Fluctuations of hpTifrom initial size ﬂuctuationstheorie.ikp.physik.tu-darmstadt.de/.../Wed/Chojnacki.pdf1based on: Wojciech Broniowski, MCh, Łukasz Obara; Phys. Rev. C80 (2009)

International Workshop XXXVIII Hirschegg

Fluctuations of 〈pT 〉 from initial sizefluctuations 1

Mikołaj ChojnackiInstitute of Nuclear PhysicsPolish Academy of SciencesKraków, Poland

Hirschegg 2010:Strongly Interacting Matter under Extreme Conditions

17-23 January 2010

1based on: Wojciech Broniowski, MCh, Łukasz Obara; Phys. Rev. C80 (2009) 051902Mikołaj Chojnacki (IFJ PAN) pT fluctuations 5 December 2009 1 / 18

Motivation Size fluctuations

MotivationSize fluctuations of the initial conditions

Events with the same number of wounded nucleons Nw may have differentshape and size.

〈r〉 = 2.95 fm 〈r〉 = 2.83 fm

Two examples2 of non-central 197Au + 197Au collision with Nw = 198.

smaller size→ larger gradients→ larger hydrodynamic flow→→ larger pT ( and vice versa )

2GLISSANDO WB, M. Rybczynski, P. Bozek, Comput. Phys. Commun. 180 (2009) 69Mikołaj Chojnacki (IFJ PAN) pT fluctuations 5 December 2009 2 / 18

Motivation Size fluctuations

MotivationSize fluctuations of the initial conditions

Events with the same number of wounded nucleons Nw may have differentshape and size.

〈r〉 = 2.95 fm 〈r〉 = 2.83 fm

Two examples2 of non-central 197Au + 197Au collision with Nw = 198.

smaller size→ larger gradients→ larger hydrodynamic flow→→ larger pT ( and vice versa )

2GLISSANDO WB, M. Rybczynski, P. Bozek, Comput. Phys. Commun. 180 (2009) 69Mikołaj Chojnacki (IFJ PAN) pT fluctuations 5 December 2009 2 / 18

Event-by-event fluctuations average size fluctuations

Event-by-event fluctuationsaverage size fluctuations

average of the transverse size in agiven event

〈r〉 =

NwXi=1

qx2

i + y2i

e-by-e average of transverse size

〈〈r〉〉 =1

Nevents

NeventsXk=1

〈r〉k

convenient measure — scaled standard deviation for set Nw

σscaled =σ (〈r〉)〈〈r〉〉

In the wounded nucleon model the σscaled is insensitive to σNN .is caused by thedifferent admixture of the binary collisions profile which is much more sensitive tofluctuations

Mikołaj Chojnacki (IFJ PAN) pT fluctuations 5 December 2009 3 / 18




〈r〉 =

NwXi=1

qx2

i + y2i


〈〈r〉〉 =1

Nevents

NeventsXk=1

〈r〉k



In the wounded nucleon model the σscaled is insensitive to σNN .is caused by thedifferent admixture of the binary collisions profile which is much more sensitive tofluctuations





〈r〉 =

NwXi=1

qx2

i + y2i


〈〈r〉〉 =1

Nevents

NeventsXk=1

〈r〉k



In the mixed model (α2 Nw + (1− α) Nbin) a moderate change with σNN is caused by thedifferent admixture of the binary collisions profile which is much more sensitive tofluctuations.


Event-by-event fluctuations fluctuating initial conditions

e-by-e hydrodynamicsfluctuating initial conditions

Instead of 100 000 events, two are enough!

Size of the initial condition for hydrodynamics (energy density profile) is scaledup and down according to the scaled variance.No e-by-e energy fluctuations (could be included).

initial central temperature is changed from 455 MeV to 466 MeV (squeezed) or 445 MeV

(stretched) profile → total energy is the same.




Instead of 100 000 events, two are enough!Size of the initial condition for hydrodynamics (energy density profile) is scaledup and down according to the scaled variance.No e-by-e energy fluctuations (could be included).






Instead of 100 000 events, two are enough!Size of the initial condition for hydrodynamics (energy density profile) is scaledup and down according to the scaled variance.No e-by-e energy fluctuations (could be included).




Results distributions of < r > and < pT >

Resultsdistributions of 〈r〉 and 〈pT 〉

The distribution of the 〈r〉 is approximately Gaussian

f (〈r〉) ∼ exp„− (〈r〉 − 〈〈r〉〉)2

2σ2(〈r〉)

«Imagine we ran simulations with fixed 〈r〉 (no size fluctuations). Then particles would havesome average momentum pT

Since hydrodynamic evolution is deterministic, pT is a (very complicated)function of 〈r〉.Now let us include fluctuations of 〈r〉. We can use Taylor expansion

pT − 〈〈pT 〉〉 =dpT

d〈r〉

˛〈r〉=〈〈r〉〉

(〈r〉 − 〈〈r〉〉) + . . .

The statistical distribution of 〈pT 〉 is

f (pT ) ∼ exp

0B@− (pT − 〈〈pT 〉〉)2

2σ2(〈r〉)“

dpTd〈r〉

”2

1CAMikołaj Chojnacki (IFJ PAN) pT fluctuations 5 December 2009 6 / 18

Results scaled variance of < pT >

Resultsscaled variance of 〈pT 〉

The full statistical distribution f (〈pT 〉) in a given centrality class is a folding of thestatistical distribution of 〈pT 〉 at a fixed initial size, centered around a certain pT ,with the distribution of pT centered around 〈〈pT 〉〉.

f (〈pT 〉) ∼Z

d2pT exp„− (〈pT 〉 − pT )2

2σ2stat

«exp

− (pT − 〈〈pT 〉〉)2

2σ2dyn

!

∼ exp

0@− (〈pT 〉 − 〈〈pT 〉〉)2

2“σ2

stat + σ2dyn

”1A

where σdyn (〈pT 〉) = σ(〈r〉) dpTd〈r〉

˛〈r〉=〈〈r〉〉

is extracted by the experimentalists.

Scaled dynamical variance

σdyn

〈〈pT 〉〉=σ(〈r〉)〈〈r〉〉

〈〈r〉〉〈〈pT 〉〉

dpT

d〈r〉

˛〈r〉=〈〈r〉〉


Results comparison with experiments

Resultscomparison with STAR data

STAR vs hydro

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STAR � sNN = 200, 130, 62, 20 GeV

viscosity vs perfect fluid

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´ 2+1D, Gaussian mix

Æ 3+1D, Glauber mix, HBozekL

scaled variation for 2+1 boost invariant hydro with bulk&shear viscosity, Glaubermixed IC (red doted circles) by Piotr Bozekperfect hydro 2+1 B-I and 3+1, Glauber mixed IC (blue symbols)overal amazing agreement when viscosity is introduced!perfect hydro mixed model overshoots data by 20%approximate scaling σdyn/〈〈pT 〉〉 ∼ 1/

√NW holds


Results connection to the EoS

Resultsconnection to the EoS3

scaled variance of 〈pT 〉 is connected to thermodynamics

σdyn

〈〈pT 〉〉=

Pε

σ(〈s〉)〈〈s〉〉 = 2

Pε

σ(〈r〉)〈〈r〉〉

where s is the entropy density, ε energy density, and P the pressure

0 100 200 300 400 5000.00

0.05

0.10

0.15

0.20

0.25

0.30

T @MeVD

P�¶

We can study this way the average properties of the equation-of-state i.e. itsstiffness

3Jean-Yves Ollitrault, Phys. Lett. B 273 (1991)Mikołaj Chojnacki (IFJ PAN) pT fluctuations 5 December 2009 9 / 18

Results connection to the EoS

Resultsconnection to the EoS3

scaled variance of 〈pT 〉 is connected to thermodynamics

σdyn

〈〈pT 〉〉=

Pε

σ(〈s〉)〈〈s〉〉 = 2

Pε

σ(〈r〉)〈〈r〉〉

where s is the entropy density, ε energy density, and P the pressure

0 100 200 300 400 5000.00

0.05

0.10

0.15

0.20

0.25

0.30

T @MeVD

P�¶

We can study this way the average properties of the equation-of-state i.e. itsstiffness

3Jean-Yves Ollitrault, Phys. Lett. B 273 (1991)Mikołaj Chojnacki (IFJ PAN) pT fluctuations 5 December 2009 9 / 18

Conclusions

Conclusions

a few percent fluctuations at the initial size of the collision explainsthe bulk of the experimental 〈pT 〉 fluctuations

viscosity lowers the fluctuations by about 20%, which helps to goexactly through the data (perfect hydro gives a bit too much)

proper scaling with the number of wounded nucleonsσdyn/〈〈pT 〉〉 ∼ 1/

√NW – proper dependence on centrality

a weak dependence on energy

our 〈pT 〉 fluctuations should be considered as the main geometricbackground for studying further effects like: (mini) jets, clusters,temperature fluctuations, etc.

average information on P/ε according to Ollitrault’s formula


Backup slides

backup slides


Bozek viscous hydrodynamics

Viscous 2+1 B-I hydrodynamicsby Piotr Bozek

[GeV/c]T

p0 0.5 1 1.5 2 2.5 3

]-2

dy)

[GeV

T d

pT

pπdN

/(2

-910

-610

-310

1

310

id. fl.vQGPvHGvQGP+vHG

+π0-5%

)-6 10× (40-50%

4

6c=0-5%

Ro

ut

[fm

]

4

6

Rsi

de

[fm

]

vHG TF=130MeV

vQGP TF=150MeV

0

10id. fl. TF=140MeV

vQGP+vHG TF=135MeV

Rlo

ng

[fm

]

0.5

1

1.5

0.2 0.3 0.4 0.5

Ro

ut/R

sid

e

kT [GeV/c]


Bozek viscous hydrodynamics

Viscous 2+1 B-I hydrodynamicsby Piotr Bozek

-m [GeV/c]Tm0 0.5 1 1.5 2

0

0.2

0.4

0.6

0.8

(10-20%)+πp (20-40%)

(10-40%)s0K

(10-40%)Λ+Λ (10-40%)Ξ+Ξ

∈/2v

b) vQGP+vHG

shear viscosity in QGP & HG η/s = 0.1bulk viscosity only in HG with ζ/s = 0.04− 0.03


Results comparison with experiments

Resultscomparison with PHENIX data

+++++ + + +

++

+++++ +

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0 50 100 150 200 250 300 3500

1

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NW

Σdy

n�<<

p T>>@%D

2+1D B-I perfect fluid hydrodynamics with wounded nucleon IC (blue crosses)and with the mixed model IC (red crosses.)


Results GLauber Initial-State Simulation AND mOre

GLISSANDOGLauber Initial-State Simulation AND mOre

The algorithm:nucleon positions generated according to the Woods-Saxon distribution,

a short-range repulsion is simulated by keeping the distance (d ≥ 0.4 fm)between the nucleons,

-10 -5 5 10x

-6

-4

-2

2

4

6

8

y

Overlapping nucleons in the transverse plane


Results Nuclear density profiles

GLISSANDONuclear density profiles

Nucleons interact if the distance d =pσNN/π.

Three models for constructing the nuclear density profile are consider:

Wounded Nucleons [Bialas, Bleszynski, Czyz, 1976],

Binary Collisions,

mixture of the two above, where α is the fraction of the binary collisions taken.

The inelastic cross-section σNN varies from 32 mb (SPS), 42 mb (RHIC) to 63 mb atthe LHC.

-7.5 -5 -2.5 2.5 5 7.5x

-7.5

-5

-2.5

2.5

5

7.5

y

Wounded Nucleons

-7.5 -5 -2.5 2.5 5 7.5x

-5

-2.5

2.5

5

7.5

y

Binary Collisions


Results Initial condition

Hydrodynamics with statistical hadronizationInitial condition

initial transverse energy density profile — Gaussian fit to GLISSANDO

ε(x , y) = ε0(Ti ) exp„− x2

2a2 −y2

2b2

«

parameters a, b and Ti depend on centrality,

eccentricity fluctuations are included,

a and b are fitted to reproduce the GLISSANDO’s 〈x〉 and 〈y〉,Ti is fitted to reproduce the correct particle multiplicity

c [%] 0-5 5-10 10-20 20-30 30-40 40-50 50-60 60-70a [fm] 2.70 2.54 2.38 2.00 1.77 1.58 1.40 1.22b [fm] 2.93 2.85 2.74 2.59 2.45 2.31 2.16 2.02Ti [MeV] 500 491 476 455 429 398 354 279


Results average transverse momentum

Resultsaverage transverse momentum

event-averaged transversed momentum 〈〈pT 〉〉

0 50 100 150 200 250 300 3500

100

200

300

400

500

600

NW

<<

p T>>@M

eVD

solid line: averaged over whole pT range,dashed line: STAR cuts 0.2 GeV < pT < 2 GeV

experimental points from STAR Collaboration Phys. Rev. C 79, 034909 (2009)extrapolated to full pT range


Documents

Fluctuations of hpTifrom initial size ﬂuctuationstheorie.ikp.physik.tu-darmstadt.de/.../Wed/Chojnacki.pdf1based on: Wojciech Broniowski, MCh, Łukasz Obara; Phys. Rev. C80 (2009)