1. 1/10Boris Tomášik: Identifying fireball fragmentation with the Kolmogorov-Smirnov test Boris Tomášik Univerzita Mateja Bela, Banská Bystrica, Slovakia Czech Technical University, Prague, Czech Republic NICA Round Table Workshop September 10, 2009 Identifying fireball fragmentation with the Kolmogorov-Smirnov test

2. 2/10Boris Tomášik: Identifying fireball fragmentation with the Kolmogorov-Smirnov test Droplets and rapidity distributions rapidity distribution in a single event y dN/dy y without droplets with droplets If we have droplets, each event will look differently

3. 3/10Boris Tomášik: Identifying fireball fragmentation with the Kolmogorov-Smirnov test The measure of difference between events Kolmogorov–Smirnov test (general intro): Are two empirical distributions generated from the same underlying probability distribution? (null hypothesis) y 0 1 D maximum distance D measures the difference of two empirical distributions

4. 4/10Boris Tomášik: Identifying fireball fragmentation with the Kolmogorov-Smirnov test Kolmogorov-Smirnov: theorems 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.

5. 5/10Boris Tomášik: Identifying fireball fragmentation with the Kolmogorov-Smirnov test Null hypothesis All events within the selected centrality class are evolve according to the same scenario and the bulk evolves smoothly from the beginning to the freeze out. (Like in hydrodynamic simulation.)

6. 6/10Boris Tomášik: Identifying fireball fragmentation with the Kolmogorov-Smirnov test DRoplet and hAdron GeneratOr for Nuclear collisions DRAGON: MC generator of (momenta and positions of) particles [BT: Computer Physics Communications 180 (2009) 1642, arXiv:0806.4770 [nucl-th]] some particles are emitted from droplets (clusters) if no droplet formation is assumed, 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

7. 7/10Boris Tomášik: Identifying fireball fragmentation with the Kolmogorov-Smirnov test Results from simulation: Q histograms RHIC simulation (but similar for NICA with droplets) Droplets with average volume 5 fm 3 All hadrons are produced by droplets Small signal also with no droplets due to resonance decays With identified species problems with small multiplicity droplets no droplets

8. 8/10Boris Tomášik: Identifying fireball fragmentation with the Kolmogorov-Smirnov test Different sizes of droplets and droplet abundance The peak at Q = 0 is visible … down to average droplet size of 2.5 fm 3 … also if not all hadrons come from the droplets

9. 9/10Boris Tomášik: Identifying fireball fragmentation with the Kolmogorov-Smirnov test 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 p t ) quantum correlations (how to simulate them) stable with resonance decays y cut

10. 10/10Boris Tomášik: Identifying fireball fragmentation with the Kolmogorov-Smirnov test Summary The Kolmogorov-Smirnov test can be used to compare rapidity distributions event-by-event in order to identify non-statistical differences between the events Try KS test – if it gives no effect, then all events are the same and we have one piece of bulk matter (null hypothesis) Advantage of the KS test is no bias on any moment of the rapidity distribution. Fireball fragmentation would lead to a clear signal with this technique Other effects on the KS test to be examined

11. Boris Tomášik: Identifying fireball fragmentation with the Kolmogorov-Smirnov test Backup: exact formulas for Q