Entropy analysis in and collisions at GeV

1. Entropy analysis in and collisions at GeV Zhiming LI (For the NA22 Collaboration)

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

3. 1. Motivation to study entropy The assumption of thermodynamic equilibrium is commonly used when discussing the system created in central heavy ion collisions: A.Białas and W.Czyż, Phys. Rev. D 61,074021(2000) A.Białas, W.Czyż and J. Wosiek, Acta Phys. Pol. B 30,107(1999) 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 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.

4. 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 2. Procedure and variables of Ma’s coincidence probability method The original definition of the standard (Shannon) entropy is: Ma’s coincidence probability method : Number of events belonging to the same configuration s.

5. Step2: Total numbers of observed coincidences of k configurations: • 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

6. If , with Standard definition

7. The advantages of Ma’s method: • Statistical error drops very fast with increasing number of configurations. • Monte Carlo shows the results are more stable for this method. （K.Fiałkowski and R. Wit, Phys. Rev. D 62,114016(2000) )

8. If a system is close to thermal equilibrium and phase space is divided small enough, scaling and additivity hold: （A.Białas and W.Czyż, Acta Phys. Pol. B 31,687(2000) ) • Scaling: 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 . • Additivity: Phase space region

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

10. 3. The results from the NA22 experiment and PYTHIA Rényi entropy in the central y region • 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!

11. 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!

12. Test of additivity in y region 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.

13. Multiplicity dependence of the Rényi entropy logarithmic scale 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! |y| < 3 |y| < 2 |y| < 1

14. 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. 4. Conclusions and outlook

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. Thanks!

16. Appendix: NA22 data sample • The NA22 experiment made use of the European Hybrid Spectrometer (EHS) in combination with the Rapid Cycling Bubble Chamber (RCBC). • 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.