Chaos, Solitons and Fractals 94 (2017) 86–94

Contents lists available at ScienceDirect

Chaos, Solitons and Fractals

Nonlinear Science, and Nonequilibrium and Complex Phenomena

journal homepage: www.elsevier.com/locate/chaos

Detrended analysis of shower track distribution in nucleus-nucleus

interactions at CERN SPS energy

P. Mali a , S.K. Manna

b , P.K. Haldar c , A. Mukhopadhyay

a , ∗, G. Singh

d

a Department of Physics, University of North Bengal, Siliguri 734013, West Bengal, India b Department of Physics, Chakdaha College, Chakdaha, Nadia 741222, West Bengal, India c Department of Physics, Dinhata College, Dinhata, Cooch Behar 736135, West Bengal, India d Department of Computer and Information Sciences, SUNY at Fredonia, Fredonia, New York 14063, USA

a r t i c l e i n f o

Article history:

Received 12 August 2016

Revised 23 November 2016

Accepted 28 November 2016

Keywords:

High-energy heavy-ion collisions

Bose–Einstein correlation

Multifractal detrended analysis

Generalized hurst exponent

Multifractal spectral function

a b s t r a c t

We have studied the charged particle density fluctuations in 16 O+Ag(Br) and 32 S+Ag(Br) interactions at

200A GeV incident energy in the laboratory frame by using the detrended methods. These methods can

extract (multi)fractal properties of the underlying distributions after filtering out the average trend of

fluctuations associated. Multifractal parameters obtained from data analysis are systematically compared

with event samples generated by the Ultra-relativistic Quantum Molecular Dynamics (UrQMD) model,

where Bose–Einstein correlation (BEC) effect is mimicked via a charge reassignment algorithm imple-

mented as an after burner. Both the experimental and the simulated data are subjected to two different

statistical techniques namely the multifractal detrended fluctuation analysis (MFDFA) and multifractal de-

trended moving average (MFDMA) analysis. The results indicate that for both the interactions considered

the pseudorapidity distributions of the shower tracks are multifractal in nature. Qualitatively, both meth-

ods of analysis and both interactions considered, result in similar behavior of multifractal parameters. We

do however notice significant quantitative differences in certain cases.

© 2016 Elsevier Ltd. All rights reserved.

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1. Introduction

The origin of large fluctuations in multiparticle distribution

within narrow phase space intervals is and has always been

an intriguing issue to the researchers working in the field of

high-energy interactions. Several speculative measures such as the

formation of a quark-gluon plasma (QGP) and its transition to a

hadronic state [1,2] , a self-similar cascade mechanism [3–5] , for-

mation of jets with self-similar branch-cascade process [6] , Bose–

Einstein interference [7] , conventional short-range correlation

[8] etc., have been proposed to interpret the observed fluctuations

that apparently lack any definite pattern. But none of the above

measures is universally capable of explaining all aspects of these

fluctuations. It is believed that the origin of intermittent type of

fluctuations for like-sign charged particles, i.e. a power-law like

increase of scaled factorial moments (SFM) with reducing phase-

space interval size, is dominated by the Bose–Einstein Correlation

(BEC) effect between identical particles [8–11] , i.e. by a symmetry

property of the underlying fields but not by their dynamics. The

experimental data do indeed show some indication of contribution

∗ Corresponding author.

E-mail address: amitabha_62@rediffmail.com (A. Mukhopadhyay).

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http://dx.doi.org/10.1016/j.chaos.2016.11.011

0960-0779/© 2016 Elsevier Ltd. All rights reserved.

rom BEC effect, and in general, introduction of BEC tends to

educe the disagreement between experiments and model predic-

ions. For a more specialized review of the subject one can see

12,13] . It is to be noted that the same physics effects can become

ore (or less) pronounced depending on the choice of the variable

n which it is displayed. In particular it is shown that the rapidity

ariable ( y ) is suitable to extract the dynamical intermittency and

uppress the BEC contribution to factorial moments [12] . On the

ther hand, when analyzed as functions of squared 4-momentum

ransfer ( Q

2 ) between identical pions, the SFM are dominated by

he BEC contribution which shadows the true (dynamical) inter-

ittency. Therefore, a power-law like behavior in terms of Q

2 nei-

her proves nor disproves dynamical intermittency. Another expla-

ation is that the intermittent behavior is a straightforward mani-

estation of short range correlations [8,9,11] . Calculations based on

A22 and UA5 data [8] show that for lower order moments, inter-

ittency can be explained reasonably well in terms of short range

orrelations. As a self-similar nature (self-affine in two dimension)

f the intermittency phenomenon is well established, there is a

atural motivation to pursue a (multi)fractal characterization of

ultiparticle production process, which was initially proposed in

14] . Therefore, it is important to address the intermittency related

ssues from the perspective of a fractal theory. Over the last three

P. Mali et al. / Chaos, Solitons and Fractals 94 (2017) 86–94 87

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four decades or so, various methods of multifractal analysis of

ultiparticle emission data have been developed [14–18] . How-

ver, multiplicity moments in limited phase-space intervals like

he G q moment [14] and the T q moment [18] are very widely used.

In a significant work the JETSET Monte Carlo code was used to

enerate e + e − → hadrons events tuned to the OPAL data at the Z 0

ass, and cumulant moments were studied both in one dimen-

ional rapidity space and in two dimensional rapidity - azimuthal

ngle plane [19] . It was shown that due to mixing of many sources

igher order cumulant moments are suppressed, thereby reducing

he effects of intermittency. This dilution in genuine q- th order

q > 2) correlations was attributed to a reduced probability of get-

ing exactly q number of particles emerging from the same source,

hich from simple combinatorial considerations was shown to

ary with the number of sources present obeying an inverse

ower-law. In high-energy interactions several models describe the

ultiparticle production as a branching process, which can suitably

arametrize the multiplicity distribution of the produced hadrons.

n another work, once again e + e − → hadrons events tuned to the

PAL data at the Z 0 mass, have been used to predict the cumu-

ant moments using the negative binomial, log-normal, modified

egative binomial and generalized negative binomial parametriza-

ion of the multiplicity distribution within limited phase-space

ntervals [20] . While such type of parametrization could describe

he lower order fluctuations reasonably well, significant deviations

rom experiment were observed in higher order moments. It was

oncluded that the observed intermittency is due to a particle

roduction mechanism that is stochastic in nature and originates

rom self-similar branching processes, and genuine higher order

orrelations ( q > 2) have to be taken into account while modeling

hem. Investigations like the above might have caused a decline in

he interest in intermittency and/or multifractality related studies

n multiparticle production. Recent years have however, seen a

esurgence in the number of works in this area of research, partic-

larly due to the introduction of new methods of analysis like the

ultifractal detrended fluctuation analysis (MFDFA) [21] , multifrac-

al detrended moving average (MFDMA) [22] method of analyzing

he time series data and the visibility graph-sandbox algorithm of

nalysis of fractional Brownian motion [23–25] . Moreover, recent

orks show that the structure function of protons at small Bjorken

ariable x , and the momentum distribution of partons inside a

ucleon can as well be interpreted from a fractal perspective

26,27] . Attempts are also being made to relate intermittency to

on-extensive thermodynamics of hadron production [28–31] .

Actually, the MFDFA method is a kind of generalization of

he detrended multifractal analysis (DMA) method developed in

32] to describe the long-range correlation in coding and decoding

NA nucleotides sequence [32,33] . On the other hand, the MFDMA

ethod is an advancement of the moving average or mobile av-

rage technique, which was initially proposed in [34] to estimate

he Hurst exponent of self-affinity signals and further extended to

he detrending moving average (DMA) method by considering the

econd-order difference between the original signal and its moving

verage function [35] . To a certain extent the MFDMA method is

imilar to the MFDFA method. However, in the MFDMA analysis

ne can select the position of the detrending moving window with

espect to the point to be detrended, which gives an extra freedom

o MFDMA over MFDFA as far as the time series data are con-

erned. On the other hand, while in MFDFA one can detrend the

ignal by a polynomial of desired order, in MFDMA the signal has

o be detrended only by the average value of the series. Though

he MFDFA method has been applied over a wide field of stochas-

ic data analysis, as for instance biophysics, geophysics, commerce,

conomics, solid state physics, high-energy interactions etc. [36–

9] , application of MFDMA method has so far been limited only

o time series analysis [50–54] , and very recently to a few high-

nergy nucleus-nucleus ( AB ) collision data [55] . In this work we

dopt both the MFDFA and the MFDMA techniques to analyze the

seudorapidity ( η) density distribution functions of singly charged

articles produced in

16 O+Ag(Br) and

32 S+Ag(Br) collisions at

lab = 200 A GeV. We also systematically compare the experiment

ith the UrQMD model [56] predictions taking the BEC effect into

ccount by using a charge reassignment algorithm [57] . The sub-

equent text of the article is organized as follows. In Section 2 we

rovide the detrended algorithms. The experimental data collection

ethod, the UrQMD model along with the charge reassignment

lgorithm that mimics BEC in the UrQMD output, are briefly

utlined in Section 3 . The results of our analysis are described in

ection 4 . Finally the article is critically summarized in Section 5 .

. The detrended multifractality

As mentioned above, we employ both the detrended fluctuation

ethod [21] and the detrended moving average [22] method to

nalyze the present set of data. In this section the methods are

riefly outlined. Let { x k : k = 1 , 2 , · · · N} be a series of N consecu-

ive measurements at a regular interval of time. A profile series Y i s computed from x k through

(i ) =

i ∑

k =1

[ x k − 〈 x 〉 ] , i = 1 , 2 , · · · , N, (1)

here 〈 x 〉 = (1 /N) ∑ N

k =1 x k is the mean value of { x k }. In the MFDFA

ethod first the profile series Y ( i ) is divided into N n = int(N/n )

on-overlapping segments of equal length n . If N is not a multiple

f n considered, a short part at the end of the profile remains as

eft over. To retain this part of the profile, the same procedure is

epeated in the reverse direction starting from the opposite end.

hereby, 2 N n segments are obtained. Then the variance of all 2 N n

egments are determined subtracting the local trend, obtained

y a best fitted polynomial of desired degree. For an arbitrary

egment ν ∈ (1, N n ) the variance is given as,

2 (ν, n ) =

1

n

n ∑

i =1

{ Y [(ν − 1) n + i ] − Y ν (i ) } 2 , (2)

hereas for ν ∈ (N n + 1 , 2 N n ) it is

2 (ν, n ) =

1

n

n ∑

i =1

{ Y [ N − (ν − N n ) n + i ] − Y ν (i ) } 2 , (3)

here Y ν ( i ) is the fitting polynomial in segment ν . 1 st , 2 nd , ���egree polynomials can be used for detrending purpose, and

ccordingly the method is said to be the first, second, ��� order

FDFA, respectively. From the variance the q th order MFDFA

uctuation function is determined as,

q (n ) =

{

1

2 N n

2 N n ∑

ν=1

[ F 2 (ν, n )] q/ 2

} 1 /q

, ∀ q � = 0

q (n ) = exp

{

1

4 N n

2 N n ∑

ν=1

ln [ F 2 (ν, n )]

}

, q = 0 . (4)

ote that the fluctuation function F q ( n ) can be defined only for

≥ m + 2 , where m is the degree of the detrending polynomial.

oreover, F q ( n ) is statistically unstable for very large n ( ≥ N /4).

n the other hand, in the MFDMA analysis the profile series (1) is

etrened by an average function

˜ y (i ) calculated over a moving

88 P. Mali et al. / Chaos, Solitons and Fractals 94 (2017) 86–94

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window of size n through

(i ) =

1

n

(n −1)(1 −θ ) � ∑

k = −� (n −1) θ Y (i − k ) , (5)

where � ξ is the largest integer not larger than ξ and ξ� is

the smallest integer not smaller than ξ . The parameter θ ∈ [0, 1]

specifies the position of the moving window. In general, the

moving average function includes (n − 1)(1 − θ ) � data points

from the backward region and � (n − 1) θ data points from the

forward region with respect to the value to be detrended, x α (say).

Here we consider θ = 0 . 5 for which the function

˜ Y (i ) is equally

extended towards either side of x α , and hence the moving window

is said to be the ‘central moving’ window. Note that for θ = 0 the

moving average function is calculated from all the n backward

points ( x k < α), while f or θ = 1 the function is calculated from all

the n forward points ( x k > α) with respect to x α , and accordingly

the detrending windows are said to be the backward moving and

forward moving windows, respectively. From the moving average

detrended signal once again one can calculate the variance for

non-overlapping segments of same length ( n ). For an arbitrary νth

segment it is given as,

F

2 (ν, n ) =

1

n

n ∑

i =1

{ Y (i ) − ˜ Y (i ) } 2 , (6)

where i satisfies the criterion: n − � (n − 1) θ ≤ i ≤ N − � (n − 1) θ and N n = � N/n − 1 . The q th order MFDMA fluctuation function

F q (n ) is then defined as,

F q (n ) =

{

1

N n

N n ∑

v =1

[ F

2 (ν, n )] q/ 2

} 1 /q

, ∀ q � = 0 ,

F q (n ) = exp

{

1

2 N n

N n ∑

v =1

ln [ F

2 (ν, n )]

}

, q = 0 . (7)

The scaling behavior of the fluctuation functions F q ( n ) and F q (n )

are examined for a set of q exponents. If the signal { x i } contains

multifractality (long-range correlation), both F q ( n ) and F q (n ) for

large values of n would follow a power-law type of scaling relation

like,

F q ∼ n

h (q ) , (8)

and the exponent h ( q ) would be a nonlinear function of q . For

stationary series h (2) is the well-defined Hurst exponent H [21] ,

and its generalized form h ( q ) is known as the generalized Hurst

exponent. The exponent h (2) = H can be used to analyze corre-

lations in the signal { x k }. The scaling exponent H = 0 . 5 means

that the signal under consideration is uncorrelated; 0.5 < H < 1

implies long-term persistence and 0 < H < 0.5 implies short-term

persistence [58] . For a monofractal series h ( q ) is independent of

q , since the variances F 2 ( ν , n ) and F

2 (ν, n ) are identical for all

the sub-series and hence Eqs. (4) and (7) yield identical values

for all q . If the scaling behaviors of small and large fluctuations

are different, there will be a significant dependence of h ( q ) on

q . For positive values of q , the fluctuation functions (4) and

(7) will be dominated by the large variance which corresponds

to the large deviations from the detrending function, whereas for

negative values of q , major contributions in those functions arise

form small fluctuations from the detrending function. Thus, for

positive/negative values of q, h ( q ) describes the scaling behavior of

the segment with large/small fluctuations.

From the knowledge of h ( q ) one can easily derive the mul-

tifractal scaling exponent τ ( q ), and consequently some other

parameters such as the multifractal (singularity) spectrum f ( α),

the generalized multifractal dimensions D q etc. In order to exam-

ine the connection between the h ( q ) exponent and the multifractal

ass exponent τ ( q ), we consider that the series { x k } is a stationary

nd normalized one. The variance of such series is given by

2 N (ν, n ) = { Y (νn ) − Y [(ν − 1) n ] } 2 . (9)

hen the fluctuation function and its scaling-law are given by

q (n ) =

[

1

2 N s

2 N s ∑

n =1

| Y (νn ) − Y [(ν − 1) n ] | q ]

1 q

∼ s h (q ) (10)

ow if we assume that the length of the series N is an integer mul-

iple of the scale n , then the above relation can be rewritten as,

N/s

n =1

| Y (νn ) − Y [(ν − 1) n ] | q ∼ s qh (q ) −1 . (11)

n the above relation the term under | · | is nothing but the sum of

x k } within an arbitrary νth segment of length n . In the standard

heory of multifractals it is known as the box probability P(ν, n )

or the series { x k }. Hence,

(ν, n ) ≡νn ∑

k =(ν−1) n +1

x k = Y (νn ) − Y ((ν − 1) n ) . (12)

he multifractal scaling exponent τ ( q ) is defined via the partition

unction Z p ( n )

P (n ) ≡N/n ∑

ν=1

|P(ν, n ) | q ∼ s τ (q ) , (13)

here q is a real parameter. From Eqs. (11) –(13) it is clear that the

ultifractal exponent τ ( q ) is related to h ( q ) through the following

elation:

(q ) = qh (q ) − 1 . (14)

or a multifractal set, τ ( q ) is also a nonlinear function of q ,

hereas for a monofractal set τ ( q ) is linear in q . The singularity

trength, also called the Hölder exponent α( q ), is obtained via a

egendre transformation:

(q ) =

∂τ (q )

∂q . (15)

he multifractal spectrum can now be obtained as,

f (α) = q α(q ) − τ (q ) . (16)

he spectral function quantifies the nature of singularity present

n the signal analyzed. For a monofractal signal f ( α) would be a

elta function centered around a particular value of α = α0 (say).

rom τ ( q ) one can derive D q as,

q ≡ τ (q )

q − 1

=

q h (q ) − 1

q − 1

. (17)

ote that, even for a monofractal signal D q depends on q .

. Experiment and simulation

The experimental data used in the present analysis have been

btained from the stacks of Ilford G5 nuclear photo-emulsion

ellicles of size 18 cm × 7 cm × 600 mum, that are hor-

zontally irradiated by the 16 O and

32 S beams, each with an

ncident momentum of 200A GeV/c in the laboratory frame from

he Super-proton Synchrotron (SPS) at CERN (Expt. no. EMU08).

eitz microscopes with a magnification of 300 ×, were used to

can the plates along the projectile tracks to find out the primary

nteractions. Koristka microscopes with a total magnification of

500 × are utilized to measure the polar ( θ ) and azimuthal

ngle ( ϕ) of tracks coming out of each event. According to the

mulsion terminology, tracks emitted from an event (also called

P. Mali et al. / Chaos, Solitons and Fractals 94 (2017) 86–94 89

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star) are classified into four categories namely, shower, grey and

lack tracks, and the projectile fragments. The present analysis

s confined only to the shower tracks which are caused by the

ingly charged particles moving with a relativistic speed v > 0.7 c .

sually the ionization of particles belonging to this category is

≤ 1.4 I 0 , where I 0 ( ≈ 20 grains/100 μm in G5 emulsion) is the

inimum ionization due to any track observed within a plate.

n elaborate discussion on emulsion experiments can be found

n [59] . Shower tracks are produced by charged mesons, mostly

ions, and the tracks are not in general confined within the plate.

or 16 O+Ag(Br) interaction our event sample size was 280 events,

nd that for the 32 S+Ag(Br) interaction was 200 events. Corre-

ponding average shower multiplicities are 〈 n s 〉 = 119 . 26 ± 3 . 59 ,

nd 〈 n s 〉 = 217 . 79 ± 6 . 16 , respectively, where n s is the total num-

er of shower tracks in an event. In an emulsion experiment the

seudorapidity variable η together with azimuthal angle φ of a

rack constitutes a convenient pair of basic variables in terms

f which the particle emission data can be analyzed. In a one-

imensional analysis the η variable is the most preferable one. The

seudorapity variable is an approximation of the dimensionless

oost parameter rapidity ( y ) which is defined as,

=

1

2

ln

[ E + p l E − p l

] , (18)

here E and p l are respectively, the energy and longitudinal mo-

entum of the particle. For light particles moving with relativistic

elocity the pseudorapidity is a convenient replacement of the

apidity variable, where energy and momentum measurements

re difficult to execute. The pseudorapidity variable of a particle is

iven by

= − ln [ tan (θ/ 2) ] , (19)

here θ is the angle of emission of the particle with respect to the

eam direction. An accuracy of δη = 0 . 1 unit could be achieved

hrough the reference primary method of angle measurement.

uclear emulsion experiments in spite of many limitations are

uperior to other big budget experiments in one respect, that they

ffer a very high angular resolution. Due to this reason nuclear

mulsion technique is very suitable to examine local structures of

luster of particles.

The attempt to describe the phenomenology of heavy-ion col-

isions is based on extrapolations from known to so far uncharted

reas of high-energy interactions by using various relativistic

icroscopic models, and therefore, it is possible to extract new

hysics from the background of conventional physics. The UrQMD

Ultra-relativistic Quantum Molecular Dynamics) model developed

y Bass et al. [56] is a microscopic transport theory based on the

ovariant propagation of hadrons along their classical trajectories

n combination with stochastic binary scatterings, color string

ormation and resonance decays. The justification of using a

ransport model like UrQMD is that it treats the final freeze-out

tage dynamically, does not make any equilibrium assumption,

nd describes the dynamics of a hadron-gas like system very well

n and out of the chemical and/or thermal equilibrium. Resonance

bsorption decays and scatterings are handled via the principle

f detailed balance. The collisions terms used in the UrQMD

odel includes more than 50 baryon and 45 meson species. In

ddition, their antiparticles are also taken into account using

harge-conjugation to assure full particle-antiparticle symmetry.

he basic input to this transport model is that, a hadron-hadron

nteraction would occur if the impact parameter b <

√

σtot /π,

here the total cross-section σ tot depends on the isospin of the

nteracting hadrons, their flavors and the center of mass energy

( √

s ) involved. The UrQMD model has been extensively tested in

he GSI Schwerionen-Synchrotron (SIS), BNL Alternating-Gradient

ynchrotron (AGS), and CERN Super Proton Synchrotron (SPS)

nergy domains and provides a robust description of hadronic and

eavy-ion physics phenomenology. Using the UrQMD code (ver-

ion: v3.3p1) [56] with default settings we simulate 16 O + Ag(Br)

nd

32 S + Ag(Br) event samples at 200A GeV/c that are five times

s large as the experiments.

Since UrQMD is a classical Monte-Carlo model, quantum statis-

ical effects like BEC are not embedded into the code. We make an

ttempt to include the BEC effect numerically in the form of the

o called ‘charge reassignment’ algorithm [57] by using the UrQMD

utput. The algorithm is very briefly described below. The UrQMD

ode provides 4-spacetime coordinates and 4-momenta of all

articles coming out of an interaction. The particle information are

ontained in an ASCII file that can be written in the OSCAR format

file ‘test.f19’ ). Each particle entry in an event contains a

erial number, the particle 4-momentum, the particle rest mass

, the final freeze-out 4-spacetime coordinates and the particle

D. As a first step of the algorithm we choose at random a meson

rom an event, call it the i th one, and assign a charge ‘sign’ i.e.,

, – or 0 to it, irrespective of its original charge with weight

actors respectively, given by p + = n + /n, p − = n −/n and p 0 = n 0 /n .

ere n + , n −, n 0 are respectively, the number of + ve, − ve and

eutral mesons in the event considered, while n (= n + + n − + n 0 )

s obviously the total number of mesons in that event. This meson,

ay the i th one, defines a new phase-space cell. In the next step,

e calculate the distances in the 4-momenta δi j (p) =

∣∣p i − p j ∣∣ and

-coordinates δi j (x ) =

∣∣x i − x j ∣∣ between the already chosen meson

nd all other mesons (indexed by j ) that are not yet assigned any

harge ‘sign’ . To each j th particle we now assign a weight factor

i j = exp

[ −1

2

δ2 i j (x ) δ2

i j (p) ] , (20)

hat characterizes the bunching probability of the particles in a

iven cell. Then we start generating uniformly distributed random

umbers r ∈ (0 , +1) . If r < P ij we reassign the same charge ‘sign’

s the i th one to the j th meson. We continue the process until ei-

her r exceeds P ij , or until all mesons in the event having same

harge ‘sign’ as the i th one are exhausted. Now we go back to our

rst step and again choose another meson at random from the

ool for which the charge reassignment has not yet been done. Ob-

iously, the weight factors p ±, 0 will now be modified, as some of

he particles present in the event are already used up. The algo-

ithm is then repeated until mesons belonging to all charge vari-

ties in the event are used up. The UrQMD model provides all pion

airs with −Q = (p i − p j ) 2 = (�E) 2 − (�p) 2 < 0 . In order to keep

he value of the factor P ij below unity, only the pion pairs having

pace-like separation: −R 2 = (x i − x j ) 2 = (�t) 2 − (�x ) 2 < 0 are

onsidered [60] . Without changing the overall set of 4-momenta,

-coordinates, or total mesonic charge of the system, we can in

his way generate clusters of identical charge states of mesons.

. Results

In Fig. 1 we present the pseudorapidity distribution (a) of a

ingle 16 O+Ag(Br) event, and (b) for all the 280 16 O+Ag(Br) events.

imilar distributions for 32 S+Ag(Br) interaction are shown in

ig. 2 . Large fluctuations observed in single event pseudorapidity

istributions may arise from statistical factors as well as from

ome non-trivial dynamics. As the single event distributions are

ummed/averaged over a large event sample, the fluctuating pat-

erns are smoothed out to result in normal distributions as shown

n the right panel of Figs. 1 and 2 , and eventually the fractal

tructure (if any) is also washed out. Therefore, it is essential to

erform the analysis on an event-by-event basis. So the fluctuation

unctions that are used in MFDFA and MFDMA method, are first

alculated for each event and then they are averaged over all the

vents. The scaling behavior of the event averaged fluctuation

90 P. Mali et al. / Chaos, Solitons and Fractals 94 (2017) 86–94

Fig. 1. Pseudorapidity distributions of shower particles in 16 O+Ag(Br) interaction at

200A GeV. (a) Pseudorapidity distribution for a single event and (b) Pseudorapidity

distribution for whole sample.

Fig. 2. The same as in Fig. 1 but for 32 S+Ag(Br) interaction at 200A GeV.

a

d

n

d

s

t

o

s

m

t

a

0

a

1

v

0

0

0

i

f

t

l

m

b

r

m

g

n

p

q

m

3

w

f

c

r

i

b

v

t

t

s

t

v

d

s

b

r

t

t

i

T

w

w

a

D

o

M

t

e

a

i

d

t

i

functions is schematically represented through a plot of Eq. (4) for

MFDFA method in Fig. 3 , and Eq. (7) for MFDMA method in Fig. 4

for q = 0 , ±2 , ±5 . We have calculated all the values of < F q ( n ) >

and 〈 F q (n ) 〉 for q = −5 to +5 in steps of 0.25, but to maintain clar-

ity all of them are not shown in these figures. In each figure the re-

sults of experiment are compared with those of the UrQMD + BEC

simulated data. Average fluctuation functions obtained from exper-

imental data show more or less similar trend as the corresponding

UrQMD + BEC simulation for both

16 O+Ag(Br) and

32 S+Ag(Br) inter-

actions, and in both MFDFA and MFDMA methods. It is to be noted

that the variation of < F q ( n ) > and 〈 F q (n ) 〉 is not linear over the

entire scale region. In each case, compared to the positive q values,

significant nonlinearity is observed in the q < 0 region, specially

at low values of the scale parameter ( n ), and the nonlinearity of

average fluctuation functions is more prominent in the MFDFA

method than in the MFDMA method. Therefore, in our calculation

we consider a range of 8 ≤ n ≤ 24 for the MFDFA method, and

6 ≤ n ≤ 24 for the MFDMA method. We keep it in mind that

these regions produce better scaling for the averaged fluctuation

functions. Moreover MFDMA method shows better scaling over

MFDFA for both the interactions considered in this analysis.

The nature of fractal structure in the particle density func-

tion can also be analyzed with the help of generalized Hurst

exponent h ( q ). We extracted the values of h ( q ) using the ln <

F q > versus ln n (MFDFA) and ln 〈 F q 〉 versus ln n (MFDMA) plots

both ranging from q = −5 to 5 in the n -region mentioned above.

The q −dependence of h ( q ) is graphically shown in Fig. 5 for the

MFDFA method for 16 O+Ag(Br) collisions and

32 S+Ag(Br) collisions.

Similar plots for the MFDMA method are shown in Fig. 6 . In

each of the figures mentioned above, the experimental results

are compared with the corresponding UrQMD+BEC simulation. It

is clear from Fig. 5 that the h ( q ) values for the simulated data

re marginally higher than those obtained from the experimental

ata in the q < 0 region, and in q > 0 region the differences are

ot statistically significant. However, for the MFDMA method as

epicted in Fig. 6 just the opposite behavior is observed. The h ( q )

pectra obtained from MFDMA is more smoothly varying than

hat in the MFDFA case for both the interactions. In each case the

bserved variation is nonlinear and h ( q ) decreases with q which

ignifies multifractal nature of the pseudorapidity distribution as

anifested by different scaling behaviors for large and small fluc-

uations. The h (q = 2) values obtained from the MFDFA method

re always very close to unity – (a) for 16 O+Ag(Br) interaction:

.9558 ± 0.0438 (Experiment), 0.9486 ± 0.0436 (UrQMD+BEC)

nd (b) for 32 S+Ag(Br) interaction: 1.0700 ± 0.0388 (Experiment),

.0998 ± 0.0513 (UrQMD+BEC). On the other hand, the h (q = 2)

alues obtained from the MFDMA method are a little higher than

.5 – (a) for 16 O+Ag(Br) interaction: 0.6165 ± 0.0407 (Experiment),

.6 6 60 ± 0.0370 (UrQMD+BEC) and (b) for 32 S+Ag(Br) interaction:

.6162 ± 0.0333 (Experiment), 0.7221 ± 0.0463 (UrQMD+BEC). It

s mentioned above that H > 0.5 indicates long-range correlation

or a stationary signal. Therefore, as the present analysis suggests,

he η distribution functions for 16 O+Ag(Br) and

32 S+Ag(Br) col-

isions are long-range correlated. The observed variation of the

ultifractal mass exponents τ ( q ) is nonlinear in each case, see the

ottom panels of Figs. 5 and 6 . In each figure the experimental

esults are plotted along with the simulated data, which show

arginal or no difference. The degree of non-linearity of τ ( q ) can

ive us an idea about the degree of multifractality. The observed

onlinearity supports the presence of multifractal pattern in the

article density distributions for both the interactions studied.

In order to study the local scaling patterns and to get a

uantitative idea about multifractality, we have computed the

ultifractal spectrum f (α) = qα(q ) − τ (q ) for the 16 O+Ag(Br) and2 S+Ag(Br) interactions, and compared the experimental results

ith the simulated one. In Figs. 7 and 8 , f ( α) is plotted with αor the MFDFA and MFDMA methods, respectively. From Fig. 7 it is

lear that in the MFDFA method, in higher α region the simulated

esult deviates significantly from the experimental one for both

nteractions. Whereas in the MFDMA method significant deviations

etween experiment and simulation are observed only at low αalues for both interactions. Moreover, in the MFDFA method due

o lack of smoothness in the h ( q ) versus q plot, some points near

he peak region of the spectrum could not be obtained. We fit the

pectrum to a 4-th degree polynomial function around the posi-

ion of its maximum at α0 and extrapolate the curves to f (α) = 0

alues on either side of the maxima. A right-skewed spectrum

enotes relatively strongly weighted high fractal exponents, corre-

ponding to fine structures [61] . The width of the spectrum defined

y W = α1 − α2 , with f (α1 ) = f (α2 ) = 0 , and the skew parameter

= (α1 − α0 ) / (α0 − α2 ) are calculated. The strength of multifrac-

ality is measured by W . If W is significantly greater than zero,

hen the η-distributions are multifractal nature. On the contrary,

f W gets close to zero the η-distribution is almost monofractal.

he spectrum parameters are presented in Table 1 . The spectrum

idth appears to be a little wider in the present case than what

e obtained earlier using the G q moments for 32 S+Ag(Br) inter-

ction [62] . We finally obtain the generalized fractal dimensions

q , and plot them against order number q in Fig. 9 . The D q values

btained from experiment and simulation for both MFDFA and

FDMA method are plotted together for easy comparison for each

ype of interaction. From the left panel diagram it is clear that the

xperimental and simulated results for 16 O+Ag(Br) interactions, are

lmost superposed on each other and practically little deviation

s seen while applying the MFDFA method. However, significant

eviations between experiment and simulation are observed in

he MFDMA case. For both interactions the D q values are less

n MFDMA compared to MFDFA. We have also noticed from the

P. Mali et al. / Chaos, Solitons and Fractals 94 (2017) 86–94 91

Fig. 3. Scaling behavior of the event averaged MFDFA fluctuation functions < F q ( s ) > for q = 0, ±2 and ±5. (a) 16 O+Ag(Br) collisions at 200A GeV and (b) 32 S+Ag(Br)

collisions at 200A GeV.

Fig. 4. The same as in Fig. 3 but for the MFDMA method.

Table 1

The centroid ( α0 ), width ( W ) and skewness ( r ) parameters of the multifractal spectra for

the studied event samples.

Data sample MFDFA MFDMA

α0 W r α0 W r

16 O+Ag/Br (Experiment) 1 .242 1 .177 1 .023 0 .830 1 .277 1 .318 16 O+Ag/Br (UrQMD+BEC) 1 .277 1 .282 1 .057 0 .840 1 .231 1 .438 32 S+Ag/Br (Experiment) 1 .321 1 .170 1 .175 0 .872 1 .530 1 .517 32 S+Ag/Br (UrQMD+BEC) 1 .389 1 .271 1 .136 0 .932 1 .469 1 .479

92 P. Mali et al. / Chaos, Solitons and Fractals 94 (2017) 86–94

Fig. 5. The generalized Hurst exponent spectra h(q) and the multifractal exponent spectra τ (q ) = qh (q ) − 1 for MFDFA method. The (statistical) error bars are smaller than

the point size. The h(q) exponents are calculated in the scale intervals 8 ≤ s ≤ 24 and 8 ≤ s ≤ 24 respectively for the (a) 16 O+Ag(Br) collisions at 200A GeV and (b) 32 S+Ag(Br)

collisions at 200A GeV.

Fig. 6. The same as in Fig. 5 but for the MFDMA method using the scale intervals 6 ≤ s ≤ 24 and 6 ≤ s ≤ 24.

s

t

v

C

i

t

t

e

o

h

o

p

m

present analysis that there are small but observable dependences

of D q on q . In previous cases while using the G q and T q moments

we observed a stronger dependence. Our general conclusions are

therefore, the experimental as well as simulated distributions

for both interactions considered in this analysis are multifractal

in nature, the multifractal spectra are slightly right skewed, the

results are however dependent on the method of analysis.

5. Discussion

We have presented a systematic study on charge particle den-

sity fluctuation in pseudorapidity distribution in the framework of

MFDFA and MFDMA techniques in

16 O+Ag(Br) and

32 S+Ag(Br) colli-

ions at 200A GeV. The results show that for both the interactions,

he pseudorapidity distributions having sharp spikes and empty

alleys, are long-range correlated and highly mulifractal in nature.

omparison in terms of multifractal parameters like the general-

zed Hurst exponents h ( q ), the mass exponents τ ( q ) and the mul-

ifractal spectral function f ( α) obtained from the experiment with

hose obtained from the UrQMD generated events where the BEC

ffect has been mimicked, reveals that the observed multifractality

f the experiment goes beyond BEC, and contribution of nontrivial

itherto unknown dynamics should be taken into account. The

bservations presented here are qualitatively consistent with our

revious findings [49,55] and also with the particle production

echanism of the UrQMD model [56] . However, quantitative

P. Mali et al. / Chaos, Solitons and Fractals 94 (2017) 86–94 93

Fig. 7. The multifractal singularity spectra for MFDFA method in (a) 16 O+Ag(Br) col-

lisions and (b) 32 S+Ag(Br) collisions at 200A GeV. Lines joining points are shown

only to guide the eye.

Fig. 8. The same as in Fig. 7 but for the MFDMA method.

Fig. 9. Order dependence of the generalized fractal dimension D q for (a) 16 O+Ag(Br)

collisions and (b) 32 S+Ag(Br) collisions at 200A GeV. In each panel results of MFDFA

and MFDMA method are compared together.

d

m

u

c

M

d

t

c

t

b

s

t

a

t

o

R

[

[

[

[

[

[

[

[

[

[

[

[

[

ifferences between the results of the MFDFA and MFDMA

ethods are prominent in some cases. From this analysis we

nderstand that as far as the multiparticle production data are

oncerned where the signal length is an important issue, the

FDFA method like the MFDMA method is not quite capable of

iscriminating the statistical noise from the signal. However, unlike

he MFDFA (first order) method the MFDMA analysis produces a

omplete and stable singularity spectrum which indicates the exis-

ence of a few events with large fluctuations (might be nontrivial)

ut the coarse fluctuations arise from the statistical noise. The ob-

ervation is grossly identical for both the interactions considered in

he present investigation. In conclusion, we argue that the MFDFA

nd MFDMA techniques may turn out to be effective tools for mul-

ifractal characterization of multiparticle data if a reliable method

f filtering out the statistical noise can properly be implemented.

eferences

[1] Van Hove L . Two problems concerning hot hadronic matter and high energy

collisions (equilibrium formation, plasma deflagration). Z Phys C 1983;21:93–8 .

[2] Gyulassy M , Kajantie K , Kurki-Suonio H , McLerran L . Deflagrations and deto-nations as a mechanism of hadron bubble growth in supercooled quark gluon

plasma. Nucl Phys B 1984;237:477–501 . [3] Bialas A , Peschanski R . Moments of rapidity distributions as a mea-

sure of short-range fluctuations in high-energy collisions. Nucl Phys B1986;273:703–18 .

[4] Bialas A , Peschanski R . Intermittency in multiparticle production at high en-ergy. Nucl Phys B 1988a;308:857–67 .

[5] Bialas A , Peschanski R . Random cascading models and the link between

long-range and short-range interactions in multiparticle production. Phys LettB 1988b;207:59–63 .

[6] Ochs W , Wosiek J . Intermittency and jets. Phys Lett B 1988;214:617–20 . [7] Carruthers P , Friedlander EM , Shih CC , Weiner RM . Phys Lett B

1989;222:487–92 . [8] Carruthers P , Sarcevic I . Phys Rev Lett 1989;63:1562 ; Erratum: Phys Rev Lett

1989;63:2612 .

[9] Capella A , Fialkowski K , Krzywicki A . Has intermittency been observed in mul-ti-particle production? Phys Lett B 1989;230:149–52 .

[10] Gyulassy M , Festschrift L . In: Giovannini A, Kittel W, editors. LVH. Singapore:World Scientific; 1990. p. 479 .

[11] Carruthers P , Eggers HC , Sarcevic I . Analysis of multiplicity moments forhadronic multiparticle data. Phys Lett B 1991;254:258–66 .

[12] Andreev IV , Biyajima M , Dremin IM , Suzuki N . Int J Mod Phys A

1995;10:3951–84 . [13] Kittel W , De Wolf EA . Soft multihadron dynamics. Singapore: World Scientific;

2005 . [14] Hwa RC . Fractal measure in multiparticle production. Phys Rev D

1990;41:1456–62 . [15] Grassberger P , Procaccia I . Dimensions and entropies of strange attractors from

a fluctuating dynamics approach. Physica D 1984;13:34–54 .

[16] Halsey TC , Jensen MH , Kadanoff LP , Procaccia I , Shriman BI . Fractal mea-sures and their singularities: the characterization of strange sets. Phys Rev A

1986;33:1141–51 ; Erratum: Phys Rev A 1986;34:1601 . [17] Paladin G , Vulpiani A . Anomalous scaling laws in multifractal objects. Phys Rep

1987;156:147–224 . [18] Takagi F . Multifractal structure of multiplicity distribution in particle collisions

at high energies. Phys Rev Lett 1994;72:32–5 .

[19] Alexander G , Sarkisyan E . The many sources effect on the genuine multihadroncorrelations. Nucl Phys Proc Suppl 2001;92:211–22 .

20] Sarkisyan E . Description of local multiplicity fluctuations and genuine multi-plicity correlations. Phys Lett B 20 0 0;477:1–12 .

[21] Kantelhardt JW , Zschiegner SA , Koscielny-Bunde E , Havlin S , Bunde A , Stan-ley HE . Multifractal detrended fluctuation analysis of nonstationary time se-

ries. Physica A 2002;316:87–114 .

22] Gu G-F , Zhou WX . Detrended moving average algorithm for multifractals. PhysRev E 2010;82:011136 .

23] Lacasa L , Luque B , Luque J , Nuño JC . The visibility graph: a new method forestimating the hurst exponent of fractional brownian motion. Eur Phys Lett

20 09;86:30 0 01 . [24] Lacasa L , Toral R . Description of stochastic and chaotic series using visibility

graphs. Phys Rev E 2010;82:036120 . 25] Tél T , Fülöp A , Vicsek T . Determination of fractal dimensions from geometrical

fractals. Physica A 1989;159:155–66 .

26] La stovi cka T . Self-similar properties of proton structure at small x . Eur Phys JC 2002;24:529–33 .

[27] Tokarev MV , Zborovsk I , Aparin A . Fractal structure of hadrons in processeswith polarized protons at SPD NICA (proposal for experiment). Phys Part Nucl

Lett 2015;12:48–58 . 28] Wilk G , Wlodarczyk . Self-similarity in jet events following from p-p collisions

at LHC. Phys Lett B 2013;727:163–7 .

29] Utyuzh OV . Fluctuations, correlations and non-extensivity in high-energy col-lisions. Wrsaw, Poland: The Andrzej Soltan Institute for Nuclear Studies, Ph.d.

thesis; 2002 . 30] Deppmann A . Properties of hadronic systems according to the non-extensive

self-consistent thermodynamics. J Phys G 2014;41:055108 . [31] Deppmann A . Thermodynamics with fractal structure, tsallis statistics and

hadrons. Phys Rev D 2016;93:054001 .

32] Peng C-K , Buldyrev SV , Havlin S , Simons M , Stanley HE , Gold-berger AL . Mosaic organization of DNA nucleotides. Phys Rev E 1994;49:

1685–1689 . [33] Ossadnik SM , Buldyrev SB , Goldberger AL , Havlin S , Mantegna RN , Peng C-K ,

et al. Correlation approach to identify coding regions in DNA sequences. Bio-phys J 1994;67:64–70 .

34] Vandewalle N , Ausloos M . Crossing of two mobile averages: a method for mea-

suring the roughness exponent. Phys Rev E 1998;58:6832 . [35] Alessio E , Carbone A , Castelli G , Frappietro V . Second order moving average

and scaling of stochastic time series. Eur Phys J B 2002;27:197–200 . 36] Matia K , Ashkenazy Y , Stanley HE . Multifractal properties of price fluctuations

of stocks and commodities. Europhys Lett 2003;61:422–8 . [37] O swiecimka , Kwapie n , Dr ozd z S . Multifractality in the stock market: price in-

crements versus waiting times. Physica A 2005;347:626–38 .

38] Mali P . Multifractal characterization of global temperature anomalies. TheorAppl Climatol 2015;121:641–8 .

39] Koscielny-Bunde E , Kantelhardt JW , Braun P , Bunde A , Havlin S . Long-term per-sistence and multifractality of river runoff records: detrended fluctuation stud-

ies. J Hydrol 2006;322:120–37 .

94 P. Mali et al. / Chaos, Solitons and Fractals 94 (2017) 86–94

[40] Veronese TB , Rosa RR , Bolzan MJA , Fernandes FCR , Sawant HS , Karlicky M .Fluctuation analysis of solar radio bursts associated with geoeffective x-class

flares. J Atmos Sol Terr Phys 2011;731:311–1316 . [41] Benicio R , Stosic T , de Figueiredo PH , Stosic BD . Multifractal behavior of wild–

land and forest fire time series in brazil. Physica A 2013;392:6367–74 . [42] Liao F , Struck DB , Macrobert M , Jan Y-K . Multifractal analysis of nonlinear com-

plexity of sacral skin blood flow oscillations in older adults. Med Biol Eng2001;49:925–34 .

[43] Esen F , Caglar S , Ata N , Ulus T , Birdane A , Esen H . Fractal scaling of laser

doppler flowmetry time series in patients with essential hypertension. Mi-crovasc Res 2001;82:291–5 .

[44] Cardenas N , Kumar S , Mohanty S . Dynamics of cellular response to hypotonicstimulation revealed by quantitative phase microscopy and multifractal de-

trended fluctuation analysis. Appl Phys Lett 2012;101:203702 . [45] Ignaccolo M , Latka M , West BJ . Detrended fluctuation analysis of scaling

crossover effects. Europhys Lett 2010;90:10 0 09 .

[46] Baroni MPMA , De Wit A , Rosa RR . Detrended fluctuation analysis of numericaldensity and viscous fingering patterns. Europhys Lett 2010;92:64002 .

[47] Zhang YX , Qian WY , Yang CB . Multifractal structure of pseudorapidity and az-imuthal distributions of the shower particles in AU+AU collisions at 200a GeV.

Int J Mod Phys A 2007;23(18):2809–16 . [48] Wang X , Yang CB . Int J Mod Phys E 2013;22(4) . 1350021–8

[49] Mali P , Sarkar S , Ghosh S , Mukhopadhyay A , Singh G . Multifractal detrended

fluctuation analysis of particle density fluctuations in high-energy nuclear col-lisions. Physica A 2015;424:25–33 .

[50] Wang Y , Wu C , Pan Z . Multifractal detrending moving average analysis on theUS dollar exchange rates. Physics A 2011;390:3512–23 .

[51] Ruan Y-P , Zhou W-X . Long-term correlations and multifractal nature in theintertrade durations of a liquid chinese stock and its warrant. Physica A

2011;390:1646–54 . [52] Wang F , Wang L , Zou RB . Multifractal detrended moving average analysis for

texture representation. Chaos 2014;24(3):033127 . [53] Shi W , Zou R , Wang F , Su L . A new image segmentation method based on

multifractal detrended moving average analysis. Physica A 2015;432:197–205 . [54] Mali P . Multifractal detrended moving average analysis of global temperature

records. J Stat Mech 2016:013201 .

[55] Mali P , Mukhopadhyay A , Singh G . Multifractal detrended moving averageanalysis of particle density functions in relativistic nuclear collisions. Physica

A 2016;450:323–32 . [56] Bass SA , Belkacem M , Bleicher M , Brandstetter M , Bravina L , Ernst C , et al. Mi-

croscopic model for ultrarelativistic heavy ion collision. Prog Nucl Part Phys1998;41:255–369 .

[57] Utyuzh OV , Wilk G , Wlodarczyk Z . Numerical modelling of Bose–Einstein cor-

relations. Phys Lett B 2001;522:273–9 . [58] Movahed MS , Jafari GR , Ghasemi F , Rahvar S , Tabar MRR . Multifractal de-

trended fluctuation analysis of sunspot time series. J Stat Mech 20 06:P020 03 . [59] Barakas WH . Nuclear research emulsions Vols. I and II. New York: Academic

Press; 1963 . [60] Bystersky M . Modeling Bose–Einstein correlation in heavy ion collision at

RHIC. Nucleonica Supplement 2004;49:S37–9 .

[61] Shimizu Y , Thurner S , Ehrenberger K . Fractals 2002;10:103 . [62] Ghosh MK , Mukhopadhyay A , Singh G . Multifractal moments of particles pro-

duced in 32 S-Ag/Br interaction at 200 GeV/c. J Phys G 2006;32 . 2293–2246