arXiv:0902.1607v1 [nucl-th] 10 Feb 2009
The Kolmogorov-Smirnov test and its use for the identification of fireball
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
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].
Lattice calculations indicate, however, that at RHIC
and LHC the transition from partonic to hadronic mat-
ter is a rapid but smooth crossover .
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 . An analysis of hydrodynamic insta-
bilities shows that such a scenario may be realistic .
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 , fluctua-
tions of mean pt, rapidity correlations [18, 19], proton
and kaon correlations , φ-meson production , 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
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 
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
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.
d =√nD =
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
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
number of pairs
Number of decimal places n
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
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
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
0 0.1 0.20.3 0.40.50.60.7 0.80.91
number of pairs
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).
0 0.10.2 0.30.4 0.50.6 0.70.80.91
number of pairs
= 1.0, R = 448.4
= 0.12, R = 198.6
= 0.11, R = 42.9
= 0.101, R = -2.2
= 0.09, R = 54.4
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-
number of pairs
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 . 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
2lnt + z
t − z
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
uµ(x) = (coshη coshηt, cosφ sinhηt,
sinφ sinhηt, sinhη coshηt),(7)
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 ) =
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.
The Monte Carlo event generator DRAGON is em-
ployed  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 . 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-
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-
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
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.
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-
R = 10
R = 2.4
0 0.1 0.20.30.40.220.127.116.11.91
Number of pairs
R = 112
R = 1.3
0 0.10.20.30.4 0.50.6 0.70.80.91
number of pairs
R = 5.0
R = 63.5
R = 1.5
R = 0.68
00.1 0.2 0.3 0.40.50.60.7 0.80.91
number of pairs
R = 17.4
R = - 1.1
R = - 0.18
number of pairs
R = 14.6
R = 6.8
R = 5.6
0 0.10.20.30.40.50.6 0.70.80.91
number of pairs
R = 41
R = - 2.1
R = - 2.5
number of pairs
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.
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-
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
R = 67.310 fm
number of pairs
R = 128.320 fm
R = 155.8
R = 4.6
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
50% R = 36.7
number of pairs
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
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
0.1 R = 35.3
0.5 R = 8.5
00.10.20.30.40.5 0.60.7 0.80.91
number of pairs
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.
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-
The work of GT, SV,
APPENDIX A: EVALUATION OF THE
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
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.
Q(D) = K0(D) = −2
Later, Smirnov  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  (found also in Numerical Recipes ):
d = D
?√n + 0.12 +0.11
A different formula with a few terms of an expansion in
1/√n has been derived by Li-Chien [33, 34]. In such an
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
?1 − (−1)k?− 4k2d2
?1 − (−1)k?+ 3
K3(d) = −
?1 − (−1)k??
15− 2?1 − (−1)k?+ 12
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-
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  is employed
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