hydrodynamic tests of fluctuating initial conditions
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Hydrodynamic Tests of Fluctuating Initial Conditions
George Moschelli&Hannu Holopainen
Transport Meeting24 January 2012
Motivation
MC-KLN: Drescher, Nara, nucl-th/0611017
IP-Glasma: Schenke, Tribedy, Venugopalan,arXiv:1206.6805, Phys. Rev. Lett. 108 (2012) 252301
Fluctuating Initial Conditions and Event-by-Event Studies
• Local Correlations
• Global Correlations
• Geometry Fluctuations
Local CorrelationsInitial State Configuration Final State Momentum
Final state momenta are correlated to initial position. • Reaction / event plane• Common origin
Influence of fluctuating ICs• Arbitrary event shapes.• Random number of sources
and source sizes.
Goal: Determine hydro response to “common origin” correlations and dependence on choice of IC.
Global Correlations
E-by-E Hydro Evolution• Ideal Hydro• Lattice EoS• Gaussian Energy Density
lumps at mixture of MC Glauber Nbin and Npart positions
• Gaussian width: 0.4 fm
Goal: Trace the evolution of fluid element correlations to freeze out.
Global Correlations
E-by-E Hydro Evolution• Ideal Hydro• Lattice EoS• Gaussian Energy Density
lumps at mixture of MC Glauber Nbin and Npart positions
• Gaussian width: 0.4 fm
Goal: Trace the evolution of fluid element correlations to freeze out.
Flow LinesSpace Velocity
• Dots at initial positions of binary collisions• Movement indicates fluid cell position and velocity • Black line: const*e2
• Blue line: const*e3
• Green dots: randomly chosen group within 0.4 fm radius
• 20-30% centrality• Nbin = 464• Npart = 176• Freeze out: T = 120 MeV
Flow LinesSpace Velocity
• 20-30% centrality• Nbin = 464• Npart = 176• Freeze out: T = 120 MeV
• Dots at initial positions of binary collisions• Movement indicates fluid cell position and velocity • Black line: const*e2
• Blue line: const*e3
• Green dots: randomly chosen group within 0.4 fm radius
Fluid-Fluid Correlations
1-p
yx
yx
,Cov
• “Emission” angle corresponds to initial spatial angle. Expectation: central (circular) collisions agree, peripheral (elliptical) collisions should deviate
• Faster dots have larger displacement
• Final velocity depends on initial position. → Angular correlations!
• Faster dots freeze out first
• Need mixed events
Average Displacement
r0,min
r0,max
• Larger average displacement in central collisions
• central collisions live longer • greater effect on common origin
correlations than vn
• Linear correlation between r0, Dr, and vFO
• Flow lines starting at different radial positions get different transverse push.
• Enhances common source correlations
• Changes <en>time
Goal: Determine a source “resolution”.
Freeze Out Time• Faster dots freeze out first• Blue: Event average 20-30% centrality• Red: single event with 464 Flow Lines
• Average flow line lifetime longest in most central collisions
Freeze Out Time
• Freeze out histograms indicate the flux of flow lines through the freeze out surface at different times.
Freeze out and Event Planes
rwnrw n
n
e
cos
nnrw
nrwnn
cossin
arctan1
Alvioli, Holopainen, Eskola, Strikman arXiv:1112.5306
Space Velocity
n = 1 w(r) = r3
n = 2 w(r) = r2
n = 3 w(r) = r3
e2
• Difference in initial eccentricities due to Glauber mixture IC vs. Nbin Flow Lines
• Freeze out changes initial and final eccentricity
• Freeze out velocity eccentricity represent a “time averaged” freeze out surface
• Final eccentricity agrees with freeze out velocity eccentricity
Goal: Study IC structure impact on time averaged velocity eccentricity.
e3
• Difference in initial eccentricities due to Glauber mixture IC vs. Nbin Flow Lines
• Freeze out changes initial and final eccentricity
• Freeze out velocity eccentricity represent a “time averaged” freeze out surface
• Final eccentricity agrees with freeze out velocity eccentricity
Goal: Study IC structure impact on time averaged velocity eccentricity.
en Distributions
Cartesian Space
Velocity Space#
Even
ts#
Even
ts
Fluctuations can differentiate initial conditions
Multiplicity Fluctuations
Fluctuations per source
Fluctuations in the number of sources
For K sources that fluctuate per event
KK
KK
K11
2
22
2
2
R
Negative binomial distribution 1 NBDkR
Schenke, Tribedy, Venugopalan, arXiv:1206.6805, Phys. Rev. Lett. 108 (2012) 252301
Gelis, Lappi, McLerran Nucl.Phys. A828, 149 (2009)
Gavin, Moschelli Phys.Rev. C79, 051902 (2009)
Negative Binomial Distribution
Fluctuations per source
Fluctuations in the number of sources
For K sources that fluctuate per event
KK
KK
K11
2
22
2
2
R
Negative binomial distribution 1 NBDkR
Schenke, Tribedy, Venugopalan, arXiv:1206.6805, Phys. Rev. Lett. 108 (2012) 252301
Gelis, Lappi, McLerran Nucl.Phys. A828, 149 (2009)
Gavin, Moschelli Phys.Rev. C79, 051902 (2009)
NBD put in by hand
Fluctuations and Correlationscorrelations = pairs - singles2
211121221 ,, pppppp r
R222121 1, NNNNddr pppp
ttt ppp
2121
2121 1, pppp dd
NNrpppp tttt
D
21
2122
2 cos12
,2
42 pppp ddnNN
rvv nnn
Multiplicity Fluctuations
Momentum Fluctuations
“Flow Fluctuations”
Gavin, Moschellinucl-th/1107.3317nucl-th/1205.1218
The next stepIC lumps from K random sources• Poisson flow line multiplicity per source
• Compare large <K> and small source size to small <K> and large source size
• Compare to “smooth” hydro
Angular Correlations
• Compare en and vn with different IC
• Radial cuts
• Momentum, vn (eccentricity) and vn{2}2-vn{4}2 fluctuations
Mixed Events• With and without aligned reaction / event planes
Summary
Can we use hydro select the right IC?
• Determine hydro response to “common origin” correlations and dependence on choice of IC.
• Trace the evolution of fluid element correlations to freeze out.
• Determine a source “resolution”.
• Study IC structure impact on time averaged velocity eccentricity.
Freeze out effects• Eccentricity fluctuations
• Event plane angle determination
Cumulant Expansion
212111212 ,, pppppp r
222 22 nnn vv
Pair Distribution:
Two-particle coefficient:
Correlated Part:
Borghini, Dinh, Ollitrault
vn factorization is a signature of flow if n = 0
• <vn>2 = reaction plane correlations
• 2n = other correlations
• vn{4} <vn>
Borghini, Dinh, Ollitrault;Voloshin, Poskanzer, Tang, Wang
D
21
2122
2 cos12
,2
42 pppp ddnNN
rvv nnn
The Soft Ridge
• Only cos D and cos 2D terms subtracted
•These terms also contain fluctuations
•Glasma energy dependence•R scale factor set in
Au-Au 200 GeV•Blast wave f (p,x)•Difference in peripheral
STAR→ALICE
refn
nref
rdydNnv
dydN
D
D
DD
21cos
22
1
2
Flow subtracted ridge
D
DD
ndydN
refn cos2 2
Four-Particle Coefficients
44321
44 cos224 nnn vnvv
4321
432111
4131211143214
,,
,
,,,
pppp
pppp
pppppppp
rr
r
42244321 24cos nnnn vvn
Voloshin, Poskanzer, Tang, WangBorghini, Dinh, and Ollitrault
Four-particle coefficient:
Four-Particle Distribution: keep only two-particle correlations
222244 Re24 nnnnn vvv
22224224
4321
Re224
cos
nnnnnnn vvv
n
vn{4} corrections
21
212 2cos1
,Re pppp ddnNN
rRPn
221
Four-particle coefficient:
Will cancel with vn{2} terms
Corrections of order ~1.2%
R• K flux tubes, assume • K varies event-by-event
Fluctuations per source
Fluctuations in the number of sources
For K sources that fluctuate per event
KK
KK
K11
2
22
2
2
R
KNK
KNNKK
222
KN
222222 KKKNN
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