Fluctuations around Bjorken flow and theonset of turbulent phenomena
Stefan Flörchinger (CERN)
work together with

Urs Achim Wiedemann (CERN)
arXiv:1106.5227
arXiv:1108.5535

Yukawa Institute Kyoto 201

Heavy Ion Collisions

101010

5

5

50

0

0-5-5-5-10-10-5

0

5

10-10-10-5

0

5

10-10-10-50

5ions are strongly Lorentz-contracted
some medium is produced after collision
medium expands in longitudinal direction and gets diluted

10Evolution in timeNon-equilibrium evolution at early times
initial state at from QCD? Color Glass Condensate? ...
thermalization via strong interactions, plasma instabilities,
particle production, ...

Local thermal and chemical equilibrium
strong interactions lead to short thermalization times
evolution from relativistic fluid dynamics
expansion, dilution, cool-down

Chemical freeze-out
for small temperatures one has mesons and baryons
inelastic collision rates become small
particle species do not change any more

Thermal freeze-out
elastic collision rates become small
particles stop interacting
particle momenta do not change any more

Fluid dynamic regime

assumes strong interaction effects leading to local equilibrium
fluid dynamic variables
thermodynamic variables: e.g. T (x), µ(x)fluid velocity uµ (x)can be formulated as derivative expansion for T µν
hydrodynamics is universal!
ideal hydro: needs equation of state p = p(T, µ) fromthermodynamics
first order hydro: needs also transport coefficients likeviscosity η = η(T, µ) from linear response theory
second order hydro: needs also relaxation times,...

Bjorken boost invariance
How does the fluid velocity look like?
3Bjorkens guess: v3 (x0 , x1 , x2 , x3 ) = xx0
leads to an invariance under Lorentz-boosts in the x3 -direction
use coordinatespτ = (x0 )2 − (x3 )2 , x1 , x2 , y = arctanh(x3 /x0 )fluid velocity
uµ = (uτ , u1 , u2 , uy ) = (p1 − (u1 )2 − (u2 )2 , u1 , u2 , 0)

thermodynamic scalars like energy density = (τ, x1 , x2 )remaining problem is 2+1 dimensional

Transverse Expansion

for central collisions = (τ, r)
initial pressure gradient leads to radial flow
 1  1
ux=f (τ, r)2ux2experimental signature not very distinct(particle momenta go in all directions)Elliptic flow

non-central collisions lead to deviations from rotation
symmetrypressure gradients larger in one direction
larger fluid velocity in this direction
more particles will fly in this direction
can be quantified in terms of elliptic flow v2C(∆φ) ∼ 1 + 2 v2 cos(2 ∆φ)A puzzle: v3 and v5(ALICE, arXiv:1105.3865, similar pictures also from CMS, ATLAS, Phenix)

quite generally, one can expandC(∆φ) ∼ 1 +∞X

2 vn cos(n ∆φ)

n=2from symmetry reasons one expects naively v3 = v5 = . . . = 0

Event-by-event fluctuations
argument for v3 = v5 = 0 is based on smooth energy density
distribution
there can be deviations from this due to event-by-event
fluctuations
for example using a Glauber model
50-5-10-5

0

5

10

this leads to sizeable v3 and v5

Velocity fluctuations
32

10-1-2-3-3-2-101

23

also the velocity field will fluctuate at the initialization time τ0take here transverse velocity for every participant to be
Gaussian distributed with width 0.1cvorticity |∂1 u2 − ∂2 u1 | and divergence |∂1 u1 + ∂2 u2 |Why are fluctuations interesting?

Hydrodynamic fluctuations: Local and event-by-event
perturbations around the average of hydrodynamical fields:
energy density 
fluid velocity uµmore general also: baryon number density nB , ...

measure for deviations from equilibrium
contain interesting information from early times
an be used to constrain thermodynamic and transport
properties
might affect other phenomena, e.g. jet quenchingSimilarities to cosmic microwave backgroundfluctuation spectrum contains info from early times
many numbers can be measured and compared to theory
detailed understanding of evolution needed
could trigger precision era in heavy ion physics

Setup for treating fluctuations
ensemble average over many events with fixed impact
parameter b is described by smooth hydrodynamical fields
¯ = hiūµ = huµ ifluctuations are added on top = ¯ + δuµ = ūµ + δuµhere we use Bjorkens simplified model (infinite extend intransverse plane)¯ = ¯(τ )ūµ = (1, 0, 0, 0)
uµ = ūµ + δuτ , u1 , u2 , uyLinearized equations for fluctuations
consider only terms linear in δ, (u1 , u2 , uy )decompose velocity field into
gradient term, described by divergenceθ = ∂1 u1 + ∂2 u2 + ∂y uy
rotation term, described by vorticity
ω1 = τ ∂2 uy −1
∂ y u2τ1∂y u1 − τ ∂1 uy
τω3 = ∂1 u2 − ∂2 u1
ω2 =θ and δ are coupled: density or sound waves
vorticity modes decouple from θ and δVorticity modes
transverse propagation (k2=ky=0)2

ωi(τ) / ωi(τ = 1 fm/c)longitudinal propagation (k1=k2=0)
5

ω3 for k1 = 1/fm
ω3 for k1 = 2/fm
ω3 for k1 = 4/fm
ω1 for k1 = 1/fm
ω1 for k1 = 2/fm
ω1 for k1 = 4/fm

ω3 for ky = 4
ω1 for ky = 14

1.5

ω1 for ky = 2
ω1 for ky = 4

3

1

20.5

10

ω3 for ky = 1
ω3 for ky = 20

246
τ [fm]

81000246810

solve equations in Fourier space ωj = ωj (τ, k1 , k2 , ky )short wavelength modes get damped by viscosity
some modes can grow, however!

Sound modes
transverse propagation (k2=ky=0)
1.5

1

0.5

0

3

2

1

-1

02

4

τ [fm]

68ky = 44-0.5

0ky = 1ky = 2θy(τ) / θy(1 fm/c)

θT(τ) / θT(1 fm/c)longitudinal propagation (k1=k2=0)5

k1 = 1/fm
k1 = 2/fm
k1 = 4/fm

10024

τ [fm]

68

oscillation for long times
short wavelength modes get damped by viscosity
also here some modes can grow!

10Linear vs. non-linear evolution

for linearized theory one can easily determine two-point
correlation function at late times from the one at early times
more difficult for non-linear evolution of fluctuations
evolution equation for n-point functions couple:“Closure problem” in fluid dynamics literatureneeds more elaborate tools:
functional techniques, numerical simulation, ...
... but here we do something else:
we map the problem to another one!

Limits of linearized theory
linear approximation works for:energy densityδ1¯
velocity field
Re  1large Reynolds number Re  1 leads to turbulence!
typical numbers:

T = 0.3 Gev, l = 5 fm, uT = 0.1c
⇒ Re ≈1≈ O(10)η/sSmall Mach numberp
u1 u1 + u2 u2 + uy uy
Ma =
1cS
turbulent motion can be described as “compression-less”
θ = ∂1 u1 + ∂2 u2 + ∂y uy = 0

sound modes decouple from vorticity, they are much faster
this does not mean that there are no sound waves presentChange of variableskinematic viscosity, essentially independent of timeην0 =s TBj (τ0 )new time variable (not laboratory time)
3t=

τ 4/3
1/3

∂t =

4τ0

 τ 1/30

τ

rescaled velocity fieldvj = τ 1/30

τuj

temperature fieldd= τ 2/30

τ
lnTTBj (τ )∂τCompression-less flow
this leads to∂t vj +2X

1vy ∂y vj + ∂j d
τ2m=1


1 2
22−ν0 ∂1 + ∂2 + 2 ∂y vj = 0.
τvm ∂m vj +index j = 1, 2, ysolenoidal constraint∂1 v1 + ∂2 v2 +1∂y vy = 0
τ2for large times τ effectively two-dimensional Navier-Stokes!

Reynolds numbers
assumetypical velocity in transverse direction vT
typical velocity in rapidity direction vy
typical transverse length scale ltypical rapidity difference ∆ykinematic viscosity ν0 = sηTdefineRe(T ) =vT l,ν0Re(y) =vy l 2 1
ν0 ∆y τ 2forl
1τ ∆yvT τ 2 ∆y 2,
ν0 l

Re(y) =vy ∆yν0forl1τ ∆yandRe(T ) =Turbulence in d = 2Kraichnan (1967):
vorticity is conserved for ν0 → 0
scaling theory of forced turbulence in d = 2
inverse cascade of energy to small wave numbers
cascade of vorticity to large wave numbers
E(k) ∼ k −3

qualitatively different to d = 3, emerges here dynamicallyDecaying turbulence in d = 2
E(k)
t=5
t=4
t=3
t=2
t=1
k

Batchelor (1969):
scaling theory of decaying turbulence in d = 2
E(t, k) = λ3 t f (k λ t)with

λ2 = h~v 2 i = const.

turbulent motion goes to smaller and smaller wave numbers

One-particle spectra

fluctuations in fluid fields modify the one-particle spectra
dN0 dδN1dδN2dδN3dN= 3 + 3 h(u1 )2 i+ 3 h(uy )2 i+ 3 h(T −Tfo )2 i3d pd pd pd pd p
explicit expressions can be derived
effect qualitatively similar to viscosity
depends only on a few numbers

Two-particle spectra
correlation function of particles with momenta p~A and p~BC(~pA , p~B ) =dN
d3 pA d3 pBdNdN
d3 pA d3 pB.

effect of hydro fluctuations on this can be calculatedparticular interesting are identical particles(e.g. pions with equal charge and spin)C(~pA , p~B ) can probe correlation functions of hydrodynamic
fields at different space-time points
characteristic power-law decay with |~pA − p~B | in turbulent
situation
allows in principle to test Kraichnans law E(k) ∼ k −3

Effects on macroscopic motion of fluid

turbulent fluctuations might affect macroscopic motion
modified equation of state
modified transport properties

anomalous, turbulent or eddy viscosity
proposed by Asakawa, Bass, Müller (2006) for plasma
turbulence and Romatschke (2007) for fluid turbulence
could become negative in d = 2 (Kraichnan (1976))
depends on detailed state of turbulence – not universal
gradient expansion needs separation of scales

more work neededSummary

fluid fluctuations are very interesting!
some modes even grow, others are damped
hydrodynamical fluctuations on expanding medium canbecome turbulent
evolution laws can be mapped to two-dimensional
Navier-Stokes equation for late times
turbulence has interesting effects on the two-particle spectrum

Backup

Little Bang vs. Big Bang
Heavy Ions

Cosmology

Bjorken modelFriedmann-Robertson-Walker

xµ = (τ, x1 , x2 , y)−11gµν = 10 (τ ),τ2

uµ0 = (1, 0, 0, 0)+ hydrodyn. fluctuations

xµ = (t, x1 , x2 , x3 )

−1

a(t)
gµν = 

a(t)




a(t)

0 (t),uµ0 = (1, 0, 0, 0)+ hydrodyn. fluctuations = 0 (τ ) + 1 (τ, x1 , x2 , y)

 = 0 (t) + 1 (t, x1 , x2 , x3 )uµ = uµ0 + uµ1 (τ, x1 , x2 , y)uµ = uµ0 + uµ1 (t, x1 , x2 , x3 )+ gravity fluctuations

Turbulence in d = 3
fully developed turbulence
Re → ∞

dissipated energy per unit time
DEd 2~ × ~v )2 = −εh~v i = −ν0 (∇dtRichardson (1922):
Big whorls have little whorls,
Which feed on their velocity;
And little whorls have lesser whorls,
And so on to viscosity.

Kolmogorov (1941):E(k) ∼ ε2/3 k −5/3

L. da Vinci (ca. 1500)

with1 2v i=2 h~Z ∞dk E(k)0