doi:10.3390/w12113263
published as Water 12, 3263 (2020) · 56 pages, 9 figures
arxiv created 2020/09/05 · openalex publication_date 2020/11/21 · arxiv updated 2020/11/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.