Turbulence and Bose-Einstein condensation Far from Equilibrium

Transcription

1 Turbulence and Bose-Einstein condensation Far from Equilibrium Dénes Sexty Uni Heidelberg Collaborators: Jürgen Berges, Sören Schlichting March, 01, Seattle Gauge Field Dynamics In and Out of Equilibrium

2 Non-equilibrium initial state Turbulent flow Thermal equilibrium

3 Kolmogorov turbulence local interactions in momentum space Constant flux in momentum space Scale invariant transport Dimensional analysis: E(k) ~ P log E(k) /3 ρ1/3 k 5/3 pump dump log k Turbulence in an incompressible fluid 3D: Radial energy density Radial energy flux Density ρ E [kg s ] P [kg m 1 s 3] [kg m 3]

4 Turbulence and condensation in scalar field theories

5 Classical statistical field theory Classical fluctuations dominate over quantum for n 1 F = {, } [, ] = O, = D 0 D 0 O t, t x P(x) t Using classical EOM 1 3 m =0 6 [Berges, Schmidt, Rothkopf 008]

6 What is a condensate? In equilibrium: d3 k 1 N =V 3 e 1 Maximum at k N N max Condensation: Condensate fraction Macroscopic occupation of the zero mode F x, y ={ x, y } In terms of point function n k =F k k N0 N n k = 3 k n0 n ' k Particle distribution: condensate = 0 d = 3 F k=0 ~V x x F k =0 = V V Independent of the volume

7 Condensation in bose gas [Berges, Sexty (01)] x,t Non relativistic scalars described by complex field Gross-Pitaevski equation: conserved particle number i i t x,t = g x,t x, t m 3 n tot = d x x,t occupation in zero mode: condensate = d 3 x x,t V

8 Non-equilibrium Bose condensation O(4) massless relativistic scalars Initial conditions: overpopulation condensate = d 3 x a x V ens [Berges, Sexty (01)]

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

10 IR resummation Strong turbulence 1/N resummation: effective vertex p = kql eff p q G q G k G l p q k l 4 eff p = R A 1 p 1 p In the IR: p 1 In the UV: eff = With one loop bubble: p = q G p G p q The vertex scales: eff s p =s r eff p with r=3 d Strong turbulence in the IR: sp = p =4 or 5 (in d=3)

11 From PI to kinetic equations Using Wigner coordinates 4 F p X = d s exp ip s F X s /, X s/ Gradient expansion, spatially homogeneous ensemble: t p X =0 F p 0 t F p X = p X F p X p X p X Define: F p X = n p X 1/ p X dp 0 neff t, p = 0 p0 p X n p X On-shell limit, only -> contributes t neff t, p = d [ 1 n p 1 n l nq n r n p nl 1 n q 1 n r ] eff p l Effective kinetic description also valid at n 1 ~ [Berges, Sexty (011)]

12 Turbulence in d=4 3 UV =d IR =d 1

13 Bose-Einstein Condensation and Thermalization of the Quark Gluon Plasma [Blaziot et al, 011] Initial CGC: Thermal eq: Q 4s 0 ~ s Q 3s n0 ~ s 4 1/ 4 n 0 e 3/ ~ 0 s eq ~T 4 neq ~T 3 /4 n0 e 3 ~1 0 Elastic processes dominate Particles pile up in the IR Overpopulation leads to emergence of condensate

14 Gauge theory turbulence Pure SU() gauge theory overpopulated initial condition n p ~ p 1.5 same as scalar UV exponent Dispersion [Berges, Schlichting, Sexty, arxiv: ]

15 Time dependence of gauge theory exponent Fit n p ~ p [0.4: 1.0]

16 Wave Turbulence F x, y ={ x, y } In terms of corrleation functions = a or A a Stationarity condition: (Collision integral vanishes) x, y =[ x, y ] p F p F p p =0 With self energy: z Scaling ansatz Classicality condition p F s, s p = s F, p z s, s p = s, p F p p

17 Lowest order contribution to self energy: V abc ~g abc abc V abc =V V 0, A, abc V abc p, q, k =g f g p q g q k g k p 0, d d d V abc x, y, z = C g A x C g A x C g A A, ac,bd ab, dc ab,cd x d with A x ~1/ g background field V0 Kinematically forbidden on shell Stationarity condition: F p p p F p =0 C ab,cd = f abe f cde f ade f cbe g d 1 x y d 1 x z

18 Classical part of stationarity condinition: 4 F F = p q k p q k V 4 p F q F k F p q F k F p F q k Scaling ansatz: F sp = s k F p sp = s sgn s p V sp, sq, sk =s v V p, q, k Transformation (swapping and rescaling) p0 q q, k0 p0 k p, k0 p0 p k k0 4 4 F F = p q k p q k V Solution: = 1 p0 p0 p0 p0 p F q F k 1 sgn sgn k0 k0 q0 q0 3 =

19 Conclusions Scalar case well understood Dual cascade Condensation Weak and strong wave exponents from kinetic theory (with resummation) Gauge theory Numerical inditcation of scaling behaviour with =3/ May be explained with background field g contribution similar to scalars

20 Scaling analysis with sunset diagram p = qkl G q G k G l p q k l 4 Classical part of the stationarity condition: 0= p qkl V p, q, k, l p q k l [ F p F q F k l 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 4 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 p q k l p F q F k F l [ Solutions: On shell limit -> dominates p p p p p p 1 0 sgn 0 0 sgn 0 0 sgn 0 q0 q0 k0 k0 l0 l0 = 1 =0 4 = 5 4 and = 3 3 ]