1/10Boris Tomášik: Identifying fireball fragmentation with the Kolmogorov-Smirnov test

Boris Tomášik

Univerzita Mateja Bela, Banská Bystrica, SlovakiaCzech Technical University, Prague, Czech Republic

NICA Round Table WorkshopSeptember 10, 2009

Identifying fireball fragmentationwith the Kolmogorov-Smirnov test

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

dN/dy

without droplets with droplets

If we have droplets, each event will look differently

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The measure of difference between events

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

y0

1

D maximumdistance

D measures the difference of two empirical distributions

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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.

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

nOnDkPDnQk

k

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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.)

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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

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Results from simulation: Q histogramsRHIC simulation (but similar for NICA with droplets)

Droplets with average volume 5 fm3

All hadrons are produced by droplets

Small signal also with no droplets due to resonance decays

With identified species problems with small multiplicity

droplets

droplets

droplets

no droplets

no dropletsno droplets

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Different sizes of droplets and droplet abundance

The peak at Q = 0 is visible

… down to average droplet size of 2.5 fm3

… also if not all hadrons come from the droplets

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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)

stable

with resonancedecaysy cut

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

Boris Tomášik: Identifying fireball fragmentation with the Kolmogorov-Smirnov test

Backup: exact formulas for Q

1

220 )2exp()1(2)(

k

k dkdK

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