how to measure specific heat using event-by-event average p t fluctuations
DESCRIPTION
How to Measure Specific Heat Using Event-by-Event Average p T Fluctuations. M. J. Tannenbaum Brookhaven National Laboratory Upton, NY 11973 USA. PHENIX Collaboration. - PowerPoint PPT PresentationTRANSCRIPT
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How to Measure Specific Heat How to Measure Specific Heat Using Event-by-Event Using Event-by-Event
Average pAverage pTT Fluctuations Fluctuations
M. J. TannenbaumBrookhaven National Laboratory
Upton, NY 11973 USA
Division of Nuclear Physics Meeting 2005 Maui, Hawaii September 20, 2005
PHENIX Collaboration
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Something New: cSomething New: cVV/T/T33
• R. Gavai, S.Gupta and S. Mukherjee, hep-lat/0412036, PRD 71, 074013 (2005) predict in “quenched QCD” at 2Tc and 3Tc that cV/T3 differs significantly from the ideal gas. Can this be measured?
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It is generally agreed that It is generally agreed that ccVV is related to temperature fluctuations is related to temperature fluctuations
• Ntot is the total number of particles on an event
• the ``temperature’’ T varies event-by-event with T and T.
• R.Korus, St.Mrowczynski, M.Rybczynski, Z.Wlodarczyk, PRC 64, 054908 (2004).
• Also see: L. Stodolsky, PRL 75, 1044 (1995) , S.A.Voloshin, V.Koch, H.G.Ritter, PRC 60, 024901 (1999).
• For a more nuanced (i.e. complicated) view see M. Stephanov, K. Rajagopal, E. Shuryak, PRD 60, 114028 (1999)
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p < 2
p=2p > 2
dN/x
dx
x =
Inclusive pInclusive pTT spectra are Gamma Distributions spectra are Gamma Distributions note: dN/xdx is shown. p shown is for dN/dx
• This is inclusive, averaged over all events. T=1/b is the Temperature parameter.
x =pT
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Event-by-Event Average pEvent-by-Event Average pTT
• If all the pTi on all events are random samples of the same distribution:
• For events with n charged particles of transverse momentum pTi, the event-by-event average transverse momentum is defined:
•By its definition <MpT>=<pT> but you must work hard to make sure that your data have this property to <<< 1%.
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What e-by-e tells you that you don’t What e-by-e tells you that you don’t learn from the inclusive averagelearn from the inclusive average
• A nice example I like is by R.Korus, St.Mrowczynski, M.Rybczynski, Z.Wlodarczyk, PRC 64, 054908 (2004).
• Suppose the temperature T~1/b varies event-by-event with T and T:
• Also see: L. Stodolsky, PRL 75, 1044 (1995) , S.A.Voloshin, V.Koch, H.G.Ritter, PRC 60, 024901 (1999).
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PHENIX MpPHENIX MpTT vs centrality vs centrality
200 GeV Au+Au 200 GeV Au+Au PRL PRL 9393, 092301 (04), 092301 (04)
• compare Data to Mixed events for random.
• Must use exactly the same n distribution for data and mixed events and match inclusive <pT> to <MpT>
• best fit of real to mixed is statistically unacceptable
• deviation expressed as:
FpT= MpTdata / MpTmixed -1 ~ few %
MpT (GeV/c)
MpT (GeV/c)
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Measures of non-random fluctuationsMeasures of non-random fluctuations
random means pT of all particles on all events are independent samples of the inclusive pT distribution (averaged over all events). non-random is the difference between measured and random.
PHENIX
STAR
CERES
NA49 Mroczynski
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For small non-random/randomFor small non-random/randomAll measures are equivalentAll measures are equivalent
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Fluctuation is a few percent of Fluctuation is a few percent of MpMpT T ::
Interesting variation with N Interesting variation with Npartpart and p and pTmax Tmax
n >3 0.2 < pT < 2.0 GeV/c 0.2 GeV/c < pT < pTmax
PHENIX nucl-ex/0310005 PRL PRL 9393, 092301 (2004), 092301 (2004)
Errors are totally systematic from run-run r.m.s variations
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Npart and pNpart and pTTmaxmax dependences explained by jet dependences explained by jet
correlations with measured jet suppressioncorrelations with measured jet suppression
20-25% centrality
Other explanations proposed include percolation of color strings E.G.Ferreiro, et al, PRC69, 034901 (2004)
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Assuming all fluctuations are from Assuming all fluctuations are from TT//TT Very small and relatively constant with Very small and relatively constant with ssNNNN
T/T
CERES tabulation H.Sako, et al, JPG 30, S1371 (04) QM2004
Where is the critical point?
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22TT//TT22Specific HeatSpecific Heat
• Korus, et al, PRC 64, 054908 (2001) discuss specific heat:
• n represents the measured particles while Ntot is all the particles, so n/Ntot is a simple geometrical factor for each experiment
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Something New: cSomething New: cVV/T/T33
• Gavai, et al, hep-lat/0412036 call this same quantity cV/T3
• These formulas with cvcV/T3 agree with Eq 9. in Korus, et al.
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Something New: cSomething New: cVV/T/T33
• R. Gavai, S.Gupta and S. Mukherjee, hep-lat/0412036, PRD 71, 074013 (2005) predict in “quenched QCD” at 2Tc and 3Tc that cV/T3 differs significantly from the ideal gas. Can this be measured?
• In PHENIX publication, PRL PRL 9393, 092301 (2004), 092301 (2004), n/Ntot~1/33, so FpT
~ 0.2% for cV/T3~15. This may be possible if we go to low pTmax out of the region where jets contribute.
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Worth TryingWorth Trying
0.2 GeV/c < pT < pTmax
ccVV/T/T33~15~15
PRL PRL 9393, 092301 (2004), 092301 (2004)
• Concentrate on pTmax < 0.6 GeV/c where jets have least effect
• Error is totally systematic---run by run variation---can be improved.
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IssuesIssues
• at fixed centrality, test for is FpT n, i.e. independent of n. Increase n by increasing solid angle, e.g. PHENIX vs STAR.
€
T2 / T
2
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What We Have LearnedWhat We Have Learned• In central heavy ion collisions, the huge correlations in p-p collisions are washed out. The remaining correlations are:
Jets Bose-Einstein correlations Hydrodynamic flow
• These correlations saturate the fluctuation measurements. No other sources of non-random fluctuations are observed. This puts a severe constraint on the critical fluctuations that were expected for a sharp phase transition but is consistent with the present expectation from lattice QCD that the transition is a smooth crossover.
• In order to see temperature fluctuations predicted by cV/T3~15 in lattice gauge calculations, present sensitivity needs to be improved by an order of magnitude by removing other known sources of correlation and improving the measurement errors.
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Some DetailsSome Details
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Statistics--What you have to rememberStatistics--What you have to remember
• The mean and standard deviation of an average of n independent trials from the same population obey the rules:
where is the mean and x (or ) is the standard deviation of the population x .
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Variance of MpVariance of MpTT
• If all the pTi are random samples of the same distribution:
• If the pTi are correlated, we find an identity with the pT-pT correlator variable of Voloshin, et al PRC 60, 024901 (1999)
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A)---Detail of MpA)---Detail of MpTT Calculation Calculation
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B)--Variance of MpB)--Variance of MpTT for T fluctuation for T fluctuation
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<N<Ntottot>/<n> for PHENIX>/<n> for PHENIX
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BACKUPBACKUP
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NA49-First Measurement of MpNA49-First Measurement of MpTT distribution distribution
NA49 Pb+Pb central measurement PLB 459, 679 (1999)
• Points=data; hist=mixed; minimal, if any, difference
• Very nice paper, gives all the relevant information
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Statistics at Work--Analytical Formula for MpStatistics at Work--Analytical Formula for MpT T for statistically independent Emissionfor statistically independent Emission
It depends on the 4 semi-inclusive parameters: b, p of the pT distribution (Gamma) <n>, 1/k (NBD), which are derived from the quoted means and standard deviations of the semi-inclusive pT and multiplicity distributions. The result is in excellent agreement with the NA49 Pb+Pb central measurement PLB 459, 679 (1999)
See M.J.Tannenbaum PLB 498, 29 (2001)
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0-5 % Centrality
Black Points = Data
Blue curve = Gamma distribution derived from inclusive pT spectra
It’s not a Gaussian…it’s a Gamma distribution!
“ “Average pAverage pTT Fluctuations” Fluctuations”
PHENIX
From one of Jeff Mitchell’s talks:
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Mortadella-NYTimes 2/10/2000Mortadella-NYTimes 2/10/2000