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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∗wangxin@iopp.ccnu.edu.cn†cbyang@mail.ccnu.edu.cn
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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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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Fractal Properties in Phase Space
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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Fractal Properties in Phase Space
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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