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Some interesting observations regarding factorial moments and generalised moments in intermittency analysis
-.
6 April 1995
PHYSICS LETTERS B
PhysicsLetters B 348 ( 199s) 297-302
Some interesting observationsregardingfactorial moments and generalisedmoments in intermittency analysis G.C. Mishra, B.K. Nan&, S.K, Nayak, D.P. Mahapatra Institute of’Physics, Bhubanmwar
751005, India
Received5 January1995 Mitor: L. Montanet -
Abstract A study of intermittency has been carriedout within the framework of the a-cascade model using both scaled ktorial as well as generalised moments. It has been found that, depending upon the multiplicity, the scaled factorial moments can sometimes leadto negativeslopeswhenlog Fq is plottedagainst-log Sr].Compared to thisthe generalised moments G, have been found to be unaffected by multiplicity.
Intermittency in multiparticle production is a very interestingtopic since investigationof fluctuationsin phasespacecan revealsomedeepaspects[ 11. This is why the techniquehas beenused for analysisof multiparticle production in high energyexperimentaldata [2]. In caseswhere intermitterlcy analysis has been applied there are indications regarding transitions betweenquarks (gluons) and hadrons.Thereforeit is all the more important,particularly in the caseof high energyheavy ion collision experimentswherethereis a possibility of QGP formation. In the analysisof intermittencyone essentiallytakes the multiparticle distribution over the specifiedpseudorapidityrangeandcalculates,asdiscussedlater,a set bf momentsG, or Fq of order q of the particle distribution.‘Thenone plots -log G, and/or log Fq versus - log 677,877 denotingthe resolutionor bin size.In the absenceof fluctuation,what is expectedis a flat behaviour with zero slopes.Howevec,intermittencyresults in finite non-zeroslopesknown as the intermittency exponents.These slopes could be related to certain physictilpropertiesof the system.For example;the one 0370~2693/95/$09.50C 1995 Eisevier ScienceB.V. Ail rights reserved SUVO370-2693(95)00143-3
correspondingto order 2 directly yields the exponent of the two particle correlationfunction which in turn behavesas an algebraicpower law. The intermittency exponentfor the order 2 must also be positive leading to the usualalgebraicdecayof the correlationfunction. Although this is expectedto be true always,therecan be problems.In the presentwork, following the prescription as given elsewhere[ 11, we have carriedout a Monte Carlo simtilation in one dimensionwherethe intermittencyexponents,in somecases,turned out to be negative,which also dependupon the total multiplicity. In the next sectionwe briefly discussthe formalism which is followed up by a sectionon the presentCalculation.At the end we summarisethe results. Let us considera typical examplewhich involves a multiplicity distributionpi given over a pseudorapidiiy interval Aq which is further divided into M bins jqi (i= 1, . . .. M). The distributionpi satisfiesthe normalization condition (1)
G.C. Mishra et ul. /Physics Letters B 348 (1995) 297-302
298
Following [I], in the usualanalysiswhat one calculates,for various67, is a set of moments
c, = ($- c (MpijR)
(q= 1,2, 3, etc.)
(2)
The strengthof the singularitiesare given in termsof a parametercu(q)=dT(q) ldq andhow denselythey are distributedis given by .fCd = qa(3) - 7(q) .
(7)
where ( > indicatesan event average.lnrermittent behaviourresults in non zero slopes when log C, is plottedagainst - log 6~. This is a consequence of the underlyingself-similarity leadingto scaleinvariance. Eq. ( I ) is only valid in the limit M t m. However, for a finite numberof bins onecan calculatethe scaled factorial momentsFp which are equivalentto C, with the samebehaviour.At this point, it also must be mentionedthat neither4 nor any of the momentsC, can be negative.Furthermore,it hasbeenpointedout thatthese momentsare relatedto certain physical propertiesof the system.For example.if a systemis selfsimilar,the two particlecorrelationin r) space
One can determineT(q) from the slope of -logG, versus -log 67. Apart from the abovementionedmomentanalysisit is alsoimportantto studytwo particlecorrelations.The two particlecorrelationfunction can be independently calculateddirectly from the given ,q-distribution.In terms of single particle densitiesp( 71) and P(Q), it is given as
R(rl:, d-(/q,
Having presentedthe basicformulaefor the analysis let us considera particular examplewhere the alpha cascademodel as shown elsewhere [ I] has heen employedfor generatingan intern&tent system.We start with a systemwith a uniform multiplicity in the pseudorapidityinterval [ 0, 1] . Sequentiallywe divide the system into two halves and enhanceby a given factor (a) the multiplicities at a given fraction (p) of the points selectedat randomin eachof the new intervals.The sequenceis thenrepeatedup to a given number of generationsV.Initially the ideawas to reproduce the resultsthereand therebyto understandthe model. Therefore the parameterschosen viz. h = 2, ~=4, a = 0.25 and/3= 0.1 arealmostthe sameasin theabove work. However,wecarriedouttwo simulations.Inonecase the Q valueswere generatedat random,uniformly distributed betweenzero and one. In the other case they wereequispacedon a linearlattice.The resultsobtained were then groupedinto 200 q-bins of width 0,005. In eachcase500 eventsweregeneratedfor the eventaveraging. Eq. (3) of 181 was usedfor the calculationof Fy. Typical total multiplicities areof the orderof 2000, The results obtainedfor various momentsF,, for the two cases,areshownin Fig. 1. One can noticethat the randomsimulationyields positiveslopesin contrastto the latticesimulationwhich showsnegativeslopes.But the results, in terms of generalisedmomentsG, arc almostidenticalin both cases(Fig. 2).
-~l)--~.
(3)
In such a casethe slope of log F2 versus -log 67 is expectedto be positive with the value (Y[ 3,4]. Usually intermittencyinvolvesa certaindistribution of probabilitiesdefinedon a supportwhich may be a simple geometricalfractal often called the substrate. The probability distributionis called the %zctul meusllre. A fractal measurepossessesinfinite types of singularities. Points correspondingto a given type of singularity usua$ form a fractal subsetof a given dimensiondependingon the type ofsinguiari!y. Therefore it is importantto study the multifractality behaviour which is doneusinga setof generalizedmoments [4-6] given as G,=
ti
$py
)
.
(4)
It is interestingto note that the order4 can be positive, Neroor negative,unlike the scaledfactorial moments discussedearlier.In generalfor an intermittentsystem
horn which one can get, for variousorders4, the generalizeddimensions
(8)
where p(r],, Q) is the two particle inclusive density r71.
C.C. Mishm et al. /Physics Letters B 348 (1995) 297-302
u &i
0.06 0.04
c
A
-0.1
v q=4
I
*
.
q=5
-0.15
+ -0.2 -0.25
.
I
0.02
-0.3 0
i
a
;
: -0.35
1
402
!I11 I III< I
L.u/LIIL_ILI_LII 0.5
1
1.5
! . t
“? -0.05 Lv 0.05 :I------
q=3
-In
(a)
Lt-
e
q=1
u
q=2
A
q=3
0.5
1 ,I,,
67
;
.
.
1
.
I,,, 1.5
1
1 . l
v q=4 * q=s
-0.4 I- III,
3
2.5
2
1 /--
299
I h&.-L&A *
2
W
2.5
3
-In
67f
L” 0.02 c
-0.1
u
q=l
u
q=2
A
q=3
v
q=4
*
q&i
tJ-l-uLul-Ll-u 3.5 1
l
v
1.5
03
LkLd--, 2
-06* 2.5
3
-In 69
q=1 q=2 q=3 q=4 q=5
v *
I
*
e
q A
l
t~~~~I1~~~tl~~~i~t~i~~~~~ 0.5 1 1.5 2
(4
J
2.5 -In
3 6~
Fig. 1. lntermittency data obtained using the a-cascade model as described in the text. (a) random simulation results (b) random simulation with multiplicity reduced by a factor of five; (c) simulation on a lattice; (d) lattice data with nadtiplicity reduced try a factor of five.
It is generallytaken to be true that Fq is more accurately determinedwhen the multiplicities are large.in view of this it is temptingto ask whetherthe resultsare invariantundera resealingof the multiplicities. This is not expectedto result in any changein the probabilities pi for all i ( = 1, . . . , M) and thereforeis not expected to yield any different result. The probability of obtaining pi for the ith bin in N events,alsodoesnot change. It only resultsin a resealingof the fluctuations.However, as shov:c in Fig. 1, this is not in generaltrue. There is a certain class of events (presently the one correspondingto the lattice simulation), where intermittency slopesare negative.In fact, the two particle pseudorapiditycorrelation as derived from the same data (before resealing)using Eq. (8), showsa downward spike (Fig. 3) at zero which is unphysicalunder
the presentcircumstances.The S-functionlike bew’l&iour, although negative,seemsto suggestpractically no correlationin accordancewith the smallnessof the correspondingintermittencyexponent.Includedin Fig. 1 are a set of data where the multiplicities have been scaleddown by a factor of 5. For the randomsimulation. surprisingly,this resuitsin a changein sign of the slopeswhich are now scaledup by a factor of 5. Comparedto this the lattice simulationdoesnot result in a changein signsof the slopes.The generalisedmoE :nts G,, are unaffected by this transformation.Thus it appears,the generalizedmamen&G, aremore reliable comparedto the scaledfactorial momentsF, which may not turn out to be correct (with the rtght sign for the intermittcncyslopes)anddependbn the multiplicities. In such cases,use of Eq. (8) leads to wr“r;
G.C. Mishra et al. /Physics Letters B 348 (1995]297-302
300
:,”Ejj----A qz3 1
12
‘I *
10
*k
q=4 q=5
* v
k
t
8 ;
.
2
* . *
0
= ,/,,]/
6
16
1.5
e 2
e 2.5
. 3
t
q
14; 12 10
L _ -E-
8
-
q=l q=2
l
$$j *
. *
q=5 l
6
=
*
4
y
v
_ A L-l.-l-I.Itl~~rtI~!frI~t :
0.5
1
.
.
A
A
(r A A
A
Prf
a
II
m
e 1.5
2
.,,,,,,, 2.5
;,,(,,,:,,,I 3
K4 Fig. 2. Intermittency lattice.
da% obtained using the generaked
moments G,, for various orderc: (q>O).
results.Insteadof the usualpower law divergenceat the origin it yields a negativespike which is absurd. This happensdue to the negativeslopeas seenfor Fz. But this is not really true andcan be avoidedusingthe following analysis. In generalthe two partic!ecorrelationfunction can be defined[4,9] as R(x) = C Pt 7j)PtTj .i
3.5
e
0
m
6
.
a 1 ,,/,,,,,(,,,,,,,,,,,(,,, 1
I .
.
.
OS
.
.
*
4
d c T
*
+X) .
(9)
Using this, the algebraicpower LXin Eq. (3) for the correlationdecaycanbe calculatedfrom the linearpart of logR(x) versuslogn [5]. In fact a is expectedto be equalto d - D2 whereD2 is the correlationdimension as givenearlierin termsof Eq. (6) . In the present ‘case,the dimensionof the support,d is equalto unity.
3.5
4
4.5
--In 61 (a) Random simulation;
(b) simulation
on a
In Fig. 4, we have presentedlog R as a function of log 811as obtainedfrom Eq. (9) for the randomand the lattice calculationswith opposite intermittency slopes(prior to any scaling).One can noticethat a! is almostthe samefor both the caseswhich is very much expected.A linear fit to the top portion yields a value 0.018. The slope is also, as expected,negative.This valueof a!leadsto a correlationdimensionDz of 0.982. This is quite consistentwith the value 0.979 obtained usingEq. (6) with 72obtainedfrom Fig. 2. It remainsto be seenwhetherexperimentaldatashow negative intermittency slopes based on which one would be temptedto ignorethe correspondingevents asjunk. In conclusion,keepingall the abovefindings, it must be mentionedthat it is better to considerthe generalizedmomentsrather than the scaledfactorial
G. C. Mishra et al. /Physics Otters B 348 (1991) 297-302
0.001
-
---
I-op*e*e
-.._--_ *e.*
. I
-0.006
-0.6
-0.4
/
I
I,
,
-0.2
,
I,,
,
0
I
0.2
j
/
,
j
,
0.4 7,--q,
Fig. 3. Two particle correlation obtainedfrom Eq. (8) of text for the lattice data.Thesereiults correspondto negativeiniermittency slopesin terms of factorial momentsFq
-5.32
i;
1-
P
-5.34 -5.36
-
-5.36
-
-5.4
-
-5.42
-
-5.44
r
-5.46
1
In R(x)
-5.46
In(x)
Fig. 4. Plot of log R(x) versuslogx. The top portion can be fitted to a straightline with negativeslope.The starsand the squaresrepresentdata correspondingto the random and the lattice simulationsrespectively.
momentswhile carryingout the intermittencyanalysis. One shouldbe cautiouswhile using Eq. (8) for a correlationanalysis.
References [ 1J A. Bialas and R. Peschanski,Nucl. Phys.B 273 ( 1986) 703. !2J B. Buschbeck,P. Lipa and R. Peschanski,Phys. Lett. B 215 (1988) 788; I.V. Ajinenko et al., Phys.Lett. B 222 ( 1989) 306;
302
G.C. k-fish et al. /Physics
R. Holynski et aI., Phys.Rev. Led. 62 i i989) 733; 1. Derado,in: HadronicMatter Collisions (Tucson,M,, 1988). edq.P. Carruthersand J. Rafelski (World Scientific,Singapore, 1989) p, 222. [31 B. Bialasand M. Gradizicki, Phys.Letc.B 247 (1990) 483. 141T. Vicsek, Fractai Growth Parnnmeua (World Scientific, Singapore,1989)
LettersB 348 (1995) 297-302 151 R.C. Hw:, Phys.Rev. D 41 (199Oj 1456, [6! C. Abajaret al., CERN-PPE/92-85. 1’7I E.L. Berger,Nud. Phys.B 8.5(1975> 61. 18 1 SP. Ratti,G. Salvadon,G. %x&i, S.Lovejoyand D. SchertTer, 2. Phys.C 6: (1994) 229. 1‘31P. Carruthers,fntern. J. hlod. Phy?.A 4 (I 9%) 5587.
6 April 1995
PHYSICS LETTERS B
PhysicsLetters B 348 ( 199s) 297-302
Some interesting observationsregardingfactorial moments and generalisedmoments in intermittency analysis G.C. Mishra, B.K. Nan&, S.K, Nayak, D.P. Mahapatra Institute of’Physics, Bhubanmwar
751005, India
Received5 January1995 Mitor: L. Montanet -
Abstract A study of intermittency has been carriedout within the framework of the a-cascade model using both scaled ktorial as well as generalised moments. It has been found that, depending upon the multiplicity, the scaled factorial moments can sometimes leadto negativeslopeswhenlog Fq is plottedagainst-log Sr].Compared to thisthe generalised moments G, have been found to be unaffected by multiplicity.
Intermittency in multiparticle production is a very interestingtopic since investigationof fluctuationsin phasespacecan revealsomedeepaspects[ 11. This is why the techniquehas beenused for analysisof multiparticle production in high energyexperimentaldata [2]. In caseswhere intermitterlcy analysis has been applied there are indications regarding transitions betweenquarks (gluons) and hadrons.Thereforeit is all the more important,particularly in the caseof high energyheavy ion collision experimentswherethereis a possibility of QGP formation. In the analysisof intermittencyone essentiallytakes the multiparticle distribution over the specifiedpseudorapidityrangeandcalculates,asdiscussedlater,a set bf momentsG, or Fq of order q of the particle distribution.‘Thenone plots -log G, and/or log Fq versus - log 677,877 denotingthe resolutionor bin size.In the absenceof fluctuation,what is expectedis a flat behaviour with zero slopes.Howevec,intermittencyresults in finite non-zeroslopesknown as the intermittency exponents.These slopes could be related to certain physictilpropertiesof the system.For example;the one 0370~2693/95/$09.50C 1995 Eisevier ScienceB.V. Ail rights reserved SUVO370-2693(95)00143-3
correspondingto order 2 directly yields the exponent of the two particle correlationfunction which in turn behavesas an algebraicpower law. The intermittency exponentfor the order 2 must also be positive leading to the usualalgebraicdecayof the correlationfunction. Although this is expectedto be true always,therecan be problems.In the presentwork, following the prescription as given elsewhere[ 11, we have carriedout a Monte Carlo simtilation in one dimensionwherethe intermittencyexponents,in somecases,turned out to be negative,which also dependupon the total multiplicity. In the next sectionwe briefly discussthe formalism which is followed up by a sectionon the presentCalculation.At the end we summarisethe results. Let us considera typical examplewhich involves a multiplicity distributionpi given over a pseudorapidiiy interval Aq which is further divided into M bins jqi (i= 1, . . .. M). The distributionpi satisfiesthe normalization condition (1)
G.C. Mishra et ul. /Physics Letters B 348 (1995) 297-302
298
Following [I], in the usualanalysiswhat one calculates,for various67, is a set of moments
c, = ($- c (MpijR)
(q= 1,2, 3, etc.)
(2)
The strengthof the singularitiesare given in termsof a parametercu(q)=dT(q) ldq andhow denselythey are distributedis given by .fCd = qa(3) - 7(q) .
(7)
where ( > indicatesan event average.lnrermittent behaviourresults in non zero slopes when log C, is plottedagainst - log 6~. This is a consequence of the underlyingself-similarity leadingto scaleinvariance. Eq. ( I ) is only valid in the limit M t m. However, for a finite numberof bins onecan calculatethe scaled factorial momentsFp which are equivalentto C, with the samebehaviour.At this point, it also must be mentionedthat neither4 nor any of the momentsC, can be negative.Furthermore,it hasbeenpointedout thatthese momentsare relatedto certain physical propertiesof the system.For example.if a systemis selfsimilar,the two particlecorrelationin r) space
One can determineT(q) from the slope of -logG, versus -log 67. Apart from the abovementionedmomentanalysisit is alsoimportantto studytwo particlecorrelations.The two particlecorrelationfunction can be independently calculateddirectly from the given ,q-distribution.In terms of single particle densitiesp( 71) and P(Q), it is given as
R(rl:, d-(/q,
Having presentedthe basicformulaefor the analysis let us considera particular examplewhere the alpha cascademodel as shown elsewhere [ I] has heen employedfor generatingan intern&tent system.We start with a systemwith a uniform multiplicity in the pseudorapidityinterval [ 0, 1] . Sequentiallywe divide the system into two halves and enhanceby a given factor (a) the multiplicities at a given fraction (p) of the points selectedat randomin eachof the new intervals.The sequenceis thenrepeatedup to a given number of generationsV.Initially the ideawas to reproduce the resultsthereand therebyto understandthe model. Therefore the parameterschosen viz. h = 2, ~=4, a = 0.25 and/3= 0.1 arealmostthe sameasin theabove work. However,wecarriedouttwo simulations.Inonecase the Q valueswere generatedat random,uniformly distributed betweenzero and one. In the other case they wereequispacedon a linearlattice.The resultsobtained were then groupedinto 200 q-bins of width 0,005. In eachcase500 eventsweregeneratedfor the eventaveraging. Eq. (3) of 181 was usedfor the calculationof Fy. Typical total multiplicities areof the orderof 2000, The results obtainedfor various momentsF,, for the two cases,areshownin Fig. 1. One can noticethat the randomsimulationyields positiveslopesin contrastto the latticesimulationwhich showsnegativeslopes.But the results, in terms of generalisedmomentsG, arc almostidenticalin both cases(Fig. 2).
-~l)--~.
(3)
In such a casethe slope of log F2 versus -log 67 is expectedto be positive with the value (Y[ 3,4]. Usually intermittencyinvolvesa certaindistribution of probabilitiesdefinedon a supportwhich may be a simple geometricalfractal often called the substrate. The probability distributionis called the %zctul meusllre. A fractal measurepossessesinfinite types of singularities. Points correspondingto a given type of singularity usua$ form a fractal subsetof a given dimensiondependingon the type ofsinguiari!y. Therefore it is importantto study the multifractality behaviour which is doneusinga setof generalizedmoments [4-6] given as G,=
ti
$py
)
.
(4)
It is interestingto note that the order4 can be positive, Neroor negative,unlike the scaledfactorial moments discussedearlier.In generalfor an intermittentsystem
horn which one can get, for variousorders4, the generalizeddimensions
(8)
where p(r],, Q) is the two particle inclusive density r71.
C.C. Mishm et al. /Physics Letters B 348 (1995) 297-302
u &i
0.06 0.04
c
A
-0.1
v q=4
I
*
.
q=5
-0.15
+ -0.2 -0.25
.
I
0.02
-0.3 0
i
a
;
: -0.35
1
402
!I11 I III< I
L.u/LIIL_ILI_LII 0.5
1
1.5
! . t
“? -0.05 Lv 0.05 :I------
q=3
-In
(a)
Lt-
e
q=1
u
q=2
A
q=3
0.5
1 ,I,,
67
;
.
.
1
.
I,,, 1.5
1
1 . l
v q=4 * q=s
-0.4 I- III,
3
2.5
2
1 /--
299
I h&.-L&A *
2
W
2.5
3
-In
67f
L” 0.02 c
-0.1
u
q=l
u
q=2
A
q=3
v
q=4
*
q&i
tJ-l-uLul-Ll-u 3.5 1
l
v
1.5
03
LkLd--, 2
-06* 2.5
3
-In 69
q=1 q=2 q=3 q=4 q=5
v *
I
*
e
q A
l
t~~~~I1~~~tl~~~i~t~i~~~~~ 0.5 1 1.5 2
(4
J
2.5 -In
3 6~
Fig. 1. lntermittency data obtained using the a-cascade model as described in the text. (a) random simulation results (b) random simulation with multiplicity reduced by a factor of five; (c) simulation on a lattice; (d) lattice data with nadtiplicity reduced try a factor of five.
It is generallytaken to be true that Fq is more accurately determinedwhen the multiplicities are large.in view of this it is temptingto ask whetherthe resultsare invariantundera resealingof the multiplicities. This is not expectedto result in any changein the probabilities pi for all i ( = 1, . . . , M) and thereforeis not expected to yield any different result. The probability of obtaining pi for the ith bin in N events,alsodoesnot change. It only resultsin a resealingof the fluctuations.However, as shov:c in Fig. 1, this is not in generaltrue. There is a certain class of events (presently the one correspondingto the lattice simulation), where intermittency slopesare negative.In fact, the two particle pseudorapiditycorrelation as derived from the same data (before resealing)using Eq. (8), showsa downward spike (Fig. 3) at zero which is unphysicalunder
the presentcircumstances.The S-functionlike bew’l&iour, although negative,seemsto suggestpractically no correlationin accordancewith the smallnessof the correspondingintermittencyexponent.Includedin Fig. 1 are a set of data where the multiplicities have been scaleddown by a factor of 5. For the randomsimulation. surprisingly,this resuitsin a changein sign of the slopeswhich are now scaledup by a factor of 5. Comparedto this the lattice simulationdoesnot result in a changein signsof the slopes.The generalisedmoE :nts G,, are unaffected by this transformation.Thus it appears,the generalizedmamen&G, aremore reliable comparedto the scaledfactorial momentsF, which may not turn out to be correct (with the rtght sign for the intermittcncyslopes)anddependbn the multiplicities. In such cases,use of Eq. (8) leads to wr“r;
G.C. Mishra et al. /Physics Letters B 348 (1995]297-302
300
:,”Ejj----A qz3 1
12
‘I *
10
*k
q=4 q=5
* v
k
t
8 ;
.
2
* . *
0
= ,/,,]/
6
16
1.5
e 2
e 2.5
. 3
t
q
14; 12 10
L _ -E-
8
-
q=l q=2
l
$$j *
. *
q=5 l
6
=
*
4
y
v
_ A L-l.-l-I.Itl~~rtI~!frI~t :
0.5
1
.
.
A
A
(r A A
A
Prf
a
II
m
e 1.5
2
.,,,,,,, 2.5
;,,(,,,:,,,I 3
K4 Fig. 2. Intermittency lattice.
da% obtained using the generaked
moments G,, for various orderc: (q>O).
results.Insteadof the usualpower law divergenceat the origin it yields a negativespike which is absurd. This happensdue to the negativeslopeas seenfor Fz. But this is not really true andcan be avoidedusingthe following analysis. In generalthe two partic!ecorrelationfunction can be defined[4,9] as R(x) = C Pt 7j)PtTj .i
3.5
e
0
m
6
.
a 1 ,,/,,,,,(,,,,,,,,,,,(,,, 1
I .
.
.
OS
.
.
*
4
d c T
*
+X) .
(9)
Using this, the algebraicpower LXin Eq. (3) for the correlationdecaycanbe calculatedfrom the linearpart of logR(x) versuslogn [5]. In fact a is expectedto be equalto d - D2 whereD2 is the correlationdimension as givenearlierin termsof Eq. (6) . In the present ‘case,the dimensionof the support,d is equalto unity.
3.5
4
4.5
--In 61 (a) Random simulation;
(b) simulation
on a
In Fig. 4, we have presentedlog R as a function of log 811as obtainedfrom Eq. (9) for the randomand the lattice calculationswith opposite intermittency slopes(prior to any scaling).One can noticethat a! is almostthe samefor both the caseswhich is very much expected.A linear fit to the top portion yields a value 0.018. The slope is also, as expected,negative.This valueof a!leadsto a correlationdimensionDz of 0.982. This is quite consistentwith the value 0.979 obtained usingEq. (6) with 72obtainedfrom Fig. 2. It remainsto be seenwhetherexperimentaldatashow negative intermittency slopes based on which one would be temptedto ignorethe correspondingevents asjunk. In conclusion,keepingall the abovefindings, it must be mentionedthat it is better to considerthe generalizedmomentsrather than the scaledfactorial
G. C. Mishra et al. /Physics Otters B 348 (1991) 297-302
0.001
-
---
I-op*e*e
-.._--_ *e.*
. I
-0.006
-0.6
-0.4
/
I
I,
,
-0.2
,
I,,
,
0
I
0.2
j
/
,
j
,
0.4 7,--q,
Fig. 3. Two particle correlation obtainedfrom Eq. (8) of text for the lattice data.Thesereiults correspondto negativeiniermittency slopesin terms of factorial momentsFq
-5.32
i;
1-
P
-5.34 -5.36
-
-5.36
-
-5.4
-
-5.42
-
-5.44
r
-5.46
1
In R(x)
-5.46
In(x)
Fig. 4. Plot of log R(x) versuslogx. The top portion can be fitted to a straightline with negativeslope.The starsand the squaresrepresentdata correspondingto the random and the lattice simulationsrespectively.
momentswhile carryingout the intermittencyanalysis. One shouldbe cautiouswhile using Eq. (8) for a correlationanalysis.
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