Relativistic heavy-ion collisions: general introduction Collective flow and hydrodynamic behaviour: an overview The theory setup: relativistic hydrodynamics Mode-by-mode hydrodynamics How (non-)linear is the hydrodynamics of heavy ion collisions? Andrea Beraudo University of Santiago de Compostela and CERN Santiago de Compostela, 16 th January 2014 1 / 39
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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
Relativistic heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview
The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics
Relativistic heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview
The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics
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
Relativistic heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview
The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics
Heavy-ion collisions: a typical event
5 / 39
Relativistic heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview
The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics
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
Relativistic heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview
The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics
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!
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
Relativistic heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview
The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics
The general setupHydro predictionsEvent-by-event fluctuations and recent developments
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
Relativistic heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview
The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics
The general setupHydro predictionsEvent-by-event fluctuations and recent developments
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
Relativistic heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview
The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics
The general setupHydro predictionsEvent-by-event fluctuations and recent developments
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
Relativistic heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview
The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics
The general setupHydro predictionsEvent-by-event fluctuations and recent developments
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
Relativistic heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview
The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics
The general setupHydro predictionsEvent-by-event fluctuations and recent developments
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
Relativistic heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview
The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics
The general setupHydro predictionsEvent-by-event fluctuations and recent developments
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
Relativistic heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview
The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics
The general setupHydro predictionsEvent-by-event fluctuations and recent developments
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
Relativistic heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview
The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics
The general setupHydro predictionsEvent-by-event fluctuations and recent developments
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
Relativistic heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview
The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics
The general setupHydro predictionsEvent-by-event fluctuations and recent developments
Elliptic flow: mass ordering
The mass ordering of v2 is a direct consequence of the hydro expansion
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)
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
Relativistic heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview
The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics
The general setupHydro predictionsEvent-by-event fluctuations and recent developments
Elliptic flow: mass ordering
The mass ordering of v2 is a direct consequence of the hydro expansion
[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
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
Relativistic heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview
The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics
The general setupHydro predictionsEvent-by-event fluctuations and recent developments
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
Relativistic heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview
The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics
The general setupHydro predictionsEvent-by-event fluctuations and recent developments
Relativistic heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview
The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics
The general setupHydro predictionsEvent-by-event fluctuations and recent developments
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
Relativistic heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview
The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics
The general setupHydro predictionsEvent-by-event fluctuations and recent developments
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
Relativistic heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview
The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics
The general setupHydro predictionsEvent-by-event fluctuations and recent developments
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
Relativistic heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview
The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics
The general setupHydro predictionsEvent-by-event fluctuations and recent developments
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
Relativistic heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview
The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics
The general setupHydro predictionsEvent-by-event fluctuations and recent developments
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
Relativistic heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview
The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics
The general setupHydro predictionsEvent-by-event fluctuations and recent developments
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
Relativistic heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview
The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics
The general setupHydro predictionsEvent-by-event fluctuations and recent developments
Flow vs eccentricity
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)
(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
Relativistic heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview
The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics
The general setupHydro predictionsEvent-by-event fluctuations and recent developments
Flow vs eccentricity
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)
(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
Relativistic heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview
The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics
The general setupHydro predictionsEvent-by-event fluctuations and recent developments
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
Relativistic heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview
The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics
The general setupHydro predictionsEvent-by-event fluctuations and recent developments
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
=2.76 TeVNNs-1bµ = 7 intL
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
=2.76 TeVNNs-1bµ = 7 intL
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
Relativistic heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview
The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics
Ideal RHDViscous RHD
Relativistic hydrodynamics
22 / 39
Relativistic heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview
The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics
Ideal RHDViscous RHD
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
Relativistic heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview
The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics
Ideal RHDViscous RHD
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
Relativistic heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview
The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics
Ideal RHDViscous RHD
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
Relativistic heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview
The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics
Ideal RHDViscous RHD
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
Relativistic heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview
The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics
Ideal RHDViscous RHD
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
Relativistic heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview
The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics
Ideal RHDViscous RHD
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
Relativistic heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview
The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics
Ideal RHDViscous RHD
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
Relativistic heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview
The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics
Ideal RHDViscous RHD
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 heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview
The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics
Ideal RHDViscous RHD
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 heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview
The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics
Ideal RHDViscous RHD
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
Relativistic heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview
The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics
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
Relativistic heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview
The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics
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
Relativistic heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview
The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics
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
Relativistic heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview
The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics
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
Relativistic heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview
The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics
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) =
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)
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
Relativistic heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview
The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics
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
Relativistic heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview
The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics
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
Relativistic heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview
The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics
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
Relativistic heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview
The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics
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
Relativistic heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview
The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics
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
Relativistic heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview
The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics
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
Relativistic heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview
The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics
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
Relativistic heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview
The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics
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
Relativistic heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview
The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics
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
Relativistic heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview
The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics
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
Relativistic heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview
The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics
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
enthalpy density (GeV/fm3) tau0+10 fm/c
-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
Relativistic heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview
The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics
The setupResults
Single-mode (linear) evolution
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
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
Relativistic heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview
The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics
The setupResults
Single-mode (linear) evolution
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
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
Relativistic heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview
The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics
The setupResults
Single-mode (linear) evolution
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
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
Relativistic heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview
The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics
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
Relativistic heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview
The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics
The setupResults
Single-mode evolution: non-linear effects
5 10 15r0.000
0.005
0.010
0.015
0.020
0.025
wBGHΤ,rLw H4LHΤ,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 15r0.00
0.02
0.04
0.06
0.08
wBGHΤ,rLw H4LHΤ,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
Relativistic heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview
The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics
The setupResults
Single-mode evolution: non-linear effects
5 10 15r
-0.006
-0.004
-0.002
0.000
wBGHΤ,rLw H6LHΤ,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
-0.03
-0.02
-0.01
0.00
0.01
0.02
wBGHΤ,rLw H6LHΤ,rLHw 1H2LL3
, 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
Relativistic heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview
The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics
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
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
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
enthalpy density (GeV/fm3) tau0+10 fm/c
-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
Relativistic heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview
The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics
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 80 100 120 140 160
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
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
enthalpy density (GeV/fm3) tau0+10 fm/c
-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
Relativistic heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview
The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics
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.
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
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
Relativistic heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview
The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics
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.
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
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
Relativistic heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview
The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics
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
Relativistic heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview
The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics
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
Relativistic heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview
The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics
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
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
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
Relativistic heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview
The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics
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
Relativistic heavy-ion collisions: general introductionCollective flow and hydrodynamic behaviour: an overview
The theory setup: relativistic hydrodynamicsMode-by-mode hydrodynamics
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