Hydrodynamics on non-commutative space: A step toward hydrodynamics of granular materials

Saitou, Mayumi; Bamba, Kazuharu; Sugamoto, Akio

doi:10.1093/ptep/ptu138

Cited by 7 publications

(4 citation statements)

References 44 publications

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“…The same observation has been made by Yoichiro Nambu in the description of an incompressible liquid (private communication, see also[36][37][38]. )…”

supporting

confidence: 69%

Analogy between the Schwarzschild solution in a noncommutative gauge theory and the Reissner-Nordström metric

Bufalo

Tureanu

2015

Phys. Rev. D

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We study modifications of the Schwarzschild solution within the noncommutative gauge theory of gravity. In the present analysis, the deformed solutions are obtained by solving the field equations perturbatively, up to the second order in the noncommutativity parameter Θ, for both exterior and interior solutions of the equations of motion for e a µ (x). Remarkably, we find that this new noncommutive solution is analogous to the Reissner-Nordström solution in the ordinary spacetime, in which the square of electric charge is replaced by the square of the noncommutativity parameter, but with opposite sign. This amounts to the noncommutative Schwarzschild radius r NCS becoming larger than the usual radius r S = 2M, instead of smaller as it happens to the Reissner-Nordström radius r RN , implying that r NCS > r S > r RN . An intuitive interpretation of this result is mentioned.

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“…The same observation has been made by Yoichiro Nambu in the description of an incompressible liquid (private communication, see also[36][37][38]. )…”

supporting

confidence: 69%

Analogy between the Schwarzschild solution in a noncommutative gauge theory and the Reissner-Nordström metric

Bufalo

Tureanu

2015

Phys. Rev. D

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“…Now we introduce the non-commutativity in space coordinates, and find the additional terms which will appear. This was carried out in [3] where the noncommutativity is introduced by replacing the Nambu-Poisson brackets with the Moyal brackets [4]. We discuss the two dimensional hydrodynamics in this paper, so the Moyal bracket becomes…”

Section: Hydrodynamics On Non-commutative Spacementioning

confidence: 99%

“…Now, the Navier-Stokes equation of the hydrodynamics on the 2 dimensional non-commutative space is given by [3] ρ Dv Dt + ∇p − η∆v = K (13)…”

Section: Hydrodynamics On Non-commutative Spacementioning

confidence: 99%

A simulation of hydrodynamics on non-commutative space

Kawamura

Kuwana

Nagata

et al. 2018

Progress of Theoretical and Experimental Physics

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A simulation of the hydrodynamics on the two dimensional noncommutative space is performed, in which the space coordinates (x, y) are non-commutative, satisfying the commutation relation [x, y] = iθ. The Navier-Stokes equation has an extra force term which reflects the non-commutativity of the space, being proportional to θ 2 . This parameter θ is related to the minimum size of fluid particles which is implied by the uncertainty principle, ∆x∆y ≥ θ/2. To see the effect of this parameter on the flow, following situation is considered. An obstacle placed in the middle of the stream, separates the flow into small slit and large slit, but the flow is joined afterwards in the down stream. For the Reynolds number 700, the behavior of the flows with and without θ is observed to differ, and the difference is seen to be correlated to the difference of the activity of vortices in the down stream. The oscillation of the flow rate at the small slit diminishes after the certain time in the usual flow when the "two attached eddies" appear. In the non-commutative flow this two attached eddies appear from the beginning and the behavior of the flows does not fluctuate largely. The irregularity in the flow existing in the beginning disappears after the certain time.

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“…In order for the Liouville theorem to hold in the N -dimensional extended phase space, the Nambu equations are defined by N − 1 Nambu Hamiltonians and the Nambu bracket, an N -ary generalization of the Poisson bracket. The structure of Nambu mechanics has impressed many authors, who have reported studies on its fundamental properties and possible applications, including quantization of the Nambu bracket [2][3][4][5][6][7][8][9][10][11]. However, the applications to date have been limited to particular systems, because Nambu systems generally require multiple conserved quantities as Hamiltonians and the Nambu bracket exhibits serious difficulties in systems with many degrees of freedom or quantization [1,2,10].…”

Section: Introductionmentioning

confidence: 99%

Hidden Nambu mechanics II: Quantum/semiclassical dynamics

Horikoshi

2019

Preprint

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Nambu mechanics is a generalized Hamiltonian dynamics characterized by an extended phase space and multiple Hamiltonians. In a previous paper [Prog. Theor. Exp. Phys. 2013, 073A01 (2013] we revealed that the Nambu mechanical structure is hidden in Hamiltonian dynamics, that is, the classical time evolution of variables including redundant degrees of freedom can be formulated as Nambu mechanics. In the present paper, we show that the Nambu mechanical structure is also hidden in some quantum or semiclassical dynamics, that is, in some cases, the quantum or semiclassical time evolution of expectation values of quantum mechanical operators including composite operators can be formulated as Nambu mechanics. We present a procedure to find hidden Nambu structures in quantum/semiclassical systems of one degree of freedom, and give two examples: the exact quantum dynamics of a harmonic oscillator and semiclassical wave packet dynamics. Our formalism can be extended to many-degrees-of-freedom systems, however, there is a serious difficulty in this case due to interactions between degrees of freedom. To illustrate our formalism, we present two sets of numerical results on semiclassical dynamics, in a one-dimensional metastable potential model and a simplified Henon-Heiles model of two interacting oscillators.

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Hydrodynamics on non-commutative space: A step toward hydrodynamics of granular materials

Cited by 7 publications

References 44 publications

Analogy between the Schwarzschild solution in a noncommutative gauge theory and the Reissner-Nordström metric

Analogy between the Schwarzschild solution in a noncommutative gauge theory and the Reissner-Nordström metric

A simulation of hydrodynamics on non-commutative space

Hidden Nambu mechanics II: Quantum/semiclassical dynamics

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