Correlations and fluctuations: generalized factorial moments

M.Smoluchowski Institute of Physics
Jagellonian University, Cracow
Address: Reymonta 4, 30-059 Krakow, Poland; e-mail:;

A systematic study of the relations between fluctuations of the extensive multiparticle variables and integrals of the inclusive multiparticle densities is presented. The generalized factorial moments are introduced and their physical meaning discussed. The effects of the additive conservation laws are analysed.


Keywords: correlations, fluctuations, moments

1 Introduction

Studies of multiparticle production is a complicated matter because of the large number of variables describing the system and thus several methods were developped to simplify the problem. Two of them turned out particularly useful in providing information: (i) studies of correlations between particles by investigation of the inclusive distributions and (ii) studies of event-by-event fluctuations. Since both of them give -in principle- a complete description of the system, they must be related. This relation was firsst discussed in [1], and reformulated in [2]. In the present paper we follow the arguments of [1, 2] and investigate further the relation between these two ways of approach to multiparticle production.

To illustrate the problem, consider the well-known formula


where is the inclusive distribution of particles and is the factorial moment of the -th order:


is the event multiplicity and denotes the multiplicity distribution.

The main interest in the formula (1) is that its L.H.S. characterizes event-by-event fluctuations of the multiplicty while its R.H.S. represents the measurement of particle distribution by -arm spectrometer. Thus (1) connects the two quantities which are defined in an entirely different manner. Needless to say, confronting such -apparently unrelated- quantities is often a source of a new insight into the problem. This is precisely the interest in investigation of further relations of this type.

Another feature of (1) which make this relation very useful in practical applications is that it connects the factorial moment of order with the inclusive density of the same order. Thus it can be used even if one does not have a complete knowledge of the system (which is of course practically always the case)111 This feature may be contrasted with the relation between and . They must be related because they both describe completely the multiplicity distribution. However, as seen from (2), to calculate one needs to know all (and vice versa)..

For , the extension of (1) to quantities other than multiplicity was proposed in [3]. It was discussed and applied by several authors (see, e.g., [4]-[7] and the review [8]). The possibility of generalization to greater than was suggested in [3] and considered in [9].

The purpose of the present paper is twofold. First, we derive an explicit formula, valid for arbitrary , which extends the the relation (1) to fluctuations of quantities other than multiplicity. This is obtained by introducing the generalized factorial moments [2], defined in terms of measurements of event-by-even fluctuations of measurable extensive quantities. Second, we investigate the physical meaning of the generalized factorial moments along the lines developped in [10], as described below.

As first discussed in [10], the factorial moments have a rather simple physical interpretation. It follows from the observation that for any distribution of multiplicity which can be represented as a superposition of Poisson distributions


one has


Thus the factorial moment of the actual multiplicity distribution measures the standard moment of the underlying distribution . One can say [10] that the factorial moment removes the statistical noise (represented by the Poisson distribution) from the corresponding moment of the distribution . This observation is easily generalized to local multiplicity fluctuations [10].

It is shown in the present paper that, in absence of other constraints, the generalized factorial moments can be used as well to remove the statistical noise from the data. This is of course to be expected. The problem must be treated more carefully, however, in presence of an additive conservation law. Indeed, the additive conservation law implies that the ”statistical noise” cannot be entirely random and therefore the simple argument of [10] must be modified. We consider in some detail the effects of conservation of charge and of transverse momentum.

The problem of removing the statistical noise from fluctuations of various quantatities observed in multiparticle production, is the long-standing one. The most popular method remains the measure, introduced in [11]. Other possibilities, and their relation to the measure were discussed in [5]-[7]. They are now commonly used in data analysis [12]-[17].

Our treatment, using the generalized factorial moments, goes beyond the previous studies in two points. On the theoretical side, our method applies to correlations of any order (until now mostly was discussed222 A related method of removing the statistical noise, applicable to arbitrary , was described in [18] and studied in [19].). On the practical side, including the effects of additive conservation laws makes the control over the reliabity of the experimental results much more solid333For the effects of charge conservation were discussed in [6]. An interesting proposal to deal with energy-momentum conservation can be found in [20]..

In the next section the generalized factorial moments are introduced and their relation to integrals of inclusive multiparticle densities explained. Their physical interpretation in case of random noise is explained in Section 3. Modifications due to conservation of a discrete quantum number are discussed on the example of charge distributions in Section 4. The effects of conservation of transverse momentum are considered in Section 5. Discussion and conclusions are given in the last section. Two Appendices explain the details of algebra.

2 Generalized factorial moments

Generalizations of (1) to fluctuations of extensive quantities other than multiplicity were found in [1] and reformulated in [2]. Here we follow the argument used in these two papers.

Consider a single-particle variable , where is the particle momentum. Consider, furthermore, the set of extensive quantities


where is the multiplicity of the event.

We want to study fluctuations of . One possible method is to consider the moments


where is the probability to find an event with particles at momenta and where we have introduced the shorthand


which will be used henceforth. It was shown in [1] that the moments (6) can be expressed by linear combinations of the integrals


These relations are, however, fairly complicated.

To find a more elegant formulation we observe that when one takes , one has and thus the simple relation (1) must hold. It is thus clear that one has to find a generalization of the factorial moments (2). Such a generalization was proposed in [2]:


where is the operator, acting on a product as follows444The definition of the operator given in [2] was incomplete. I would like to thank Andrzej Kotanski for pointing out the error.


where the indices need not be different. It follows from (10) that, in particular,


From (10) one can also deduce how acts on a product . The result is


where, again, the indices need not be different.

Using (9) and (10) we obtain for


We shall now prove that


where all indices are different from each other.

The proof goes by induction. For we have with . Thus (14) is satisfied. Suppose now that it is valid for a given . Multiplying both sides of (14) by we thus have




where in the first term in the R.H.S. one of the variables from is identical to one of , whereas in the second term all variables differ from each other. Noting that, as seen from (12), the first term is identical to


one sees that it cancels in (15) and thus R.H.S. of (15) equals to the second term in the R.H.S. of (16). But this term is just what is needed to complete the proof.

It follows from (14) that


where is the -particle inclusive density. This follows almost directly from definition of inclusive density


if one notices that the number of terms in the R.H.S. of (14) equals .

Eq. (18) represents a generalization of (1) (which is obtained from (18) by putting ). It has the same two attractive features: it connects the fluctuations of the extensive variables with the integral of the inclusive density and it involves only moments of finite order.

3 Physical interpretation of the generalized factorial moments

To discuss the physical interpretation of the generalized factorial moments, we again consider the momentum space split into s-bin and the rest. To determine the generalized factorial moment in the s-bin, we need the distribution of the number of particles and of the variable in it. Following the lines of [10], we demand that in each bin particles are distributed as randomly as possible. In absence of any constraints this implies in the s-bin the distribution


where is the average multiplicity in the s-bin and is the momentum distribution of one particle on this bin, with representing a collection of parameters on which this distribution may depend.

To obtain the actual distribution we have to weight (20) by the underlying probability distribution :


From (21) we deduce the inclusive particle density in the s-bin


Introducing (22) into (18) and using (20) we obtain for the generalized factorial moment in the s-bin




is the average of at fixed . is the average of at fixed and .

Eq. (23) is the generalization we were seeking for. It shows the physical interpretation of the generalized factorial moments: they remove the statistical noise from the moments of of the underlying distribution . For we of course recover the well-known result [10].

4 Fluctuations of charge

As already explained in the introduction, the argument present in the previous section fails in case when the variable satisfies an additive conservation law. In this section we discuss fluctuations of charge. The discussion applies to any discrete, additive quantum number.

We again select an s-bin from the momentum space and consider the distribution of particles inside and outside of this bin.

The first problem is to define the distribution describing the ”statitical noise”. Following the idea of [10], we take the normalized product of Poisson distributions (expressing the independent particle emission and thus introducing no correlations). This is supplemented by the Kronecker -symbol to satisfy the conservation law:


where denote the number of produced charges in the s-bin, those outside of the s-bin. is the net charge in the initial state. We take . The normalization factor is given by (see Appendix 1)


where denotes the Bessel function,




The observed distribution of particles is thus


where is the ”dynamical” distribution, free of the statistical noise555The selection of statistical noise in the form of (25) is not unique. Another natural possibility is to consider independent emission of particle pairs (supplemented by independent emission of positive particles). The two methods differ mostly by the treatment of the initial charge . Without further information about dynamics it is difficult to decide which of them describes better the physics of the problem..

It is important to observe that, contrary to the naive expectation, the parameters do not represent the average values of , respectively. The explicit formulae for these average values are given in Appendix 1. In particular, the average values of the number of produced positive and negative particles in the the s-bin are given by


where we have denoted by the average at fixed , and by the average over (with the probability distribution ).

The explicit formulae for the generalized factorial moments of the charge and the (standard) factorial moments of the multiplicity in the s-bin are derived in Appendix 1. They read:


Eqs. (30) and (31) simplify substantially in the interesting limit of very large number of produced particles, . In this limit, using the asymptotic expansion of the two Bessel functions, one obtains up to the terms of order


The first term corresponds to the standard interpretation of the generalized factorial moments, as expressed by (23). One sees from (32) that the correction to this result (induced by the conservation law) vanishes, at fixed , with the inverse power of the total multiplicity of produced particles. In this case the correction is expected to be small at high energies (particularly for the central heavy ion collisions)666For the relative correction may be large if .. We thus conclude that the generalized factorial moments can indeed provide a useful tool for eliminating the statistical noise from the event-by-event fluctuations of multiplicity () and charge () even in presence of the conservation law.

5 Fluctuations in transverse momentum

In this section we discuss fluctuations of the tranverse momentum, as an example of a continuous variable subject to an additive conservation law.

Selecting one bin in rapidity (s-bin) and dividing the available momentum phase-space into this bin and the rest, we write the transverse momentum distribution in the form


where denotes the two-dimensional transverse momentum vector and is the total transverse momentum of the considered phase-space region. The uncorrelated ”statistical noise” in the form of Poisson distribution for multiplicity multiplied by a gaussian distribution of transverse momenta is supplemented by the function insuring the transverse momentum conservation777 This is a commonly used form of the ”statistical noise” (see, e.g., [20, 21] ). Note that, since is a general (positive) function, the assumed gaussian forms in (33) do not restrict seriously the observed distribution of particles (as both the average values, and dispersions, of the Gaussians can fluctuate according to the distribution ). and by the normalization factor .

It is also natural to take


Replacing the function in (33) by its Fourier transform we obtain for the inclusive distribution in the s-bin


Using (18) we thus have for the generalized factorial moment in the bin




(note that ), and


where is the average value of at fixed and . The function depends obviously on the choice of the variable . Generally, if is a polynomial in of order , then is a polynomial in of the same order888E.g., for we have , for we obtain . multiplied by .

If the number of particles is large, the integral over can be evaluated by the saddle point method. We write




The saddle point equation gives and thus (34) implies . Furthermore , where and


Consequently we obtain


To evaluate this integral in the limit of large , one can expand in powers of and observe that, since the coefficient in the exponent increases with increasing , the dominant term at large is that with the lowest power of under the integral (the terms with odd powers of do not contribute). A detailed discussion is given in Appendix 2. Here we anly summarize the results.

If one obtains


i.e. we recover the formula (23). Thus, if the average value of in the -bin does not vanish, the standard interpretation of the generalized factorial moments remains valid, even in presence of the conservation law. The corrections to this result vanish as . They are given in (62) of Appendix 2.

This is not the case if and . For even, , one obtains


which shows that now the factorial moments are related to the average value rather than to . For a fixed and large these moments tend to zero. For a finite ratio , however, they may be large. The corrections to (44) and the formula for odd are given in Appendix 2.

6 Discussion

We have reconsidered relations between the event-by-event fluctuations of extensive multiparticle variables and the integrals of the inclusive distributions [1]. Our main result is the formula (18) which expresses the integral of the -particle inclusive distribution in terms of the generalized factorial moment of the -th order.

The physical meaning of the generalized factorial moments is discussed. It is shown that they remove the random uncorrelated statistical noise from the data, the result known already from previous investigations [10, 2]. When an additive conservation law is at work, however, the statistical noise cannot be uncorrelated and, consequently, its removal can at best be approximate. The exact formulae were derived for charge and transverse momentum conservation. The corrections were evaluated in the limit of very large total number of produced particles, relevant for collisions at high incident energy.

In short, the results presented in this paper formulate a systematic approach to investigation of fluctuations of extensive variables. They seem to be particularly useful for studies aiming to uncover the local structure of the multiparticle system. To take just one example, interpretation of the data on transverse momenta (or energies) in small rapidity bins will be much more transparent if presented in terms of the generalized factorial moments.

It was shown in [1] that the relation between fluctuations and correlations can be extended to other extensive variables. This, however, goes beyond the scope of the present investigation.

7 Appendix 1

Replacing the Kronecker symbol by its integral representation one obtains




and is the normalization factor.

Thus the generating function for the distribution is


From this it is not difficult to derive, by the standard methods, the average multiplicities. One obtains


where and are defined in (28).

Similarly, one obtains


It follows that


The inclusive distribution in the bin is






is the number of ways one can select out of positive and out of negative particles in a given order.

To calculate the generalized factorial moment in the s-bin


we observe that