Bose condensation and turbulence Dnes Sexty Uni Heidelberg 2011, - PowerPoint PPT Presentation

Bose condensation and turbulence Dénes Sexty Uni Heidelberg 2011, Heidelberg Thermalization in Non-Abelian Plasmas

Classical-statistical field theory n  p ~ 1 1  ≫ n ≫ 1 n  p  1  over-populated classical-particles quantum Classical statistical field theory Kinetic regime Effectively 2 to 2 Using 1/N resummation Wave turbulence exponents Strong turbulence exponents

What is a condensate? 3 k N = V ∫ d 1 = 0 In equilibrium: Maximum at  k − − 1 3 2  e N  N max Condensation: N 0 Condensate fraction Macroscopic occupation of the zero mode N 3  k  n 0  n' k n k = Particle distribution: F  x, y ={ x  ,  y } In terms of 2point function n k = F  k  k ⇒ F  k = 0 ~ V =  ∫ d  3 x  x  2 = F  k = 0  condensate Independent of the volume V V

Condensation in bose gas  x ,t  Non relativistic scalars described by complex field i ∂ t  x ,t =  − ∂ i 2   x ,t  2 2 m  g ∣  x ,t  ∣ Gross-Pitaevski equation: 3 x ∣  x ,t  ∣ n tot = ∫ d 2 conserved particle number occupation in zero mode: condensate = ∣ ∫ d 3 x  x ,t  ∣ 2 V

Non-equilibrium Bose condensation O(4) massless relativistic scalars Initial conditions: overpopulation = 〈  ∫ d 〉 ens  3 x  a  x  2 condensate V

Turbulent cascade Conserved charge Stationary power law solution ∂ k n k = 0 with k-independent flow 2->2 dominates: particle number effectively conserved Dual cascade: particles to IR energy to UV Particle flow Energy flow  IR = d  1 or  IR = d  2  UV = d − 2 or  UV = d − 3 / 2 − n k ~ k

Gauge theory turbulence Pure SU(2) gauge theory overpopulated initial condition − 1.5 n p ~ p same as scalar UV exponent Dispersion

Kolmogorov Turbulence F  x , y ={ x  ,  y } In terms of corrleation functions a = a or A   x , y =[ x  ,  y ] Stationarity condition:    p  F  p − F  p  p = 0 (Collision integral vanishes)  p  With self energy: − 2 − F  , p  z  , s p = ∣ s ∣ F  s Scaling ansatz 2 −  , p  z  ,s p = ∣ s ∣  s F  p ≫ p  Classicality condition

 p = ∫ qkl G  q  G  k  G  l   4   p  q  k  l  Classical part of the stationarity condition: 0 = ∫ p qkl V  p,q ,k ,l   4   p  q  k  l  [ F  p  F  q  F  k  l  2  F  p  F  q  k  F  l  F  p  q  F  k  F  l  Zakharov transformation:  p  F  q  F  k  F  l  ] swapping momenta l ' = p ; p' = l ; k ' = k ; l ' = l F  p  F  q  F  k  l ⇒ p  F  q  F  k  F  l  0 = ∫ p q k l V  p ,q , k ,l  2   4   p  q  k  l  p  F  q  F  k  F  l  [ 1  ∣  ] q 0 ∣  ∣ k 0 ∣  ∣ l 0 ∣    p 0 sgn  p 0 p 0 sgn  p 0 p 0 sgn  p 0 Solutions: q 0 k 0 l 0 =− 1 = 5 3 and = 4 On shell limit = 0 3 2->2 dominates

IR resummation – Strong turbulence 1/N resummation: effective vertex  p = ∫ kql  eff  p  q  G  q  G  k  G  l   4   p  q  k  l   With one loop bubble:  eff  p = R  p  1  A  p   1   p = ∫ q G  p  G  p − q  The vertex scales:  p ≫ 1 In the IR: 2r  eff  p  with r = 3 − d  eff  s p = s  sp = p   eff = In the UV: = 4 or 5 (in d=3) Strong turbulence in the IR:

From 2PI to kinetic equations Using Wigner coordinates 4 s exp − ip  s F p  X = ∫ d   F  X  s / 2 , X − s / 2  Gradient expansion, spatially homogeneous ensemble: ∂ t  p  X = 0 F  X  p  X    X  F p  X − p 2 p 0 ∂ t F p  X = p Define: F p  X = n p  X  1 / 2  p  X  ∞ dp 0 n eff  t , p = ∫ 0 2  2 p 0  p  X  n p  X  On-shell limit, only 2->2 contributes ∂ t n eff  t , p = ∫ d  2  2 [  1  n p  1  n l  n q n r − n p n l  1  n q  1  n r  ]  eff  p  l   n ≫ 1 Effective kinetic description valid at

Turbulence in d=4  UV = d − 3  IR = d  1 2

Conclusions Scalar case well understood Dual cascade Condensation Weak and strong exponents from kinetic theory (with resummation) Gauge theory Gauge fixing necessary UV exponent 3/2 condensation?

Time dependence of gauge theory exponent