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Entropy analysis in and and collisions at GeV collisions at GeV

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(For the NA22 Collaboration)

22sp pK

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Motivation to study entropy.Procedure and variables which are used to

measure the entropy.The results from the NA22 experiment and the

PYTHIA Monte Carlo model.Conclusions and outlook.

OUTLINE

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The assumption of thermodynamic equilibrium is commonly used when discussing the system created in central heavy ion collisions:

TE

VNES

VN

1),,(

,

1. Motivation to study entropy

The real difficulty is the measurement of entropy. Here, we use the coincidence probability method proposed some time ago by Ma.

Before applying it to heavy ion collisions, we will test the sensitivity of this method by applying it to the more basic hadron-hadron collisions.

Temperature T: slope of the transverse momentum distribution

Number of particles N: direct measurement in the experiment

Energy E: measure the energies of all the particles produced

A.Białas, W.Czyż and J. Wosiek, Acta Phys. Pol. B 30,107(1999) A.Białas and W.Czyż, Phys. Rev. D 61,074021(2000)

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2. Procedure and variables of Ma’s coincidence probability method

The original definition of the standard (Shannon) entropy is:

j

jj ppS ln

Step 1: For every event, a certain phase space region in rapidity is divided into M bins of equal size.Then, an event is characterized by the number of particles falling into each bin ( ), i.e.,

, where i = 1, ……, M }{ ims

:sn Number of events belonging to the same configuration s.

im

Ma’s coincidence probability method :

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Step3: Coincidence probability of k configurations:

Step4: The Rényi entropy is then given by:

The Shannon entropy S is formally equal to the limit of

)1()1(

kNNN

NC k

k

1

ln

k

CH k

k

)1( kH k

).1()1( knnnN ss

ssk

Step2: Total numbers of observed coincidences of k configurations:

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sss

s

ks

kk

kppp

kHS ln)ln(

1

1limlim

11

s

ks

ss

s

s

s

sssssss

k

p

kN

kn

N

n

N

n

kNNN

knnn

kNNN

knnnC

1

1

1

1

)1()1(

)1()1(

)1()1(

)1()1(

If , with N

np s

s 1 kns

)ln(1

1ln

1

1

s

kskk p

kC

kH

Standard definitio

n

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1. Statistical error drops very fast with increasing number of configurations.

2. Monte Carlo shows the results are more stable for this method.

The advantages of Ma’s method:

（ K.Fiałkowski and R. Wit, Phys. Rev. D 62,114016(2000) )

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If a system is close to thermal equilibrium and phase space is divided small enough, scaling and additivity hold:

lMSlMSlMHlMH kk ln)()(ln)()(

Scaling:

Additivity:

)()()()()()( 2121 RSRSRSRHRHRH kkk

Phase space region 21 RRR

（ A.Białas and W.Czyż, Acta Phys. Pol. B 31,687(2000) )

Here M and lM are numbers of bins in two discretizations.

If scaling holds, one should observe a linear relation if is plotted as a function of , where .

)(MH k

yln Myy /

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It is suggested that the dependence of Rényi entropies on particle multiplicity N carries important information on the produced system.

( A.Białas and W.Czyż, Acta Phys. Pol. B 31,2803(2000) )

~ N

For an equilibrated system with no strong long-range correlations.

~ ln N

For a non-equilibrated system or a superposition of sub-systems with different properties.

)(MH k

)(MH k

3. The results from the NA22 experiment and PYTHIA

• Flattening (No scaling, no TE).

Strong correlations!

• PYTHIA does not fully agree with the data.

Discrepancy!

• Random model gives a nearly straight-line relation.

No correlations!

GeVspKp 22, Rényi entropy in the central y region

Rényi entropy in the non-central y region

• Flatten, no linear relationship

• PYTHIA overestimates the data in the peripheral regions Weak correlations in these regions!

• In two adjacent regions, a clear difference is observed! Strong correlations exist between two adjacent regions!

•In two non-adjacent regions, the difference is small! Weak correlations for widely separated rapidity intervals!

•PYTHIA still somewhat overestimates the data.

Test of additivity in y region

Multiplicity dependence of the Rényi entropy

|y| < 3

|y| < 2

|y| < 1

linear scale

• A logarithmic rather than a linear relation is observed here!

Consistent with no thermal equilibrium in this system!

• PYTHIA agrees quite well with the data here!

logarithmic scale

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4. Conclusions and outlook

Both the scaling and the additivity properties are not generally valid and the multiplicity dependence is logarithmic rather than linear. All of these confirm the expectation that thermal equilibrium is not reached in the hadron-hadron collisions at

PYTHIA Monte Carlo model, in general, agrees with the data. However, significant deviations exist. In particular, the model overestimates the values in the peripheral rapidity regions, presumably due to too weak correlations. This shows that Rényi entropies provide a sensitive measure of multiparticle correlations.

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RHIC has already collected data on gold-gold and pp collisions at 200GeV. It would be interesting to investigate the entropy properties on this high-temperature, high-density system, which may create the long expected QGP.

Our results presented here should provide a valuable guide to the interpretation of the future results from the high energy heavy ion collisions.

Appendix: NA22 data sample

1. The NA22 experiment made use of the European Hybrid Spectrometer (EHS) in combination with the Rapid Cycling Bubble Chamber (RCBC).

2. Charged-particle momenta are measured over the full solid angle with an average resolution varying from 1-2% for tracks reconstructed in RCBC and 1-2.5% for tracks reconstructed in the first lever arm, to 1.5% for tracks reconstructed in the full spectrometer. Ionization information is used to identify and exclude protons up to 1.2 GeV/c and electrons (positrons) up to 200 MeV/c. All unidentified tracks are given the pion mass.

3. In our analysis, events are accepted when the measured and reconstructed charged-particle multiplicity are the same, no electron is detected among the secondary tracks and the number of badly reconstructed tracks is 0. After all necessary rejections, a total of 44,524 inelastic, non-single-diffractive events is obtained. Acceptance losses are corrected by a multiplicity-dependent event weighting procedure.