Boris Tomášik Czech Technical University, Prague, Czech Republic

1. Boris Tomášik: The use of Kolmogorov-Smirnov test in event-by-event analysis The use of Kolmogorov-Smirnov test in event-by-event analysis • Boris Tomášik • Czech Technical University, Prague, Czech Republic • Univerzita Mateja Bela, Banská Bystrica, Slovakia • in collaboration with • Ivan Melo (University of Žilina)‏ • Giorgio Torrieri (FIAS Frankfurt)‏ • Sascha Vogel (Universität Frankfurt)‏ • Marcus Bleicher (Universität Frankfurt)‏ • Quark Matter 2009

2. Boris Tomášik: The use of Kolmogorov-Smirnov test in event-by-event analysis Fragmentation of the fireball – why and when? Consequences of fireball fragmentation: single event rapidity distributions The Kolmogorov-Smirnov (KS) test: general introduction Application of the KS test in case of fireball fragmentation A discussion of other possible effects on the KS test Summary Outline

3. Boris Tomášik: The use of Kolmogorov-Smirnov test in event-by-event analysis At the phase transition Case 1: first order phase transition Rapid passage through the phase transition leads to spinodal decomposition (known also in classical physics) Scenario possible if nucleation rate < expansion rate may be relevant for nuclear collisions observed in multifragmentation Fragmentation of the fireball: why and when Example: van der Waals isotherm p slow expansion: equilibrium trajectory fast expansion spinodal V

4. Boris Tomášik: The use of Kolmogorov-Smirnov test in event-by-event analysis Spinodal fragmentation scenario is irrelevant at RHIC and LHC. Case 2: Rapid cross-over The bulk viscosity suddenly grows near Tc [K. Paech, S. Pratt, Phys. Rev. C 74 (2006) 014901, D. Kharzeev, K. Tuchin, JHEP 0809:093 (2008). F.Karsch, D.Kharzeev, K.Tuchin, Phys. Lett. B 663 (2008) 217] bulk viscosity as a function of T [Kharzeev, Tuchin] Fragmentation at the cross-over

5. Boris Tomášik: The use of Kolmogorov-Smirnov test in event-by-event analysis (s)QGP expands easily Bulk viscosity singular atcritical temperature System becomes rigid Inertia may win and fireball will fragment Fragments evaporatehadrons Bulk-viscosity-driven fragmentation ... and freeze-out

6. Boris Tomášik: The use of Kolmogorov-Smirnov test in event-by-event analysis rapidity distribution in a single event Droplets and rapidity distributions dN/dy dN/dy y y without droplets with droplets If we have droplets, each event will look differently

7. Boris Tomášik: The use of Kolmogorov-Smirnov test in event-by-event analysis Kolmogorov–Smirnov test (general intro): Are two empirical distributions generated from the same underlying probability distribution? The measure of difference between events 1 D measures the difference of two empirical distributions D maximum distance 0 y

8. Boris Tomášik: The use of Kolmogorov-Smirnov test in event-by-event analysis How are the D's distributed? Smirnov (1944):If we have two sets of data generated from the same underlying distribution, then D's are distributed according to This is independent from the underlying distribution! For each t=D we can calculate For events generated from the same distribution, Q's will be distributed uniformly. Kolmogorov-Smirnov: theorems

9. Boris Tomášik: The use of Kolmogorov-Smirnov test in event-by-event analysis DRAGON: MC generator of (momenta and positions of) particles [BT: Computer Physics Communications, in press, arXiv:0806.4770 [nucl-th]] some particles are emitted from droplets (clusters) if no droplet formation is assumedm, then similar to THERMINATOR droplets are generated from a blast-wave source (tunable parameters) tunable size of droplets: Gamma-distributed or fixed droplets decay exponentially in time (tunable time, T) no overlap of droplets also directly emitted particles (tunable amount) chemical composition: equilibrium, resonances rapidity distribution: uniform or Gaussian possible OSCAR output DRoplet and hAdron GeneratOr for Nuclear collisions

10. Boris Tomášik: The use of Kolmogorov-Smirnov test in event-by-event analysis Results from simulation: Q histograms RHIC simulation Droplets with average volume 5 fm3 All hadrons are produced by droplets Small signal also in events with no droplets due to correlations from resonance decays With identified species problems with small multiplicity droplets no droplets droplets droplets no droplets no droplets

11. Boris Tomášik: The use of Kolmogorov-Smirnov test in event-by-event analysis Different sizes of droplets and droplet abundance The peak at Q = 0 is visible … down to average droplet size of 5 fm3 … also if not all hadrons come from the droplets

12. Boris Tomášik: The use of Kolmogorov-Smirnov test in event-by-event analysis The effect of momentum conservation • Toy model simulation: • N subsystems – momentum is exactly 0 within each subsystem • This leads to a dip in the Q histogram at small Q • This generates a histogram which looks as if the events were correlated with each other • NB: other effects which • may influence the KS test: • string fragmentation (weaker than droplets) • jets (high pt) • quantum correlations (how to simulate them)

13. Boris Tomášik: The use of Kolmogorov-Smirnov test in event-by-event analysis Due to spinodal decomposition or sudden rise of the bulk viscosity the fireball can fragment at the phase transition The Kolmogorov-Smirnov test can be used to compare rapidity distributions event-by-event in order to identify non-statistical differences between the events Fireball fragmentation would lead to a clear signal with this technique Other effects on the KS test to be examined Try KS test – if it gives no effect, then all events are the same and we have one piece of bulk matter Summary

14. Boris Tomášik: The use of Kolmogorov-Smirnov test in event-by-event analysis Backup: exact formulas for Q