Nambu Dynamics and Hydrodynamics of Granular Material

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.

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.
Hamiltonian dynamics is given in the two dimensional phase space (x, p), with a single Hamiltonian H;ẋ Nambu generalized this to D-dim phase space (x 1 , x 2 , · · · , x D ) with D−1 Hamiltonians (H 1 , H 2 , · · · , H D−1 );ẋ where the r.h.s. is Jacobian, called "Nambu bracket": Liouville theorem that the phase space volume is unchanged gives incompressible fluid. The peculiar "advective term", an origin of non-linearity in hydrodynamics, appears naturally in Nambu dynamics; Hamiltonians are equal to the stream functions in hydrodynamics which implies ∇ · v = 0, and the velocity points along the intersection line of all the stream functions be constant. Navier-Stokes equation for incompressive fluid reads where p is pressure, µ is shear viscosity with dimension [Pas][s]. We also use ν = µ/ρ with dimension [m 2 /s]. V (x) is an external potential of force. The r.h.s. (force term) can also be written by using Nambu brackets: In his last work, Nambu proposed "the quantization of space" (2011, 2013), or the introduction of non-commutative space in hydrodynamics [2]. His intension is to understand the empirical "Bode law", that is, the radius of the n-th planet follows R n = a 2 n + b.

Quantization of Nambu's hydrodynamics (Part 1)
To quantize the Nambu bracket, we can use the Moyal product, familiar for non-commutative space. For D=2, 3 hydrodynamics, the Moyal products are given by Therefore, the quantization is carried out by replacing and The Planck constant in our case is the minimum area θ 2 in 2D and minimum volume θ 3 in 3D of a granular particle which are identified as the particle size. Then, the additional force K appears, as the finite size effect or the quantum effect of fluid particle, where (ψ 1 (x), ψ 2 (x)) are stream functions. Next, we perform the hydrodynamic simulation in 2D [KaKuNaSaSu(2018)] [4]. Fluid flows from the left to the right inside a canal; with a table-shaped obstacle making a upper small and a lower large slits.  3) Characteristics of the 1st and the 2nd movies: [movie 1] (commutative space with zero particle size) Before1000[s], vortices are generated periodically; at 1025 [s] "blue" (anticlockwise vortex) region and "red" (clockwise vortex) region merges to form "two attached eddies"; at 2000[s] flow is stabilized.
[movie 2] (non-commutative space with non-zero particle size) No active periodic production of vortices just after the obstacle, but is active at down stream; "two attached eddies" appear from the beginning; after 1300[s] flow is stabilized.

Quantization of Nambu's hydrodynamics (Part 2)
It is important to note that Nambu's (hydro-)dynamics in D-dim phase space is equal to the dynamics of (D-2)-dim extended object (D-2)-brane; D=2: particle; D=3: string [Nambu(1980) For example, 3D hydrodynaics in (X, Y, Z) phase space can be described by Its equation of motion reads Nambu dynamics: This is a generalization of the usual 2D phase space (x, p) case of S 2 = pdx − ψ 1 dt.
Here, time t in 2D becomes "area A" in 3D, namely, World sheet of string sweeps in (Y, Z), moving in t and parametrized by (0 ≤ σ ≤ 2π); X is the momentum.
Path integration with weights, e i S 2 and e i θ 3 S 3 , gives quantum mechanics: , and δ δC 23 (σ) The derivative in 3D is replaced by the functional derivative (in tangential t and normal direction µ): Now, we have the uncertainty relation in 2D and 3D (see Appendix 2)): Now, QED and its analog in 3D hydrodynamics read the non-relativistic 3D acton Eq.(15) can be generalized relativistically as S ′ 3 = p µν dX µ ∧ dX ν + · · · , including µ, ν = 0.
In deriving the actions, they used the Hamilton-Jacobi (H-J) formalism. The reason is as follows: The H-J is a method to give two actions, differing by total derivatives (or p-forms). They are usually identical, giving the canonical transformation, but sometimes differ due to the non-integrable phase factors. Therefore, Clebsch theory and gauge theories (with p-forms) in general fits to the H-J. This is the Schild gauge string theory, coupled to Kalb-Ramond field 4 ): where m can be (θ 3 ) 1/3 and the string tension θ −2/3 3 . Our uncertainty relation to give the minimum size for a granular particle is equal to the space-time uncertainty principle claimed by Yoneya [Yoneya(1997)] [6].
Nambu studied the flow along velocity (v), while Clebsch studied the flow along vorticity (ω).
This duality is probably "T-duality" (translational mode ↔ winding mode) in string theory; suggesting a duality in hydrodynamics.

Hybrid model: vortex as string field, and the hydrodynamics as local field
This model was given 40 years ago by Nambu (1977) and by one of the authors (A.S).
[Su(1979), Seo-Okawa-Su(1979,1981] [8]. This Nambu's relativistic hydrodynamics be Kalb-Ramod theory coupled with vorticity source: where W µν is Kalb-Ramond field (1973) (now called B-field or anti-symmetric tensor field), and V µ (x): fluid velocity, and ω µν (x): vorticity source. Then, the realtivistic fluid velocity satisfies the continuity relation naturally, 3 µ=0 ∂ µ V µ = 0. • A filament of vortex line, extending in Z direction with circulation γ, gives so that • Kalb-Ramond field is a "gauge field" in string. This gauge symmetry is called "Kalb-Ramond symmetry": The hybrid model describes the interaction of vortex strings through the fluid field (K-R field): This is very similar to QED in which electric current couples to photon. That is, in hydrodynamics, the interaction of vortices with fluid can be represented by the coupling of current of string to K-R field: Understanding of power spectrum of high energy cosmic ray (having Knee and Ankle) can be applicable to power spectrum of vortices?
• An example of string field in the ordinary case: Then, the wave equation reads • String has, in addition to the translational mode, the k-th oscillation mode in the i-th direction with occupation number {n i k }. The wave functional becomes whereÂ p,{n i k } andÂ p,{ñ i k } are annihilation operator of string itself of right-and left-moving, respectively. Here, we have to choose a proper string theory (not necessary to follow the usual examples) so as to reproduce the vortex production rate in a given hydrodynamics.

Distribution function of vortices inside turbulence
A nucleon is composed of quarks and gluons (partons); having momentum fraction x relative to P of the nucleon. Patrons are probed by photon γ, with momentum Q. If Q 2 is increased, the finer structure reveals: A parent turbulence (like quark) moves in z direction (i.e. in the frame where the observer moves in -z direction very fast), the momentum in z direction is dominantly large, but the other transverse momenta in (x, y) directions are small. Then, the vortex (daughter patrons) inside a parent turbulence (with momentum P ), has the momentum p = xP + +p ⊥ , (p ⊥ ) 2 = Q 2 . Distribution function D S ′ S (x, Q 2 ) of string S ′ inside S ′ with momentum fraction x at the energy scale Q 2 satisfies where the splitting function is given in Figure 6. Using the light-cone field theory of string by Mandelstam (1973) and Kaku-Kikkawa (1974) [10], we obtain which differs from the usual distribution functions of quarks and gluons, ⋆ Note: The concept itself is applicable to the turbulence or other vortex phenomena, but we have to choose the more proper string theory (not necessary the usual ones), so that it may reproduce the realistic phenomena in a given hydrodynamics.

(Simple-minded) inclusion of shear viscosity
Shear viscosity η, (η/ρ = ν) introduces diffusion in velocity field:v = ∆v (constitutive equation). To include the thermal fluctuations at finite temperature the "constitutive equation" is relaxed by the fluctuation of force, following the Gaussian distribution with width proportional to T :v In thermodynamics, ξ plays the same role as the momentum p in quantum mechanics, and hence We can include the dissipation effects also in string theory, following for example [Aibara et al. (2019)] [11]. In the particle theory we have under the constraint ofẋ = v, or The corresponding equation in string is with a constraint, representing where ∂C tµ ∂A (x) = D tµ (x), and C tµ = X t ∧ X µ .

Conclusions
(1) Nambu's hydrodynamics in non-commutative space can be the hydrodynamics of granular material.
(2) The non-commutativity (or quantization) of space can be introduced, by 1) Moyal product, 2) vortex as string and its coupling to Kalb-Ramond field, or 3) string field theory.
(3) In applying the string field theory, 1) evolution equation describing the change of energy scale, and 2) Landau's way of derivative expansion of energy momentum tensor, can be applied.
(4) Note however: We have to find a proper string theory so as to reproduce a given hydrodynamics.
(5) Force from the stress tensor should be re-examined.

• Appendix 3): Interaction between vortices
Without viscosity v = ∇ × ψ, we have a constant ∇ · ψ, and ω = ∇ × v = −∆ψ. Then, in hydrodynamics while in QED: Now we can understand that attraction and repulsion is the same for electric charge and vortex, but is reversal for electric current.
In the hybrid model of hydrodynamics, vortex interaction is mediated by the Kalb-Ramond field: with the components Now, the vortex interaction is correctly reproduced: The vortex interaction is mediated by e(x) which are the Kalb-Ramond fields or the stream functions: e i (x) = W 0i = ψ i (x).
• As we know that the massless modes of closed string in our simple case are dilation (D), graviton (G) and Kalb-Ramond (W) fields.
in the first quantized notation, that is, • Appendix 4): Different shape of granular particle To differentiate the shape of particle, 1) consider the particle as a rigid body, and 2) impose the symmetry. For 1), introduce the body fixed frame and Euler angles, and specify the configuration of the frame, relative to the space fixed frame. For 2), impose the invariance under the "point group G" of the particle; SO(3) for a sphere, tetrahedral group T for a tetrahedron, and octahedral group O for a cube. The flow of six variables (velocities and Euler angles), or the flow of four stream functions, being invariant under the point group G; the additional two stream functions are Hamiltonian and angular momentum squared of the fluid particle as rigid body, which also follows the Nambu dynamics.
The usual form T ij = −pδ ij + η ∂ i v j + ∂ j v i − 2 3 (∇v) + ζδ ij (∇v) is surely violated. Can we modify it and understand the following phenomena in granular hydrodynamics?
• (Open Problem): Various oscillation modes in string give various types of strain; is this useful to understand the above phenomena? For example, 1) flow of sands in hourglass, 2) arc structure of sands clogging in a funnel, 3) Reynolds "dilatancy": decrease of density by the increase of strain, 4) Brazilian-nuts effect e.t.c. (see [14]).
• (Open Problem): Various oscillation modes in string give various types of strain; is this useful to understand the above phenomena ?