How (non-)linear is the hydrodynamics of heavy ion …personalpages.to.infn.it/~beraudo/santiago.pdfHeavy-ion collisions: exploring the QCD phase-diagram QCD phases identi ed through

Relativistic heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview

The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics

How (non-)linear is the hydrodynamics of heavyion collisions?

Andrea Beraudo

University of Santiago de Compostela and CERN

Santiago de Compostela, 16th January 2014

1 / 39



Outline

Heavy-ion collisions: general introduction

Phenomenological overview: soft observables andhydrodynamics

Theory setup: Relativistic HydroDynamics (RHD)

Linear and non-linear rensponse to initial fluctuations: aperturbative framework1

1Stefan Floerchinger, Urs Achim Wiedemann, A.B., Luca Del Zanna,Gabriele Inghirami and Valentina Rolando, arXiv:1312.5482 [hep-ph].

2 / 39



Heavy-ion collisions: exploring the QCD phase-diagram

QCD phases identified through the orderparameters

Polyakov loop 〈L〉 ∼ e−β∆FQ energycost to add an isolated color charge

Chiral condensate 〈qq〉 ∼ effectivemass of a “dressed” quark in a hadron

Region explored at LHC: high-T/low-density (early universe, nB/nγ∼10−9)

From QGP (color deconfinement, chiral symmetry restored)

to hadronic phase (confined, chiral symmetry breaking2)

NB 〈qq〉 6=0 responsible for most of the baryonic mass of the universe: only

∼35 MeV of the proton mass from mu/d 6=0

2V. Koch, Aspects of chiral symmetry, Int.J.Mod.Phys. E6 (1997)3 / 39



Heavy-ion collisions: a typical event

Valence quarks of participant nucleons act as sources of strong colorfields giving rise to particle production

Spectator nucleons don’t participate to the collision;

Almost all the energy and baryon number carried away by the remnants

4 / 39



Heavy-ion collisions: a typical event

5 / 39



Heavy-ion collisions: a cartoon of space-time evolution

Soft probes (low-pT hadrons): collective behavior of the medium;

Hard probes (high-pT particles, heavy quarks, quarkonia): producedin hard pQCD processes in the initial stage, allow to perform atomography of the medium 6 / 39



The general setupHydro predictionsEvent-by-event fluctuations and recent developments

Hydrodynamics and heavy-ion collisions

The success of hydrodynamics in describing particle spectra in heavy-ioncollisions measured at RHIC came as a surprise!

The general setup and its implications

The main predictions

Radial flowElliptic flow

Recent developments (fluctuating initial conditions)

Flow in central collisionsHigher flow harmonicsEvent-by-event flow measurements

What can we learn?

Equation of State (EOS) of the produced matterInitial conditionsQGP viscosity

7 / 39




Hydrodynamics: the general setup

Hydrodynamics is applicable in a situation in which λmfp L

In this limit the behavior of the system is entirely governed by theconservation laws

∂µTµν = 0︸︷︷︸four−momentum

, ∂µjµB = 0︸︷︷︸baryon number

,

where

Tµν =(ε+P)uµuν−Pgµν , jµB =nB uµ and uµ=γ(1, ~v)

Information on the medium is entirely encoded into the EOS

P = P(ε)

The transition from fluid to particles occurs at the freeze-outhypersuface Σfo (e.g. at T = Tfo)

E (dN/d~p) =

∫Σfo

pµdΣµ exp[−(p · u)/T ]8 / 39




Hydro predictions: radial flow (I)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.71

10

102

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

10-2

10-1

1

1/(

2π)

d2N

/ (

mT d

mT d

y)

[c

4/G

eV

2]

mT - m0 [GeV/c2] mT - m0 [GeV/c2]

1/(

2π)

d2N

/ (

mT d

mT d

y)

[c

4/G

eV

2]

STAR preliminary

Au+Au central

√sNN = 200 GeV

STAR preliminary

p+p min. bias

√sNN = 200 GeV

π-π-

K-

p

π-π-

K-

p

dN

mT dmT∼ e−mT/Tslope ≡ e−

√p2

T +m2/Tslope

Tslope(∼ 167 MeV) universal in pp collisions;

Tslope growing with m in AA collisions: spectrum gets harder!

9 / 39




Hydro predictions: radial flow (II)

Physical interpretation:

Thermal emission on top of a collective flow

0.1

0.2

0.3

0.4

0.5

0 0.5 1 1.5 2

SPS Pb+Pb: NA49 preliminary

AGS Au+Au: E866

SPS S+S: NA44

π+

h−

K±

p

φ Λ

Λ−

Ξ−

d

Particle mass (GeV/c2)

mT in

vers

e sl

ope

(GeV

) 1

2m〈v2

⊥〉 =1

2m⟨

(v⊥th + v⊥flow )2⟩

=1

2m〈v2

⊥th〉+1

2mv2⊥flow

=⇒ Tslope = Tfo +1

2mv2⊥flow

Radial flow gets larger going from RHIC to LHC!

10 / 39




Hydro predictions: radial flow (II)

Physical interpretation:

Thermal emission on top of a collective flow

(GeV/c)T

p0 0.5 1 1.5 2 2.5 3

-1dy

) (G

eV/c

)T

N/(

dp2 d

-110

1

10

210

310

0-5% most centralALICE Preliminary

= 2.76 TeVNNsALICE, Pb-Pb,

= 200 GeVNNsSTAR, Au-Au, not feed-down corrected)p (STAR

= 200 GeVNNsPHENIX, Au-Au,

S0K -π -

K p

1

2m〈v2

⊥〉 =1

2m⟨

(v⊥th + v⊥flow )2⟩

=1

2m〈v2

⊥th〉+1

2mv2⊥flow

=⇒ Tslope = Tfo +1

2mv2⊥flow

Radial flow gets larger going from RHIC to LHC!

10 / 39




Hydro predictions: elliptic flow

xφ

y

In non-central collisions particle emissionis not azimuthally-symmetric!

The effect can be quantified through theFourier coefficient v2

dN

dφ=

N0

2π(1 + 2v2 cos[2(φ− ψRP )] + . . . )

v2 ≡ 〈cos[2(φ− ψRP )]〉

v2(pT ) ∼ 0.2 gives a modulation 1.4 vs0.6 for in-plane vs out-of-plane particleemission!

11 / 39




Hydro predictions: elliptic flow

[GeV/c]T

p0 1 2 3 4 5 6

2v

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4 = 2.76 TeVNNsPb-Pb events at

Centrality 20-40%πKpΞΩ

VISH2+1

/s=0.20)η(MCKLN, πKpΞΩ ALICE preliminary

In non-central collisions particle emissionis not azimuthally-symmetric!

The effect can be quantified through theFourier coefficient v2

dN

dφ=

N0

2π(1 + 2v2 cos[2(φ− ψRP )] + . . . )

v2 ≡ 〈cos[2(φ− ψRP )]〉

v2(pT ) ∼ 0.2 gives a modulation 1.4 vs0.6 for in-plane vs out-of-plane particleemission!

11 / 39




Elliptic flow: physical interpretation

xφ

y

Matter behaves like a fluid whose expansion is driven by pressuregradients

(ε+ P)dv i

dt=

vc− ∂P

∂x i(Euler equation)

Spatial anisotropy is converted into momentum anisotropy;

At freeze-out particles are mostly emitted along the reaction-plane.12 / 39




Elliptic flow: mass ordering

The mass ordering of v2 is a direct consequence of the hydro expansion

2A

niso

trop

y P

aram

eter

v

(GeV/c)T

Transverse Momentum p

0 1 2 3 4

0

0.1

0.2

0.3

10%-40%

(ToF)±π (dE/dx)±π

±h 0SK

(ToF)±K (dE/dx)±K

(ToF)pp+ (dE/dx)p

Λ+Λ

0 1 2 3 40

0.1

0.2

0.3 0%-10%

1 2 3 4

40%-80%

Particles emitted according to athermal distribution∼exp[−p ·u(x)/Tfo] in the localrest-frame of the fluid-cell;

Parametrizing the fluid velocity as

uµ ≡ γ⊥(cosh Y ,u⊥, sinh Y ),

one gets (vz≡ tanh Y =z/t)

p ·u = γ⊥[m⊥ cosh(y−Y )− p⊥ ·u⊥]

Dependence on mT at the basis ofmass ordering at fixed pT

13 / 39






0

0.1

0.2

0.3 0-20%pbarπ− K−

ch−

v 2

20-40% 40-60%

0

0.1

0.2

0.3

0 2 4

0-20%pπ+ K+

ch+

0 2 4

20-40%

0 2 4

40-60%

pT (GeV/c)





p ·u = γ⊥[m⊥ cosh(y−Y )− p⊥ ·u⊥]


13 / 39






[GeV/c]T

p0 1 2 3 4 5 6

2v

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4 = 2.76 TeVNNsPb-Pb events at

Centrality 20-40%πKpΞΩ

VISH2+1

/s=0.20)η(MCKLN, πKpΞΩ ALICE preliminary





p ·u = γ⊥[m⊥ cosh(y−Y )− p⊥ ·u⊥]


13 / 39




Event by event fluctuations

Due to event-by-event fluctuations (e.g. of the nucleon positions)the initial density distribution is not smooth and can display higherdeformations, each one with a different azimuthal orientation.

Higher harmonics (m > 2) contribute to the angular distribution

dN

dφ=

N

2π

(1 + 2

∑m

vm cos[m(φ− ψm)]

)of the final hadrons, where for each event,

vm = 〈cos[m(φ− ψm)]〉 and ψm =1

marctan

∑i pi

T sin(mφi )∑i pi

T cos(mφi ) 14 / 39




Event-by-event fluctuations: experimental consequences

Fluctuating initial conditions give rise toa:

Non-vanishing v2 in central collisions;

Odd harmonics (v3 and v5)

Hydro can reproduce also higher harmonicsb

aALICE, Phys.Rev.Lett. 107 (2011) 032301bB: Schenke et al., PRC 85, 024901 (2012)

15 / 39




Event-by-event fluctuations: experimental consequences

Fluctuating initial conditions give rise toa:

Non-vanishing v2 in central collisions;

Odd harmonics (v3 and v5)

Hydro can reproduce also higher harmonicsb

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.5 1 1.5 2 2.5 3pT [GeV]

η/s=0.08 v2 20-30% v3 20-30% v4 20-30% v5 20-30% PHENIX v2 PHENIX v3 PHENIX v4

aALICE, Phys.Rev.Lett. 107 (2011) 032301bB: Schenke et al., PRC 85, 024901 (2012)

15 / 39




Modelling the initial conditions: Glauber-MC approach

Generate Nconf configurations, each configuration obtainedextracting from a Woods-Saxon distribution

the coordinates of the A nucleons of nucleus A;the coordinates of the B nucleons of nucleus B.

For each configuration re-write the nucleon coordinates wrt thecenter-of-mass of each nucleus;

Given a configuration, extract a possible impact parameter from thedistribution dP = 2πbdb, with b < bmax =20 fm;

Nucleons i and j collide if (xi−xj )2 +(yi−yj )

2 < σNN/π

If at least one collision occurs...keep b and store the info;Else extract a different b and repeat.

Final events can be organized in centrality classes according toNpart (or Ncoll or a combination of the two).

16 / 39




Glauber-MC initial conditions: results

Taking a smeared energy-density distribution around each participant

ε(x , y , τ0) =K

2πσ2

Npart∑i=1

exp

[− (x − xi )

2 + (y − yi )2

2σ2

]

0 100 200 300 400N

part

0.0001

0.001

0.01

1/N

ev (

dNev

/dN

part)

Glauber-MC

Au-Au @ 200 GeVσ

NN=42 mb

edens(tau0,x,y) (GeV/fm3)

-10 -8 -6 -4 -2 0 2 4 6 8 10

x

-10

-8

-6

-4

-2

0

2

4

6

8

10

y

0

50

100

150

200

250

300

17 / 39




Characterizing the initial conditions

For each event the initial density distribution can be characterized interms of complex eccentricity coeffients

εn,me imΨn,m ≡ −

∫d~r r ne imφε(~r , τ0)∫

d~r r nε(~r , τ0)≡ −r

n cos(mφ)+ ir n sin(mφ)r n

whose orientation and modulus are given by

Ψn,m =1

matan2 (−r n sin(mφ),−r n cos(mφ))

and

εn,m =

√r n⊥ cos(mφ)2 + r n

⊥ sin(mφ)2

r n⊥

= −rn cos[m(φ−Ψn,m)]

r n

18 / 39




Connecting initial conditions to hadron spectra

Hydrodynamics is expected to propagate the initial eccentricity of thedensity distribution into the final azimuthal anisotropy of hadron spectra

Averages: 〈εn,m〉 −→ 〈vm〉

Probability distributions: P(εn,m) −→ P(vm)

Correlations, e.g. 〈εn,mεn′,m′〉 −→ 〈vmvm′〉

Basic question

To what extent vm ∼ εn,m and ψm ∼ Ψn,m?in particular with realistic initial conditions involving several modes,

which can give rise to non-linear effect...

19 / 39




Connecting initial conditions to hadron spectra

Hydrodynamics is expected to propagate the initial eccentricity of thedensity distribution into the final azimuthal anisotropy of hadron spectra

Averages: 〈εn,m〉 −→ 〈vm〉

Probability distributions: P(εn,m) −→ P(vm)

Correlations, e.g. 〈εn,mεn′,m′〉 −→ 〈vmvm′〉

Basic question

To what extent vm ∼ εn,m and ψm ∼ Ψn,m?in particular with realistic initial conditions involving several modes,

which can give rise to non-linear effect...

19 / 39




Flow vs eccentricity

0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40ǫ2

0.01

0.02

0.03

0.04

0.05

0.06

v 2

c(ǫ2 ,v2 ) =0.985

C2 =0.159

sBC η/s=0.16

0−5 %

(b)

0.1 0.2 0.3 0.4 0.5 0.6ǫ2

0.02

0.04

0.06

0.08

0.10

v 2

c(ǫ2 ,v2 ) =0.990

C2 =0.148

sBC η/s=0.16

20−30 %

(b)

(Results from Niemi, Denicol, Holopainen and Huovinen, Phys.Rev.C 87(2013) 054901)

Strong correlation for the n =2, 3 harmonics

Mild correlation for the n = 4 harmonic only in central events.

20 / 39





0.05 0.10 0.15 0.20 0.25 0.30ǫ3

0.005

0.010

0.015

0.020

0.025

0.030

v 3

c(ǫ3 ,v3 ) =0.906

C3 =0.100

sBC η/s=0.16

0−5 %

(b)

0.05 0.10 0.15 0.20 0.25 0.30 0.35ǫ3

0.005

0.010

0.015

0.020

0.025

0.030

v 3

c(ǫ3 ,v3 ) =0.955

C3 =0.088

sBC η/s=0.16

20−30 %

(b)



Mild correlation for the n = 4 harmonic only in central events.

20 / 39





0.05 0.10 0.15 0.20ǫ4

0.002

0.004

0.006

0.008

0.010

v 4

c(ǫ4 ,v4 ) =0.511

C4 =0.039

sBC η/s=0.16

0−5 %

(b)

0.05 0.10 0.15 0.20 0.25 0.30 0.35ǫ4

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.016

v 4

c(ǫ4 ,v4 ) =0.195

C4 =0.033

sBC η/s=0.16

20−30 %

(b)



Mild correlation for the n = 4 harmonic only in central events.20 / 39




P(εn) vs P(vn)

0 0.1 0.2 0.3 0.4ε

n

0.1

1

10

P(ε n)

ε2

ε3

ε4

ε5

Au-Au collisionsσ

NN=42 mb

Npart

≥ 300

0 0.1 0.2 0.3 0.4 0.5ε

n

0.01

0.1

1

10

P(ε n)

ε2

ε3

ε4

ε5

Au-Au collisionsσ

NN=42 mb

200 ≤ Npart

≤ 299

Event-by-event flow measurements allow to connect probabilitydistribution of

initial fluctuations (εm ≡ ε2,m)

different flow harmonics (ATLAS coll., JHEP 1311 (2013) 183)

allowing one to put constraints on the initial state.

21 / 39




P(εn) vs P(vn)

2v0 0.1 0.2

) 2p(

v

-210

-110

1

10

|<2.5η>0.5 GeV, |T

p

centrality:0-1%5-10%20-25%30-35%40-45%60-65%

ATLAS Pb+Pb

=2.76 TeVNNs-1bµ = 7 intL

3v0 0.05 0.1

) 3p(

v

-110

1

10

|<2.5η>0.5 GeV, |T

p

centrality:0-1%5-10%20-25%30-35%40-45%

ATLAS Pb+Pb


4v0 0.01 0.02 0.03 0.04 0.05

) 4p(

v

1

10

210

|<2.5η>0.5 GeV, |T

p

centrality:0-1%5-10%20-25%30-35%40-45%

ATLAS Pb+Pb


Event-by-event flow measurements allow to connect probabilitydistribution of

initial fluctuations (εm ≡ ε2,m)

different flow harmonics (ATLAS coll., JHEP 1311 (2013) 183)

allowing one to put constraints on the initial state.21 / 39



Ideal RHDViscous RHD

Relativistic hydrodynamics

22 / 39




Relativistic hydrodynamics: the ideal case

In the absence of non-vanishing conserved charges (nB = 0), theevolution of an ideal fluid is completely described by the conservation ofthe ideal energy-momentum tensor:

∂µTµν = 0, where Tµν = Tµνeq = (ε+ P)uµuν − Pgµν

It is convenient to project the above equations

along the fluid velocity (uν∂µTµν = 0)

Dε = −(ε+ P)Θ, (with D ≡ uµ∂µ and Θ ≡ ∂µuµ)

and perpendicularly to it (∆αν∂µTµν = 0, with ∆αν≡gαν − uαuν)

(ε+ P)Duα = ∇αP (with ∇α ≡ ∆αµ∂µ),

which is the relativistic version of the Euler equation (fluidacceleration driven by pressure gradients)

23 / 39




Viscous hydrodynamics

Better flow measurements required the introduction of viscous correctionsto the energy-momentum tensor in order to reproduce the data:

Tµν = Tµνeq + Πµν = Tµν

eq + πµν − Π∆µν ,

where we have isolated the traceless (πµµ = 0) shear viscous tensor πµν .The condition uµΠµν=uµπ

µν=0 (Landau frame) defines the fluid velocity

uµuνTµν = uµuνTµνeq = ε (T

00= T

00

eq = ε in the LRF)

Projecting along uν :

Dε = −(ε+ P + Π)Θ + πµν∇<µuν>,

after replacing ∇µuν −→ ∇<µuν>≡ 12 (∇µuν+∇νuµ)− 1

3 ∆µνΘ

Projecting along ∆αν :

(ε+ P + Π)Duα = ∇α(P + Π)−∆αν ∂µπ

µν

24 / 39











00= T

00

eq = ε in the LRF)




3 ∆µνΘ


(ε+ P + Π)Duα = ∇α(P + Π)−∆αν ∂µπ

µν

24 / 39











00= T

00

eq = ε in the LRF)




3 ∆µνΘ


(ε+ P + Π)Duα = ∇α(P + Π)−∆αν ∂µπ

µν

24 / 39




Fixing the viscous tensor: first order formalism

A way to fix the viscous tensor is through the 2nd law ofthermodynamics, imposing ∂µsµ ≥ 0.

Using the ideal result for theentropy current sµ = suµ and employing the thermodynamic relations

Ts = ε+ P and T ds = dε

one gets

∂µsµ = uµ∂µs + s ∂µuµ =1

T[Dε+ (ε+ P)Θ] ≥ 0

EmployingDε = −(ε+ P + Π)Θ + πµν∇<µuν>,

one gets

∂µsµ =1

T[−ΠΘ + πµν∇<µuν>] ≥ 0

which is identically satisfied if (relativistic Navier Stokes result)

Π = −ζΘ and πµν = 2η∇<µuν>,

where ζ and η are the bulk and shear viscosity coefficients.

25 / 39





A way to fix the viscous tensor is through the 2nd law ofthermodynamics, imposing ∂µsµ ≥ 0. Using the ideal result for theentropy current sµ = suµ and employing the thermodynamic relations


one gets


T[Dε+ (ε+ P)Θ] ≥ 0


one gets

∂µsµ =1




where ζ and η are the bulk and shear viscosity coefficients.

25 / 39





A way to fix the viscous tensor is through the 2nd law ofthermodynamics, imposing ∂µsµ ≥ 0. Using the ideal result for theentropy current sµ = suµ and employing the thermodynamic relations


one gets


T[Dε+ (ε+ P)Θ] ≥ 0


one gets

∂µsµ =1




where ζ and η are the bulk and shear viscosity coefficients. 25 / 39




Relativistic causal theory: second order formalism

The naive relativistic generalization of the Navier Stokes equationsviolates causality! This pathology can be cured including viscouscorrections into the entropy current, of second order in the gradients:

sµ = sµeq + Qµ = suµ −(β0Π2 + β2παβπ

αβ) uµ

2T

One gets then (Df ≡ f ):

T∂µsµ = Π[−Θ− β0Π− T Π ∂µ(β0uµ/2T )

]+ παβ [∇<αuβ> − β2παβ − Tπαβ ∂µ(β2uµ/2T )] ≥ 0,

which is satisfied if Π≈ζ[−Θ− β0Π] and παβ≈2η[∇<αuβ> − β2παβ].

One has then to evolve also the components of the viscous tensor (6independent equations, due to uµπ

µν =0 and πµµ =0)

Π ≈ − 1

ζβ0[Π + ζΘ] and παβ ≈ −

1

2ηβ2[παβ − 2η∇<αuβ>],

whera τΠ ≡ ζβ0 and τπ ≡ 2ηβ2 play the role of relaxation times.

26 / 39




Relativistic causal theory: second order formalism

The naive relativistic generalization of the Navier Stokes equationsviolates causality! This pathology can be cured including viscouscorrections into the entropy current, of second order in the gradients:

sµ = sµeq + Qµ = suµ −(β0Π2 + β2παβπ

αβ) uµ

2T

One gets then (Df ≡ f ):

T∂µsµ = Π[−Θ− β0Π− T Π ∂µ(β0uµ/2T )

]+ παβ [∇<αuβ> − β2παβ − Tπαβ ∂µ(β2uµ/2T )] ≥ 0,

which is satisfied if Π≈ζ[−Θ− β0Π] and παβ≈2η[∇<αuβ> − β2παβ].One has then to evolve also the components of the viscous tensor (6independent equations, due to uµπ

µν =0 and πµµ =0)

Π ≈ − 1

ζβ0[Π + ζΘ] and παβ ≈ −

1

2ηβ2[παβ − 2η∇<αuβ>],

whera τΠ ≡ ζβ0 and τπ ≡ 2ηβ2 play the role of relaxation times. 26 / 39




Numerical implementation: the ECHO-QGP code

We employed for our numerical studies the ECHO-QGP code

Some references...

An italian project (MIUR & INFN): L. Del Zanna, V. Chandra,G. Inghirami, V. Rolando, A. Beraudo, A. De Pace, G. Pagliara,A. Drago and F.Becattini: Eur.Phys.J. C73 (2013) 2524based on the astrophysical code ECHO: L. Del Zanna et al.,(2007) Astron.Astrophys.,473,11ECHO-QGP webpage: http://www.astro.unifi.it/echo-qgp/

The main features:

Possibility to run both with Cartesian and Bjorken(τ≡√

t2 − z2, η≡ 12 ln t+z

t−z ) coordinates,both in (2+1)D and in (3+1)D;in the ideal or viscous case;with any EOS and initial condition supplied by the user

27 / 39



The setupResults

Mode-by-mode hydrodynamics

Original proposal presented in Phys.Rev. C88 (2013) 044906(Stefan Floerchinger and U.A. Wiedemann)

Results obtained through full RHD calculations presented inarXiv:1312.5482 [hep-ph] and displayed in this talk

28 / 39



The setupResults

Mode-by-mode hydrodynamics: the general idea

For each event the system is initialized via a full set of hydrodynamicfields on a τ0-hypersurface (w being the enthalpy density):

hi (τ0, r , ϕ, η) =(w , ur , uφ, uη,Π, πµν

)

For each event one can express hi in terms of a smooth backgroundhBG

i , obtained averaging over a large sample of events, and a

fluctuating term hi . One can write for instance:

w = wBG(1+w), ur = urBG+

1√2

(u−+u+), uφ =i√2r

(u−−u+)

We propose the following expansion for the evolution of hi (τ) ontop of the evolved backgroud hBG

j (τ):

hi (τ, r , ϕ) =

Zr′,ϕ′Gij (τ, τ0, r , r

′, ϕ− ϕ′) hj (τ0, r′, ϕ′)

+1

2

Zr′,r′′,ϕ′,ϕ′′

Hijk (τ, τ0, r , r′, r ′′, ϕ−ϕ′, ϕ−ϕ′′) hj (τ0, r

′, ϕ′) hk (τ0, r′′, ϕ′′)+O(h3)

29 / 39



The setupResults




)For each event one can express hi in terms of a smooth backgroundhBG




1√2

(u−+u+), uφ =i√2r

(u−−u+)


j (τ):

hi (τ, r , ϕ) =


′, ϕ− ϕ′) hj (τ0, r′, ϕ′)

+1

2

Zr′,r′′,ϕ′,ϕ′′


′, ϕ′) hk (τ0, r′′, ϕ′′)+O(h3)

29 / 39



The setupResults




)For each event one can express hi in terms of a smooth backgroundhBG




1√2

(u−+u+), uφ =i√2r

(u−−u+)


j (τ):

hi (τ, r , ϕ) =


′, ϕ− ϕ′) hj (τ0, r′, ϕ′)

+1

2

Zr′,r′′,ϕ′,ϕ′′


′, ϕ′) hk (τ0, r′′, ϕ′′)+O(h3)

29 / 39



The setupResults

Mode-by-mode hydrodynamics: the strategy

From the exact numerical solution (with ECHO-QGP)

both for the average background hBGi (τ0) −→

full hydrohBG

i (τ)

and for fluctuating initial conditions hi (τ0) −→full hydro

hi (τ)

we will show that the expansion

hi (τ, r , ϕ) =


′, ϕ− ϕ′) hj (τ0, r′, ϕ′)

+1

2

Zr′,r′′,ϕ′,ϕ′′


′, ϕ′) hk (τ0, r′′, ϕ′′)+O(h3)

actually holds and in particular that

the dominant response to initial fluctuations is (in most cases) linear

non-linearities (important in some cases) can be consistentlyinterpreted as higher-order corrections within our perturbativeexpansion and quantitatively reproduced

30 / 39



The setupResults

Mode-by-mode hydrodynamics: density perturbations

We will focus on the hydrodynamic propagation of initial fluctuating

density distributions, parametrized in terms of the coefficients w(m)l of a

Fourier-Bessel expansion (k(m)l ≡z

(m)l /R, with z

(m)l the l th-zero of Jm)

w(τ0, r , ϕ)=wBG (τ0, r)

1+

∞Xm=−∞

w (m)(τ0, r)e imϕ

!, w (m)(τ0, r)=

∞Xl=1

w(m)l Jm

“k

(m)l r

”

The goal: understanding which of the initial Fourier modes in theexpansion of w(τ0, r , ϕ) contribute to the various azimuthal harmonics

w (m)(τ, r) ≡∫ 2π

0

dϕe−imϕw(τ, r , ϕ)

at a later time, by analyzing the full RHD outcomes by ECHO-QGP for

wBG(τ0) −→ wBG(τ) and w(τ0) −→ w(τ) ≡ wBG(τ) [1 + w(τ)]

31 / 39



The setupResults

Mode-by-mode hydrodynamics: density perturbations

We will focus on the hydrodynamic propagation of initial fluctuating

density distributions, parametrized in terms of the coefficients w(m)l of a

Fourier-Bessel expansion (k(m)l ≡z

(m)l /R, with z

(m)l the l th-zero of Jm)

w(τ0, r , ϕ)=wBG (τ0, r)

1+

∞Xm=−∞

w (m)(τ0, r)e imϕ

!, w (m)(τ0, r)=

∞Xl=1

w(m)l Jm

“k

(m)l r

”The goal: understanding which of the initial Fourier modes in the

expansion of w(τ0, r , ϕ) contribute to the various azimuthal harmonics

w (m)(τ, r) ≡∫ 2π

0

dϕe−imϕw(τ, r , ϕ)

at a later time, by analyzing the full RHD outcomes by ECHO-QGP for

wBG(τ0) −→ wBG(τ) and w(τ0) −→ w(τ) ≡ wBG(τ) [1 + w(τ)]

31 / 39



The setupResults

Propagation and interaction of different Fourier modes

From the perturbative expansion for the hydrodynamic fluctuations andperforming a Fourier decomposition of the response functions

G(τ, τ0, r , r′,∆ϕ) =

1

2π

∞Xm=−∞

e im∆ϕG(m)(τ, τ0, r , r′)

H(τ, τ0, r , r′, r ′′,∆ϕ′,∆ϕ′′) =

1

(2π)2

∞Xm′,m′′=−∞

e i(m′∆ϕ′+m′′∆ϕ′′)H(m′,m′′)(τ, τ0, r , r′, r ′′)

one obtains for the mth harmonic of the enthalpy fluctuations

w (m)(τ, r) =

Zr′G(m)(τ, τ0, r , r

′) w (m)(τ0, r′)

+1

2

Zr′,r′′

1

2π

Xm′,m′′

δm,m′+m′′H(m′,m′′)(τ, τ0, r , r′, r ′′) w (m′)(τ0, r

′) w (m′′)(τ0, r′′)+. . .

A single m-mode at τ0 can give rise at time τ to:

an m-mode at linear order

a 0 and 2m-mode at quadratic order

a 3m-mode and corrections to the m-mode at cubic order...

32 / 39



The setupResults



G(τ, τ0, r , r′,∆ϕ) =

1

2π

∞Xm=−∞


H(τ, τ0, r , r′, r ′′,∆ϕ′,∆ϕ′′) =

1

(2π)2

∞Xm′,m′′=−∞

e i(m′∆ϕ′+m′′∆ϕ′′)H(m′,m′′)(τ, τ0, r , r′, r ′′)


w (m)(τ, r) =


′) w (m)(τ0, r′)

+1

2

Zr′,r′′

1

2π

Xm′,m′′


′) w (m′′)(τ0, r′′)+. . .





32 / 39



The setupResults



G(τ, τ0, r , r′,∆ϕ) =

1

2π

∞Xm=−∞


H(τ, τ0, r , r′, r ′′,∆ϕ′,∆ϕ′′) =

1

(2π)2

∞Xm′,m′′=−∞

e i(m′∆ϕ′+m′′∆ϕ′′)H(m′,m′′)(τ, τ0, r , r′, r ′′)


w (m)(τ, r) =


′) w (m)(τ0, r′)

+1

2

Zr′,r′′

1

2π

Xm′,m′′


′) w (m′′)(τ0, r′′)+. . .





32 / 39



The setupResults



G(τ, τ0, r , r′,∆ϕ) =

1

2π

∞Xm=−∞


H(τ, τ0, r , r′, r ′′,∆ϕ′,∆ϕ′′) =

1

(2π)2

∞Xm′,m′′=−∞

e i(m′∆ϕ′+m′′∆ϕ′′)H(m′,m′′)(τ, τ0, r , r′, r ′′)


w (m)(τ, r) =


′) w (m)(τ0, r′)

+1

2

Zr′,r′′

1

2π

Xm′,m′′


′) w (m′′)(τ0, r′′)+. . .





32 / 39



The setupResults



G(τ, τ0, r , r′,∆ϕ) =

1

2π

∞Xm=−∞


H(τ, τ0, r , r′, r ′′,∆ϕ′,∆ϕ′′) =

1

(2π)2

∞Xm′,m′′=−∞

e i(m′∆ϕ′+m′′∆ϕ′′)H(m′,m′′)(τ, τ0, r , r′, r ′′)


w (m)(τ, r) =


′) w (m)(τ0, r′)

+1

2

Zr′,r′′

1

2π

Xm′,m′′


′) w (m′′)(τ0, r′′)+. . .




a 3m-mode and corrections to the m-mode at cubic order... 32 / 39



The setupResults

Setting the initial conditions

The fluctuations being real sets the constraint w(−m)l =(−1)m(w

(m)l )∗.

Parametrizing the weights as w(m)l = |w (m)

l |e−imψ(m)l allows one to recast

the expansion for the initial enthalpy density into the form (reminiscentof the harmonic decomposition of azimuthal single-particle distributions)

w(τ0, r , ϕ) = wBG (τ0, r)

1 +

∞Xl=1

w(0)l J0

“k

(0)l r”

+2∞X

m=1

∞Xl=1

|w (m)l | cos

hm(ϕ− ψ(m)

l )iJm

“k

(m)l r

”!,

which will be then evolved via the full hydrodynamic equations.

In the following we will study the evolution of a selected set of

Fourier-Bessel modes, exploring for the weights w(m)l the typical range of

values provided by a sample of Glauber-MC initial conditions

33 / 39



The setupResults

Setting the initial conditions

The fluctuations being real sets the constraint w(−m)l =(−1)m(w

(m)l )∗.

Parametrizing the weights as w(m)l = |w (m)

l |e−imψ(m)l allows one to recast

the expansion for the initial enthalpy density into the form (reminiscentof the harmonic decomposition of azimuthal single-particle distributions)

w(τ0, r , ϕ) = wBG (τ0, r)

1 +

∞Xl=1

w(0)l J0

“k

(0)l r”

+2∞X

m=1

∞Xl=1

|w (m)l | cos

hm(ϕ− ψ(m)

l )iJm

“k

(m)l r

”!,

which will be then evolved via the full hydrodynamic equations.

In the following we will study the evolution of a selected set of

Fourier-Bessel modes, exploring for the weights w(m)l the typical range of

values provided by a sample of Glauber-MC initial conditions

33 / 39



The setupResults

Initialization with a single-mode

We start considering the evolution of a single (m =2, l =1) mode on topof an average background

w(τ0,~r) = wBG (τ0, r)[1 + 2|w (2)

1 |J2

(k

(2)1 r)

cos(

2(ϕ− ψ(2)1 ))]

We will explore values of |w (2)1 | typical of central collisions

0.0 0.2 0.4 0.6 0.8 w

1H2L

w

1H2L*0

50

100

150

200Nevents

2000 events, b=2fm, m=2, l=1

0.0 0.2 0.4 0.6 0.8 1.0 1.2 w

1H2L

w

1H2L*0

20

40

60

80

100

120

Nevents

2000 events, b=4 fm, m=2, l=1

34 / 39



The setupResults

Initialization with a single-mode

We start considering the evolution of a single (m =2, l =1) mode on topof an average background

w(τ0,~r) = wBG (τ0, r)[1 + 2|w (2)

1 |J2

(k

(2)1 r)

cos(

2(ϕ− ψ(2)1 ))]

We will explore values of |w (2)1 | typical of central collisions

0.0 0.2 0.4 0.6 0.8 w

1H2L

w

1H2L*0

50

100

150

200Nevents

2000 events, b=2fm, m=2, l=1

0.0 0.2 0.4 0.6 0.8 1.0 1.2 w

1H2L

w

1H2L*0

20

40

60

80

100

120

Nevents

2000 events, b=4 fm, m=2, l=1

34 / 39



The setupResults

Single-mode (linear) evolution

0

20

40

60

80

100

120

140

x

y

enthalpy density (GeV/fm3) tau0

-15 -10 -5 0 5 10 15

-15

-10

-5

0

5

10

15

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x

y

enthalpy density (GeV/fm3) tau0+5 fm/c

-15 -10 -5 0 5 10 15

-15

-10

-5

0

5

10

15

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

x

y


-15 -10 -5 0 5 10 15

-15

-10

-5

0

5

10

15

We evolve an initial condition with w(2)1 =0.5 at τ =0.6 fm/c, with

η/s =0.08

After subtracting the background one can follow the evolution ofthe m =2 mode

Varying the weight of the initial perturbation and rescaling the

result by w(2)1 one can verify that the evolution is to very good

accuracy linear,

even for late times!

35 / 39



The setupResults


2 4 6 8 10 12 14r

0.01

0.1

1

10

100

wBGHΤ,rL @GeVfm3D

Τ0=.6 fmc

Τ=1.6 fmc

Τ=2.6 fmc

Τ=3.6 fmc

Τ=4.6 fmc

Τ=5.6 fmc

Τ=10.6 fmc

2 4 6 8 10 12r

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

w H2LHΤ,rL

Τ0=.6 fmc

Τ=1.6 fmc

Τ=2.6 fmc

Τ=3.6 fmc

Τ=4.6 fmc

Τ=5.6 fmc


η/s =0.08




accuracy linear,


35 / 39



The setupResults


2 4 6 8 10 12 14r

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

w H2LHΤ= Τ0+5 fmc,rL

w

1

H2L=0.6

w

1

H2L=0.5

w

1

H2L=0.4

w

1

H2L=0.25

w

1

H2L=0.1

2 4 6 8 10 12 14r

-0.1

0.0

0.1

0.2

0.3

0.4

w H2LHΤ= Τ0+5 fmc,rLw 1

H2L

w

1

H2L=0.6

w

1

H2L=0.5

w

1

H2L=0.4

w

1

H2L=0.25

w

1

H2L=0.1


η/s =0.08




accuracy linear,


35 / 39



The setupResults


5 10 15r

-0.2

-0.1

0.0

0.1

0.2

0.3

w H2LHΤ= Τ0+10 fmc,rL

w

1

H2L=0.6

w

1

H2L=0.5

w

1

H2L=0.4

w

1

H2L=0.25

w

1

H2L=0.1

5 10 15r

-0.4

-0.2

0.0

0.2

0.4

w H2LHΤ= Τ0+10 fmc,rLw 1

H2L

w

1

H2L=0.6

w

1

H2L=0.5

w

1

H2L=0.4

w

1

H2L=0.25

w

1

H2L=0.1


η/s =0.08




accuracy linear, even for late times!35 / 39



The setupResults

Single-mode evolution: non-linear effects

5 10 15r

-0.020

-0.015

-0.010

-0.005

0.000

wBGHΤ,rLw H0LHΤ,rL @GeVfm3D, Τ=Τ0+10 fmc

w

1

H2L=0.6

w

1

H2L=0.5

w

1

H2L=0.4

w

1

H2L=0.25

w

1

H2L=0.1

5 10 15r

-0.08

-0.06

-0.04

-0.02

0.00

wBGHΤ,rLw H0LHΤ,rLHw 1H2LL2

, Τ=Τ0+10 fmc

w

1

H2L=0.6

w

1

H2L=0.5

w

1

H2L=0.4

w

1

H2L=0.25

w

1

H2L=0.1

A m =0 mode arises at quadratic order in w(2)1 from δ0,2−2

A m =4 mode arises at quadratic order in w(2)1 from δ4,2+2

A m =6 mode arises at cubic order in w(2)1 from δ6,2+2+2

36 / 39



The setupResults


5 10 15r0.000

0.005

0.010

0.015

0.020

0.025


w

1

H2L=0.6

w

1

H2L=0.5

w

1

H2L=0.4

w

1

H2L=0.25

w

1

H2L=0.1

5 10 15r0.00

0.02

0.04

0.06

0.08


, Τ=Τ0+10 fmc

w

1

H2L=0.6

w

1

H2L=0.5

w

1

H2L=0.4

w

1

H2L=0.25

w

1

H2L=0.1




36 / 39



The setupResults


5 10 15r

-0.006

-0.004

-0.002

0.000


w

1

H2L=0.6

w

1

H2L=0.5

w

1

H2L=0.4

w

1

H2L=0.25

w

1

H2L=0.1

5 10 15r

-0.03

-0.02

-0.01

0.00

0.01

0.02


, Τ=Τ0+10 fmc

w

1

H2L=0.6

w

1

H2L=0.5

w

1

H2L=0.4

w

1

H2L=0.25

w

1

H2L=0.1




36 / 39



The setupResults

Interaction between different modes

We evolve and initial condition containing two modes, (m =2, l =2) and

(m =3, l =1), with all possible combinations of weights |w (2)2 |=0.1, 0.25

and |w (3)1 |=0.1, 0.25 and phases ψ

(2)2 =0 and ψ

(3)1 =−0.2.

0

20

40

60

80

100

120

140

x

y


-15 -10 -5 0 5 10 15

-15

-10

-5

0

5

10

15

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x

yenthalpy density (GeV/fm3) tau0+5 fm/c

-15 -10 -5 0 5 10 15

-15

-10

-5

0

5

10

15

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

x

y


-15 -10 -5 0 5 10 15

-15

-10

-5

0

5

10

15

m =1 (δ1,3−2) and m =5 (δ5,3+2) harmonics arise from the interference ofthe two initial modes

They display the expected scaling behavior

Their phases are consistent with the expectation 3ψ(3)1

37 / 39



The setupResults




and |w (3)1 |=0.1, 0.25 and phases ψ

(2)2 =0 and ψ

(3)1 =−0.2.

0 20 40 60 80 100 120 140 160

x

y


-15 -10 -5 0 5 10 15

-15

-10

-5

0

5

10

15

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x

yenthalpy density (GeV/fm3) tau0+5 fm/c

-15 -10 -5 0 5 10 15

-15

-10

-5

0

5

10

15

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

x

y


-15 -10 -5 0 5 10 15

-15

-10

-5

0

5

10

15




37 / 39



The setupResults




and |w (3)1 |=0.1, 0.25 and phases ψ

(2)2 =0 and ψ

(3)1 =−0.2.

5 10 15r

-0.006

-0.004

-0.002

0.000

0.002

0.004

0.006

wBGHΤ,rLw H1LHΤ,rLÈw 2H2L

w

1H3LÈ @GeVfm3D

Re

Im

2 4 6 8 10 12 14r

-0.010

-0.005

0.000

0.005

0.010

wBGHΤ,rLw H5LHΤ,rLÈw 2H2L

w

1H3LÈ @GeVfm3D

Re

Im




37 / 39



The setupResults




and |w (3)1 |=0.1, 0.25 and phases ψ

(2)2 =0 and ψ

(3)1 =−0.2.

2 4 6 8 10 12r

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

Arg@w H1LHΤ,rLD for ΨH2L=0.0, ΨH3L=-0.2

2 4 6 8 10 12r

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

Arg@w H5LHΤ,rLD for ΨH2L=0.0, ΨH3L=-0.2




37 / 39



The setupResults

Relevance for realistic initial conditions

Embed a single (m = 2, l = 1) mode (w(2)1 =0.5)

On top of the usual wBG

On top of wBG, but together with all other m 6=2 modes from arealistic Glauber-MC initialization

The assumption of a predominantly linear response on top of a suitablychosen background is applicable for realistic initial conditions that display

strong fluctuations

38 / 39



The setupResults






strong fluctuations

38 / 39



The setupResults





5 10 15

-0.2

-0.1

0.0

0.1

0.2

0.3

w H2LHΤ,rL

Τ=2.6 fmcΤ=5.6 fmc

Τ=10.6 fmc

single mode w

1

H2L=0.5

w

1

H2L=0.5 in fluctuating event


strong fluctuations

38 / 39



The setupResults

Conclusions

We have provides evidence from full numerical solutions that thehydrodynamical evolution of initial density fluctuations in heavy ioncollisions can be understood order-by-order in a perturbative series indeviations from a smooth and azimuthally symmetric backgroundsolution

to leading linear order, modes with different azimuthal wavenumbers do not mix

deviations from a linear response to the initial fluctuations can bequantitatively understood as quadratic and higher order corrections.

We plan to perform a more systematic study in the future

investigating the role of viscosity

extending the analysis to a wider set of centrality classes

39 / 39



The setupResults

Conclusions

We have provides evidence from full numerical solutions that thehydrodynamical evolution of initial density fluctuations in heavy ioncollisions can be understood order-by-order in a perturbative series indeviations from a smooth and azimuthally symmetric backgroundsolution

to leading linear order, modes with different azimuthal wavenumbers do not mix

deviations from a linear response to the initial fluctuations can bequantitatively understood as quadratic and higher order corrections.

We plan to perform a more systematic study in the future

investigating the role of viscosity

extending the analysis to a wider set of centrality classes

39 / 39

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How (non-)linear is the hydrodynamics of heavy ion …personalpages.to.infn.it/~beraudo/santiago.pdfHeavy-ion collisions: exploring the QCD phase-diagram QCD phases identi ed through