Hydrodynamics on non-commutative space is studied based on a formulation of hydrodynamics by Y. Nambu in terms of Poisson and Nambu brackets. Replacing these brackets by Moyal brackets with a parameter θ, a new hydrodynamics on non-commutative space is derived. It may be a step toward to find the hydrodynamics of granular materials whose minimum volume is given by θ. To clarify this minimum volume, path integral quantization and uncertainty relation of Nambu dynamics are examined.

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.

On the basis of the intimate relation between Nambu dynamics and the hydrodynamics, the hydrodynamics on a non-commutative space (obtained by the quantization of space), proposed by Nambu in his last work, is formulated as “hydrodynamics of granular material”. In Part 1, the quantization of space is done by Moyal product, and the hydrodynamic simulation is performed for the so obtained two dimensional fluid, which flows inside a canal with an obstacle. The obtained results differ between two cases in which the size of a fluid particle is zero and finite. The difference seems to come from the behavior of vortices generated by an obstacle. In Part 2 of quantization, considering vortex as a string, two models are examined; one is the “hybrid model” in which vortices interact with each other by exchanging Kalb-Ramond fields (a generalization of stream functions), and the other is the more general “string field theory” in which Kalb-Ramond field is one of the excitation mode of string oscillations. In the string field theory, Altarelli-Parisi type evolution equation is introduced. It is expected to describe the response of distribution function of vortex inside a turbulence, when the energy scale is changed. The behaviour of viscosity differs in the string theory, being compared with the particle theory, so that Landau theory of fluid to introduce viscosity may be modified. In conclusion, the hydrodynamics and the string theory are almost identical theories. It should be noted, however, that the string theory to reproduce a given hydrodynamics is not a usual string theory.

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