July 212
Gross-Pitaevsy / Nonlinear Schrödinger equation iψ t + 2 ψ ψ 2 ψ = iˆf ψ No forcing / damping ψ = N exp( in t) Integrals of motion H = ( ψ 2 + ψ 4 /2 ) d 2 r N = ψ 2 d 2 r f n forcing damping condensate, N Notations over-condensate fluctuations N = ψ 2 n N = ψ 2 n = N N = ψ 2 d 2 = inverse cascade f
t = : N = 58, n = 16 t = 15 : N = 751, n = 2
different regimes N ➃ ➂ ➁ ➀ t 2 t 1/2 1 ε 1 t run A run B run C run D ɛ = Nλ 2 forcing /4.1
Bogolyubov dispersion relation: Ω 2 = 2N 2 + 4, c = 2N Compare quadratic and cubic terms in Hamiltonian H 2 = Ω n = N 1/2 n X H 3 = V 123 ψ 1 ψ 2 ψ 3 δ( 1 + 2 3) 1, 2, 3 X V 123 2 n 1n 2 δ( 1 + 2 3)δ(Ω 1 + Ω 2 Ω 3) 1, 2, 3 V 2 n 2 c 3 c n2 N 1/2 Effective nonlinearity parameter is small, H 3 H 2 n N. But: Wea turbulence theory also requires random phases!
Consider interactions between the condensate and (, ) pair. Denote N ψ ψ = C ib/2. dn dt db dt dc dt = 1 2 Steady state: dn dt = B = 2N n n sin(2φ φ φ ) = 4(2n + 2n N 2 )C + 8N n n 2N 2 (n + n ) = (N + 2 2n 2n )B B =, C = N n n, n = n = 2 /2, and 2φ φ φ = π
Condensate fluctuates around its average: N = N + M, n = n + m dm dm = B = dt 2dt db dt = 4( N + 2 )C 4 N p 2 m Ω = 2Ω = 2 2 N 2 + 4 dc dt = ( N + 2 )B 25.2 3 ω 1, L=2π ω 2, L=2π ω 1, L=4π ω 2, L=4π n 2 fft(n) 2.1 ω 2 15 N = 1518 N = 486.1.2.3.4.5 t - t' 2 3 ω 2 4 6 8 N 1/2
And then THE SURPRISES START
: breadown of symmeries N = 219 N = 771 N = 1166 N = 422
: breadown of symmeries -1-3 π angle 3π/2 2π -7 n n n π/2-5 π/2 π angle 3π/2 2π -7 t = 75-3 -5 t = 65-3 -5-1 t = 25-3 -7-1 t = 15 n -1-5 π/2 π angle 3π/2 2π -7 π/2 π angle 3π/2 2π
of n Spectra n 2-2 -6-2 -3 t = 15 max min - 1/4 1 4 16 64 n 2-2 -6-2 -3 t = 25 max min - 1/4 1 4 16 64 n 2-2 -6-2 -3 t = 65 max min - 1/4 1 4 16 64 n 2-2 -6-2 -3 t = 75 max min - 1/4 1 4 16 64 Flux lines in Fourier space
1 t = 75 max min.8 < δψ1 δψ*2 >.6.4.2 -.2 -.4 1 2 3 4 5 r / rpump 6 Correlation function of the over-condensate fluctuations 7 8
Instability-driven force iψ t + 2 ψ ψ 2 ψ = iˆf ψ Random force iψ t + 2 ψ ψ 2 ψ = i ˆF
The more we drive the system, the more ordered it becomes. What happens if we drive it even further?
The more we drive the system, the more ordered it becomes. What happens if we drive it even further?
We consider turbulence in the Gross-Pitaevsy model and study the creation of a coherent condensate via an inverse cascade originated at small scales. The growth of the condensate leads to a spontaneous breadown of symmetries of small-scale over-condensate fluctuations: first, statistical isotropy is broen, then series of phase transitions mar the change of symmetry from the two-fold to three-fold to four-fold. We describe respective anisotropic spectral flux flows in -space. At the highest condensate level reached, we observe a short-range positional and long-range orientational order (lie in a hexatic phase). In other words, the longer one pumps the system the more ordered it becomes. The phase transitions happen when the driving term corresponds to an instability and does not occur when pumped by a random force. We thus demonstrate for the first time non-universality of an inverse-cascade turbulence with respect to the nature of small-scale forcing.