Elliptic Flow from Color Glass Condensate

如果为大量固定质量的流体微元他就是有限控制体，少量流体微团（但足够表现流体宏观量）的就是流动微元（fluid
element），但是体积和面积会变化。之所以守恒和不守恒是因为空间位置固定的微元或者控制体可以应用质量守恒
方程来推导，空间位置随流因为质量固定不能应用与质量守恒因此成为非守恒形式。
1.2 动量方程
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第七章明渠恒定非均匀流由于产生明渠均匀流的条件非常严格，自然界中的水流条件很难满足，故实际中的人工渠道或天然河道中的水流绝大多数是非均匀流。

明渠非均匀流的特点是底坡线、水面线、总水头线彼此互不平行(如图7-1所示)。

产生明渠非均匀流的原因很多，例如明渠横断面的几何形状或尺寸的沿流程改变，粗糙度或底坡沿流程改变，在明渠中修建水工建筑物(闸、桥梁、涵洞等)，都能使明渠水流发生非均匀流。

明渠非均匀流中也存在渐变流和急变流，若流线是接近于相互平行的直线，或流线间夹角很小、流线的曲率半径很大，这种水流称为明渠非均匀渐变流。

反之，则为明渠非均匀急变流。

图7-1本章首先分析和讨论明渠非均匀流的一些基本概念和明渠急变流(水跃和水跌)，然后讨论明渠非均匀渐变流水深(或水位)沿程变化的基本方程，最后着重研究水面曲线变化规律，并进行水面线计算。

而本章的重点是明渠非均匀流中水面曲线变化的规律及其计算方法。

在实际工程中，例如，在桥渡勘测设计时，为了预计建桥后墩台对河流的影响，便需算出桥址附近的水位标高；在河渠上修建水电站，为了确定由于水位抬高所造成的水库淹没范围，亦要进行水面曲线的计算。

因明渠非均匀流的水深沿程变化，即h=f(s)，为了不致引起混乱，将明渠均匀流的水深称为正常水深，以h0表示。

§7-1 明渠水流的三种流态明渠水流有的比较平缓，象灌溉渠道中的水流和平原地区江河中的流动。

如果在明渠水流中有一障碍物，便可观察到障碍物上水深降低，障碍物前水位壅高能逆流上传到较远的地方(见图7-2a)；而明渠水流有的则非常湍急，像山区河道中的水流，过坝下溢的水流，跌水、瀑布和险滩地的水流。

如遇障碍物仅在石块附近隆起，障碍物上水深增加，障碍物干扰的影响不能问上游传播(见图7-2b)。

上述两种情况表明，明渠水流存在两种不同的流态。

它们对于所产生的干扰波(Disturbance Wave)的传播，有着不同的影响。

障碍物的存在可视为对水流发生的干扰，下面分析干扰波在明渠中传播的特点。

Rayleigh-Benard-convection（瑞利-贝纳尔-对流）Rayleigh–Bénard convectionis a type of natural convection,occurring in a plane offluid heated from below, in which the fluid develops a regular pattern of convection cells known as Bénard cells.Rayleigh–Bénard convection is one of the most commonly studied convection phenomena because of its analytical and experimental accessibility.[1]The convection patterns are the most carefully examined example of self-organizing nonlinear systems[1].Buoyancy is responsible for the appearance of convection cells. The initial movement is the upwelling of warmer liquid from the heated bottom layer.[2] This upwelling spontaneously organizes into a regular pattern of cells.Perfect Conditions, Perfect Pattern (Almost!)Under the right conditions, convection cells will take the shape of hexagons.Why don't we see hexagon-shaped clouds in the sky?Take a look at the picture to the below, and notice the small glitch in the pattern. It was later discovered that there was a tiny dent in the copper plate under the fluid. This tells us thatthe pattern is very sensitive to the bottom surface. Think about our earth - it's surface has millions of dents and bumps in the form of mountains, valleys, canyons, and more. All of these surface features affect the convection patterns in the atmosphere.Fluid in MotionThis picture shows a time lapse view of Rayleigh-Benard cells. The picture was taken over ten seconds, so the aluminum flakes in the fluid look like long trails instead of small particles. This helps to visulaize how the fluid is moving: up through the center of the cell, then spreading out and sinking at the edges of the cell.Development of convectionThe experimental set-up uses a layer of liquid, e.g. water, between two parallel planes. The height of the layer is small compared to the horizontal dimension.At first, the temperature of the bottom plane is the same as the top plane.The liquid will then tend towards an equilibrium, where its temperature is the same as its surroundings. (Once there, the liquid is perfectly uniform: to an observer it would appear the same from any position. This equilibrium is also asymptotically stable: after a local, temporary perturbation of the outside temperature, it will go back to its uniform state, in line with the second law of thermodynamics).Then, the temperature of the bottom plane is increased slightly yielding a flow of thermal energy conducted through the liquid. The system will begin to have a structure of thermal conductivity: the temperature, and the density and pressure with it, will vary linearly between the bottom and top plane. A uniform linear gradient of temperature will be established. (This system may be modelled by statistical mechanics).Once conduction is established, the microscopic random movement spontaneously becomes ordered on a macroscopic level, forming Bénard convection cells, with a characteristic correlation length.Convection featuresConvection cells in a gravity fieldSimulation of Rayleigh–Bénard convection in 3D.（Simulation of 3D Rayleigh-Bèrnard convection with Rayleigh number 10^4 and Prandtl number 1. Temperature is mapped onto the colors of the spectrum, and streamlines are shown in white.）The rotation of the cells is stable and will alternate from clock-wise to counter-clockwise horizontally; this is an example of spontaneous symmetry breaking.Bénard cells are metastable. This means that a small perturbation will not be able to change the rotation of the cells, but a larger one could affect the rotation; they exhibit a form of hysteresis. Moreover, the deterministic law at the microscopic level produces a non-deterministic arrangement of the cells: if the experiment is repeated, a particular position in the experiment will be in a clockwise cell in some cases, and a counter-clockwise cell in others.Microscopic perturbations of the initial conditions are enough to produce a(non-deterministic) macroscopic effect. This inability to predict long-range conditions and sensitivity to initial-conditions are characteristics of chaotic or complex systems (i.e., the butterfly effect).If the temperature of the bottom plane was to be further increased, the structure would become more complex in space and time; the turbulent flow would become chaotic.Convective Bénard cells tend to approximate regular right hexagonal prisms, particularly in the absence of turbulence[3][4][5], although certain experimental conditions can result in the formation of regular right square prisms[6] or spirals[7].The Rayleigh–Bénard InstabilitySince there is a density gradient between the top and the bottom plate, gravity acts trying to pull the cooler, denser liquid from the top to the bottom. This gravitational force is opposed by the viscous damping force in the fluid. The balance of these two forces is expressed by a non-dimensional parameter called the Rayleigh number.The Rayleigh Number is defined as:whereTu is the Temperature of the top plateTb is the Temperature of the bottom plateL is the height of the container.g is the acceleration due to gravity.ν is the kinematic viscosity.α is the Thermal diffusivityβ is the Thermal expansion coefficientAs the Rayleigh number increases, the gravitational forces become more dominant.At a critical Rayleigh number of 1708, the instability sets in, and convection cells appear.The critical Rayleigh number can be obtained analytically for a number of different boundary conditions by doing a perturbation analysis on the linearized equations in the stable state.[8] The simplest case is that of two free boundaries, which Lord Rayleigh solved in 1916.[9] and obtained Rc = 27⁄4 π4 ≈ 657.51.[10] In the case of a rigid boundary at the bottom, and a free boundary at the top (which is the situation in an kettle without a lid), the critical Rayleigh number comes out as Rc = 1,100.65Effects of surface tensionIn case of a free liquid surface in contact with air,buoyancy and surface tension effects will also play a role in how the convection patterns develop.Liquids flow from places of lower surface tension to places of higher surface tension. This is called the Marangoni effect.When applying heat from below, the temperature at the top layer will show temperature fluctuations. With increasing temperature, surface tension decreases.Thus a lateral flow of liquid at the surface will take place, from warmer areas to cooler areas. In order to preserve a horizontal (or nearly horizontal) liquid surface, cooler surface liquid will descend. This down-welling of cooler liquid contributes to the driving force of the convection cells. The specific case of temperature gradient-driven surface tension variations is known as thermo-capillary convection, or Bénard–Marangoni convection.贝纳尔—瑞利对流模型（Benard Rayleigh convective model）是由贝纳尔和瑞利分别通过实验研究提出的有关热对流作用形成机理及其影响因素的、从定性到定量的实验。

Methods for analyzing anisotropic flow in relativistic nuclear collisionsA.M.Poskanzer 1and S.A.Voloshin 2,*1Nuclear Science Division,Lawrence Berkeley National Laboratory,Berkeley,California 947202Physikalisches Institut der Universita¨t Heidelberg,Heidelberg,Germany ͑Received 20May 1998͒The strategy and techniques for analyzing anisotropic flow ͑directed,elliptic,etc.͒in relativistic nuclear collisions are presented.The emphasis is on the use of the Fourier expansion of azimuthal distributions.We present formulas relevant for this approach,and in particular,show how the event multiplicity enters into the event plane resolution.We also discuss the role of nonflow correlations and a method for introducing flow into a simulation.͓S0556-2813͑98͒04109-0͔PACS number ͑s ͒:25.75.LdI.INTRODUCTION:SUBJECT AND TERMINOLOGYRecently,the study of collective flow in nuclear collisions at high energies has attracted increased attention of both theoreticians and experimentalists ͓1͔.There are several rea-sons for this:͑i ͒the observation of anisotropic flow at the AGS ͓2–7͔and at the SPS ͓8–11͔,͑ii ͒progress in the theo-retical understanding of the relation between the appearance and development of flow during the collision evolution,and processes such as thermalization,creation of the quark-gluon plasma,phase transitions,etc.͓12–18͔,͑iii ͒the study of mean field effects ͓14,19͔,͑iv ͒the importance of flow for other measurements such as identical and nonidentical two-particle correlation analyses ͓20–23͔,͑v ͒the development of new techniques suitable for flow studies at high energies ͓24–26,1,9͔.Although all forms of flow are interrelated and represent only different parts of one global picture,usually people discuss different forms of collective flow,such as longitudinal expansion,radial transverse flow,and aniso-tropic transverse flow of which the most well established are directed flow ͓24,19͔and elliptic flow ͓27͔.The study of flow in ultrarelativistic nuclear collisions has just begun and it is very important to define the subject and establish the terminology.At high energies the longitudinal flow is well decoupled from transverse flow.The flow ͑po-lar ͒angles observed at low energies are relatively large and a rotation of the coordinate system was done in order to ana-lyze the event shape in the plane perpendicular to the main axis of the flow ellipsoid.At high energies the flow angles become very small,␪flow Ϸ͗p x ͘/p beam Ӷ1,so that not doing such a rotation ‘‘induces’’an elliptic anisotropy of the order of the square of the flow angle and usually can be neglected.This means that at high energies one does not have to rotate to the flow axis to study the flow pattern ͓27͔,but one can use the plane transverse to the beam axis.Thus we discuss only anisotropic transverse flow from the particle azimuthal distributions at fixed rapidity or pseudorapidity.It appears very convenient to describe the azimuthal distributions by means of a Fourier expansion ͓28,25,24͔,and below we characterize different kinds of anisotropies as corresponding to different harmonics ͑in analogy and/or contrast to the de-scription at low energies,where the three-dimensional event shape is characterized using multipole terminology ͒.Aniso-tropic flow corresponding to the first two harmonics plays a very important role and we use special terms for them:di-rected and elliptic flow,respectively.The word ‘‘directed’’comes from the fact that such flow has a direction,and the word ‘‘elliptic’’is due to the fact that in polar coordinates the azimuthal distribution with nonzero second harmonic represents an ellipse.II.CORRELATIONS WITH RESPECTTO AN EVENT PLANEStrategy .In this section we summarize an approach to the study of anisotropic flow which is particularly suitable at high ͑AGS/SPS/RHIC ͒energies.It uses the Fourier expan-sion of azimuthal distributions,introduced in this way for the analysis in Ref.͓25͔.The essence of the method is to first estimate the reaction plane.The estimated reaction plane we call the event plane.The Fourier coefficients in the expan-sion of the azimuthal distribution of particles with respect to this event plane are evaluated.Because the finite number of detected particles produces limited resolution in the angle of the measured event plane,these coefficients must be cor-rected up to what they would be relative to the real reaction plane.This is done by dividing the observed coefficients by the event plane resolution,which is estimated from the cor-relation of the planes of independent subevents ͓29͔͑sub-groups of the particles used for the event plane determina-tion ͒.The resolution obtained from the subevents can be converted to that for the full event by means of the multi-plicity dependence of the resolution which will be described below.Also,if the detector does not have full azimuthal acceptance,the acceptance bias has to be removed.Fourier expansion .The quantity under study in the most general case is the triple differential distribution.In this,the dependence on the particle emission azimuthal angle mea-sured with respect to the reaction plane can be written in a form of Fourier seriesE d 3Nd 3p ϭ12␲d 2N p t dp t dy ͩ1ϩ͚n ϭ1ϱ2v n cos ͓n ͑␾Ϫ⌿r ͔͒ͪ,͑1͒*On leave from Moscow Engineering Physics Institute,Moscow,115409,Russia.PHYSICAL REVIEW C SEPTEMBER 1998VOLUME 58,NUMBER 3PRC 580556-2813/98/58͑3͒/1671͑8͒/$15.001671©1998The American Physical Societywhere⌿r denotes the͑true͒reaction plane angle,and the sine terms vanish due to the reflection symmetry with respect to the reaction plane.The main advantage of the Fourier method is that the Fourier coefficients,evaluated using ob-served event planes,can be corrected for the event plane resolution caused by thefinite multiplicity of the events.This correction always raises the value of the coefficients.The great importance of this is that then the results for particles in a certain phase space region may be compared directly to theoretical predictions,or to simulations unfiltered for the detector acceptance,and for which the reaction plane has been taken to be the plane containing the theoretical impact parameter.Note the factor of2in front of each v n coeffi-cient.We propose to use it because in this case the meaning of the coefficients v n becomes transparent͓25͔,v n ϭ͗cos͓n(␾Ϫ⌿r)͔͘,where͗͘indicates an average over all particles in all events.For the particle number distribution, the coefficient v1is͗p x/p t͘and v2is͗(p x/p t)2Ϫ(p y/p t)2͘.Estimation of the reaction plane.The method uses the anisotropicflow itself to determine the event plane.It also means that the event plane can be determined independently for each harmonic of the anisotropicflow.The eventflow vector Q n and the event plane angle⌿n from the n th har-monic of the distribution are defined by the equationsQ n cos͑n⌿n͒ϭX nϭ͚i w i cos͑n␾i͒,͑2͒Q n sin͑n⌿n͒ϭY nϭ͚i w i sin͑n␾i͒,͑3͒or⌿nϭͩtanϪ1͚i w i sin͑n␾i͚͒i w i cos͑n␾i͒ͪͲn.͑4͒The sums go over the i particles used in the event plane determination and the w i are weights.In general the weights are also optimized to make the reaction plane resolution the best that is possible.Sometimes it can be done by selecting the particles of one particular type,or weighting with trans-verse momentum of the particles,ually the weights for the odd and even harmonic planes are different.Optimal weights are discussed in footnote2of͓17͔.For symmetric collisions reflection symmetry says that particle distributions in the backward hemisphere of the center of mass should be the same as in the forward hemisphere if the azimuthal angles of all particles are shifted by␲.This explains why for the odd harmonics the signs of the weights are reversed in the backward hemisphere while for the even harmonics the signs of the weights are not reversed.Note that the event plane angle⌿n determined from the n th harmonic is in the range0р⌿nϽ2␲/n.For the case of nϭ1,Eqs.͑2͒–͑4͒are equivalent to obtaining⌿1for numberflow from͓29͔Qϭ͚w p t/͉p t͉͑5͒where the sum is over all the particles.The case of nϭ2is equivalent to the event plane determined from the transverse sphericity matrix͓24͔.Acceptance correlations.Biases due to thefinite accep-tance of the detector which cause the particles to be azimuth-ally anisotropic in the laboratory system can be removed by making the distribution of event planes isotropic in the labo-ratory.We know of several different methods to remove the effects of anisotropy which have been used͑sometimes in a combination with each other͒.Each of them has some advan-tages along with disadvantages.The simplest one is to recenter͓3,4,10,30͔the distribu-tions(X n,Y n)͓Eqs.͑2͒and͑3͔͒by subtracting the(X n,Y n) values averaged over all events.The main disadvantage of this method is that it does not remove higher harmonics from the resulting distribution of⌿n.If such harmonics are present then the method requires additionalflattening of the event plane distribution by one of the other methods.The second,one of the most commonly used methods,is to use the distribution of the particles themselves as a measure of the acceptance͓3,4,9͔.One accumulates the laboratory azi-muthal distribution of the particles for all events and uses the inverse of this as weights in the above calculation of the event planes.The limitation of this approach is that it does not take into account the multiplicityfluctuations around the mean value.The third method is to use mixed events͓3,4,9͔. Correlations with the raw event planes are stored in histo-grams and correlations with event planes from previous events are also stored.The real correlations are then divided by the mixed event correlations to remove the acceptance correlations.This third method suffers from the problem that,if one uses only one mixed event for each real event,the errors areͱ2larger.If one uses many mixed events for each real event,the errors decrease as n mix1/4instead ofͱmix be-cause the same events are being used n mix times͓31͔.The fourth methodfits the unweighted laboratory distribution of the event planes,summed over all events,to a Fourier ex-pansion and devises an event-by-event shifting of the planes needed to make thefinal distribution isotropic͓3,4͔.1In all these methods one has to check that the event plane distri-butions indeed become isotropic.Particle distributions with respect to the event plane.The next step is to study the particle distributions with respect to the event planes.Note that for a given n the corresponding Fourier coefficient v n can be evaluated using the reaction planes determined from any harmonic m,with nуm,if n is a multiple of m.If nϾm,the sign of v n is determined rela-tive to v m.That is,thefirst harmonic plane can be used,in principle,to evaluate all v n.The second harmonic plane can be used to evaluate v2,v4,etc.For the event plane evalu-ated from the m th harmonic the Fourier expansion is d͑wN͒d͑␾Ϫ⌿m͒ϭ͗wN͘2␲ͩ1ϩ͚kϭ1ϱ2v km obs cos͓km͑␾Ϫ⌿m͔͒ͪ.͑7͒1The equation for the shift is͑see Appendix in Ref.͓3͔͒n⌬⌿nϭ͚iϭ1i max2i͓Ϫ͗sin͑in⌿n͒͘cos͑in⌿n͒ϩ͗cos͑in⌿n͒͘sin͑in⌿n͔͒.͑6͒We have usually taken i maxϭ4/n for nϭ1,2.1672PRC58A.M.POSKANZER AND S.A.VOLOSHINWriting the equation in terms of km instead of n insures,for instance,that when mϭ1all terms are present,but when mϭ2only the even terms are present.The quantity w is a weight which could be p t of the particle if one studies trans-verse momentumflow,or just unity,if one studiesflow of the number of particles.The coefficients v n obs are evaluated by͗cos͓n(␾Ϫ⌿m)͔͘.The quantity(␾Ϫ⌿m)has a lowest order periodicity of2␲/m.To graph the distribution one can shift the negative values to positive by adding2␲,and thenfold the distribution with this periodicity using the module function.When a particle has been used in the calculation of an event plane,the auto-correlation effect in its distribution with respect to this plane is removed by recalculating that plane without this particle͓29͔.This is easily done if one saves the sums of sines and cosines from Eqs.͑2͒–͑4͒,and subtracts the contribution of the particle and its weight from these sums.This method of removing autocorrelations as-sumes that contributions from conservation of momentum are small.The event plane resolution.The coefficients in the Fourier expansion of the azimuthal distributions with respect to the real reaction plane are then evaluated by dividing by the event plane resolution͓29,32,25,3,4,26͔,v nϭv n obs/͗cos͓km͑⌿mϪ⌿r͔͒͘.͑8͒The mean cosine values are less than one and thus this cor-rection always increases theflow coefficients.The resolution depends both on the harmonic used to de-termine the event plane m and the order of the calculated coefficient n.It is generally true that better accuracy for the determination of v n is achieved by using the event plane (⌿n)determined from the same harmonic(mϭn,kϭ1) because the resolution deteriorates as k increases͑see be-low͒.For example,better accuracy for v2can be achieved using⌿2even when the ellipticflow is somewhat weaker than the directedflow.To calculate the resolution we start with the distribution of m(⌿mϪ⌿r),which can be written as͓25͔dPd͓m͑⌿mϪ⌿r͔͒ϭ͵v mЈd v mЈ2␲␴2ϫexpͩϪv m2ϩv mЈ2Ϫ2v m v mЈcos͓m͑⌿mϪ⌿r͔͒2␴2ͪ.͑9͒The parameter␴,which to second order inflow is common for all m,is inversely proportional to the square-root of N, the number of particles used to determine the event plane␴2ϭ12N͗w2͗͘w͘2.͑10͒The integral͑9͒can be evaluated analytically͓24,25͔,and then the event plane resolution can be expressed as ͗cos͓km͑⌿mϪ⌿r͔͒͘ϭͱ␲2ͱ2␹m exp͑Ϫ␹m2/4͒ϫ͓I͑kϪ1͒/2͑␹m2/4͒ϩI͑kϩ1͒/2͑␹m2/4͔͒,͑11͒where␹mϵv m/␴(ϭv mͱ2N for numberflow͒and I␯is the modified Bessel function of order␯.This resolution function is plotted in Fig.1.Please note that␹used in this paper isͱ2 larger than that used by Ollitrault.2Note that͗cos͓m(⌿mϪ⌿r)͔͘is a correction for the recon-struction of the signal if the event plane is derived using the flow of the same harmonic(kϭ1).In the case when the harmonic orders do not coincide,for example,when one uses the event plane derived from thefirst harmonic͑directed flow͒and studies the second harmonic͑ellipticflow͒of the particle distribution with respect to this plane͓the mϭ1,k ϭ2term in Eq.͑7͔͒the correction would be͗cos͓2(⌿1Ϫ⌿r)͔͘.For practical use all such functions can be calculated numerically as in the subsection on Approximations below.The resolution correction used in earlier times at the Bevalac͓28͔is close to the present curve for values of␹1ϭv1ͱ2N greater than about1.5,which was generally true at the Bevalac.However,using the old procedure at the higher beam energies of the AGS or SPS would greatly overesti-mate the resolution and make theflow values too small.The correlation betweenflow angles of independent sets of particles.If one constructs the event planes in two differ-ent windows,͑a͒and͑b͒,or from two random subevents,3 the corresponding correlation function also can be written analytically.But more important in this case is the simple relation for such correlations,͗cos͓n͑⌿m aϪ⌿m b͔͒͘ϭ͗cos͓n͑⌿m aϪ⌿r͔͒͘ϫ͗cos͓n͑⌿m bϪ⌿r͔͒͘.͑12͒2The parameter␹used in this paper is a factor ofͱlarger than the one used in Refs.͓1,26͔;it is equivalent to the parameter␰˜in Ref.͓25͔.3The random subevents can be made to have equalmultiplicity using the CERN library routine SORTZV.FIG.1.The event plane resolution for the n th(nϭkm)har-monic of the particle distribution with respect to the m th harmonic plane,as a function of␹mϭv m/␴.PRC581673METHODS FOR ANALYZING ANISOTROPIC FLOW IN...Here,the assumption is made that there are no other corre-lations except the ones due to flow,or that such other corre-lations can be neglected.͑For example,two-particle correla-tions due to resonance decays should scale with multiplicity as 1/N and usually can be neglected.͒If this is not true,then special precautions have to be made to avoid or correct for such correlations.See the subsection below on Nonflow cor-relations .Note that the correlation of two planes ͑the distribution in ⌿aϪ⌿b )is not well represented by a Fourier expansion.For a strong correlation it should be Gaussian.For no flow the correlation between two random planes of the same order should be a triangular distribution.If one takes the absolutevalue of ⌿n a Ϫ⌿n band folds this distribution about the angle ␲/n ,the triangle becomes flat.This does not affect the͗cos ͓n (⌿n a Ϫ⌿n b)͔͘value needed for Eq.͑12͒,but makes it easier to see a real flow effect by the non-flatness of the distribution in the graph.For the correlation between angles determined from two subevents of different harmonics one can write similar rela-tions.For example,the correlation between ⌿1a and ⌿2bis͗cos ͓2͑⌿1a Ϫ⌿2b ͔͒͘ϭ͗cos ͓2͑⌿1aϪ⌿r ͔͒͘ϫ͗cos ͓2͑⌿2b Ϫ⌿r ͔͒͘.͑13͒This expression is proportional to v 12v 2/␴3and can be rather small in magnitude,but it can be useful for the determination of the relative orientation of the flow of different orders ͑see also the subsection on Approximations below ͒.On the other hand,the relative orientation also can be determined using Eq.͑7͒with k Ͼ1.The determination of the event plane resolution .The above relations permit the evaluation of the event plane reso-lution directly from the data.For example,if one knows the correlation between two equal multiplicity subevents ͑where the resolution of each is expected to be the same ͒then from Eq.͑12͒the resolution of each of them is͗cos ͓n ͑⌿m a Ϫ⌿r ͔͒͘ϭͱ͗cos ͓n ͑⌿m a Ϫ⌿m b͔͒͘,͑14͒where,as before,n ϭkm ,and k is not necessarily equal to 1.If the subevents are correlated,then the term inside the square-root is always positive.4Note that the event plane resolution determined in such a way is the event plane reso-lution of the subevents.If one wants to use the full event ͑all detected particles from the event ͒to determine the event plane,then the full event plane resolution can be calculated from the sub-event resolution using Eq.͑11͒or the approxi-mations below,Eqs.͑23͒and ͑24͒,taking into account that the multiplicity of the full event is twice as large as the multiplicity of the sub-event.Because ␹m ϭv m /␴is propor-tional to ͱN ,in a case of low resolution where the curves in Fig.1are linear,this reduces to͗cos ͓n ͑⌿m Ϫ⌿r ͔͒͘ϭͱ2͗cos ͓n ͑⌿m a Ϫ⌿r ͔͒͘.͑15͒If the subevents are not ‘‘equal,’’or if you have only corre-lations between particles in different windows,and the reso-lution in each window can be different,then one needs atleast three windows to determine the event plane resolution in each of them.In this case,for example,the resolution in the first window is determined as ͓3,4͔͗cos ͓n ͑⌿m a Ϫ⌿r ͔͒͘ϭͱ͗cos ͓n ͑⌿m a Ϫ⌿m b ͔͒͗͘cos ͓n ͑⌿m a Ϫ⌿m c͔͒͗͘cos ͓n ͑⌿m b Ϫ⌿m c͔͒͘.͑16͒Approximations .We want to discuss two limits.In a case ofweak flow,␹m Ӷ1(v m Ӷ␴,which also means that the event plane resolution is low ͒one can expand the exponent under the integral in Eq.͑9͒before the integration.It will follow that in this limitdPd ͑m ͑⌿m Ϫ⌿r ͒͒ϰ1ϩͱ␲2␹mcos ͓m ͑⌿m Ϫ⌿r ͔͒.͑17͒Then the event plane resolution can be estimated analyticallyto be ͑the case k ϭ1)͗cos ͓m ͑⌿m Ϫ⌿r ͔͒͘Ϸͱ␲8␹m.͑18͒In this case the resolution is linear in ␹and,since ␴is the same for all m ,the resolution for different m scales with v m .Combining Eqs.͑14͒,͑10͒,and ͑18͒one can estimate the flow signal directly from the correlation between two sub-eventsv m Ϸͱ͑4/␲͒͗w 2͘/͗w ͘2ͱ͗cos ͓m ͑⌿m a Ϫ⌿m b͔͒͘/N sub ,͑19͒where N sub is the multiplicity of the subevents.Keeping the second order terms in the expansion of Eq.͑9͒it can be shown that͗cos ͓2m ͑⌿m Ϫ⌿r ͔͒͘Ϸ␹m24Ϸ2␲͕͗cos ͓m ͑⌿m Ϫ⌿r ͔͖͒͘2.͑20͒The latter relation for k ϭ2is needed as a resolution correc-tion for v 2in the case where the event plane was determined by the directed flow (m ϭ1).The approximate relations ͑18͒–͑20͒are rather accurate for ␹m Ͻ0.5.Using the above approximations the correlation between flow angles of subevents of different harmonics (m ϭ1and m ϭ2)can be written as͗cos ͓2͑⌿1a Ϫ⌿2b ͔͒͘ϷϮ2␲͗cos ͑⌿1a Ϫ⌿1b͒͘ϫͱ͗cos ͓2͑⌿2a Ϫ⌿2b͔͒͘.͑21͒As stated above,the sign of the left-hand side shows the relative orientation of the flow of the different harmonics,while the right-hand side after the Ϯsign is always positive.Note that all quantities in this equation are evaluated from4For small amounts of flow,fluctuations and/or nonflow correla-tions can cause this term to be negative when the total p t is required to be zero.1674PRC 58A.M.POSKANZER AND S.A.VOLOSHINthe data independently,so the comparison of the magnitudes is an additional consistency check.In the second limiting case of strongflow␹mӷ1,one can expand the cosine in the exponent of Eq.͑9͒and get approxi-matelydP d…m͑⌿mϪ⌿r…Ϸ␹mͱ2␲expͫϪ␹m2…m͑⌿mϪ⌿r͒2…2ͬ,͑22͒which also can be used to calculate the reaction plane reso-lution.The case of strongflow is important mostly for rela-tively low energy collisions and we will not discuss it in detail.Note that in this case͑and only in this case͒the dis-tribution of m(⌿mϪ⌿r)can be described by a Gaussian.Here we present interpolation formulas of Eq.͑11͒for the most needed cases of kϭ1,2͗cos͓m͑⌿mϪ⌿r͔͒͘ϭ0.626657␹mϪ0.09694␹m3ϩ0.02754␹m4Ϫ0.002283␹m5,͑23͒͗cos͓2m͑⌿mϪ⌿r͔͒͘ϭ0.25␹m2Ϫ0.011414␹m3Ϫ0.034726␹m4ϩ0.006815␹m5.͑24͒The interpolation formulas are valid for␹mϽ3.In order to use the above equations to go from the subev-ent resolution to the full-event resolution,one has to set the equation equal to the subevent resolution and take the root to obtain␹m.One multiplies this byͱ2because␹m is propor-tional toͱ,and then evaluates the full-event resolution from the same equation.Our routines forfinding the roots of these equations use an iterative method,and the routines for propagating the errors involve a calculation for the change in the result with a smallfinite change in the input. Another approximate method is to evaluate␹mϭv m/␴of the full event from the fraction of events where the correla-tion of the planes of the subevents is greater than␲/2͓26͔,N events͑m͉⌿m aϪ⌿m b͉Ͼ␲/2͒N total ϭeϪ␹m2/42.͑25͒This fraction,and therefore the equation,is only accuratewhen␹m is small.III.EVENT-BY-EVENT ANALYSISOF AZIMUTHAL DISTRIBUTIONSHere we study the anisotropies in the azimuthal distribu-tions of particles within a relatively large rapidity͑pseudo-rapidity͒window without determining an event plane.Finite multiplicities used in each event for the evaluation of theeventflow vector Q n,defined in Eqs.͑2͒and͑3͒,causes bothfinite event plane resolution and alsofluctuations in the vector magnitude Q n.In the case of zeroflow͗Q n2͘ϭ͗N͘.͑In this section for simplicity we assume w iϭ1.)Anisotropic flow,which shifts the vector Q n in each event in the͑ran-dom͒flow direction by a value v n N,results effectively inbroadening the distribution in Q n.Keeping thefirst non-vanishing contribution fromflow one has͗Q n2͘ϭ͗N͘ϩv¯n2͗N2͘.͑26͒This equation can be used to estimate theflow signal v¯n, which is the average of v n over the rapidity region used for the Q n calculation.Another method͓25͔involvesfitting the distribution in r nϭQ n/N to the theoretical formula,which has v¯n͑more exactly its absolute value͒as a parameter.In this method the distribution of r n has to befitted by the functiondPr n dr nϭ1␴2expͩϪv¯n2ϩr n22␴2ͪI0ͩr n v¯n␴2ͪ,͑27͒where I0is the modified Bessel function,v¯n is the parameter of interest,and␴is the parameter related tofinite multiplic-ityfluctuations.The distribution͑27͒was derived using the central limit theorem,requiring the particle multiplicity N to be large.Note that in principle Eq.͑26͒does not have this limitation,although effectively the‘‘signal to noise ratio’’is proportional to N and one would need N large to apply the equation to data.Note also that both methods do not require the determi-nation of the event plane and can be performed within one ͑pseudo͒rapidity window,but they do require this window to be rather wide in order to have a relatively large number of particles.The distribution͑27͒can be used to select events with larger or smaller values offlow.IV.NONFLOW CORRELATION CONTRIBUTIONThe methods described so far in this paper are correct when correlations induced byflow dominate all others.By all others we mean correlations due to momentum conserva-tion͓29͔,long-and short-range two-and many-particle cor-relations͑due to quantum statistics,resonances,mini and real jet production,etc.͒.Below,we discuss the contribution of such nonflow correlations to the correlation of the event planes determined from two independent subevents.Very of-ten the contribution of nonflow correlations scales as1/N, where N is the multiplicity of particles used to determine the event plane.But one should remember that the contribution due to momentum conservation increases with the fraction of particles detected,and that the relative contribution of Bose-Einstein correlations can be independent of multiplicity͑the later can be important if the subevents are formed by random division of all particles from the same phase space region into two subgroups,where the particles contributing to the different subevents can be very close to each other in phase space͒.The study of the effects of nonflow correlations͑from the point of view of aflow analysis͒in real data is rather complicated,so we start with the analysis of Monte-Carlo events.Nonflow correlations in Monte Carlo generated events. When the true reaction plane is known͑as it is in any gen-erated event͒the contribution of nonflow correlations can be studied by analyzing correlations along the axis perpendicu-lar to the reaction plane(y axis͒.For example,let us con-sider the correlation between Q a and Q b,the vectors͓see Eq.PRC581675METHODS FOR ANALYZING ANISOTROPIC FLOW IN...͑5͔͒defined by two independent subevents.One can think of these vectors as the total transverse momentum of all par-ticles of the sub-event ͓which is the case if the transverse momentum is used as a weight in Eq.͑5͔͒.Then if there are no other correlations except flow͗Q a Q b ͘ϭ͗Q a ͗͘Q b ͘ϭ͗Q x a ͗͘Q x b͘,͑28͒the very relation our first method is based on.It was assumed here that the two Q vectors are totally uncorrelated except that both of them are correlated with the reaction plane.5If this is not true and there exist other correlations,then their contribution in first order would be the same to the correla-tions of x components and y components.Then͗Q x a Q x b ͘ϭ͗Q x a ͗͘Q x b ͘ϩ͗Q x a Q x b͘nonflowϷ͗Q x a ͗͘Q x b ͘ϩ͗Q y a Q y b͘nonflow ϭ͗Q x a ͗͘Q x b ͘ϩ͗Q y a Q y b ͑͘29͒͗Q x a ͗͘Q x b ͘Ϸ͗Q x a Q x b ͘Ϫ͗Q y a Q y b͘.͑30͒Analyzing the correlation ͗Q y a Q y b͘…and in particular͗sin ͓n (⌿n a Ϫ⌿r )͔sin ͓n (⌿n bϪ⌿r )͔͘…one can estimate the cor-rections to the formulas ͑8͒and ͑12͒–͑15͒.Nonflow correlations in real data .The direct application of the above described method to real data is not possible.What can be done is the analysis of similar correlations us-ing,instead of ⌿r ,the event plane derived from the second harmonic,where as the analysis of different models shows,the contribution of nonflow effects is significantly less.Then with the ͑second harmonic ͒event plane resolution known one can carry out the above analysis.There exists in the literature other methods for estimating and accounting for nonflow correlations,see,for example Refs.͓30,33͔.Here we briefly describe the method ͓33͔,which was applied to the data of the WA93Collaboration ͓34͔.It was proposed ͓33,34͔to characterize the nonflow correlation contribution by the value of the parameterc ϭ͗Q a Q b ͘Ϫ͗Q x a ͗͘Q x b͘ͱ͗͑Q a ͒2͘ͱ͗͑Q b ͒2͘Ϸ͗Q a Q b ͘Ϫ͗Q x a ͗͘Q x b͘N,͑31͒where N is the subevent multiplicity and,as in Sec.III,for simplicity we assume w i ϭ1.The parameter c can strongly depend on the particular choice of subevents,but if the non-flow contribution is dominated by two-particle correlations,it is largely independent of multiplicity ͓33͔.The nonflow correlation changes the distribution of event flow vectors,and in particular Eq.͑26͒now reads͗Q n 2͘Ϸ͗N ͘ϩv ¯n 2͗N 2͘ϩ2c ͗N ͘.͑32͒If the parameter c is relatively large,it can bias the resultsderived from an application of Eq.͑26͒to the data.However,the nonflow correlations contribute to the shape of the dis-tribution in Q n ͓Eq.͑27͔͒in a different way than flow.Theyproduce mostly a change in ␴,the ‘‘Gaussian width’’of thedistribution ͓Eq.͑10͔͒,which is modified to␴2ϭ12N͑1ϩ2c ͒,͑33͒and the parameter c can be directly extracted from the data.The analysis ͓34͔of S ϩS and S ϩAu data gave ͑for their particular subevent selection ͒c Ϸ0.034Ϯ0.025which is not negligible compared with v 2Ϸ0.04–0.05found in this analy-sis at multiplicities of about 30–50.Another possible conse-quence of nonflow contributions could be the change in the shape of the distribution of the difference of flow angles of subevents ͓33͔.V.SIMPLE WAY TO INTRODUCE FLOW IN A MONTE-CARLO EVENT GENERATORSometimes in order to investigate different detector ef-fects or the reliability of the method,one needs to introduceanisotropic flow into a Monte Carlo event generator.It can be done by changing the azimuthal angle of each particle ͑and consequently changing the density in the azimuthal angle space ͒␾→␾Јϭ␾ϩ⌬␾,͑34͒where⌬␾ϭ͚nϪ2n v ˜nsin ͓n ͑␾Ϫ␺0͔͒,͑35͒v ˜n are the n parameters of the transformation,and ␺0is the direction of the added flow.The v ˜n can be functions of ra-pidity and transverse momentum and,in particular,the v ˜n for n odd should reverse sign in the backward hemisphere.One can check that such a change in the azimuthal angle results in the required change in the distribution.To first order in v ˜n ,v n ϵ͗cos ͓n (␾ЈϪ⌿r )͔͘ϭv ˜n .Small higher order correc-tions (ϳv ˜n 2),if needed,can also be easily calculated,for example,numerically.VI.DISCUSSIONHigher harmonics (n у3).Note that the flow analysis methods presented in this paper are valid for all harmonic orders of anisotropy.The higher harmonics look at the event with higher symmetry on a finer scale.It should be empha-sized that the study of anisotropic flow corresponding to har-monic orders of n у3has interesting aspects.For instance,one would expect large differences between the theoretical predictions of hydro-and cascade-type models in the higher harmonics of the particle azimuthal distributions.Also there are physics processes which could lead to nonzero higher harmonics.It is widely accepted that one of the main reasons for pion ‘‘antiflow,’’directed flow in the direction opposite to that of the protons,is caused by pion shadowing by co-moving nucleons.Such shadowing could affect pion higher harmonic azimuthal distributions,and affect them differently at different rapidities.Another effect is the transition from out-of-plane elliptic flow ͑squeeze-out ͒which is very impor-5Strictly speaking there is one more assumption here,namely that the strength of the flow does not fluctuate event by event.1676PRC 58A.M.POSKANZER AND S.A.VOLOSHIN。