Spectrum in Gross-Pitaevskii turbulence . . . . . Kyo Yoshida - PowerPoint PPT Presentation

Workshop on New Perspectives in Quantum Turbulence: experimental visualization and numerical simulation Nagoya . . Spectrum in Gross-Pitaevskii turbulence . . . . . Kyo Yoshida University of Tsukuba 11th Dec, 2014 Kyo Yoshida (University of Tsukuba) Spectra in GP turbulence 11th Dec, 2014 1 / 25

Table of contents . .. Quantum fluid (Introduction) 1 . .. Closure Approximation 2 Kyo Yoshida (University of Tsukuba) Spectra in GP turbulence 11th Dec, 2014 2 / 25

. .. Quantum fluid (Introduction) 1 . .. Closure Approximation 2 Kyo Yoshida (University of Tsukuba) Spectra in GP turbulence 11th Dec, 2014 3 / 25

Quantum field equation Hamiltonian of interacting bosonic fields ( 4 He, Rb etc.) ˆ ψ ( x , t ) ψ † � 2 � � ψ + g � 2 m ∇ 2 ˆ ψ † ˆ ψ † ˆ ψ † ˆ − ˆ ψ − µ ˆ ˆ ψ ˆ ˆ H = d x ψ 2 µ : chemical potential, g : coupling constant Heisenberg equation � � 2 i � ∂ ˆ � ψ 2 m ∇ 2 + µ ψ † ˆ ψ + g ˆ ˆ ψ ˆ ∂t = − ψ ψ = ψ + ˆ ˆ ψ := � ˆ ψ ′ , ψ � Order parameter ψ ( x , t ) ψ � = 0 for temperature T < T c . The order parameter contains information of superfluid component or Bose-Einstein condensate. Kyo Yoshida (University of Tsukuba) Spectra in GP turbulence 11th Dec, 2014 4 / 25

Gross-Pitaevskii equation The order parameter ψ ( x ) ( x := { x , t } ) obeys Gross-Pitaevskii (GP) equation ∂tψ ( x ) = − � 2 i � ∂ 2 m ∇ 2 ψ ( x ) − µψ ( x ) + g | ψ ( x ) | 2 ψ ( x ) . Transformation of variables v ( x ) := � n ( x ) e iϕ ( x ) , � ψ ( x ) = m ∇ ϕ ( x ) Equations of motion for Quantum fluid ∂ ∂ ∂tn ( x ) = − ∇ · ( n ( x ) v ( x )) , ∂t v ( x ) = − v ( x ) · ∇ v ( x ) − ∇ p q ( x ) , − � 2 ∇ 2 � n ( x ) p q ( x ) := − µ m + gn ( x ) . 2 m 2 � m n ( x ) Kyo Yoshida (University of Tsukuba) Spectra in GP turbulence 11th Dec, 2014 5 / 25

Constants of motion n and Energy ¯ Number of particles ¯ E n := 1 � d x | ψ ( x ) | 2 , ¯ V ¯ E := E K ( t ) + E I ( t ) , d x � 2 E K ( t ) := 1 � 2 m | ∇ ψ ( x ) | 2 , V E I ( t ) := 1 � d x g 2 | ψ ( x ) | 4 = 1 � d x g 2[ n ( x )] 2 , V V E K ( t ) : kinetic energy, E I ( t ) : interaction energy Kyo Yoshida (University of Tsukuba) Spectra in GP turbulence 11th Dec, 2014 6 / 25

Quantum fluid Differences between quantum fluid and ordinary fluid obeying Navier-Stokes equation are No dissipation, Quasi-pressure term p q ( x ) , n = 0 No vorticity, ω ( x ) := ∇ × v ( x ) = 0 where n ( x ) � = 0 , C Vortex line for n ( x ) = 0 with a quantized circulation. � d l · v ( x ) = 2 π � m k ( k ∈ Z ) . C . . Is the quantum fluid turbulence similar to the ordinary fluid turbulence? . . . . . Kyo Yoshida (University of Tsukuba) Spectra in GP turbulence 11th Dec, 2014 7 / 25

Numerical simulation of GP equation Fourier transform of ψ � d x e − i k · x ψ ( x ) , ψ k ( t ) := GP equation with external force and dissipation in Fourier space representation. ∂ � ∂tψ k = − i ξ 2 k 2 ψ k + i µψ k − i g δ ( k + p − q − r ) ψ ∗ p ψ q ψ r p , q , r + D k + f k D k : dissipation, f k : external force Kyo Yoshida (University of Tsukuba) Spectra in GP turbulence 11th Dec, 2014 8 / 25

Quantum and ordinary fluid turbulences cf. High vorticity region of a classical fluid turbulence. Simulation with 1024 3 grid Low density region of a quantum fluid points. (Kaneda and Ishihara turbulence. Simulation with 512 3 grid (2006)) points. (Yoshida and Arimitsu (2006)) Kyo Yoshida (University of Tsukuba) Spectra in GP turbulence 11th Dec, 2014 9 / 25

Spectra in Numerical Simulations Simulations with various kinds of D k and f k . Spectrum of quantity X . � F X ( k ) ∝ k ′ δ ( k − | k ′ | ) � X ( k ′ ) X ∗ ( k ′ ) � Kobayashi and Tsubota (2005) F w ( k ) ∼ k − 5 / 3 ( w = P [ √ n v ] , P pjojection onto solenoidal component). Yoshida and Arimitsu (2006) F n ( k ) ∼ k − 3 / 2 , F ψ ( k ) ∼ k − 2 / 3 . Proment, Nazarenko and Onorato (2009) F ψ ( k ) ∼ k − 1 or k − 2 , depending on D k and f k . Scaling law of the Spectra in GP turbulence is unsettled. Kyo Yoshida (University of Tsukuba) Spectra in GP turbulence 11th Dec, 2014 10 / 25

Theoretical approach Doublet representation � ψ k ( t ) � ψ + − k 2 k ( t ) � � � � � 1 0 � := e − L k t L k := i 2 m + µ , . ψ − ψ ∗ k ( t ) − k ( t ) 0 − 1 GP equation in Fourier space ∂ � ∂tψ α δ k − p − q − r M αβγζ kpqr ( t ) ψ β p ( t ) ψ γ q ( t ) ψ ζ k ( t ) = g r ( t ) . pqr � � d 3 k / (2 π ) 3 , δ k = (2 π ) 3 δ ( k ) and � = 1 . where k := kpqr ( t ) := (e − L k t ) αα ′ ˜ M α ′ β ′ γ ′ ζ ′ (e L p t ) β ′ β (e L q t ) γ ′ γ (e L r t ) ζ ′ ζ , M αβγζ kpqr  − i for ( α, β, γ, ζ ) ∈ { (+ , − , + , +) , (+ , + , − , +) , (+ , + , + , − ) } 3   ˜ M αβγζ i kpqr := for ( α, β, γ, ζ ) ∈ { ( − , + , − , − ) , ( − , − , + , − ) , ( − , − , − , +) } . 3  0 otherwise  Kyo Yoshida (University of Tsukuba) Spectra in GP turbulence 11th Dec, 2014 11 / 25

Weak wave turbulence theory When | ∂ ∂t ψ ± k | ≪ | L k ψ ± k | , � ψ ± d x ψ + k e i k · x + L k t . k ( t ) ∼ const. in time , ψ ( x ) ∼ Correlation function k ψ β − k ′ � = Q αβ � ψ α k δ k − k ′ , Spectrum � k ′ δ ( k ′ − k ) Q + − F ( k ) = k ′ , Weak wave turbulence (WWT) theory In the energy-transfer range, � − 1 / 3 � ln k F ( k ) ∼ k − 1 . k b In the particle-number-transfer range, F ( k ) ∼ k − 1 / 3 . Kyo Yoshida (University of Tsukuba) Spectra in GP turbulence 11th Dec, 2014 12 / 25

Strong turbulence GP turbulence Weak wave turbulence (WWT) region: | ∂ ∂t ψ ± k | ≪ | L k ψ ± k | , | ∂ ∂t ψ ± k | ≫ | L k ψ ± Strong turbulence (ST) region: k | . For the ordinary fluid turbulence, which is essentially strong turbulence, some spectral closure approximations are availiable. F u ( k ) ∝ k − 5 / 3 in the energy-transfer range (Kolmogorov spectrum). . . The aim of the present study is to derive the spectrum F ψ ( k ) of GP turbulence not only for the WWT region but for the strong turbulence (ST) region by means of a spectral closure approximation . (K. Yoshida and T. Arimitsu, J. Phys. A: Math. Theor. 46 335501 (2013)) . . . . . Kyo Yoshida (University of Tsukuba) Spectra in GP turbulence 11th Dec, 2014 13 / 25

. .. Quantum fluid (Introduction) 1 . .. Closure Approximation 2 Kyo Yoshida (University of Tsukuba) Spectra in GP turbulence 11th Dec, 2014 14 / 25

Closure approximation Unclosed hierarchy of moments, d d dt � ψ � = gM � ψψψ � , dt � ψψ � = gM � ψψψψ � . Approximate M � ψψψψ � as a function of lower order terms, gM � ψψψψ � = g 2 F [ Q ( t, s ) , G ( t, s )] + O ( g 3 ) Correlation function � ψ α k ( t ) ψ β − k ′ ( t ′ ) � = Q αβ k ( t, t ′ ) δ k − k ′ , Response function � � δψ α k ( t ) = G αβ k ( t, t ′ ) δ k − k ′ . δf β k ′ ( t ′ ) where δf ( t ′ ) is the infinitesimal disturbance added at time t ′ . Kyo Yoshida (University of Tsukuba) Spectra in GP turbulence 11th Dec, 2014 15 / 25

Invariance under global phase transformation For simplicity, let us assume that the statistical quantities are invariant under the global phase transformation, ψ α k ( t ) → e α i θ ψ α k ( t ) . Then, by introducing Q k ( t, t ′ ) and G k ( t, t ′ ) , we have n ( t − t ′ ) Q k ( t, t ′ ) , n ( t − t ′ ) Q ∗ Q + − k ( t, t ′ ) = e − 2i g ¯ Q − + k ( t, t ′ ) = e 2i g ¯ − k ( t, t ′ ) , n ( t − t ′ ) G k ( t, t ′ ) , n ( t − t ′ ) G ∗ G ++ k ( t, t ′ ) = e − 2i g ¯ G −− k ( t, t ′ ) = e 2i g ¯ − k ( t, t ′ ) , and otherwise 0 . Kyo Yoshida (University of Tsukuba) Spectra in GP turbulence 11th Dec, 2014 16 / 25

Procedures for the closure approximation (i) Expand Q and G in functional power series of the solutions Q (0) and G (0) for the zeroth-order in g . ∞ ∞ Q = Q (0) + G = G (0) + � g i Q ( i ) ( Q (0) , G (0) ) , � g i G ( i ) ( Q (0) , G (0) ) , i =1 i =1 ∞ ∞ ∂Q ∂G � g i A ( i ) ( Q (0) , G (0) ) , � g i B ( i ) ( Q (0) , G (0) ) . ∂t = ∂t = i =0 i =0 (ii) Invert these expansions to obtain Q (0) and G (0) in functional power series of Q and G . ∞ ∞ Q (0) = Q + G (0) = G + � g i C ( i ) ( Q, G ) , � g i D ( i ) ( Q, G ) . i =1 i =1 (iii) Substitute these inverted expansions into the primitive expansions of d Q/ d t and d G/ d t to obtain the renormalized expansions. ∞ ∞ ∂Q ∂G � g i E ( i ) ( Q, G ) , � g i F ( i ) ( Q, G ) . ∂t = ∂t = i =0 i =0 (iv) Truncate these renormalized expansions at the lowest nontrivial order. Kyo Yoshida (University of Tsukuba) Spectra in GP turbulence 11th Dec, 2014 17 / 25