Nonlocal Wave Turbulence in QCD - Institute for Nuclear Theory · Nonlocal Wave Turbulence in QCD ... B. Svetitski (1988) m2 d3k |k| n k T q^ m2 q^ 2 ... Yacine Mehtar-Tani INT -

Yacine Mehtar-TaniINT, University of Washington

Equilibration Mechanisms in Weakly and Strongly Coupled Quantum Field Theory August 27, 2015

INT, Seattle

Nonlocal Wave Turbulence in QCD

(In preparation)

http://www.int.washington.edu/PROGRAMS/15-2c

Yacine Mehtar-Tani INT - 15 -2c Program 2015

Wave Turbulence ( I )

• Similarity with fluid turbulence: inviscid transport of conserved quantities from large to small scales through the so-called transparency window or inertial range

2

• Out-of-equilibrium statistics of random non-linear waves

• Examples:

Atmospheric Rossby waves

Water surface gravity and capillary waves

Waves in plasmas

Nonlinear Schrödinger equation (NL Optics, BEC)


Wave Turbulence ( II )

• Waves are excited by external processes. Driven turbulence: Open system with source and sink → away from thermodynamical equilibrium

• Steady states characterized by constant fluxes P and Q rather than temperature and thermodynamical potentials

• Kolmogorov-Obukhov (KO41) theory relies on Locality of interactions: Only eddies (waves) with comparable sizes (wavelengths) interact. Steady state power spectra in momentum space depend on the fluxes and not on the pumping and dissipation scales

• Weak (Wave) Turbulence Theory: Kinetic description in the limit of weak nonlinearities (No theory for strong turbulence)

3

V. E. Zakharov, V. S. L’vov, G. Falkovich (Springer- Verlag, 1992)


• Far-from equilibrium evolution of anisotropic particle distribution in momentum space (Initial conditions in HIC)

• Chromo-Weibel Instabilities : Anisotropy (hard modes) induces exponentially growing soft modes - transverse magnetic and electric fields which help restoring isotropy- (early stage: as in abelian plasmas) turning to a linear growth due to nonlinear interactions inherent to non-abelian plasmas

4

P. Arnold, G. D. Moore (2005)

E. S. Weibel (1959) S. Mrowczynski (1993)

P. Arnold, J. Lenaghan, G. Moore, L. Yaffe (2005) A. Rebhan, P. Romatschke, and M. Strickland (2005)

D. Bödeker, K. Rummukainen (2005)

Turbulence in early stages of Heavy Ion Collisions


Turbulence in early stages of Heavy Ion Collisions

• Hard-loop simulations (large scale separation between hard modes and soft excitations) : Nonlinear interactions develop a turbulent cascade in the UV with exponent 2

5

P.Arnold, G. D. Moore (2005) A. Ipp, A. Rebhan, M. Strickland (2011)

k�2


Weak Turbulence in Kinetic Theory

• Can one understand this power spectrum from first principles?

• Note: from A. H. Mueller, A. I. Shoshi, S. M. H. Wong (2006):

Turbulence in QCD is nonlocal ⇒

• Caveats (in this work):

Homogeneous and isotropic system of gluons

Energy injection with constant rate at : Dispersion relation

Weak nonlinearities in the classical limit (high occupancy):

6

n(k) � k�1

�(k) � |k|

P kf � m

1 � n(k) � 1

g2

k�2

g2 � 1 and


Elastic 2 to 2 process (4-waves interactions)

• Elastic gluon-gluon scattering

7

1 k

2 3

• Two constant of motion: particle number and energy ⇒ Two fluxes

P �E =

�d3k |k| n(k)Q �N =

�d3k n(k)

�

�tnk =

1

2

�

k1,k2,k3

1

2�(k)|M12�3k|2 �(

�

i

ki)�(�

i

�i) F[n]

M12�3k �

F[n] � [nk1nk2

nk + nk1nk2

nk3� nk1

nk3nk � nk2

nk3nk] � n3


Kolmogorov-Zakharov (KZ) Spectra

8

P � Q � n � n3 � n � P1/3 � Q1/3

n(k) �Q1/3

k4/3n(k) �

P1/3

k5/3

• Dimensional analysis determines uniquely the out-of-equilibrium steady state (KZ) power spectra if the interactions are local in momentum space

• From collision integral: nonlinear 4-wave interactions

particle cascade energy cascade

• Same exponents for scalar theories in the absence of condensation


Dual cascade: Fjørthoft argument

• Direction of fluxes: Injection of energy at and dissipating at

9

kf

k� � kf � k+

• Direct energy cascade: If energy was dissipating at low momenta then particles would dissipate at a faster rate than the pumping rate!

Q� �P

k�� P

kf� Q

Direct energy cascade

Inverse particle cascade

kfk� k+

n(k)

k�4/3

k�5/3

P � kfQ

Fjørthoft (1953)

If no damping: condensation

Are KZ spectra in QCD physically relevant?


Small angle approximation

• Fokker-Plank equation: Diffusion and drag

11

�

�tnk � q

4k2

�

�kk2

��

�knk +

n2k

T�

�

t � q2 � s

• KZ spectra are not stationary solutions of the collision integral (contrary to non-relativistic Coulomb scattering! A. V. Kats, V. M. Kontorovich, S. S. Moiseev, and V.

E. Novikov (1975) )

• diverges in the IR for and for in the UV

k � q � m• Coulomb interaction is singular at small momentum transfer

q n � k�5/3 n � k�4/3

Diffusion coefficient Screening mass Effective temperature

|Mk1�23|2 � �2 s2

t2

L. D. Landau (1937) B. Svetitski (1988)

m2 ��

�d3k

|k|nk T� �

q

�m2q � � �2

�d3kn2

k


Steady state solutions

12

• Thermal fixed point:

• Non-thermal fixed point (inverse particle cascade):

A � 1

2T�

�1 +

�

1 +16Q

qT�

�

�

�tnk � q

4k2

�

�kk2

��

�knk +

n2k

T�

�+F� D

Forcing Dumping

• No (homogeneous) steady state solution for the energy cascade

n(k) �A

k>

T�k

• Warm cascade behavior:

2-D Optical turbulence: S. Dyachenko, A.C. Newell, A. Pushkarev, V.E. Zakharov (1992)

Boltzmann equation: D. Proment, S. Nazarenko, P. Asinari, and M. Onorato (2011)

T�k � µ


0.01 0.1 1 10k

-0.00001

-8x10-6

-6x10-6

-4x10-6

-2x10-6

0

2x10-6

4x10-6

6x10-6

E an

d N flu

xes

Numerical simulation of FK equation with forcing

13

0.01 0.1 1k

0.0001

0.001

0.01

0.1

1

10

100

1000

n(k)

kf

Q(k)

P(k)

• The occupation number (left) and, the energy and particle number fluxes (right) at late times

A

k

kf

Inelastic processes in the small angle approximation


Effective 3 waves interactions (1 to 2 scatterings)

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• LPM regime: many scatterings can cause a gluon to branch with the rate

• Bethe-Heitler regime for

kd�

dk�

�

tf(k)� �

�q

k

tf(k) �k

k2�

�k

qtf

tf(k) < �mfp � m2/q

k

k

R. Baier, Y. Dokshitzer, A. H. Mueller, S. Peigné, D. Schiff (1995) V. Zakharov (1996)

J. F. Gunion and G. Bertsch (1982)

kd�

dk�

�

�mfp

formation time:


Effective 3 waves interaction (1 to 2 scatterings)

16

�

�tnk � 1

k3

��

0

dqK(k + q, q)F(k + q, q) �

�k

0

dqK(k, q)F(k, q)

�

F(k, q) � nk+qnk + (nk+q � nk)nq � n2

K(k, q) � ��

q(k + q)7/2

k1/2q3/2

nk �P1/2

q1/4 k7/4

• Direct energy cascade (if interactions are local!)

k + q

k

q

k � q

k

q

k + q

k

k

q q

k � q

- - +

R. Baier, Y. Dokshitzer, A. H. Mueller, D. Schiff, D. T. Son (2000) P. Arnold, G. D. Moore, L. G. Yaffe (2002)

P. Arnold, G. D. Moore (2005)


Locality of interactions

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• Assume a power spectrum and require the energy flux to be independent of

n � k�x

• We obtain and

• The above integral diverges:

x = 7/4

k

P =��

q

�1

0

dz(1 � z)x + zx � 1

zx+1/2(1 � z)x+3/2ln

1

z

P = �

⇒ Effective 3 waves Interaction is nonlocal in momentum space

and the KZ spectrum cannot be realized


Gradient expansion of the collision integral ( k ≪ kf )

• The collision integral is dominated by strongly asymmetric branchings

• In the regime: To the left of the source

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k � kf

�

�tnk � 1

k3

��

0

dqK(k + q, q)F(k + q, q)

�a � �

�q

k7/2[T� � kn(k)]

k + q

k

q q

k + q

k

-

• Late times (steady state) solution is thermal (no fluxes) :

n(k) � T�k

• In agreement with Mueller, Shoshi, Wong (2006) Abraao York, Kurkela, Lu, Moore (2014)

kf

n(k)


Gradient expansion of the collision integral ( k ≫ kf )

• In the regime: To the right of the source. We perform a gradient expansion around

• We obtain an isotropic diffusion equation in “4-D”

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k � kf

k � q

�

�tn(k) � qinel

4k3

�

�kk3 �

�kn(k)

qinel =��

q

��

0

dq�

qn(q)

• with the inelastic diffusion coefficient (in the LPM regime)

kf

n(k)


Gradient expansion of the collision integral ( II )

20

�

�tn(k) � qinel

4k3

�

�kk3 �

�kn(k)

• Recall that 3-D diffusion conserves number of particles:

Its fixed point (inverse particle cascade):

N �

�dkk2 n(k)

E �

�dkk3 n(k)

n(k) �1

k

n(k) �1

k2

• 4-D diffusion conserves energy:

Its fixed point (direct energy cascade):


0.1 1k

1x10-5

1x10-4

1x10-3

1x10-2

1x10-1

1

n(k)

Numerical simulation with forcing (full solution)

21

kf

nKZ � k�7/4

n � k�2

Parametric estimate:

n(k) �P

qk2�

P1/3 k1/3f

k2

q � k3fn2

Nonlocal turbulent spectrum: hard gluons in the inertial range interact dominantly with gluons at the forcing scale (energy gain)

⇓

kmax(t) ��

qinelt


Interplay between elastic and inelastic processes ( I )

22

• For a spectrum falling faster than 1/k one can neglect the drag term in the elastic part. Then, the collision integral in the UV reads

�

�tn(k) � qinel

4k3

�

�kk3 �

�kn(k) +

qel

4k2

�

�kk2 �

�kn(k)

• Steady state solution: n(k) �1

k2��

� B

4k3��

�

�kk3��

�kn(k)

B = qinel + qelwhere

0 < � < 1

� =2

1 + qinel/qeland


Interplay between elastic and inelastic processes ( II )

23

0.1 1k

0.01

0.1

1

10

n(k)

n � k�2

kf

n � k�2+�

A. Ipp, A. Rebhan, M. Strickland (2011)

Exponent is computed self-consistently

• Elastic processes reduce slightly the exponent at asymptotically late times. At intermediate times k -2 spectrum is observed: balance between the drag and diffusion terms?

� � 0.24


Summary

Wave turbulence in QCD is different from scalar theories. It is characterized by nonlocal interactions in momentum space: Kolmogorov-Zakharov spectra are not physically relevant

Inelastic processes dominates the dynamics with a direct energy cascade

To the right of the forcing scale: Kinetic theory predicts a steady state spectrum (in the LPM and BH regimes) in agreement with Hard Loop simulations

To the left of the forcing scale the system appears to be thermalized: warm cascade

Outlook: mass corrections, anisotropic fluxes, strong turbulence in the presence of strong fields (on the lattice): different exponents?

24

� k�2

Documents

Nonlocal Wave Turbulence in QCD - Institute for Nuclear Theory · Nonlocal Wave Turbulence in QCD ... B. Svetitski (1988) m2 d3k |k| n k T q^ m2 q^ 2 ... Yacine Mehtar-Tani INT -