FRACTAL PROPERTIES OF PARTICLES IN PHASE SPACE FROM URQMD MODEL

April 19, 2013 16:59 WSPC/143-IJMPE S0218301313500213

International Journal of Modern Physics EVol. 22, No. 4 (2013) 1350021 (8 pages)c© World Scientific Publishing Company

DOI: 10.1142/S0218301313500213

FRACTAL PROPERTIES OF PARTICLES IN

PHASE SPACE FROM URQMD MODEL

X. WANG∗ and C. B. YANG†

Institute of Particle Physics, Central China Normal University,

Wuhan 430079, P. R. China∗[email protected]†[email protected]

Received 11 January 2013Revised 20 February 2013Accepted 21 February 2013Published 23 April 2013

Nonstatistical dynamical fluctuations by means of the detrended fluctuation analysis(DFA) and multifractal DFA are studied. We used a two-dimensional algorithm forthe analyses. By choosing different particles generated by UrQMD code, we show thatdifferent particles all have good scaling behaviors with bin size. The correlation betweendifferent identified particles are also discussed.

Keywords: Detrended fluctuation analysis (DFA); multifractal DFA; phase space;UrQMD model.

PACS Number(s): 25.75.Gz, 21.65.Qr, 21.60.Cs, 21.60.Ev, 21.60.Ka, 21.90.+f

1. Introduction

Fractals and multifractals are ubiquitous in natural and social sciences.1 Detrended

fluctuation analysis (DFA)2,3 and multifractal DFA4(MFDFA) are methods pro-

posed to investigate the fractal and multifractal properties of observable quantities.

In high-energy heavy-ion collisions, the single particle density distribution in

the final states exist fluctuations. These fluctuations have statistical and dynami-

cal origins. The dynamical fluctuations can reveal some physics mechanism in the

collisions. If the fluctuations under consideration are self-similar at all scales, the

normalized factorial moments behave as an inverse power of the bin size and that

phenomenon is known as “intermittency”. The intermittency is a phenomenon of

having probability higher than statistical one for high density in a narrow region

in phase space. There has been several speculative pictures to interpret the inter-

mittency results in high-energy interactions.5 For example, the intermittency phe-

nomenon can be explained in terms of the ordinary Bose–Einstein correlation.6,7

†Corresponding author.

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http://dx.doi.org/10.1142/S0218301313500213

mailto:[email protected]

mailto:[email protected]


X. Wang & C. B. Yang

Incorporating the Bose–Einstein correlation into numerical modeling certainly re-

duces the mismatch between the observation and the Monte Carlo simulation.8

The intermittency may as well be due to a QCD parton shower cascading process

of particle emission,9 or it may even be due to a nonthermal phase transition.10

Large fluctuations in the final state particle density, particularly in nucleus–nucleus

collisions, may be also an outcome of a transition from the exotic quark–gluon

plasma to the ordinary hadronic phase.11,12 Until now, the picture about the origin

of fluctuation is neither complete nor very clear. There are still some unresolved

issues related to this phenomenon for further scrutinizing.

The existence of dynamical components in particle density fluctuations has been

confirmed in many high-energy interactions. DFA and MFDFA can be good meth-

ods to study the nonstatistical dynamical fluctuations in the single particle density

distribution. It can remove the statistical fluctuations and retain dynamical ones.

The one-dimensional MFDFA has been investigated in Ref. 13 by using high-energy

nuclear emulsion experimental data. It studied the properties of particle fluctuations

in rapidity or in the pseudorapidity space. Because the actual process of particles

production takes place in three dimensions, the effects of fluctuation are reduced or

they can even be completely washed out14 when the dynamics is projected into a

lower dimension. So, in principle one should investigate the fractal properties of a

colliding system in three-dimensional phase space. In this paper, as an extension of

the one-dimension methods, we use the DFA and MFDFA15 to study the dynamical

fluctuations of particle distributions in two-dimensional phase space.

This paper is organized as follows. In Sec. 2, we will discuss how to prepare the

data to be used in this paper. In Sec. 3, we explicitly represent the algorithm for

the two-dimensional DFA and the two-dimensional MFDFA. Then, in Sec. 4, we

will show the DFA and MFDFA results for different particles generated from the

UrQMD code.17 Finally, Sec. 5 will give a brief summary.

2. UrQMD Model and Data to be Analyzed

The UrQMD is a microscopic transport model based on the covariant propaga-

tion of all hadrons on classical trajectories in combination with stochastic binary

scatterings, color string formation and resonance decay.16 In this paper, we use

the vesion-3.1 (Ref. 17), which incorporate hydrodynamical evolution for Au+Au

collisions at√SNN = 200 GeV. When generating data, we choose the impact pa-

rameter to be 2.3 fm, corresponding to centrality 0 ∼ 5%. Some of the parameters

in the code are set as follows: the CTOption(45) parameter is 1 to enable Hydro

mode; CTParam(61)=0.1, CTParam(63)= 0.5 and CTParam(66)=2 to make the

hydro mode work appropriately. We collected 2655 events for our analysis. The

UrQMD can provide the four-coordinates and the four-momenta of all particles.

In our analysis, the phase space is spanned by pseudorapidity(η) and azimuthal

angle(ψ). Then, we will investigate fluctuation properties in the η − ψ space.

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Fractal Properties in Phase Space

3. Two-Dimensional DFA and MFDFA Methods

3.1. Two-dimensional DFA

The two-dimensional DFA is a straightforward extension of the one-dimensional

DFA and consists of the following steps:

Step 1: Consider a self-similar (or self-affine) two-dimensional surface, which

is denoted by a two-dimensional array X(i, j), where i = 1, 2, . . . ,M and j =

1, 2, . . . , N . The surface is partitioned into Ms×Ns disjoint square segments of the

same size s× s, where Ms = [M/s] and Ns = [N/s]. Each segment can be denoted

by Xν,ω such that Xν,ω(i, j) = X(l1+ i, l2+ j) for 1 ≤ i, j ≤ s, where l1 = (ν− 1)s

and l2 = (ω − 1)s.

Step 2: For each segment Xν,ω identified by ν and ω, the cumulative sum

uν,ω(i, j) is calculated as follows:

uν,ω(i, j) =

i∑

k1=1

j∑

k2=1

Xν,ω(k1, k2) , (1)

where 1 ≤ i, j ≤ s. Note that uν,ω itself is a surface.

Step 3: The trend of the constructed surface uν,ω can be determined by fitting it

with a prechosen bivariate polynomial function u. Of course, the obtained detrended

fluctuations depend on the choice of the detrending functions. The simplest function

could be a plane. One can adopt one of the following detrending functions as an

example to test the validation of the methods:

uν,ω(i, j) = ai+ bj + c , (2)

uν,ω(i, j) = ai2 + bj2 + c, (3)

uν,ω(i, j) = aij + bi+ cj + d, (4)

uν,ω(i, j) = ai2 + bj2 + ci+ dj + e, (5)

uν,ω(i, j) = ai2 + bj2 + cij + di + ej + f, (6)

where 1 ≤ i, j ≤ s and a, b, c, d, e and f are free parameters to be determined. These

parameters can be estimated easily through simple matrix operations, derived from

the least squares method.

Once a detrending function is chosen, one can obtain the residual matrix

εν,ω(i, j) = uν,ω(i, j)− uν,ω(i, j) . (7)

The detrended fluctuation function F (ν, ω, s) for the segment Xν,ω is defined via

the sample variance of the residual matrix εν,ω(i, j) as follows:

F 2(ν, ω, s) =1

s2

s∑

i=1

s∑

j=1

[εν,ω(i, j)]2 . (8)

Note that the mean of the residual is nonzero due to the detrending procedure.

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Step 4: The overall detrended fluctuation is calculated by averaging F 2(ν, ω, s)

over all the segment, that is,

F 2(s2) =1

MsNs

Ms∑

ν=1

Ns∑

ω=1

F 2(ν, ω, s) , (9)

Step 5: Varying the value of s in the range from smin ≃ 6 to smax ≃min(M,N)/4, one can determine the scaling relation between the overall detrended

fluctuation function F (s) and the size scale s, which reads

F (s) ∝ sH , (10)

where H is the Hurst index of the surface, which can be related to the fractal

dimension by D = 3 −H . If there is no dynamical fluctuation, H = 1/2. So that

departure from H = 1/2 is an indication of dynamical fluctuations. The choice of

smin ≃ 6 to smax ≃ min(M,N)/4 is made empirically to ensure a better scaling

relation and at least two partitions in each direction.

3.2. Two-dimensional MFDFA

Analogous to the generalization of one-dimensional DFA to one-dimensional

MFDFA, the two-dimensional MFDFA can be described similarly, such that the

two-dimensional DFA serves as a special case of the two-dimensional MFDFA. The

two-dimensional MFDFA follows the same first three steps as in the two-dimensional

DFA and has one revised step.

Step 4: The overall detrended fluctuation is calculated by averaging over all the

segments in the following way:

Fq(s) =

{

1

MsNs

Ms∑

ν=1

Ns∑

ω=1

[F (ν, ω, s)]q

}1/q

, (11)

where q can take any real value except for q = 0. When q = 0, one can get the

following equation by taken the limit q → 0 in the last equation:

F0(s) = exp

{

1

MsNs

Ms∑

ν=1

Ns∑

ω=1

ln[F (ν, ω, s)]

}

. (12)

The same fifth step as for DFA can be used for MFDFA. One can hope to get

a scaling behavior as

Fq(s) ∼ sh(q) . (13)

4. Results for Identified Particles from UrQMD Data

Following the above steps, we can induce the DFA and MFDFA results of identified

particles produced by UrQMD event generator by using the parameters we have

discussed in the last section. The pseudorapidity and azimuthal space we study is

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from ηmin = −6 to ηmax = 6 and in all azimuthal angle range. The phase space is

divided by M , N = 200 sections. We have used the detrending equations (2) and

(3) in the last section and found that the results have almost invisible difference.

So, we show in the following the results with the use of function (2) as an example

analysis. We only choose nine kinds of particles to explore. They are K0, K+, K−,

π0, π+, π−, proton, Λ, neutron. The DFA and MFDFA results for K0, K+, K− are

almost overlapped. So do for π0, π+, π− and for proton, neutron. This means that

the property of DFA and MFDFA is almost the same for particles in an isospin

multiplet. So, we only show the results of K, π, Λ, N with k including K0, K+,

K−, π including π0, π+, π− and N for proton, neutron.

Now we will first show the DFA results for identified particles with the use of fit

Eq. (2). Figure 1 shows the relation of the detrended fluctuation function F (s) and

box size s in a log–log plot for particles with any pT . One can see that F (s) has

good scaling properties with s. The Hurst indices for the results shown in Fig. 1

are tabulated in Table 1. One can see that the Hurst index is almost the same

for all identified particles. One can also plot the DFA of identified particle types

with some transverse momentum cut. Now we use four pT cuts as corresponding

0.7 1 1.5

1

2

3

4

5

lns

lnF(s

)

K N

Fig. 1. DFA results for identified final state particles with any pT by using the fit Eq. (2).

Table 1. Hurst index of different identi-fied particles.

Particle H Particle H

K 2.023 π 2.026Λ 1.989 N 2.001

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(a) (150–250MeV); (b) (250–350MeV); (c) (350–450MeV) and (d) (450–600MeV).

The corresponding DFA results are shown in Fig. 2. The Hurst indices shown in

Fig. 2 are tabulated in Table 2. One can see that the F (s) results also exhibit good

scaling with s for all the cases. There is no big difference compared with the Hurst

index of identified particles which has no PT cut. When using MFDFA algorithm in

analyzing the UrQMD data, we choose q from −6 to 6. No PT cut is applied in this

analysis. We use Eq. (13) to calculate h(q) in different q. The results are shown in

Fig. 3 for identified particles. The numerical values of h(q) are exhibit in Table 3. We

can see that as q increases, the corresponding h(q) decreases. When q is larger than

zero, h(q) for different particles have tiny difference as can be seen in Table 3. One

can see from Eq. (11) that if q is positive, the large variance F (ν, ω, s) will domain

Fq(s). On the contrary, for negative value of q, the small variance F (ν, ω, s) will

domain Fq(s). So, Fig. 3 tells us that the scaling properties for different particles

are very different at small fluctuations, while those at the large fluctuations has

tiny difference.

1

2

3 (a) (b)

0.5 1 1.5

1

2

3 (c)

1 1.5

(d)

lns

lnF(s

)

K

N

Fig. 2. DFA results of identified particles with four transverse momentum cuts. (a) (150–250 MeV); (b) (250–350 MeV); (c) (350–450 MeV) and (d) (450–600 MeV).

Table 2. Hurst index of different identified particles withdifferent transverse momentum cut.

Particle π K N Λ

(150–250 MeV) 2.020 1.939 1.761 1.627(250–350 MeV) 2.021 1.978 1.793 1.602(350–450 MeV) 2.021 1.976 1.843 1.663(450–600 MeV) 2.023 1.997 1.917 1.775

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6 4 2 0 2 4 6

2

3

4h(q

)

K N

Fig. 3. MFDFA results of particles.

Table 3. Hurst index of different particles with q

from −6 to 6.

Particle K π Λ N

−6 3.457 2.754 3.682 2.559−5 3.359 2.714 3.566 2.467−4 3.231 2.666 3.396 2.358−3 3.062 2.606 3.128 2.246−2 2.828 2.522 2.710 2.153−1 2.483 2.359 2.300 2.0880 2.158 2.147 2.113 2.0461 2.059 2.058 2.037 2.0192 2.028 2.029 1.998 2.0033 2.011 2.016 1.969 1.9914 2.000 2.009 1.946 1.9835 1.992 2.005 1.925 1.9756 1.984 2.002 1.907 1.969

5. Conclusion

Two-dimensional algorithms of DFA and MFDFA are presented in this paper. These

methods are used to analyze the data from the UrQMD generator. The phase space

of particles is spanned by pseudorapidity and azimuthal angle. Our results indicate

the presence of dynamical fluctuations and the exist of correlations between particle

momentum. More detailed investigation is needed.

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Acknowledgments

This work was supported in part by the National Natural Science Foundation of

China under Grant No. 11075061 and by the Programme of Introducing Talents of

Discipline to Universities under No. B08033.

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Documents

FRACTAL PROPERTIES OF PARTICLES IN PHASE SPACE FROM URQMD MODEL