1/13Boris Tomášik: The use of Kolmogorov-Smirnov test in event-by-event analysis

Boris Tomášik

Czech Technical University, Prague, Czech RepublicUniverzita 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

The use of Kolmogorov-Smirnov test in event-by-event analysis

2/13Boris Tomášik: The use of Kolmogorov-Smirnov test in event-by-event analysis

Outline

1. Fragmentation of the fireball – why and when?

2. Consequences of fireball fragmentation:

single event rapidity

distributions

3. The Kolmogorov-Smirnov (KS) test:

general introduction

4. Application of the KS test in case of

fireball fragmentation

5. A discussion of other possible effects on the KS test

6. Summary

3/13Boris Tomášik: The use of Kolmogorov-Smirnov test in event-by-event analysis

Fragmentation of the fireball: why and when

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 ifnucleation rate < expansion rate

may be relevant for nuclear collisions observed in multifragmentation

pslow

expansion:

equilibrium trajectory

fast e

xpansion

spinodal

V

Example:van der Waalsisotherm

4/13Boris Tomášik: The use of Kolmogorov-Smirnov test in event-by-event analysis

Fragmentation at the cross-overSpinodal 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]

5/13Boris Tomášik: The use of Kolmogorov-Smirnov test in event-by-event analysis

Bulk-viscosity-driven fragmentation

(s)QGP expands easily

Bulk viscosity singular atcritical temperature

System becomes rigid

Inertia may win and fireball will fragment

Fragments evaporatehadrons

... and freeze-out

6/13Boris Tomášik: The use of Kolmogorov-Smirnov test in event-by-event analysis

Droplets and rapidity distributions

rapidity distribution in a single event

y

dN/dy

y

dN/dy

without droplets with droplets

If we have droplets, each event will look differently

7/13Boris Tomášik: The use of Kolmogorov-Smirnov test in event-by-event analysis

The measure of difference between events

Kolmogorov–Smirnov test (general intro):Are two empirical distributions generated from the same underlying probability distribution?

y0

1

D maximumdistance

D measures the difference of two empirical distributions

8/13Boris Tomášik: The use of Kolmogorov-Smirnov test in event-by-event analysis

Kolmogorov-Smirnov: theoremsHow 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.

)()exp()1(11)( 2/122

nOnDkPDnQk

k

9/13Boris Tomášik: The use of Kolmogorov-Smirnov test in event-by-event analysis

DRoplet and hAdron GeneratOr for Nuclear collisions

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

10/13Boris Tomášik: The use of Kolmogorov-Smirnov test in event-by-event analysis

Results from simulation: Q histogramsRHIC 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

droplets

droplets

no droplets

no dropletsno droplets

11/13Boris 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

12/13Boris 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)

13/13Boris Tomášik: The use of Kolmogorov-Smirnov test in event-by-event analysis

Summary

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

Boris Tomášik: The use of Kolmogorov-Smirnov test in event-by-event analysis

Backup: exact formulas for Q

1

220 )2exp()1(2)(

k

k dkdK

...)()()()()( 3210 dKdKdKdKdQ )/( 2121 nnnnDd