Multifractal analysis of light scattering-intensity fluctuations
Citation for published version (APA):Shayeganfar, F., Jabbari-Farouji, S., Movahed, M. S., Jafari, G. R., & Tabar, M. R. R. (2009). Multifractal analysis of light scattering-intensity fluctuations. Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, 80(6), 061126-1/8. [061126]. https://doi.org/10.1103/PhysRevE.80.061126
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10.1103/PhysRevE.80.061126 Document status and date: Published: 01/01/2009
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Multifractal analysis of light scattering-intensity fluctuations
F. Shayeganfar,1S. Jabbari-Farouji,2M. Sadegh Movahed,3G. R. Jafari,3and M. Reza Rahimi Tabar1,4 1
Department of Physics, Sharif University of Technology, P.O. Box 11365-9161, Tehran, Iran 2
Theory of Polymer and Soft Matter Group, Department of Applied Physics, Eindhoven University of Technology, P. O. Box 513, 5600 MB Eindhoven, The Netherlands
3Department of Physics, Shahid Beheshti University, G.C., Evin, Tehran 19839, Iran 4Institute of Physics, Carl von Ossietzky University, D-26111 Oldenburg, Germany
共Received 24 June 2009; published 18 December 2009兲
We provide a simple interpretation of non-Gaussian nature of the light scattering-intensity fluctuations from an aging colloidal suspension of Laponite using the multiplicative cascade model, Markovian method, and volatility correlations. The cascade model and Markovian method enable us to reproduce most of recent empirical findings: long-range volatility correlations and non-Gaussian statistics of intensity fluctuations. We provide evidence that the intensity increments⌬x共兲=I共t+兲−I共t兲, upon different delay time scales , can be
described as a Markovian process evolving in. Thus, the dependence of the probability density function
p共⌬x,兲 on the delay time scale can be described by a Fokker-Planck equation. We also demonstrate how
drift and diffusion coefficients in the Fokker-Planck equation can be estimated directly from the data.
DOI:10.1103/PhysRevE.80.061126 PACS number共s兲: 02.50.Fz, 78.35.⫹c
I. INTRODUCTION
As shown by most recent empirical studies on the huge amount of data set, the light scattering-intensity time series from higly interacting colloidal systems and inhomogeneous media are characterized by several “universal” features 关1–3兴: volatilities strong correlation and non-Gaussian
prob-ability distribution function共PDF兲 on the small time scales. The PDF’s shape changes from almost Gaussian at large time scales to fat tails at fine scales 关3兴. Many authors recently
aim at proposing simple, discrete, or continuous time models that are able to account for similar observations 关4,5兴.
Among all the proposed models, one can distinguish several streams from the simplest Brownian process, which consti-tutes the main tool used by practitioners, to the class of “het-eroskedastic” nonlinear processes as proposed in 关6兴.
Recently, an interesting method has been introduced by Ghashghaie et al.关7兴, which is known as Markovian method.
It has turned out that this method can be successfully applied to fluctuating time series, such as fluid turbulence关8,9兴,
char-acterization of rough surfaces 关10兴, finance 关11兴, etc. 共see
关12兴 for more details and applications兲. In the Markovian
method one can derive a Fokker-Planck 共FP兲 equation for describing the evolution of the probability distribution func-tion of stochastic properties of given time series. As shown in 关7兴, the conditional probability density of the increments
of a stochastic field共for example, the increments in the ve-locity field in turbulent flow兲 satisfies the Chapman-Kolmogorov 共CK兲 equation, even though the velocity field itself contains long-range nondecaying correlations. As is well known, satisfying the CK equation is a necessary con-dition for any fluctuating data to be a Markovian process over the relevant length 共or time兲 scales 共Markovian time scale兲. Hence, one has a way of analyzing stochastic phe-nomena in terms of the corresponding FP and CK equations. Here, we provide two complementary points of view to un-derstand the “multiscaling” and non-Gaussian nature of scat-tered light intensity fluctuations from a non-equilibriumm
aging colloidal system关1,2兴 and propose a simple
multifrac-tal “stochastic volatility” model that captures very well the above-mentioned features of the intensity fluctuations.
The paper is organized as follows. In Sec. II we give a review on multifractal processes and the cascade model. We introduce the related notions of multiscaling, scale invari-ance, cascade process, and self-similarity kernel. In Sec.III, we use the multifractal random walk共MRW兲 as a stochastic volatility model and derive multifractal exponentsqfor gen-eral case with arbitrary well-known Hurst exponent. Section
IVis devoted to a brief summary of the most important theo-rems on Markovian processes and their application to the analysis of empirical data. We estimate a Langevin equation to describe the fluctuations of light scattering-intensity in Sec.V. In Sec.VIvolatility and magnitude correlation func-tion were presented. The last secfunc-tion is devoted to summary and conclusions.
II. MULTIFRACTAL PROCESSES AND CASCADE MODEL
In this section we briefly discuss the related notions of multifractality and multiplicative cascade model. Most of the ideas and concepts that we recall below have been intro-duced in the field of fully developed turbulence关4兴.
A. Multifractal process and extended self-similarity
Let 兵I共t兲⬅x共t兲其 be the intensity fluctuations time series and consider its statistics over a certain time scale, which is defined as
⌬x共兲 = x共t +兲 − x共t兲. 共1兲 Let us denote M共q,兲 the order q absolute moment of inten-sity fluctuations,
M共q,兲 = 具兩⌬x共兲兩q典. 共2兲 We will say that the process is scale invariant if the scale behavior of the absolute moment M共q,兲 has a power-law
behavior. Let us callq the exponent of power law, i.e., M共q,兲 ⯝ Nqq, 共3兲 where Nqis a prefactor. The process is called monofractal if q is a linear function versus q and multifractal if q is a nonlinear function of q. To check the estimated scaling ex-ponent q 关Eq. 共3兲兴 with original time series, we use the
extended self-similarity共ESS兲 method 关13,14兴.
In the ESS method, we rely on the scaling behavior of Sq共兲 with respect to the specific order of structure function, namely, S3共兲 as
Sq共兲 ⬃ S3共兲q. 共4兲
For any Gaussian process, the exponent in the above equa-tion is given by q= q/3 关13,14兴. Any deviation from this
relation can be interpreted as a deviation from Gaussianity. Multifractality has been introduced in the context of fully developed turbulence in order to describe the spatial fluctua-tion of the fluid velocity at very high Reynolds number关15兴.
As suggested by recent studies 关7,16–19兴, multifractality is
likely to be a pertinent concept to account for the fluctuation in complex systems. We use this concept here for analyzing the fluctuation of time series of light scattering intensity from an aging colloidal suspension of Laponite clay with a con-centration of 3.2 wt % measured at a certain aging time. Here we only focus on the general stochastic behavior of light intensity fluctuations. The evolution of the statistical proper-ties on aging time will be presented in a following work. The typical intensity fluctuation is plotted in Fig.1. The details of experimental setup are given in关2兴.
B. Multiplicative cascade model
Multifractality is a notion that is often related to an un-derlying multiplicative cascading process. In the context of deterministic function the situation is rather clear since the analyticity of the q spectrum is deeply connected to the self-similarity properties of the function 关20,21兴. A process
x共t兲 is called self-similar with exponent H if ∀
⬎0,−H⌬x共兲 is the same process as ⌬x共兲. Define P 共⌬x兲
to be the PDF of⌬x共兲. The process x共t兲 is self-similar with an exponent H if its PDF satisfies关22,23兴
P共⌬x兲 = HP共H⌬x兲. 共5兲 Then, the moments at scale and T =are related by
M共q,兲 = Nq
冉
T
冊
qH
共6兲 with Nq= M共q,T兲. Therefore one has a monofractal process withq= qH. In order to account for multifractality, one has to generalize this classical definition of self-similarity. This can be done by introducing a weaker notion, as originally proposed in the field of fully developed turbulence by Casta-ing et al. 关24兴. According to Castaing’s definition of
self-similarity, a process is self-similar if the increment’s prob-ability density functions at scalesand T =0are related by
the relationship关24兴
P共⌬x兲 =
冕
G,T共u兲e−uPT共e−u⌬x兲du, 共7兲 where the self-similarity kernel G,T depends only on /T. We note that this definition generalizes the Eq. 共6兲 thatcor-responds to the “trivial” case G,T共u兲=␦(u − H ln共/T兲). This
equation basically states that the probability density function Pcan be obtained through a “geometrical convolution” be-tween the kernel G,Tand PT. A simple argument shows that the logarithm of the Fourier transform of the kernel G,Tcan be written as F,T共k兲=ln G,T共k兲=F共k兲ln共/T兲. Thus, from Eq. 共7兲, one can easily show that the q order absolute
mo-ments at scales and T are related by关24,25兴
M共q,兲 = G,T共− iq兲M共q,T兲 = M共q,T兲
冉
T冊
F共−iq兲 , 共8兲 so that Nq= M共q,T兲 andq= F共−iq兲. A nonlinearqspectrum implies that F is nonlinear and thus that G is different from a Dirac delta function. For example, the simplest nonlinear case is the so-called logarithmic-normal model that corre-sponds to a parabolic q function and thus to a function G that is Gaussian 关24兴.
Let us now make a link between the multiplicative cas-cade model and Castaing equation. This can be easily done if one consider discrete scales tn= 2−nT. Let us suppose the lo-cal variation of the process⌬tnx at scale tn is obtained from the variation at scale T as
⌬tnx共t兲 =
冉
兿
i=1 nWi
冊
⌬Tx共t兲, 共9兲 where Wi are random positive factors. This is the cascade paradigm. Realizations of such processes can be constructed using orthonormal wavelet bases as discussed in关26兴. UsingEq. 共9兲 one can prove immediately the Castaing equation,
i.e., Eq. 共7兲.
III. SIMPLE SOLVABLE MULTIFRACTAL MODEL
In this section our aim is to build a simple solvable model based on multiplicative cascade model and employ it to fit
Time (arb.units)
Int
en
si
ty
(
k
Hz
)
100 200 300 400 500 0 20 40 60 80FIG. 1. 共Color online兲 Typical light scattering-intensity fluctua-tions measured from an interacting colloidal suspension of Laponite with concentration 3.2 wt % as a function of time. In this paper we express intensities in terms of the detector count rate in kHz.
SHAYEGANFAR et al. PHYSICAL REVIEW E 80, 061126共2009兲
the experimental data of light scattering fluctuations. Multi-plicative cascading processes关26兴 consist of writing Eq. 共9兲,
starting from some “coarse” scale= T, and then iterating it toward finer scales using an arbitrary fixed scale ratio, e.g., =1/2. Such processes can be constructed rigorously using, for instance, orthonormal wavelet bases关26兴. However, these
processes have fundamental drawbacks: they do not lead to stationary increments and they do not have continuous dila-tion invariance properties. Indeed, they involve a particular arbitrary scale ratio, i.e., Eq.共3兲 holds only for the discrete
scales tn=0 n
T. We first build a discretized version x⌬t共t = k⌬t兲 of this process. Let us note that the limit process x共t兲=lim⌬t→0x⌬tis well defined关4兴. It is shown that different
quantities 共q order moments, increment correlation, etc.兲 converge when⌬t→0 关4兴. We rewrite Eq. 共6兲 at the smallest
scale so ⌬x⌬t共k⌬t兲=⑀⌬t共k兲W⌬t共k兲, where ⑀⌬t and W⌬t共k兲 = e⌬t共k兲 are Gaussian and logarithmic-normal variables,
re-spectively, i.e.,
x⌬t共t兲 =
兺
k=1 t/⌬t
⑀⌬t共k兲e⌬t共k兲, 共10兲
where ⌬t共k兲 is the logarithm of the stochastic variance. More specifically, we will choose⑀⌬tto be a Gaussian white noise independent ofand of variance2⌬t. This choice for the process⌬tis introduced in关4兴 and dictated by the
cas-cade picture. It corresponds to a Gaussian stationary process where its covariance can be written as
具⌬t共k1兲⌬t共k2兲典 = 02ln⌬t共兩k1− k2兩兲. 共11兲
Here, ⌬t is chosen in order to mimic the correlation struc-ture observed in cascade models with an integral time scale T, ⌬t共k兲 =
冦
T 兩k兩 + 1 for 兩k兩 ⱕ T/⌬t − 1 1 otherwise.冧
共12兲 ⌬t’s are correlated up to the time scale T and theirvari-ance 02ln共T/⌬t兲 goes to infinity when ⌬t approaches zero. Direct computation shows that we need to choose the follow-ing relations关4兴:
具⌬t共k兲典 = − r Var„⌬t共k兲… = − r0
2ln共T/⌬t兲 共13兲
with r = 1 and Var(x共t兲)=2t. This computation builds
“MRW” process x共t兲 关4兴. We can build MRWs with
corre-lated increments by just replacing the white noise⑀⌬twith a fractional Gaussian noise
⑀⌬t共H兲= B
H共关k + 1兴⌬t兲 − BH共k⌬t兲, 共14兲 where BH共t兲 is a fractional Brownian motion with the so-called Hurst exponent H and of variance 2t2H 关choosing r
= 1/2 in Eq. 共13兲兴.
The qth moment of light scattering-intensity time series can be computed directly from experimental data and the cascade model. In cascade model, we construct the incre-ments of the model as x⌬t共t+兲−x⌬t共t兲, which does not de-pend on t and is the same law as x⌬t共兲. It has been proven that the moments of x共兲⬅x⌬t→0+共兲 can be expressed as 关4兴
具x共兲2p典 = 2p共2p兲! 2pp!
冕
0 l 共u1兲2H−1du1¯冕
0 l 共up兲2H−1dup ⫻兿
i⬍j 共ui− uj兲0 2 . 共15兲Using this expression, Eq. 共6兲 becomes
M共2p,兲 = K2p
冉
T冊
pH−2p共p−1兲02 , 共16兲 where K2pis given by K2p= Tp2p共2p − 1兲 ! !冕
0 1 u12H−1du1¯冕
0 1 up 2H−1 dup ⫻兿
i⬍j 兩ui− uj兩0 2 . 共17兲It is worth to note that K2pis nothing else but the moment of order 2p of the random variable x共t兲. From the above expres-sion, we thus obtain
2p= 2pH − 2p共2p − 1兲
02
2 . 共18兲
The correspondingq spectrum is thus the parabola q= qH − q共q − 1兲02
2 . 共19兲
Let us suppose the MRWs with variance Var共x⌬t兲=2t, then
the spectrum of the MRW x共t兲 is 关4兴
q=q
2− q共q − 1兲 02
2 . 共20兲
To find the relation betweenq and02, we illustrated the scaling behavior of the moments M共q,兲 and corresponding extended self-similarity exponents in Fig.2. The upper panel of Fig.2indicates the log-log plot of structure function, Eq. 共3兲 versus . In the lower left panel, we show the scaling exponent q as a function of q. Filled circle symbols have been directly computed by the experimental light intensity time series. The solid line in this panel corresponds to a monofractal process with the same Hurst exponent as our data set. The value of the Hurst exponent of underlying data set has been determined by detrended fluctuation analysis and is equal to H = 0.92⫾0.02 关27兴. The long-dashed curve
corresponds to Eq.共19兲 with the value of 0= 0.077⫾0.054,
which is determined by multiplicative cascade model关28兴.
The lower right panel showsqversus q for light scatter-ing data set 共filled circle symbols兲 and a Gaussian process 共solid line兲. It appears that light scattering-intensity data rep-resented by I共t兲 are a multifractal process with continuous dilation invariance properties. As shown in the lower panel of Fig. 2, the fitting formula for q 共solid line兲 derived by multifractal analysis is in agreement with experimental data in an acceptable confidence level. We note that due to the smallness of 0, deviation from monofractality will appear
for large q’s. We confirm this observation with direct estima-tion of PDF of increments and show that they have deviaestima-tion from Gaussian distribution共see below兲.
In the cascade picture, the total number of data points is 2m 共i=1, ... ,2m兲, where m is the total number of cascade steps. Also, the time series ⌬x共兲 can be described by the
form of logarithmic-normal cascade-type multiplicative pro-cess which has been introduced in Ref. 关29兴 and with the
probability density function given by Eq.共7兲, with 2= m 0 2.
As illustrated in Fig. 3, Eq. 共7兲 accounts very well for the
evolution of the probability density function of the incre-ments. This shows that the smaller the scale, the fatter the tails of the probability density function of⌬x共t兲. These find-ings are also in agreement with the previous results关3兴.
IV. MARKOVIAN NATURE OF DATA SET
In Ref.关12兴, it has been demonstrated that the
mathemat-ics of stochastic processes is a useful tool for empirical in-vestigations of the time-scale dependence of the PDF p共⌬x,兲 of a given time series, namely, the intensity fluctua-tions on time scale . It was shown how the equations gov-erning the underlying stochastic process can be extracted di-rectly from the empirical data, provided that several experimental data obey the Markovian process. In particular, it is possible to derive a partial differential equation, the Fokker-Planck equation, which describes the evolution of the probability density function p共⌬x,兲 in the scale variable. Hence, the mathematics of Markovian processes yields a complete description of the stochastic process underlying the evolution of the PDFs from Gaussian distributions at large scales to the leptokurtic 共fat兲 PDFs at small scales. Here, we show how the existence of a Markovian process can be checked empirically and how the Fokker-Planck equation can be calculated directly from the data set.
In what follows we summarize the notions and theorems which will be important for the statistical analysis of light intensity fluctuations measured in our experiment by Mar-kovian method. For further details on the MarMar-kovian pro-cesses we refer the reader to Refs. 关12,30兴. Fundamental
quantities related to the Markovian processes are conditional probability density functions. Given the joint probability density p共x2, t2; x1, t1兲 for finding the intensity x2⬅x共t2兲 at
time scale t2 and x1 at time scale t1 with t1⬍t2, the
condi-tional PDF p共x2, t2兩x1, t1兲 is defined as p共x2,t2兩x1,t1兲 =p共x2,t2;x1,t1兲
p共x1,t1兲 , 共21兲 where p共x2, t2兩x1, t1兲 denotes the conditional probability
den-sity for the intenden-sity x2at time scale t2given x1at time scale
t1.
Higher-order conditional probability densities can be de-fined in an analogous way as follows:
p共xN,tN兩xN−1,tN−1; . . . ;x1,t1兲 =
p共xN,tN;xN−1,tN−1; . . . ;x1,t1兲
p共xN−1,tN−1; . . . ;x1,t1兲
. 共22兲 The smaller scales tiare nested inside the larger scales ti+ 1 共with the common reference point t兲.
The stochastic process in= tN− tN−1is a Markovian pro-cess if the conditional probability densities fulfill the follow-ing relations: τ M(q ,τ ) 100 101 102 103 101 103 105 107 109 1011 q = 0.5 q = 1.0 q = 1.5 q = 2.0 q = 2.5 q = 3.0 q = 4.0 q ξq 0 2 4 6 8 10 0 2 4 6 8 10 12 Monofractal data (H=0.92) Fitting Formula (Eq. (19)) Light scattering data
q ζq 2 4 6 8 10 0.5 1 1.5 2 2.5 3
3.5 Light scattering data Gaussian data
1/3
FIG. 2. 共Color online兲 Upper panel shows the scaling behavior
of structure function M共q,兲 for various values of q. Lower left
panel corresponds to the scaling exponent q of light
scattering-intensity data共filled circle symbol兲 as a function of moment order
q. In this panel, solid lines are given for a monofractal process with
Hurst exponent equal to H = 0.92. Long-dashed line is given by Eq. 共19兲. Lower right panel indicatesqversus q共the exponents derived
using the extended self-similarity method兲. Solid line shows the
exponents for a Gaussian process,q= q/3.
∆x/σ
τP
τ(∆
x)
-5 0 5
FIG. 3. 共Color online兲 Continuous deformation of the intensity
increments’ PDFs across scales 共from bottom to top兲
= 2000, 1000, 500, 250, 150, 25 共406 s兲. Solid lines for each ’s are given by Eq.共7兲 and symbols are directly computed by data set. Long-dashed curve corresponds to the Gaussian probability density function. Curves are shifted in vertical direction for clarity of presentation.
SHAYEGANFAR et al. PHYSICAL REVIEW E 80, 061126共2009兲
p共xN,tN兩xN−1,tN−1; . . . ;x1,t1兲 = p共xN,tN兩xN−1,tN−1兲 共23兲 with t1⬍t2⬍t3⬍ ¯ ⬍tN. As a consequence of Eq. 共22兲, each N-point probability density p共xN, tN; xN−1, tN−1; . . . ; x1, t1兲 can be determined as a product
of conditional probability density functions, p共xN,tN;xN−1,tN−1; . . . ;x1,t1兲
= p共xN,tN兩xN−1,tN−1兲 ¯ p共x2,t2兩x1,t1兲p共x1,t1兲. 共24兲
As Eqs. 共23兲 and 共24兲 indicate, knowledge of p共x,t兩x0, t0兲
共for arbitrary scales t and t0 with t0⬍t兲 is sufficient to
gerate the entire statistics of the underlying fluctuations en-coded in the N-point probability density, namely, p共xN, tN; xN−1, tN−1; . . . ; x1, t1兲.
To investigate whether the underlying signal共or its incre-ments兲 is a Markovian process, one should test Eq. 共24兲. But
in practice, it is beyond the current computational capability for large values of N. For N = 3共three points or events兲, how-ever, the condition will be
p共x3,t3兩x2,t2;x1,t1兲 = p共x3,t3兩x2,t2兲, 共25兲
which should hold for any value of t2 in the interval t1⬍t2
⬍t3. A process is then Markovian if Eq.共25兲 is satisfied for
a certain time separation t3− t2, in which case, we define the Markovian time scale as tMarkov= t3− t2. For simplicity, we let t2− t1= t3− t2. Thus, to compute tMarkov, we use a fundamental theory of probability according to which we write any three-point PDF in terms of the conditional probability functions as
p共x3,t3;x2,t2;x1,t1兲 = p共x3,t3兩x2,t2;x1,t1兲p共x2,t2;x1,t1兲.
共26兲 Using the properties of Markovian processes, Eq.共26兲 can be
written as follows:
pMar共x3,t3;x2,t2;x1,t1兲 = p共x3,t3兩x2,t2兲p共x2,t2;x1,t1兲. 共27兲 In order to check the condition for the data being a Markov-ian process, we must compute the three-point joint PDF through Eq.共26兲 and compare the result with Eq. 共27兲. One
can write Eq. 共27兲 as an integral equation, which is well
known as the CK equation
p共x3,t3兩x1,t1兲 =
冕
dx2p共x3,t3兩x2,t2兲p共x2,t2兩x1,t1兲. 共28兲The simplest way to determine tMarkov is using the well-known Chapman-Kolmogorov equation, which can be written as K共t3− t1兲=兩p共x3, t3兩x1, t1兲
−兰dx2p共x3, t3兩x2, t2兲p共x2, t2兩x1, t1兲兩, for given x1 and x3, in
terms of, for example, t3− t1 and considering the possible
errors in estimating K. It is obvious that, for the value of tMarkov= t3− t2= t2− t1, the quantity K vanishes or at least is
nearly zero共achieves a minimum兲 关12,31兴.
Up to now we only showed how one can estimate the Markovian time scale for data set over which time series behaves as a Markovian process. In the next section we will turn to deriving master and stochastic equations governing
the evolution of probability density function of intensity fluctuations.
V. LANGEVIAN EQUATION: EVOLUTION EQUATION TO DESCRIBE THE INTENSITY OF LIGHT SCATTERING
FLUCTUATIONS
The Markovian nature of the intensity of light scattering fluctuations enables us to derive a master equation, a Fokker-Planck equation, for the evolution of the PDF p共x,t兲 in terms of time t. The Chapman-Kolmogorov equation, formulated in differential form, yields the following Kramers-Moyal ex-pansion关30兴: tp共x,t兲 =n=1
兺
⬁冉
− x冊
n 关Dn共x,t兲p共x,t兲兴, 共29兲 where Dn共x,t兲 are called the Kramers-Moyal coefficients. For Markovian processes the conditional probability density fulfills a master equation which can be put into the form of a Kramers-Moyal expansion as follows: tp共x,t兩x0,t0兲 =n=1
兺
⬁冉
− x冊
n 关Dn共x,t兲p共x,t兩x0,t0兲兴. 共30兲The Kramers-Moyal coefficients Dn共x,t兲 are defined as
Dn共x,t兲 = lim ⌬t→0Mn共x,t,⌬t兲, 共31兲 where Mn共x,t,⌬t兲 = 1 n !⌬t
冕
−⬁ ⬁ 共x⬘
− x兲n p共x⬘
,t −⌬t兩x,t兲dx⬘
. 共32兲 For a general stochastic process, all Kramers-Moyal coeffi-cients are different from zero. According to the Pawula theo-rem, however, the Kramers-Moyal expansion stops after the second term, provided that the fourth-order coefficient D4共x,t兲 vanishes. In that case, the Kramers-Moyal expansion reduces to a Fokker-Planck equation 共also known as the backward or second Kolmogorov equation兲 关30兴, tp共x,t兩x0,t0兲 =
再
− xD1共x,t兲 + 2 x2D2共x,t兲冎
p共x,t兩x0,t0兲. 共33兲 The coefficients D1 and D2 are known as drift and diffusioncoefficients, respectively. We note that the probability den-sity p共x,t兲 has to obey the same equation with a different initial condition 关12兴. The Fokker-Planck equation describes
the probability density function of a stochastic process gen-erated by the Langevin equation 共we use the Ito definition兲 关30兴,
tx共t兲 = D1共x,t兲 +
冑
D2共x,t兲f共t兲, 共34兲 where f共t兲 is a Langevin force, i.e.,␦-correlated white noise with a Gaussian distribution具f共t兲f共t⬘
兲典=2␦共t−t⬘
兲.To check the multifractal nature of time series, we check the Markovian nature of the increments, which is defined by ⌬x共兲=x共t+兲−x共t兲, for intensity fluctuations. According to the mentioned procedure, we determine the Markovian time scales for the increments and calculate the Kramers-Moyal coefficients. The Fokker-Planck equation for probability function of the increment is given by关12兴
− p共⌬x,兲 =
再
− ⌬xD1共⌬x,兲 + 2 ⌬x2D2共x,兲冎
p共⌬x,兲, 共35兲 where the negative sign of the left-hand side of Eq. 共35兲 isdue to the direction of the cascade toward smaller time scales . The corresponding Langevin equation can be read as
−
⌬x共兲 = D1共⌬x,兲 +
冑
D2共⌬x,兲f共兲, 共36兲where f共兲 is the same as random function in Eq. 共34兲. Drift
and diffusion coefficients of increment are formulated as 关32–34兴
D1共⌬x,兲 ⯝ − H⌬x,
D2共⌬x,兲 ⯝ b⌬x2. 共37兲
Using Eq.共35兲 we obtain the evolution of structure function
as follows: −
具兩⌬x共兲兩q典 = q具兩⌬x共兲兩q−1D1共⌬x,兲典
+ q共q − 1兲具兩⌬x共兲兩q−2D
2共⌬x,兲典. 共38兲
Substituting Eq.共37兲 into Eq. 共38兲 we find
−
具兩⌬x共兲兩q典 = 关qH + bq共q − 1兲兴具兩⌬x共兲兩q典. 共39兲 The above equation implies scaling behavior for the structure function, so that
M共q,兲 ⬅ 具兩⌬x共兲兩q典 = 具兩x共t +兲 − x共t兲兩q典 ⬃q. 共40兲 According to Eqs. 共39兲 and 共40兲, the corresponding
scal-ing exponent can be read as
q= Hq − bq共q − 1兲. 共41兲 As mentioned before, for monofractal and multifractal pro-cesses the exponent q has linear and nonlinear behaviors with respect to q, respectively. We must point out that the exponent H is nothing except the Hurst exponent of time series关35兴.
Satisfying the Chapman-Kolmogorov共CK兲 equation con-firms that the signal of light scattering-intensity fluctuations is Markovian process. The coefficients D1共⌬x,兲 and D2共⌬x,兲 are estimated as
D1共⌬x,兲 = − 共0.86 ⫾ 0.19兲⌬x, 共42兲
D2共⌬x,兲 = − 共0.034 ⫾ 0.027兲⌬x2. 共43兲
Consequently, using Eqs. 共41兲–共43兲, the scaling exponents
will be
q=共0.86 ⫾ 0.19兲q − 共0.034 ⫾ 0.027兲q共q − 1兲. 共44兲 The scaling exponentsq 共for small values of q兲 derived by the Markovian approach 关Eq. 共44兲兴 are in agreement with
those given by Eq.共19兲 and as well as with direct
computa-tional from time series.
VI. VOLATILITY AND MAGNITUDE CORRELATION FUNCTIONS
As recalled in the Introduction, the scattered light inten-sity time series are correlated and their amplitude 共“local volatilities”兲 possesses power-law correlations. Let us show that our model satisfies these two properties. The increment correlation function is defined by
Cq共l,t,兲 = 具兩x⌬t共l +兲 − x⌬t共l兲兩q兩x⌬t共t +兲 − x⌬t共t兲兩q典, 共45兲 where, for 共∀兩l−t兩⬎兲, the correlation is zero. Let us study the correlation function of the squared increments. Since the increments are stationary, we can choose t = 0. Thus, we need to compute, in the limit ⌬t→0, the following correlation function that corresponds to a lag l between increments of size :
Cq共l,兲 = 具兩x⌬t共l +兲 − x⌬t共l兲兩q兩x⌬t共兲 − x⌬t共0兲兩q典. 共46兲 From the results of Ref.关4兴 and for 0ⱕl⬍T,0ⱕ+ l⬍T, we find Cq共l,兲 =4
冕
l l+ du冕
0 dv共u − v兲q202 . 共47兲A direct computation shows that
冕
l l+ du冕
0 dv共u − v兲q20 2 = 1 共1 − q2 0 2兲共2 − q2 0 2兲关共l +兲 2−q202 +共l −兲2−q20 2 − 2l2−q20 2 兴, 共48兲 and consequently Cq共l,兲 = A关共l +兲2−q 2 0 2 +共l −兲2−q20 2 − 2l2−q20 2 兴, 共49兲 whereA=4tq202/关共1−q2 0 2兲共2−q2 0 2兲兴.For 0ⱕlⰆ, one gets Cq共l,兲 ⬃ A2
冉
l T冊
2q冉
T冊
−q202 . 共50兲The correlation function for fixed values of l and T behaves as
Cq共l,兲 ⬃q, 共51兲 where the exponent is given by关26,36兴
q= − q202. 共52兲 In the upper panel of Fig. 4, we plotted the correlation function of light intensity fluctuations. The lower panel of Fig. 4 indicates the exponent of correlation function for small value of for various values of q accompanying with
SHAYEGANFAR et al. PHYSICAL REVIEW E 80, 061126共2009兲
the theoretical prediction given by Eq. 共52兲. This figure is
also another confirmation of the reliability of0determined
by multiplicative cascade model mentioned in Sec.III.
VII. CONCLUSION
In this paper we checked the multifractal nature of the light scattering-intensity time series. We showed how the mathematical framework of cascade modeling and Markov-ian processes can be applied to develop a successful
statisti-cal description of the intensity fluctuations. We characterized the non-Gaussian nature of the light scattering-intensity time series, using a multiplicative model. Also noting to the Mar-kovian nature of fluctuations, we demonstrated that the prob-ability density function of increment fluctuations satisfies a Fokker-Planck equation, which encodes the Markovian prop-erty of light intensity fluctuations in a necessary way. We computed the Kramers-Moyal coefficients for the field I共t+兲−I共t兲 and determined their corresponding Langevin equations.
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