obeying GP equation (Quantum fluid turbulence). Some results ... ξ: Healing length ( â¼ 0.5 ËA in Liquid 4He ). Hereafter,Ë· is .... 3.6 PDF of the density field.
11/Sep/2006, IUTAM Symposium NAGOYA 2006 “Computational Physics and New Perspectives in Turbulence”
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Numerical Simulation of Quantum Fluid Turbulence
Kyo Yoshida and Toshihico Arimitsu
START:.
0 Abstract
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Gross-Pitaevskii (GP) equation describes the dynamics of low-temperature superfluids and Bose-Einstein Condensates (BEC). We performed a numerical simulation of turbulence obeying GP equation (Quantum fluid turbulence). Some results of the simulation are reported.
Outline 1 Background (Statistical theory of turbulence) 2 Quantum Fluid turbulence 3 Numerical Simulation .
1 Background (Statistical Theory01234 of Turbulence) 56789 Characteristics of (classical fluid) turbulence as a dynamical system are • Large number of degrees of freedom • Nonlinear ( modes are strongly interacting ) • Non-equilibrium ( forced and dissipative ) Why quantum fluid turbulence ? • Another example of such a dynamical system. Another test ground for developing the statistical theory of turbulence. – What are in common and what are different between classical and quantum fluid turbulence? .
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2 Quantum fluid turbulence 2.1 Dynamics of order parameter
ˆ t) Hamiltonian of locally interacting boson field ψ(x, ˆ = H
Z
"
# h ¯ g dx −ψˆ† ∇2 ψˆ − µψˆ† ψˆ + ψˆ† ψˆ† ψˆψˆ 2m 2 2
µ : chemical potential,
g: coupling constant
Heisenberg equation ∂ ψˆ i¯ h =− ∂t
Ã
2
!
¯ h ∇2 + µ ψˆ + g ψˆ† ψˆψˆ 2m
ψˆ = ψ + ψˆ0 ,
ˆ ψ = hψi
ψ(x, t): Order parameter
.
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Gross-Pitaevskii (GP) equation µ
∂ψ i¯ h ∂t
=
−
µ
=
g¯ n,
¶ ¯ h ∇2 + µ ψ + g|ψ|2 ψ, 2m n = |ψ|2
¯· : volume average. Normalization ˜= x
x , L
t˜ =
g¯ n t, h ¯
ψ ψ˜ = √ n ¯
Normalized GP equation ∂ ψ˜ ˜ 2 ψ, ˜ ˜ 2 ψ˜ − ψ˜ + |ψ| i = −ξ˜2 ∇ ∂ t˜
µ ¯ h ξ= √ , 2mg¯ n
˚ in Liquid 4 He ) ξ: Healing length ( ∼ 0.5A Hereafter, ˜· is omitted. .
ξ ˜ ξ= L
¶
01234 vortex line 2.3 Quantum fluid velocity and quantized 56789
p ψ(x, t) = ρ(x, t)eiϕ(x,t) ,
∂ ρ + ∇ · (ρv) = ∂t ∂ v + (v · ∇)v ∂t
v(x, t) = 2ξ 2 ∇ϕ(x, t)
0
= −∇pq
Ã
4
pq = 2ξ ρ − 2ξ
4∇
2√
√
ρ
ρ
!
ρ : Quantum fluid density v: Quantum fluid velocity Quantized vortex line (ρ = 0) ρ=0
Z
C
ω dl · v
= =
∇ × v = 0 (for ρ 6= 0) (2πn)2ξ
2
C
(n = 0, ±1, ±2 · · · )
.
3 Numerical simulation
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3.1 Dissipation and Forcing GP equation (in wave vector space) i
∂ ψk ∂t
=
ξ 2 k 2 ψk − ψk +
Z
dpdqdrδ(k + p − q − r)ψp∗ ψq ψr
−iνk2 ψk +iαk ψk • Dissipation – The dissipation term acts mainly in the high wavenumber range ( k ∼> 1/ξ ). • Forcing (Pumping of condensates) α (k < k ) f αk = 0 (k ≥ kf )
– α is determined at every time step so as to keep ρ¯ almost constant. .
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3.2 Simulation conditions • (2π)3 box with periodic boundary conditions.
• An alias-free spectral method with a Fast Fourier Transform. • A 4th order Runge-Kutta method for time marching. • Resolution kmax ξ = 3. • ν = ξ2. N
kmax
ξ
ν(×10−3 )
kf
∆t
ρ¯
128
60
0.05
2.5
2.5
0.01
0.998
256
120
0.025
0.625
2.5
0.01
0.999
512
241
0.0125
0.15625
2.5
0.01
0.998
.
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3.3 Energy Energy density per unit volume E E kin
=
E int
=
=
E wc
=
Eq
=
E kin + E int
Z Z Z 1 dxξ 2 |∇ψ|2 = dkξ 2 k2 |ψk |2 = dkE kin (k) V Z Z Z 1 1 dx(ρ0 )2 = dx|ρ0k |2 = dkE int (k) (ρ0 = ρ − ρ¯) 2V 2 E kin
E wi
=
Z
= Z
E wi + E wc + E q µ
Z
1 1 1 √ dx|wi |2 = dk|wki |2 = dkE wi (k) w= √ ρv 2V 2 2ξ Z Z Z 1 1 dx|wc |2 = dk|wkc |2 = dkE wc (k) 2V 2 Z Z Z 1 √ √ dxξ 2 |∇ ρ|2 = dkξ 2 k2 |( ρ)k |2 = dkE q (k) V
.
¶
3.4 Energy in the simulation
10
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E Ekin Eint wi Ewc E q E
1 0.1
E wi 0.01
E wc
0.001 1e-04
0
5
10
15
20
25
30
35
t
Kobayashi and Tsubota (J. Phys. Soc. Jpn. 57, 3248(2005))
• E wc > E wi . Different from KT. • Dissipation and forcing are different from those of KT.
.
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3.5 Energy spectrum 1
1
Ekin(k) Eint(k)
-5/3
k 0.1
0.1
0.01
k-3/2
0.01
0.001
k4/3
0.001
1e-04
1e-04
1e-05
1e-05
1e-06
1e-06
1e-07
1
10
100
Ewi(k) Ewc(k) q E (k)
1000
1e-07
k-5/3
1
10
100
k
k
• E int ∼ k−3/2 .
– Consistent with the weak turbulence theory. (Dyachenko et. al. Physica D 57 96 (1992))
• E kin ∼ k4/3 . • E wi ∼ k−5/3 is not observed.
– E wi ∼ k−5/3 is observed in KT. Difference in the forcings? .
1000
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3.6 PDF of the density field 2
ρ(x, t) = |ψ(x, t)| , In the weak turbulence theory,
p
ρ(x, t) = |ψ(x, t)| |δρ| ¿ ρ.
ρ(x, t) = ρ + δρ(x, t), 1.2
0.8 0.7
1
0.6 0.5 P/sqrt n
Pn
0.8 0.6
0.4 0.3
0.4
0.2 0.2 0
0.1 0
1
2
3
4 n
5
6
7
8
0
0
0.5
1
1.5 /sqrt n
The turbulence is not weak?
.
2
2.5
3
3.7 Low density region
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N = 512 ξ = 0.0125 ρ < 0.0025
.
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4 Frequency spectrum |ψ |2 + |ψ 2 k,ω k,−ω | Ψk (ω) := |ψk,ω |2
(ω 6= 0)
,
(ω = 0)
ψk,ω
1 := 2π
Z
t0 +T
t0
In the weak turbulence theory, it is assumed that Ψk (ω) ∼ δ(ω − Ωk ), 2500
p Ωk := ξk 2 + ξ 2 k2 . k=6 k=8 k=10
Ψk(ω)
2000 1500 1000 500 0
0
5
10 ω / Ωk
15
20
The assumption is not satisfied, i.e., the turbulence is not weak. .
dt ψk (t)e−iω(t−t0 ) .
5 Summary
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Numerical simulations of Gross-Pitaevskii equation with forcing and dissipation are performed up to 5123 grid points. • E int (k) ∼ k−3/2 . – The scaling coincides with that in the weak turbulence theory. However, it is found that the turbulence is not weak, i.e., |δρ| ∼ ρ and Ψk (ω) 6= δ(ω − Ωk ) . – A possible scenario for the explanation of the scaling is to introduce the time scale of decorrelation τ (k) ∼ Ω−1 k . Closure analysis (DIA, LRA)? • E wi (k) ∼ k−5/3 is not so clearly observed. – The present result is different from that in Kobayashi and Tsubota (2005). Presumably, the forcing in the present simulation injects little to E wi of the system.
.
