2. , CZE ISMD2005 Zhiming LI 11/08/2005 1 and collisions at GeV Entropy analysis in and collisions at GeV Zhiming LI (For the NA22 Collaboration)

3. , CZE ISMD2005 Zhiming LI 11/08/2005 2  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

4. , CZE ISMD2005 Zhiming LI 11/08/2005 3 The assumption of thermodynamic equilibrium is commonly used when discussing the system created in central heavy ion collisions: 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)

5. , CZE ISMD2005 Zhiming LI 11/08/2005 4 2. Procedure and variables of Ma’s coincidence probability method The original definition of the standard (Shannon) entropy is:  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 Number of events belonging to the same configuration s. Ma’s coincidence probability method :

6. , CZE ISMD2005 Zhiming LI 11/08/2005 5  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  Step2: Total numbers of observed coincidences of k configurations:

7. , CZE ISMD2005 Zhiming LI 11/08/2005 6 If, with Standard definitio n

8. , CZE ISMD2005 Zhiming LI 11/08/2005 7 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) )

9. , CZE ISMD2005 Zhiming LI 11/08/2005 8 If a system is close to thermal equilibrium and phase space is divided small enough, scaling and additivity hold:  Scaling:  Additivity: Phase space region （ 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.

10. , CZE ISMD2005 Zhiming LI 11/08/2005 9 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.

11. 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! Rényi entropy in the central y region

12. 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!

13. 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

14. 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

15. , CZE ISMD2005 Zhiming LI 11/08/2005 14 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.

16. , CZE ISMD2005 Zhiming LI 11/08/2005 15  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.

17. 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.