Novel effect induced by spacetime curvature in quantum hydrodynamics

Koide, T.; Kodama, T.

doi:10.1016/j.physleta.2019.05.044

Cited by 14 publications

(36 citation statements)

References 46 publications

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“…We further found that, even in curved geometries, it is still possible to consider Brenner's modification of the NSF equation which has the similar form to the quantum potential [37].…”

Section: Discussionmentioning

confidence: 87%

“…There are two different interpretations to understand the role of the κ term. In quantum mechanics, it is known that the Schrödinger equation can be expressed in the form of hydrodynamics by replacing the pressure with the so-called quantum potential [37,47]. The κ term formally corresponds to this [35,41], but we consider here different origins: the quantum potential term is caused by quantum fluctuations while the κ term is induced by thermal fluctuations.…”

Section: B Role Of κ Termmentioning

confidence: 99%

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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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“…We further found that, even in curved geometries, it is still possible to consider Brenner's modification of the NSF equation which has the similar form to the quantum potential [37].…”

Section: Discussionmentioning

confidence: 87%

Section: B Role Of κ Termmentioning

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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“…One can confirm that the paradox for the angular uncertainty relation does not exist in this inequality. Moreover, SVM is applicable to quantum systems in curved geometry [42]. The above uncertainty relation is applicable to curved geometries and hence its generalization to hydrodynamics may be useful to investigate the properties of, for example, highly dense quark-hadron matter in binary neutron star mergers which are described in general relativistic hydrodynamics [88,89].…”

Section: Discussion and Concluding Remarksmentioning

confidence: 99%

“…In the previous section, we implicitly assumed that the particle trajectory is smooth and hence the virtual trajectory defined by Equation (6) is always differentiable. As was mentioned in the introduction, there are several proposals to formulate the variation of non-differentiable trajectories [3][4][5][6][7][8][9][10][11][12][13][14][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43]. In this work we use the method proposed by Yasue [14] who introduced this idea to extend the formulation of Nelson's stochastic mechanics [44].…”

Section: General Setup For Stochastic Variationmentioning

confidence: 99%

Uncertainty Relations in Hydrodynamics

Matos

Kodama

Koide

2020

Water

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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.

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“…(The basic theory of the stochastic calculus of variations had been developed by Yasue [13]. For very recent progress in stochastic variational principle, see [6,14,15].) Being beautiful, the Nelson stochastic quantization is not necessarily useful in applications to multi-dimensional systems.…”

Section: Introductionmentioning

confidence: 99%

Maupertuis-Hamilton least action principle in the space of variational parameters for Schrödinger dynamics; A dual time-dependent variational principle

Takatsuka

2020

J. Phys. Commun.

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Time-dependent variational principle (TDVP) provides powerful methods in solving the timedependent Schröinger equation. As such Kan developed a TDVP (Kan 1981 Phys. Rev. A 24, 2831) and found that there is no Legendre transformation in quantum variational principle, suggesting that there is no place for the Maupertuis reduced action to appear in quantum dynamics. This claim is puzzling for the study of quantum-classical correspondence, since the Maupertuis least action principle practically sets the very basic foundation of classical mechanics. Zambrini showed within the theory of stochastic calculus of variations that the Maupertuis least action principle can lead to the Nelson stochastic quantization theory (Zambrini 1984 J. Math. Phys. 25, 1314. We here revisit the basic aspect of TDVP and reveal the hidden roles of Maupertuis-Hamilton least action in the Schrödinger wavepacket dynamics. On this basis we propose a dual least (stationary) action principle, which is composed of two variational functionals; one responsible for 'energy related dynamics' and the other for 'dynamics of wave-flow'. The former is mainly a manifestation of particle nature in wave-particle duality, while the latter represents that of matter wave. It is also shown that by representing the TDVP in terms of these inseparably linked variational functionals the problem of singularity, which is inherent to the standard TDVPs, is resolved. The structure and properties of this TDVP are also discussed.By contrast, the TDVP gives far more practical methodologies. Indeed, there have been proposed various elaborated formalisms such as those of the so-called Dirac-Frenkel, MaLachlan [16], Kan [17], Kramer and Saraceno [18], and so on. Broeckhove, Lathouwers, Kesteloot and van Leuven [19] gave a unified account over these methods. These methods and the variants are widely used in many fields of physics and chemistry. (For more recent progress in other general theories of TDVP, we refer to [20][21][22][23][24][25][26] .) Yet, it is known that the TDVP generally faces a divergence problem in handling nonlinear variational parameters, which arises from the inversion of possibly singular matrices.An overwhelming advantage of TDVP is that it transforms the Schrödinger partial differential equation to a set of coupled ordinary differential equations over the space of variational parameters. In this context, Kan [17] determinedly states in his seminal paper that 'any time-dependent description of quantum systems derived from the variational principle is equivalent to Hamilton's description of a classical system'. On the other hand, Kan has explicitly shown that there does not exist Legendre transformation in his TDVP. Lack of the Legendre transformation suggests that there is no place for the Maupertuis reduced (or abbreviated) action [27, 28] to appear, and therefore the Maupertuis-Hamilton least action principle will not have direct significance in TDVP. It is not comfortable however to accept this assertion, since the least action principle of Maupertu...

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Novel effect induced by spacetime curvature in quantum hydrodynamics

Cited by 14 publications

References 46 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 Relations in Hydrodynamics

Maupertuis-Hamilton least action principle in the space of variational parameters for Schrödinger dynamics; A dual time-dependent variational principle

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