Generalization of uncertainty relation for quantum and stochastic systems

Koide, T.; Kodama, T.

doi:10.1016/j.physleta.2018.04.008

Cited by 21 publications

(82 citation statements)

References 71 publications

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“…To reproduce this coefficient, for example, we can consider the variation of the entropy dependence in the pressure P , as is discussed in Refs. [35,41]. Moreover the equation corresponding to the energy conservation is not obtained.…”

Section: Discussionmentioning

confidence: 99%

Variational formulation of compressible hydrodynamics in curved spacetime and symmetry of stress tensor

Koide

Kodama

2020

J. Phys. A: Math. Theor.

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Hydrodynamics of the non-relativistic compressible fluid in the curved spacetime is derived using the generalized framework of the stochastic variational method (SVM) for continuum medium. The fluid-stress tensor of the resultant equation becomes asymmetric for the exchange of the indices, differently from the standard Euclidean one. Its incompressible limit suggests that the viscous term should be represented with the Bochner Laplacian. Moreover the modified Navier-Stokes-Fourier (NSF) equation proposed by Brenner can be considered even in the curved spacetime. To confirm the compatibility with the symmetry principle, SVM is applied to the gaugeinvariant Lagrangian of a charged compressible fluid and then the Lorentz force is reproduced as the interaction between the Abelian gauge fields and the viscous charged fluid.

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Section: Discussionmentioning

confidence: 99%

Variational formulation of compressible hydrodynamics in curved spacetime and symmetry of stress tensor

Koide

Kodama

2020

J. Phys. A: Math. Theor.

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“…In SVM, we consider the optimization of the nondifferentiable trajectory for a (virtual) particle [16]. To describe such a trajectory as Brownian motion, we introduce stochastic differential equations (SDE's), which are obtained from Eqs.…”

Section: Stochastic Variationmentioning

confidence: 99%

“…On the other hand, the stochastic momenta p + i t and p − i t contribute to the stochastic Newton (or Newton-Nelson) equation on an equal footing as is shown in Ref. [16] and thus it is natural to define the standard deviation of the quantum-mechanical momentum by the average of the two contributions,…”

Section: Eigenstate Of Angular Momentummentioning

confidence: 99%

“…Although such an approach gives an alternative point of view for quantum behaviors [14], one could argue that quantum hydrodynamics does not offer an advantage over the formalism of operators acting in Hilbert spaces of quantum states. However, with the help of the stochastic variational method (SVM) [15,16], quantum hydrodynamics provides a useful methodology to study quantum behaviors in arbitrary canonical coordinates. In fact, one of the present authors developed a systematic procedure to define the uncertainty relation within the framework of quantum hydrodynamics with Cartesian coordinates [16].…”

Section: Introductionmentioning

confidence: 99%

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Uncertainty relation for angle from a quantum-hydrodynamical perspective

Gazeau

Koide

2020

Annals of Physics

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We revisit the problem of the uncertainty relation for angle by using quantum hydrodynamics formulated in the stochastic variational method (SVM), where we need not define the angle operator. We derive both the Kennard and Robertson-Schrödinger inequalities for canonical variables in polar coordinates. The inequalities have state-dependent minimum values which can be smaller than /2 and then permit a finite uncertainty of angle for the eigenstate of the angular momentum. The present approach provides a useful methodology to study quantum behaviors in arbitrary canonical coordinates.

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“…In this subsection we derive the first new set of hydrodynamic equations, by transforming the velocity field U into a newly defined velocity field U v . We assume that the flow mean mass velocity field U can be written in terms of the new velocity field called the mean volume velocity U v [8,9,[32][33][34] as…”

Section: Re-casted Navier-stokes Equations-imentioning

confidence: 99%

Recasting Navier–Stokes equations

et al. 2019

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Classical Navier-Stokes equations fail to describe some flows in both the compressible and incompressible configurations. In this article, we propose a new methodology based on transforming the fluid mass velocity vector field to obtain a new class of continuum models. We uncover a class of continuum models which we call the re-casted Navier-Stokes. They naturally exhibit the physics of previously proposed models by different authors to substitute the original Navier-Stokes equations. The new models unlike the conventional Navier-Stokes appear as more complete forms of mass diffusion type continuum flow equations. They also form systematically a class of thermomechanically consistent hydrodynamic equations via the original equations. The plane wave analysis is performed to check their linear stability under small perturbations, which confirms that all re-casted models are spatially and temporally stable like their classical counterpart. We then use the Rayleigh-Brillouin scattering experiments to demonstrate that the re-casted equations may be better suited for explaining some of the experimental data where original Navier-Stokes equations fail.Öttinger [11] also proposed earlier a substitute for the classical Navier-Stokes in his phenomenological GENERIC formalism. In the GENERIC formulation, Öttinger [11] demonstrated that by assuming the row and the column, which are associated with the mass density in the friction matrix to be identically zero, leads to the conventional Navier-Stokes equations. A more general form of friction matrix that includes the basic fact that particles operate diffusive motions, leads to a revised set of transport equations that contain two velocities, which are very similar in nature to the volume and mass velocities. Durst et al [12] based their arguments on that the absence of mass diffusion terms in the continuity equation contradicts constitutive relations for momentum and heat diffusion: when the fluid properties changes spatially in the presence of momentum and heat diffusions, there should also be present the mass diffusion. They later derived the Extended Navier-Stokes equations [12,13,17] based on the mass-diffusion controlled formalism. A late suggestion to substitute the Navier-Stokes is given by Svärd [18].Generally, there are a number of experimental data that standard Navier-Stokes fail to predict. Shock wave structure predictions are a ubiquitous example of Navier-Stokes failure [4,6,19,20]. Experimental data of water flows in carbon nanotubes or confined pores is another research topic showing large deviations from the classical Hagen-Poiseuille equation [21,22]. A convincing theoretical interpretation of this data is still lacking. Leaving aside non-linear configurations, conventional Navier-Stokes equations also fail to describe some of the linear flow problems accurately. For example, it is unsuccessful in describing the actual spectrum shapes in the Rayleigh-Brillouin scattering problem [23][24][25][26][27][28].In the current work, we provide new insights into the ...

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Generalization of uncertainty relation for quantum and stochastic systems

Cited by 21 publications

References 71 publications

Variational formulation of compressible hydrodynamics in curved spacetime and symmetry of stress tensor

Variational formulation of compressible hydrodynamics in curved spacetime and symmetry of stress tensor

Uncertainty relation for angle from a quantum-hydrodynamical perspective

Recasting Navier–Stokes equations

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