Uncertainty Relations in Hydrodynamics

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.

Possible enhancements of collective flow anisotropy induced by uncertainty relation for fluid element

“…In this section, we briefly summarize the properties of the stochastic motion of the fluid element in the NSF equation, following the review papers [8,12,13].…”

“…See the discussion in section 3.3 of [8] for details. Using this condition, we can show that the two Fokker-Planck equations are reduced to the same equation of continuity for the number density ρ,…”

“…This method is called the stochastic variational method (SVM) [6]. As for other variational approaches to formulate hydrodynamics, see [7,8] and references therein. We can then show that the velocity fluctuation of the fluid element is characterized by an uncertainty relation that is analogous to the one in quantum mechanics [9][10][11].…”

“…These parameters are absorbed into the definitions of the transport coefficients in hydrodynamics. See the discussion in [8] for details. Although it is not explicitly expressed, ε and ρ depend on the position of the fluid element ( )…”

“…The first term on the right-hand side is the additional term which does not appear in the NSF equation. This term is important to discuss the surface effect and quantum fluctuation [8,11,12,[16][17][18]. In the present work, however, we simply ignore this term by setting α B = 0, leading to κ = 0.…”