fourier2d jpg

Classification of initial state granularity via 2dFourier expansion

C.E. Coleman-Smith1,H. Petersen2, R.L. Wolpert3

E-mail: [email protected] Department of Physics, Duke University, Durham, NC 27708-0305, USA2 Department of Physics, Duke University, Durham, NC 27708-0305, USA2 Frankfurt Institute for Advanced Studies, 60438 Frankfurt am Main, Germany3Department of Statistical Science, Duke University, Durham, NC 27708-0251,USA

.

Abstract.A new method to quantify fluctuations in the initial state of heavy ion

collisions is presented. The initial state energy distribution is decomposed with aset of orthogonal basis functions which include both angular and radial variation.The resulting two dimensional Fourier coefficients provide additional informationabout the nature of the initial state fluctuations compared to a purely angulardecomposition. We apply this method to ensembles of initial states generated byboth Glauber and Color Glass Condensate Monte-Carlo codes. In addition initialstate configurations with varying amounts of fluctuations generated by a dynamictransport approach are analyzed to test the sensitivity of the procedure. Theresults allow for a full characterization of the initial state structures that is usefulto discriminate the different initial state models currently in use.

PACS numbers: 25.75.-q, 25.75.Ag, 24.10.Lx, 02.30.Mv

Submitted to: J. Phys. G: Nucl. Phys.

Classification of initial state granularity via 2d Fourier expansion 2

Ultra-Relativistic nearly-ideal fluid dynamics has proven to be a very successfultool for modeling the bulk dynamics of the hot dense matter formed during a heavy ioncollision [1, 2, 3, 4, 5, 6]. The major uncertainty in determining transport propertiesof the QGP, such as the ratio of shear viscosity to entropy, lies in the specification ofthe initial conditions of the collision. The initial conditions have been mainly assumedto be smooth distributions that are parametrized implementations of certain physicalassumptions (e.g., Glauber/CGC). Within the last 2 years the importance of includingfluctuations in these distributions has been recognized, leading to a whole new set ofexperimental observations of higher flow coefficients and their correlations [7, 8, 9, 10].On the theoretical side there has been a lot of effort to refine the previously schematicmodels with fluctuation inducing corrections and to employ adynamical descriptionsof the early non-equilibrium evolution [11, 12, 13, 14].

Hydrodynamical simulations can take these fluctuations into account bygenerating an ensemble of runs each with a unique initial condition, so-called event byevent simulations. This is in contrast to event averaged simulations where an ensembleof fluctuating initial conditions is generated, and then a single initial conditioncorresponding to this set’s ensemble average is subject to evolution. Event by eventmodeling has proven to be essential for correctly describing all the details of the bulkbehavior of heavy ion collisions [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25].

The two main models for the generation of hydrodynamic initial conditions arethe Glauber [26, 27, 28, 12] and color glass condensate (CGC) models [29, 30, 31,32, 33, 34]. The Glauber model samples a Woods-Saxon nuclear density distributionfor each nucleus.

Color glass condensate models are ab initio calculations motivated by the idea ofgluon saturation of parton distribution functions at small momentum scales x. In CGCmodels the gluon distribution for each nucleon is computed and the nuclear collisionis modeled as interactions between these coherent color fields. Each of these modelsgenerates spatial fluctuations whose details depend on the assumptions made in thespecific implementation. Glauber fluctuations come from Monte-Carlo (MC) samplingthe nuclear density distribution. CGC fluctuations arise similarly with additionalcontributions from the self interaction of the color fields.

We present a method for generating a 2d decomposition of fluctuations in theinitial state energy density of a heavy-ion collision. We apply this framework to theensembles of initial states generated by UrQMD [35, 36, 37], by an MC Glauber codeof Qin [12], and by the MC-KLN code of Drescher and Nara [29, 30, 38] which isa based on CGC ideas. The events compared are generated for Au+Au collisionsat√s = 200 AGeV at two impact parameters b = 2, 7 fm. We introduce summary

statistics for the Fourier expansion and show how the radial information exposes cleardifferences between the fluctuations generated by Glauber and CGC based codes. Wenote that the cumulant expansion of Teaney and Yan [10] provides an alternative 2ddecomposition of the initial state density. The cumulant basis used therein has manyappealing properties and the familiar 1d moments can be readily obtained from it.The norms wwhich arise from the expansion we propose below have similar invarianceproperties and as we shall show very naturally reveal the underlying roughness of theevent.

Since colliding nuclei are strongly Lorentz contracted along the beam axis, weneglect the longitudinal dimension. A priori we expect fluctuations in both radial andazimuthal directions. Experiments make measurements in a 2d transverse momentumspace, and further analysis of the final state may reveal correlations with quantities


derived from a fully 2d decomposition of the initial state. Recent work [39] hasshown that events can be constructed which have identical angular Fourier momentsε2, ε3 but dramatically different energy distributions leading to different final stateflow coefficients. The structure of such events is not sensitive to a purely azimuthaldecomposition.

We seek a two dimensional Fourier expansion on a disk of radius r0 > 0. Theorthogonality of Bessel functions of the first kind is the key to this decomposition∫ r0

0

Jα( rr0λα,n

)Jα( rr0λα,n′

)rdr =

r202δnn′ [Jα+1(λα,n)]2, ∀n, n′ ∈ Z,∀α ∈ R, (1)

where λα,n is the nth positive zero of Jα(x). For integer α, Jα(−x) = (−1)αJα(x)and so [Jα+1(λα,n)]2 = [J|α|+1(λα,n)]2 and λα,n = λ−α,n. It follows that the functions

φm,n(r, θ) :=1

J|m|+1

(λm,n

)Jm( rr0λm,n

)eimθ (2)

form a complete orthonormal set on this disk (under the uniform measure r dr dθ/πr20).As such any well behaved (square-integrable and vanishing at the boundary of the disk)function f on this disk admits the convergent expansion

f(r, θ) =∑m,n

Am,nφm,n(r, θ), (3)

in terms of these basis functions, with generalized Fourier coefficients Am,n ∈ C givenby

Am,n =1

πr20

∫f(r, θ)φ?m,n(r, θ)rdrdθ. (4)

A simple scaling law describes the dependence of coefficients Am,n on r0, whichthus is arbitrary so long as the support of f is contained in the ball of radius r0. Theparameter r0 sets the maximum radial position of features in the event that will beresolved in the decomposition. Throughout the paper we use r0 = 10 fm, sufficientfor all the distributions we consider. We center the coordinate system at the center ofmass of the distributions f .

The basis functions φm,n are solutions to Bessel’s equation on the unit disk [40],eigenfunctions of the Laplacian with Dirichlet bc and eigenvalues −(λm,n/r0)2. Thusfor sufficiently smooth f ,

−∇2f(r, θ) =∑m,n

λ2m,nr20

Am,nφm,n.

The values r0/λm,n constitute characteristic length scales for the associated Fouriercomponents Am,n. Higher orders of m and n are associated with larger λm,n (seeFig: 1), corresponding to shorter length scales, and are associated with smaller-scalefeatures (see [41, 42] for a similar perspective in 1d). Other boundary conditions(Neumann, for example) lead to similar countable sets of basis functions.

Higher values of the angular indices ±m correspond to higher numbers of zerocrossings in the angular components of basis functions, or “lumpiness” in rotation,while higher values of the radial indices n are associated with more roughness asone moves closer or farther from the center of mass. The first few basis functions


|m| (angular)n

(rad

ial)

2

4

6

8

2 4 6

0

200

400

600

800

1000

1200

Figure 1: The eigenvalues λ2m,n of the negative Dirichlet Laplacian on the unit disk areplotted as a function of |m| (angular) and n (radial). These can be interpreted as aninverse length scale for the coefficients Am,n. Asymptotically |λm,n| � π(n+|m|−1/4)as n,m→∞.

Figure 2: Plots of the first few real (left) and imaginary (right) components ofφm,n(r, θ). The angular coefficient m ∈ [0, 5] increases from left to right, the radialcoefficient n ∈ [1, 3] increases from bottom to top.

x (arb)

y (a

rb)

−1.0 0.0 0.5 1.0

−1.

00.

00.

51.

0

x (arb)

y (a

rb)

−1.0 0.0 0.5 1.0

−1.

00.

00.

51.

0

m (angular)

n (r

adia

l)

1

2

3

4

5

6

7

8

−5 0 5

0

2

4

6

8

10

12

14

16

18

Figure 3: First: A typical UrQMD ensemble average subtracted event. Second: thereconstructed event generated by (3) after applying the decomposition. Third: Theabsolute values of the Fourier coefficients |Am,n| for this event.

are plotted in Fig: 2. The lumpy shape of the basis functions suggests that thisrepresentation will be very useful for characterizing the hot and cold spot structuresin the initial state of a heavy ion reaction.

A typical UrQMD event (after subtracting the ensemble average) is shown in


Fig: 3 alongside its decomposition and the resulting Fourier coefficients Am,n. Herewe characterize the event in terms of coefficients m ∈ [−8, 8] and n ∈ [1, 8], sufficientto capture a detailed image of the original initial state distribution (further termscontribute very little additional information because differences between the originaland reconstructed distributions are dominated by numerical noise for coefficients withorder higher than |m|max = 8, nmax = 8). Let us now explore different ways to extractuseful information from this decomposition.

Using the orthogonality of the basis functions along with the Laplacian we canderive simple expressions for norms of the function f to be expanded in the frequencydomain. The L2(f) norm, a measure of the total mass of f , is given by Plancherel’stheorem as:

L2(f) := 〈f, f〉1/2 =[∑

|Am,n|2]1/2

, (5)

where 〈a, b〉 = 1πr20

∫ r00a(r, θ)b?(r, θ)rdrdθ is the inner product for functions on the

disk. The Sobolev H1(f) norm gives a measure of roughness, or of how ‘wobbly’ thefunction is across the disk:

H1(f) :=〈(−`2∇2 + I)f, f〉1/2

=

[∑(`2λ2m,nr20

+ 1)|Am,n|2

]1/2, (6)

where ` is a characteristic length scale introduced to maintain unit consistency (weuse ` = 1 fm). A variation on the Sobolev norm gives the angular variation M1(f)which quantifies angular gradients,

M1(f) := 〈∂2θf, f〉1/2 =[∑

m2|Am,n|2]1/2

. (7)

We use these below to quantify roughness features of the events considered. Notethat although the individual coefficients {Am,n} are not invariant under translationor rotation of the coordinate system, the quantities L2(f) and H1(f) are invariant(and moreover scale simply with changes in length scale r0), while M1(f) is invariantto rotations.

To illustrate the usefulness of our proposed method, we apply it to three examplemodels: UrQMD, MC-Glauber and MC-KLN, which may be viewed as representativeof the models currently in use.

We consider a set of 100 events generated by each code. UrQMD includesBoltzmann hadronic transport before the hydro begins, while the MC-Glauber codeincludes simple streaming transport of the nucleons after interaction; both of thesewill introduce added spatial fluctuation in the energy density. The Glauber code alsoincludes KNO scaling of the multiplicity fluctuations per binary collision. For allmodels the initial condition was computed in a 200 point grid in the transverse planeat the center of the collision along the beam axis. To explore the centrality dependenceof the analysis we consider events at two impact parameters b = 2, 7 fm. All eventsare generated for Au+Au collisions at

√sNN = 200 AGeV. To study the fluctuations

generated by these events the ensemble averaged event is computed for each modeland subtracted from each event in the ensemble before applying the decomposition.

We generated a series of ensembles of UrQMD events by averaging successivelylarger samples of independent raw events together before subtracting out the ensemble


average. See [43] for more details on this process and its influence on the ellipticityand triangularity of the UrQMD initial state.

As the number Na of events over which we average increases, the scale of thefluctuations will be diminished at approximate rate 1/

√Na. However this rescaling

due to averaging events will not substantially change the shape of any particularfluctuation. We have examined events with Na = {1, 2, 5, 10, 25}. While only theNa = 1 case represents the true UrQMD output the diminishing scale fluctuation inthis sequence of data sets will be used to illustrate different aspects of roughness inour analysis. Any statistic proposed as a measure of event fluctuations should beinvariant under this averaging procedure, which affects all frequencies equally, andinstead should be sensitive to smoothing procedures that filter higher frequencies.

2 4 6 8

020

040

060

080

0

b (fm)

H1

Na = 1

Na = 2

Na = 5Na = 10

Na = 25

Na = 1

Na = 2

Na = 5Na = 10Na = 25

MC−GlaubUrQMDMC−KLN

2 4 6 8

050

100

150

200

b (fm)

M1

Na = 1

Na = 2

Na = 5

Na = 10

Na = 25

Na = 1

Na = 2

Na = 5Na = 10Na = 25


2 4 6 8

010

2030

4050

60

b (fm)L 2

Na = 1

Na = 2

Na = 5

Na = 10

Na = 25

Na = 1

Na = 2

Na = 5Na = 10

Na = 25


Figure 4: (color online) The ensemble averages of the Sobolev norm H1 (left) theangular variation M1 (center) and the L2 norm (right). The labels on the UrQMDglyphs give the number of averagings used to generate the ensemble members. TheMC-Glauber and MC-KLN results are plotted with an artificial offset in b to permiteasier comparison.

In Fig: 4 the ensemble averages of each quantity (H1, M1, L2) are plotted foreach set of events. Considering first the UrQMD results it is clear that each tendsto zero as Na increases as expected, confirming that each is quantifying fluctuations.Fluctuations are lower at the larger impact parameter b but the relative ordering ofthe models is preserved. For all norms we see good agreement between the Na = 2UrQMD events and the MC-Glauber code.

The L2 norm reflects the mean square fluctuation, theH1 norm in addition reflectsthe mean square spatial gradient of the fluctuations, and the M1 semi-norm quantifiesmean square angular variation. The MC-KLN results show the largest H1 values whilehaving M1 values comparable to UrQMD Na = 1, the MC-KLN L2 is somewhat lessthan UrQMD. The relative ordering of the Glauber and UrQMD events is the sameacross all norms. The UrQMD and Glauber events are rather similar in the scale andnature of their fluctuations, UrQMD produces slightly larger fluctuations. The MC-KLN model produces fluctuations on a scale comparable to a hypothetical UrQMDNa = 3/2. However the large H1 and comparable M1 indicates that these eventsexhibit larger radial gradients than the other models. This may be attributed to thevery rapid spatial falloff of the gluon density near the edges of the nucleons.

The norms defined and plotted above, however informative, are clearly quitesensitive to the overall scale of the fluctuations. This is shown most clearly in theseparation over Na for the UrQMD results in Fig: 4. We propose the following quantity(the “roughness ratio”) as an overall scale invariant measure of the roughness in an


event:

R2 =H2

1

L22

− 1 =〈−`2∇2f, f〉〈f, f〉

=`2∑λ2m,n|Am,n|2

r20∑|Am,n|2

, (8)

a weighted average of the scale-free squared characteristic inverse lengths (λm,n`/r0)2,weighted by the squared coefficients |Am,n|2. This is invariant to rescaling of f orto Euclidean transformation, but smoothing operations that preferentially reducehigh-frequency components of f will reduce R2 (but never below its minimum of(λ0,1`/r0)2 ≈ 0.05783).

To illustrate that R truly measures the degree of roughness in an event we havehave successively applied a finite difference smoothing operator (a five point Laplacianstencil) to the event shown in Fig: 3, in effect melting the event. We present the H1,L2 and R measures resulting from this process in Table: 1. All of the norms decreasewith the number of smoothing iterations, however as we have shown above the H1 andL2 norms do not exhibit an overall scale invariance which is desirable in an explicitmeasure of roughness. The L2 norm gives the most natural measure of the overallscale of fluctuations in an event. The roughness ratio R provides an explicitly scaleinvariant measure of the gradients, or roughness, in an event.

Nf H21 L2

2 R2

0.00 3855.68 1754.63 1.2032.00 3275.55 1547.44 1.12

128.00 2159.49 1108.36 0.951024.00 244.82 173.38 0.414096.00 27.87 24.80 0.12

Table 1: The H21 , L2

2 and R2 measures of a typical UrQMD event as a function ofthe number Nf of filtering passes, using a five-point finite difference smoothing filter.The event we smoothed is shown in Fig: 3.

H12 L2

2 − 1

P(H

12L 22

−1)

0.00

00.

005

0.01

00.

015

50 100 150 200 250 300

b=2 fm

UrQMD N=1UrQMD N=5UrQMD N=25MC−KLNMC−Glaub

H12 L2

2 − 1

P(H

12L 22

−1)

0.00

00.

005

0.01

0

50 100 150 200 250 300

b=7 fm

UrQMD N=1UrQMD N=5UrQMD N=25MC−KLNMC−Glaub

Figure 5: (color online) The distribution ofR2 (left, right) for UrQMD Na = {1, 5, 25},MC-KLN and MC-Glauber. The left figure shows events at b = 2 fm, the right figureshows b = 7 fm.

In Fig: 5 we show the distribution of the roughness ratio R for UrQMD, MC-Glauber and KLN events at two centralities. It is invariant to the overall scale of thefluctuations in the event, the curves for each of the Na UrQMD classes fall neatly ontop of each other. Further we observe that the MC-Glauber and UrQMD events have


very similar distributions, this is reasonable given that the UrQMD events originatefrom sampling Woods-Saxon profiles as well. The distributions for the KLN eventsclearly separate from the Glauber curves. Given the scale invariance of this quantity weconclude that the shape and extent of the fluctuations in KLN events is fundamentallydifferent from that in equivalent Glauber events.

We have presented a new method for characterizing the fluctuations in the initialstate of heavy ion collisions. The method is simple and general, it can be applied aseasily to theoretical models as to the output of event generators. The R measure wehave introduced is invariant under changes in coordinates, rotations and the overallscale of the distributions. We have shown the ability of this measure to quantify thebroad differences among the physical models we considered. The radial informationincluded provides additional insights into the nature of fluctuations which are notreadily attainable by considering quantities derived from angular decompositionsalone.

In future work we will examine how this measure passes through thehydrodynamical evolution to the hadronic final-state of the collision. Even if theobservables developed here cannot be measured in detectors they provide a usefulbasis for apples-to-apples comparison of initial state models.

The authors are happy to provide sample analysis code to interested parties.The authors would like to thank G.Y. Qin for providing the MC-Glauber code. Thiswork was supported by U.S. Department of Energy grant DE-FG02-05ER41367 withcomputing resources from the OSG EngageVo funded by NSF award 075335, by NSFgrants DMS–1228317 and PHY–0941373, and by NASA AISR grant NNX09AK60G.Any opinions, findings, and conclusions or recommendations expressed in this materialare those of the authors and do not necessarily reflect the views of the NSF or NASA.H.P. acknowledges support by the Helmholtz-Gemeinschaft for a HYIG grant VH-NG-822.

Appendix A. Angular Decomposition

To connect our analysis with commonly used quantities we quote in Table: A1 thevalues of ε2 and ε3 computed for events without ensemble average subtraction.

b (fm) E[〈ε2〉] E[〈ε3〉]UrQMD, Na = 1 2 0.096± 0.005 0.079± 0.004MC-KLN 2 0.084± 0.005 0.046± 0.003MC-GLAUB 2 0.089± 0.005 0.070± 0.004UrQMD, Na = 1 7 0.271± 0.010 0.117± 0.006MC-KLN 7 0.343± 0.010 0.111± 0.006MC-GLAUB 7 0.231± 0.009 0.099± 0.006

Table A1: The ensemble average values of 〈ε2〉 and 〈ε3〉 for each of the models atimpact parameters b = 2, 7 fm.

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