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arXiv:0902.1607v1 [nucl-th] 10 Feb 2009

The Kolmogorov-Smirnov test and its use for the identification of fireball

fragmentation

Ivan Meloa, Boris Tom´ aˇ sikb,c, Giorgio Torrierid, Sascha Vogele,

Marcus Bleichere, Samuel Kor´ onyb, Mikul´ aˇ s Gintnera,b

a ˇZilinsk´ a Univerzita, Univerzitn´ a 1, 01026ˇZilina, Slovakia

bUniverzita Mateja Bela, Tajovsk´ eho 40, 97401 Bansk´ a Bystrica, Slovakia

cFaculty of Nuclear Science and Physics Engineering,

Czech Technical University in Prague, Bˇ rehov´ a 11, 11519 Prague, Czech Republic

dFrankfurt Institute of Advanced Studies, Johann Wolfgang Goethe Universit¨ at,

Ruth-Moufang-Str. 1, 60438 Frankfurt am Main, Germany

eInstitut f¨ ur theoretische Physik, Johann Wolfgang Goethe Universit¨ at,

Max-von-Laue-Str.1, 60438 Frankfurt am Main, Germany

(Dated: February 10, 2009)

We propose an application of the Kolmogorov-Smirnov test for rapidity distributions of individual

events in ultrarelativistic heavy ion collisions. The test is particularly suitable to recognise non-

statistical differences between the events. Thus when applied to a narrow centrality class it could

indicate differences between events which would not be expected if all events evolve according to the

same scenario. In particular, as an example we assume here a possible fragmentation of the fireball

into smaller pieces at the quark/hadron phase transition. Quantitative studies are performed with

a Monte Carlo model capable of simulating such a distribution of hadrons. We conclude that the

Kolmogorov-Smirnov test is a very powerful tool for the identification of the fragmentation process.

PACS numbers: 02.50.-r, 24.10.Pa, 24.60.Ky, 25.75.Gz

I.INTRODUCTION

The highly excited matter created in ultrarelativis-

tic nuclear collisions expands very fast. It is commonly

accepted that a deconfined phase has been reached in

Au+Au collisions at RHIC [1, 2, 3, 4], while the onset of

deconfinement has been advocated at SPS energies [5, 6].

While in lattice QCD calculations a static thermody-

namic medium is assumed, in heavy ion collisions the

situation is vastly different. Here, the longitudinal ex-

pansion dynamics leads to a rapid passage from the de-

confined to the confined phase. A system which under-

goes the phase transition quickly may not follow the usual

equilibrium scenario. In fact, for a first-order phase tran-

sition, the high temperature phase may survive down to

temperatures drastically below the transition tempera-

ture, i.e. the system supercools. If the expansion rate

is faster than the nucleation rate of bubbles of the new

phase, the system reaches the point of spinodal instabil-

ity1. Beyond such a point, entropy is gained if the system

separates into two phases and so it becomes mechanically

unstable. Spinodal fragmentation connected with nuclear

liquid/gas phase transition has been identified in heavy

ion collisions at few hundred MeV per nucleon [7, 8], and

it has been proposed that it might be the actual scenario

at ultrarelativistic energies as well [9, 10]. Fragmentation

assumes that at the phase transition the system decays

into droplets of smaller size. These droplets then emit

1This is the inflection point of the dependence of entropy on an

extensive variable, see e.g. [7, 8].

hadrons.

Lattice calculations indicate, however, that at RHIC

and LHC the transition from partonic to hadronic mat-

ter is a rapid but smooth crossover [11].

odal decomposition seems irrelevant scenario in this case.

On the other hand, conformal symmetry is broken close

to the phase transition and as a consequence the bulk

viscosity—being negligible otherwise—shows a peak here

[12, 13, 14]. Bulk viscosity acts against the expansion

and slows it down.As a result, if the system previ-

ously accumulated kinetic energy due to expansion, it

may fragment [14]. An analysis of hydrodynamic insta-

bilities shows that such a scenario may be realistic [15].

Hence, it appears that fragmentation may happen in

ultrarelativistic nuclear collisions. Many kinds of observ-

ables might be sensitive to it. Most notable are multiplic-

ity fluctuations in varying rapidity windows [16], fluctua-

tions of mean pt[17], rapidity correlations [18, 19], proton

and kaon correlations [18], φ-meson production [18], and

two-pion femtoscopy [14, 20, 21].

In this paper we inspect event-by-event fluctuations of

rapidity distributions. If final state hadrons are emitted

from droplets, their velocities will be close to those of

the droplets. Thus, clustering would appear in their mo-

mentum distribution. Moreover, in each event clusters

will have different velocities. An important contribution

to clustering will also come from the resonance decays

and we shall investigate this effect. Thus, the momen-

tum distribution will vary from event to event. Specif-

ically, here we shall compare rapidity distributions from

different events and look for differences due to fireball

fragmentation. To this end, we employ the Kolmogorov-

Smirnov (KS) test which can be used for identification

Thus spin-

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FIG. 1: Construction of the two empirical cummulative dis-

tribution functions, one with thin solid (blue) line and one

with thick dashed (red) line. The maximum distance between

them is D.

of non-statistical differences between two empirical dis-

tributions [22, 23]. An important advantage of the KS

test is its independence from the underlying distribution

of the measured quantity.

In the next Section we shortly introduce the KS test.

Then, we illustrate its sensitivity by a couple of toy sim-

ulations. For more realistic studies we generate artificial

data with the help of the event generator DRAGON [24]

which is very briefly introduced in Section IV. Results

obtained with these data are presented in Section V. We

conclude in Section VII. In the Appendix we review the

evaluation of the cummulative distribution function for

the Kolmogorov distribution.

II. THE KOLMOGOROV-SMIRNOV TEST AND

HOW TO USE IT

Let us start by explaining the technical part of the

problem. One has two empirical distributions in variable

x, which can be rapidity, pt, or yet something else2. (In

the present work we work with rapidities.) The multiplic-

ities may differ. The question we want to ask is, whether

the two empirical distributions are the same in the sense

that they would correspond to the same underlying the-

oretical single-particle probability density, and there are

no correlations between particles in one event.

Practically, the quantity x is measured for each par-

ticle in an event. The empirical cummulative distribu-

tion function (ECDF) is constructed so that a step of

the height 1/ni (ni is the multiplicity of the event) is

made on all positions of measured x’s (Fig. 1). This is

done for both events of a pair. Subsequently, one finds

the maximum vertical distance between the two ECDF’s

2

Note that for a cyclic variable, e.g. azimuthal angle φ, the

KS test cannot be applied and instead a modification known

as Kuiper test [25, 26]must be employed.

and introduces

d =√nD =

?

n1n2

n1+ n2D(1)

where D is the distance of two ECDF’s and n1, n2are the

multiplicities of the two data sets. The procedure of the

test is illustrated in Figure 1. The cummulative distribu-

tion function of the Kolmogorov distribution concerns the

case of events generated from the same underlying theo-

retical probability distribution for the quantity x, ρ(x).

It will be expressed with the help of the function Q(d) as

P(d′< d) = 1 − Q(d). (2)

where P(d′< d) is the probability that we find a dif-

ference d′smaller than d. An important feature of this

approach is that Q(d) does not depend on the particu-

lar shape of the theoretical distribution ρ(x). Unfortu-

nately, the general form of Q(d) valid for any multiplic-

ities and distances d is not suitable for practical evalua-

tion. Usable approximate expressions for Q(d) are sum-

marised in the Appendix. It follows that if many pairs of

events would be drawn from a set of events generated all

from the same underlying probablility distribution and

for each pair the quantities d and Q(d) would be de-

termined, then the Q′s would be distributed uniformly.

Deviation from uniform distribution, particularly an en-

hanced population of low Q’s (large d’s) indicates that

the events are not drawn independently from the same

underlying distribution. In this way the KS test will be

used here.

Note that the KS test does not identify the physical

origin of the difference between the events. It is a robust

way to identify that there is a difference, however, the

origin must be singled out by other means. In addition

to fireball fragmentation, these can be fluctuations of the

initial state of the fireball evolution, final resonance de-

cays, conservation laws, and quantum correlations. A de-

tailed investigation of these will be pursued in subsequent

papers. The important message of the KS test is, that

it can disprove the usual paradigm that data from many

collisions (within the same centrality class) are produced

by basically identical fireballs.

It is important to realise that the number of significant

decimal figures to which the quantity x (rapidity here) is

measured may also influence the result of the KS test if

it is applied on a large number of pairs of events. This is

illustrated in Figure 2. The peak at Q → 1 increases with

lowering the number of decimal places taken into account.

The explanation is trivial but instructive. Only rapidities

between 0 and 1 were generated. Within 105events, each

having multiplicity around 200, there are about 2 × 107

particles. Hence, if their values of x are given to less than

8 figures, we are guaranteed to have repeating values of x

in our sample. For 6 given figures we expect each value to

appear on average 20 times, for 4 figures it is 2000 times

and for 2 figures we are even at 200,000 times! Clearly,

this effect correlates the events since it artificially chooses

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6

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9000

10000

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4

0.5448

Number of decimal places n

2

FIG. 2:

pairs out of 105events generated from a uniform distribution

between 0 and 1 in the variable x (rapidity) and with multi-

plicities distributed according to Poisson distribution with the

mean 200. Different histograms correspond to rapidity data

truncated after 2, 4, and 6 decimal places. The histogram

with 8 significant figures is identical to that with 6 figures.

Histograms of Q’s from the KS test applied on 105

x from a finite number of possible values. According to

its construction, the D’s will acquire smaller values on

average.

We have also checked that the normal fluctuation of the

number of entries in a bin of the Q-histogram is equal to

the square root of the number of entries. We thus propose

the use of quantity

R =N0−Ntot

B

σ0

=N0−Ntot

?

B

Ntot

B

,(3)

where N0 is the number of pairs in the first bin of the

Q-histogram (next to Q = 0), Ntotis the total number of

pairs, B is the number of bins of the Q-histogram, and

σ0=?Ntot/B is the expected variance of the number of

the order 1; values considerably bigger than that indicate

non-statistical differences between the events.

entries in the first bin. The modulus of R should be of

III.THE SENSITIVITY OF THE TEST

In this section we study how the proposed method

works in case of clear cut examples. First, we gener-

ate samples of “events” where one half of all events is

generated according to Gaussian distribution with the

width σ = 0.1. For the second half of events we keep the

same width and vary the mean: we have samples with

the mean shifted with respect to the other half by 2σ,

1σ, 0.5σ, 0.1σ, and 0.01σ. In Figure 3 we observe how

the KS procedure recognises the difference of 0.1σ pretty

well if the average multiplicity is 512, and how the res-

olution power decreases when lowering the multiplicity

h512h512

512

128

32

Entries

Mean

RMS RMS

99998

0.3876

0.3063 0.3063

Q

0 0.1 0.20.3 0.40.50.60.7 0.80.91

number of pairs

8000

10000

12000

14000

16000

18000

20000

22000

24000

Entries

Mean

99998

0.3876

<n>

100200300400500

R

0

100

200

300

400

σ

0.01

0.1

0.5

σ

1

σ

σ

FIG. 3: (Color online) Large plot: the Q histograms from

the KS test on event samples consisting from two classes of

events. One class was generated from Gaussian distribution

with the mean 0 and the width σ = 0.1. The other class is gen-

erated from Gaussian distribution with the same width, but

the mean is shifted by 0.1σ. The multiplicities of the events

are 32 (black dotted histogram), 128 (blue dashed), and 512

(red solid). Smaller inset plot: the dependence of the parame-

ter R on the multiplicity of the events for the difference of the

means equal to 0.01σ (black circles), 0.1σ (magenta squares),

0.5σ (blue triangles), and 1σ (red upside down triangles).

h3h3

Entries

Mean

RMS RMS

99998

0.2525

0.3253 0.3253

Q

0 0.10.2 0.30.4 0.50.6 0.70.80.91

number of pairs

10000

20000

30000

40000

50000

Entries

Mean

99998

0.2525

= 1.0, R = 448.4

σ

= 0.12, R = 198.6

σ

= 0.11, R = 42.9

σ

= 0.101, R = -2.2

σ

= 0.09, R = 54.4

σ

FIG. 4: The Q-histograms from event samples consisting of

two classes of events where rapidities were generated from

Gaussian profiles with the same mean and the multiplicity

was distributed Poissonian with the mean 512. The width in

one class of events was 0.1. The width in the second class

has been varied; different histograms correspond to different

widths. The values of R are written in the legend.

to 32. As seen in the smaller inset plot, for smaller dis-

tances between the two Gaussian means the difference is

not recognised, while for larger distance the difference is

resolved by the test for all multiplicities.

In Figure 4 we explore the effect of a variable width.

One half of events was simulated with Gaussian distribu-

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RMS RMS

99995 99995

0.1483 0.1483

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number of pairs

0

10000

20000

30000

40000

50000

60000

70000

80000

90000

Entries

Mean 0.05128

(droplets, multiplicity/droplet)

(16,128)

(32,64)

(64,32)

(128,16)

(256,8)

(512,4)

FIG. 5: The effect of the number of Gaussian sources on the

Q-histogram. In the legend, left is the number of Gaussian

sources distributed uniformly between -1 and 1, right is the

average number of pions from each source. The width of each

Gaussian source was 0.707.

tion with the width of 0.1 and the other one with the same

mean but a different width. The widths are 1.0, 0.12,

0.11, 0.101, and 0.09. The multiplicity was Poissonian-

distributed with the mean of 512. We observe that except

for the cases where the widths differ by ten per cent or

less the difference is picked up by the procedure.

Finally, we test a case which is closest to the fireball

fragmentation scenario that we want to explore in detail.

In Figure 5 we show the results from a simulation, where

each event consists from superposition of many Gaus-

sian distributions. The width of all these distributions

is 0.707 and is motivated by the typical rapidity spread

of the pion rapidity at a realistic freeze-out temperature.

The means of the Gaussians are generated from a uni-

form distribution between –1 and 1. We test cases with

16, 32, 64, 128, 256, and 512 Gaussians per event, which

emit on average 128, 64, 32, 16, 8, 4 particles per Gaus-

sian, respectively, so that the total multiplicity is always

2048. We observe, that even in the least favorable simula-

tion with a large number of small droplets, the difference

between events is clearly visible.

IV.MONTE-CARLO DROPLET GENERATOR

Realistic events samples on which the KS test are ap-

plied were generated with the help of the Monte Carlo

event genarator DRAGON [24]. Here we provide very

brief overview of its capabilities.

DRAGON assumes that the fireball decays into

droplets which are distributed according to the blast-

wave model. Thus their distribution in position and ve-

locity is given by

SD(x,v) ∝ H(η)Θ(R − r)δ(τ − τ0)δ(4)(v − u(x)), (4)

where we use polar coordinates r and φ, the space-time

rapidity and longitudinal proper time

η =

1

2lnt + z

t − z

?

(5)

τ =

t2− z2,(6)

as coordinates in the space-time. The fireball has a trans-

verse radius R and τ0is the Bjorken proper time of the

decay. The four-velocity of the droplet v is given by the

local flow velocity at the position where the droplet is

created,

uµ(x) = (coshη coshηt, cosφ sinhηt,

sinφ sinhηt, sinhη coshηt),(7)

with

ηt=

√2ρ0r

R

,(8)

where ρ0is a model parameter. (The model is designed

so that it can simulate azimuthally non-symmetric fire-

balls, but we do not explore such a possibility here.) The

function H(η) specifies the space-time rapidity distribu-

tion. It can be uniform or Gaussian. For the present

investigation we use the uniform distribution in rapidity.

The volumes of the droplets are random according to

a gamma distribution

P2(V ) =

1

bΓ(2)

V

b

exp(−V/b) ,(9)

with a model parameter b.

hadrons exponentially in time, so the times of emission of

the droplets are distributed in the rest frame of the emit-

ting droplet according to exp(−τ/RD), where RDis the

radius of the droplet. A droplet emits hadrons according

to thermal distribution with a temperature Tk, until it

uses up all of its mass. The mass of the droplet is de-

termined according to its volume and the energy density

which is set to 0.7GeV fm−3.

Hadrons may be emitted from the droplets or produced

in the remaining space between them. The relative abun-

dance of those emitted from droplets is specified as a

model parameter. Hadrons emitted from the bulk are

generated according to the blast-wave emission function

[27, 28, 29]

The droplets decay into

S(x,p)d4x =2s + 1

(2π)3mtcosh(y − η) exp

× Θ(R − r)H(η)δ(τ − τ0)dτ τ dη rdrdφ.

Here the factor (2s + 1) denotes spin degeneracy.

Resonances are included in the simulation. They decay

according to the standard two-body or three-body kine-

matics. Probabilities of production of individual species

are given by the statistical model with a chemical freeze-

out temperature Tchand chemical potentials for baryon

number and strangeness.

?

−pµuµ

Tk

?

(10)

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V.FLUCTUATING RAPIDITY

DISTRIBUTIONS

The Monte Carlo event generator DRAGON is em-

ployed [24] to simulate realistic data on which the KS

test is performed. We use the test on data generated for

RHIC Au+Au collisions at√s = 200AGeV and FAIR

Au+Au at√s = 7.6AGeV. For the data analysis we

accept hadrons within the rapidity interval [–0.5,0.5].

For RHIC, we have generated events with uniform ra-

pidity distribution in the interval [–3,3]. The total hadron

multiplicity was set to dN/dy = 1000. The chemical com-

position is determined by the following choice of parame-

ters: Tch= 155 MeV, µB= 26 MeV [30]. We neglect the

strangeness chemical potential. The list of resonances in-

cludes mesons up to a mass of 1.5 GeV/c2and baryons

up to 2 GeV/c2. The geometry of the decaying fireball is

given by the radius R = 10 fm and τ0= 9 fm/c. The dy-

namical state of the fireball is set by the kinetic freeze-out

temperature Tk= 150 MeV and the transverse expansion

gradient ηf = 0.6. We set the volume parameter of the

droplets b to the value of 10 fm3. As a first benchmark,

complementary samples of 10,000 events are generated:

one with all particles being emitted from droplets, the

other with all particles being emitted from the bulk fire-

ball.

As a second benchmark test we generate 10,000 events

at the FAIR energy of√s = 7.6A GeV where no parti-

cles are emitted from droplets. In this case the chemical

freeze-out parameters are set to the corresponding values

Tch= 140 MeV, µB = 375 MeV, and µS = −53 MeV.

The kinetic freeze-out temperature is Tk= 140 MeV and

the transverse expansion of the fireball is characterised

by ηf = 0.4. Here, the rapidity distribution is Gaussian

with a width of 0.7 and the total hadron multiplicity

is 1,500. The transverse radius of the fireball and its

Bjorken lifetime are 9 fm and 8 fm/c, respectively.

In figure 6 we show the difference between the Q-

histograms from events with and without droplets. For

the RHIC energy, one observes a characteristic enhance-

ment towards small Q values in the case of particle emis-

sion from droplets for all investigated particle species ex-

cept (anti-)protons. (For the setting without droplets

(i.e. only bulk emission) this low Q enhancement is

strongly suppressed. Quantitatively, this is reflected in

a factor of 10 difference of the extracted R values. The

RHIC results without droplet formation are also in line

with the results obtained at FAIR energies, showing that

the KS test does not produce falsly positive results when

going to smaller samples with a different rapidity distri-

bution.

Resonance decays also have a clustering effect on the

decay products. Therefore, a signal of clustering is also

seen in the set of events without droplets. In case of all

hadrons, these are mainly ρ’s and ∆’s. If we limit our

analysis to pions only, then there is correlation due to the

ρ. To test this hypothesis, one can perform the KS test

with protons only, since there is no resonance that would

decay into two baryons. The drawback of using protons

only is limited statistics in two ways. Firstly, their total

multiplicity is lower, e.g. there are only 10 to 50 protons

in the acceptance per RHIC event. Therefore, one ob-

serves fewer pairs at small Q and a peak at Q close to 1

in case there are no droplets (see appendix). Secondly,

if the droplets are small, protons are a less ideal probe

because a droplet may not have enough energy to emit

more than one proton and the correlation is gone then.

An alternative solution is to use pions of the same charge.

These are more abundant than protons and no strong ef-

fect of resonances is seen here. Note, however, that we

have not included the effect of pair wave function sym-

metrisation which leads to Bose-Einstein correlations.

Note also, that resonances not only introduce corre-

lations, they can also weaken the correlations due to

droplets. Resonance decay products obtain some momen-

tum due to higher mass of the mother resonance. Thus,

the velocities of decay products will be more smeared

around the velocity of the droplet which emitted the res-

onance than the velocities of hadrons emitted from the

droplet directly.

The influence of the size of the droplets is studied in

Figure 7. The parameters of the simulation are kept the

same as in the previous case, but the volume parameter

b varied to values 5, 10, 20, 50 fm3. All particles are

emitted from droplets. We do the KS test with charged

hadrons. As expected from previous analysis (with b =

10fm3), a dominant low Q peak emerges for all droplet

volumes down to 5 fm3.From this we conclude that

even small size droplets can be detected with the analysis

method presented here.

As a final physics benchmark of the KS test we ex-

plore the effect of changing droplet fraction of the to-

tal multiplicity. A systematic study of how the percent-

age of hadrons emitted from droplets affects the result is

presented in Figure 8. Here, the size of the droplets is

fixed to b = 10fm3and only the percentage of hadrons

from droplets is varied. Even if only one quarter of all

hadrons comes from the droplets and the rest from the

gas in between them, the signal is well visible and the

KS test can discriminate between the formation and the

non-formation of droplets.

VI.RESOLUTION

Finally, we address the experimentally crucial ques-

tion whether the droplet signal in the Q-histogram

stays recognisable if the rapidities are measured with fi-

nite resolution. We investigate this issue in Figure 9.

Events simulated with DRAGON for a volume param-

eter b = 10fm3and with 50 per cent of hadrons emit-

ted from droplets are taken and the generated rapidities

are smeared by a Gaussian with widths of 0.1, 0.5, and

1, to mimick the finite resolution of experimental mea-

surements. The events with smeared rapidities are then

processed with the KS test. We observe a gradual weak-

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R = 112

0.4087

hadrons

h3 h3

Entries 99992

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R = 1.3

0.445

0.3001 0.3001

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0 0.10.20.30.4 0.50.6 0.70.80.91

number of pairs

8000

9000

10000

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0.445

Charged hadrons

R = 63.5

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R = 1.5

R = 0.68

0.4892

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10000

10500

11000

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+

π

R = 17.4

h3h3

Entries 99992

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R = - 1.1

R = - 0.18

0.4897

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00.1 0.20.30.40.50.60.70.80.91

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9500

10000

10500

11000

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-π

R = 14.6

h3 h3

Entries 99992

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RMS RMS

R = 6.8

R = 5.6

0.4646

0.2972 0.2972

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0 0.10.20.30.40.50.6 0.70.80.91

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10000

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12000

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-π

+

π

R = 41

h3h3

Entries 99992

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R = - 2.1

R = - 2.5

0.5123

0.2944 0.2944

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8500

9000

9500

10000

10500

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11500

12000

12500

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Mean 0.5123

pp and

R = - 0.22

FIG. 6: (Color online) The Q-histograms resulting from simulations of realistic hadronic final states with the help of DRAGON.

Solid (red) histograms correspond to simulation of RHIC Au+Au collisions with droplets. Dashed (blue) histograms are from

simulations for RHIC without droplets. Dotted (brown) histograms show the results of simulations for nuclear collision at

FAIR without fragmentation. Different panels show results obtained for all hadrons, charged hadrons, π+, π−, charged pions,

protons and antiprotons. The values of R are indicated in the panels.

ening of the signal strength at low Q. For smearing by

0.5 units of rapidity the peak height becomes comparable

with the peak resulting from resonance decays only (cf.

Figure 6). For even poorer resolution, the peak can not

be regarded as an unambiguous signal for droplet forma-

tion. The resolution, however, is usually on the level of

∆y ≈ 0.1.

VII.CONCLUSIONS

The Kolmogorov-Smirnov test is a powerful tool in

searching for non-statistical differences between events.

The test itself is more general than investigated here, and

applications will be presented in following papers. The

logic of its use is the following: select a class of events

which are “as identical as possible”, in particular in a

very narrow centrality class. Conventional scenarios pre-

dict that each event would evolve according to the same

scenario and the final distributions of hadrons would be

identical in all events. The KS test is able to detect de-

viations from this scenario. If an effect is observed, it

remains to be studied what phenomenon leads to posi-

tive results.

As a currently widely discussed topic we focussed the

present investigation on the possible decay of the fireball

into smaller droplets. The present study showed that

the KS test is perfectly suited for this task and allows

Page 7

7

h50h50

Entries 99992

Mean

RMS RMS

R = 67.310 fm

0.3827

0.3061 0.3061

Q

00.10.20.30.40.5 0.60.70.80.91

number of pairs

6000

8000

10000

12000

14000

16000

18000

20000

22000

24000

26000

Entries 99992

Mean

R = 128.320 fm

0.3827

V50

R = 155.8

3

50 fm

3

3

R = 4.6

3

5 fm

FIG. 7:

droplets: b = 5, 10, 20, 50 fm3. The values of R are shown

in the legend. The Q-histograms are obtained from samples

of 10,000 events where all hadrons have been emitted from

droplets. Only charged hadrons have been taken in construct-

ing these histograms.

Q-histograms corresponding to various sizes of

h100h100

Entries

Mean

RMS RMS

50% R = 36.7

99992

0.4445

0.3001 0.3001

Q

00.1 0.20.30.40.50.60.70.80.91

number of pairs

8000

9000

10000

11000

12000

13000

14000

15000

16000

17000

Entries

Mean

99992

0.4445

100% R = 67.3

75% R = 57.1

25% R = 26.5

FIG. 8: Q-histograms obtained from samples of 10,000 events

where droplets with a volume parameter of b = 10fm3are

present.The percentage of hadrons coming from droplets

is varied: 25%, 50%, 75%, and 100%. The histograms are

constructed with charged hadrons only. The values of R are

listed in the legend.

to extract a prominent signal. The signal is robust even

if only a small amount of the hadrons come from the

droplets and survives realistic final rapidity resolution.

Thus, the test can be also used in a negative way: if its

application on data yields only limited or no signature of

non-statistical event-by-event fluctuations of the rapidity

distributions, this puts limits on the scenarios assuming

fireball fragmentation.

The investigation of the signal of other effects (includ-

ing a comparison to full Monte Carlo transport simula-

tion) in the KS test deserves separate studies and shall

h3h3

0.1 R = 35.3

Entries 99992

Mean

RMS RMS

0.5 R = 8.5

0.4683

0.2966 0.2966

Q

00.10.20.30.40.5 0.60.7 0.80.91

number of pairs

9000

10000

11000

12000

13000

Entries 99992

Mean 0.4683

y smeared by Gauss of width

0.0 R = 36.7

1.0 R = 1.1

FIG. 9: The influence of finite rapidity resolution. Charged

hadrons were generated with the Monte Carlo event genarator

DRAGON with b = 10fm3and 50% of hadrons generated

from droplets. Before the analysis, rapidities were smeared

with Gaussian distribution with the width 0.1 (long-dashed

line), 0.5 (dash-dotted), 1.0 (short-dashed). Solid line shows

the result with non-smeared data.

be performed in subsequent papers.

Acknowledgments

The work of BT, IM, SK, and MG has been sup-

ported by VEGA 1/4012/07.

and MB was (financially) supported by the Helmholtz

International Center for FAIR within the framework

of the LOEWE program (Landesoffensive zur Entwick-

lung Wissenschaftlich-¨ okonomischer Exzellenz) launched

by the State of Hesse. BT acknowledges support from

MSM 6840770039 and LC 07048 (Czech Republic). We

thank Drs. J.R. Brown and M.E. Harvey for valuable dis-

cussions.

The work of GT, SV,

APPENDIX A: EVALUATION OF THE

KOLMOGOROV-SMIRNOV DISTRIBUTION

Throughout this paper we use the two-sample two-

sided (Kolmogorov-)Smirnov test3.

distribution function of the difference D for the one-

sample test in case n → ∞ was derived by Kolmogorov

The cummulative

3The Kolmogorov one-sample test refers to a comparison of one

empirical cummulative distribution function based on data with

a smooth thoretical distribution function. This is distinguished

from the two-sample (Smirnov) test where two data samples are

compared with each other. One-sided and two-sided tests refer

simply to the difference of two cumulative distribution functions

or to its absolute value, respectively.

Page 8

8

[22]:

Q(D) = K0(D) = −2

∞

?

k=1

(−1)kexp?−2k2D2?.(A1)

Later, Smirnov [23] proved that the same result applies

for the two-sample test for n1,n2 → ∞ under replace-

ment D → d = D?n1n2/(n1+ n2). We have checked

tion even for n’s around 200. It is therefore desirable to

obtain formulas valid in non-asymptotic case.

A simple solution is to replace the quantity d =√nD

in eq. (A1) with the following one, originally due to

that such an asymptotic case is not a good approxima-

Stephens [31] (found also in Numerical Recipes [32]):

d = D

?√n + 0.12 +0.11

√n

?

(A2)

A different formula with a few terms of an expansion in

1/√n has been derived by Li-Chien [33, 34]. In such an

expansion

Q(d) = K0(d) + K1(d) + K2(d) + K3(d) + ... . (A3)

The leading order term has been displayed in eq. (A1).

The following three terms are

K1(d) =

4d

3√n

∞

?

∞

?

2d

27n3/2

k=1

(−1)kk2exp?−2k2d2?

(−1)k

(A4)

K2(d) =

1

9n

k=1

?

k2−1

2

?1 − (−1)k?− 4k2d2

??

?4

?

k2−1

2

?1 − (−1)k?+ 3

?

+ 8k4d4

?

exp?−2k2d2?

(A5)

K3(d) = −

∞

?

k=1

(−1)kk2

k2+22

5

−3

2

?1 − (−1)k??

3k2+88

− k2d2

15− 2?1 − (−1)k?+ 12

?

+ 8k4d4

?

exp?−2k2d2?.(A6)

These relations were derived for one-sample test.

checked, however, that they are much easier to han-

dle and give better results when tested on samples of

statistically identical events (see below) than approxi-

mations to two-sample distributions for non-asymptotic

cases [35, 36]. Therefore, we decided to use these rela-

tions although we note that a revision of the formulae for

two-sample tests is desirable.

In practical calculations it turns out that it is sufficient

to cut off the expansions in eqs. (A1), (A4–A6) at k = 4.

To illustrate this point, we compare in Fig. 10 the his-

tograms of Q’s calculated from eq. (A2) with histograms

based on eq. (A3) with the cut-off at k = 2, 3 and 4 re-

spectively. We show the results for 105pairs of events

chosen randomly out of 105simulated events. Each sim-

We

ulated event is represented by ‘rapidities’ uniformly gen-

erated on (0,1) with Poissonian distributed multiplicities

with mean values of 50, 200 and 1000. Thus we calculate

Q(d) using the expansion (A3) up to the term K3 and

evaluate the sums in eqs. (A1), (A4–A6) up to the fourth

order in k.

The Li-Chien approximation truncated after k = 4

may lead to negative Q for d > 1.94. Such a pair would

fall into the last Q-bin. To fix the problem for this value

of d an approximation due to Marsaglia [37] is employed

Q(d) = 2exp

??

−2.000071−0.331

√n

−1.409

n

?

nd2

?

(A7)

.

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Entries

Mean

RMS RMS

99998

0.5224

0.2974 0.2974

number of pairs

9000

9500

10000

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11000

11500

12000

12500

13000

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Mean

99998

0.5224

multiplicity 50

Stephens

Li-Chien, k = 2

LI-Chien, k = 3

Li-Chien, k = 4

h3h3

Entries

Mean

RMS RMS

99998

0.5049

0.2904 0.2904

Q

00.10.2 0.30.40.50.60.70.80.91

number of pairs

9000

9500

10000

10500

11000

11500

12000

12500

13000

Entries

Mean

99998

0.5049

multiplicity 1000

Stephens

Li-Chien, k = 2

Li-Chien, k = 3

Li-Chien, k = 4

h3h3

Entries

Mean

RMS RMS

99998

0.5116

0.2936 0.2936

number of pairs

9000

9500

10000

10500

11000

11500

12000

12500

13000

Entries

Mean

99998

0.5116

multiplicity 200

Stephens

Li-Chien, k = 2

Li-Chien, k = 3

Li-Chien, k = 4

FIG. 10: Stephens’ approximation according to eq. (A2) vs.

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