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Statistical Ensembles with Fluctuating Extensive Quantities M.I. Gorenstein1,2 and M. Hauer3 1 Bogolyubov Institute for Theoretical Physics 14-b, Metrolohichna str., Kiev, 03680, Ukraine 2Frankfurt Institute for Advanced Studies Johann Wolfgang Goethe - Universität Frankfurt Ruth-Moufang-Str. 1, D-60438 Frankfurt am Main, Germany 3Helmholtz Research School 8 Johann Wolfgang Goethe - Universität Frankfurt 0 Ruth-Moufang-Str. 1, D-60438 Frankfurt am Main, Germany 0 2 Wesuggestanextensionofthestandardconceptofstatisticalensembles. Namely,weintroducea classofensembleswithextensivequantitiesfluctuatingaccordingtoanexternallygivendistribution. n As an example the influence of energy fluctuations on multiplicity fluctuations in limited segments a of momentum space for a classical ultra-relativistic gas is considered. J 8 PACSnumbers: 24.10.Pa,24.60.Ky,25.75.-q 2 ] h Successful application of the statistical model to in the TL corresponding results for the scaled vari- t hadron production in high energy collisions (see, e.g., ance are different in different ensembles. Hence the - l recent papers [1] and references therein) has stimu- equivalence of ensembles holds for mean values in the c u lated investigations of properties of statistical ensem- TL, but does not extend to fluctuations. n bles of relativistic hadronic gases. Whenever possi- Statisticalmechanicsis usually formulatedthrough [ ble, one prefers to use the grand canonical ensem- the following steps. Firstly, one fixes the system’s ex- ble (GCE) due to its mathematical convenience. The 1 tensive quantities: volume V, energy E, momentum v canonical ensemble (CE) [2] should be applied when P~, and conserved charges {Q }. Secondly, one pos- 9 the number of carriersof a conservedchargesis small i tulates that all microstates have equal probability of 1 (of the order of 1), such as strange hadrons [3], 2 being realized. This defines the MCE. The CE intro- antibaryons [4], or charmed hadrons [5], in other- 4 duces temperature T. Each set of microstates with wise large systems. The micro-canonical ensemble 1. fixed energy E (a macrostate) is weighted with the (MCE) [6] has been used to describe small systems 0 Boltzmann factor e−E/T. The probability to find a (with total number of produced particles less than 8 macrostate with energy E in the CE is then propor- or equal to 10) with additionally fixed energy, and 0 tional to the number of all microstates with energy v: momentum, e.g. elementary particle collisions or an- E times the Boltzmann factor e−E/T. To define the nihilation. In these cases, calculations performed in i GCE, one makes a similar construction for conserved X different statistical ensembles yield different results. charges{Q }andintroduceschemicalpotentials{µ }. r Hence, ensembles are not equivalent, and systems are i i a Canonicalorgrandcanonicalobservablescanthenbe ‘far away’ from the thermodynamic limit (TL). obtainedbyaveragingovertheCEenergydistribution Measurement of hadron multiplicity distributions or the GCE (joint-) distribution of both energy and P(N) in relativistic nucleus-nucleus collisions opens conserved charges. another interesting field of investigations. The aver- Fluctuations in statistical systems, e.g., multiplic- age number of produced hadrons ranges from 102 to ity distributions P(N) in relativistic gases [7, 8, 9], 104, and mean multiplicities (of light hadrons) ob- are sensitive to conservation laws obeyed by the sys- tained within GCE, CE, and MCE approach each tem,andthereforetofluctuationsofextensivequanti- other. Onerefersheretothethermodynamicalequiv- ties. For calculation of multiplicity distributions, the alence of statistical ensembles and uses the GCE for choice of statistical ensemble is then not a matter of fitting experimentally measured mean multiplicities. convenience, but a physical question. Fluctuations of However,the number of particles fluctuates event-by- extensive quantities A~ ≡(V,E,P~,{Q }) around their i event. Thesefluctuationsareusuallyquantifiedbythe average values depend not on the system’s physical ratio of variance to mean value of a multiplicity dis- properties, but rather on external conditions. One tributionP(N),thescaledvariance,andareasubject can imagine a huge variety of these conditions, thus, of current experimental activities. In statistical mod- MCE, CE, GCE, or pressure ensembles [10] are only elsthereisaqualitativedifferenceinthepropertiesof some special examples. mean multiplicity and scaled variance of multiplicity A more general statistical ensemble can be defined distributions. Itwasrecentlyfound[7, 8,9]thateven by an externallygiven distributionof extensive quan- 2 tities, P (A~). All microstates with a fixed set A~ In the TL the MCE distribution (3) converges to a α are taken to be equiprobable. Thus, the probability Gaussian, P (N;A~)offindingthesysteminamacrostatewith fixmecde A~ and additionally fixed multiplicity N is given P (N;E)∼= 1 exp −(N −hNi)2 . mce by the ratio of the number of microstates with fixed 2πωmcehNi (cid:20) 2ωmcehNi (cid:21) N and A~ to the number of microstates with fixed A~. (5) p The construction of multiplicity distributions in such The Normal form of distributions is a general feature an ensemble proceeds in two steps. Firstly, the MCE multiplicity distribution, P (N;A~), at fixed values of all statistical ensembles in the TL [11]. mce of the extensive quantities A~ is calculated. Secondly, The GCE energy distribution is equal to: this result is averaged over the external distribution Pα(A~), Pgce(E)= 1 exp −E Zmce(V,E) , (6) Z (V,T) T gce (cid:18) (cid:19) P (N) = dA~ P (A~) P (N;A~) . (1) α α mce where the GCE partition function Z (V,T) gce Z is defined by the normalization condition, The ensemble defined by Eq. (1), the α-ensemble, in- dE P (E) = 1. We consider a system gce cludesthestandardstatisticalensemblesasparticular withfixedlargevolumeV. As anillustrativeexample cases. wRe will use the following asymptotic form of energy distribution, Let us illustrate above statements for a simple system of non-interacting massless particles, neglecting 2 1 E−E the effects of quantum statistics (Boltzmann approx- P (E)= exp − , (7) α imation). For a particular realization of Eq. (1) we 2π ωE α2E " 2(cid:0)ωE α2(cid:1)E# choose: p where E =3gVT4/π2 is the GCE energy expectation Pα(N)= dE Pα(E) Pmce(N;E) (2) value and ωE = (E2 − E2)/E = 4T is the scaled Z variance of GCE energy fluctuations. This choice for to calculate the multiplicity distribution P (N) in P (E) resultsin asimple correspondenceto the GCE α α the presence of an energy distribution P (E). We and MCE in the large volume limit. In Eq. (7), α is α will firstly discuss the MCE multiplicity distribution a dimensionless tuneable parameter for the width of P (N;E) and then solve the integral (2) for a par- the distribution. In the MCE limit α → 0, Eq. (7) mce ticular choice of P (E) in the large volume limit. becomes a Dirac δ-function, δ(E −E). For α = 1, α The MCE multiplicity distribution is given by [8], Eq. (7) results in the GCE energy fluctuations (6) in the TL [11]. The physical interpretation of the GCE 1 E−1 xN energy fluctuations corresponds to an ‘infinite heat P (N;E) = , (3) mce Zmce(V,E) (3N −1)! N! bath’. The values, 0 < α < 1, would correspond to a ‘large, but finite’ heat bath. The case of α > 1 we where x ≡ gVE3/π2, and g is the particle’s would like to denote as ‘strong’ energy fluctuations. degeneracy factor. The MCE partition function Lastly we note that the MCE distribution Z (V,E) is defined by the normalization condi- mce ∞ Pmce(N;E) can be conveniently presented in terms tion, P (N;E) = 1. In the TL, x ≫ N=1 mce of the GCE distributions [11], P (N;E) ≡ mce 1 in Eq.(3), one finds from the raw moments hNki P≡ ∞ Nk P (N;E) : Pgce(N,E)/Pgce(E). At fixed volume V, the GCE N=1 mce distribution P (E) is given by Eq. (7) with α = 1, gce Px 1/4 hN2i−hNi2 1 while the joint GCE energy andmultiplicity distribu- hNi∼= 27 , ωmce ≡ hNi ∼= 4 . (4) tion Pgce(N,E) is given by a bivariate normal distri- (cid:16) (cid:17) bution in the large volume limit [11], Additionally it follows from the equivalence of statis- ticalensemble thatGCE andMCE valuesfor average P (N,E)∼= 1 exp − 1 multiplicity are equal to each other, hNimce ∼= N = gce 2πV σ2σ2 (1−δ2) "2V(1−δ2) E N gVT3/π2, where the temperature T of the GCE is found from E = E = 3gVT4/π2. Momentum spec- (∆E)2p 2δ ∆E∆N (∆N)2 × − + , (8) tra in the MCE also converge to GCE Boltzmann σE2 σEσN σN2 !# spectraunderthislimit. Multiplicityfluctuations,ex- pressedbythe scaledvariances,arehoweverdifferent, where ∆E =E−E, ∆N =N−N, E =VκE, N = 1 ω =1 and ω =1/4. VκN, σ2 = κE,E = κEω , σ2 = κN,N = κNω , gce mce 1 E 2 1 E N 2 1 gce 3 andthecorrelationcoefficientδ =σ /(σ σ ),with p 2 EN E N a = 0, MCE σEN =κE2,N. The relevant cumulants κ, for both 4π- Da 1.8 a = 0.5 integratedparticleyieldsandforyieldsinlimitedmo- w 1.6 a = 1, GCE 1.4 a = 1.5 mentum windows ∆p for the ultra-relativistic Boltz- 1.2 mann gas are given in the Appendix. One finds then 1 for the multiplicity distribution: 0.8 0.6 P (N,E) gce Pα(N)= dE Pα(E) 0.4 P (E) Z gce 0.2 2 1 N −N 0 ∼= exp − , (9) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 (2πωαN)1/2 " (cid:0)2ωα N(cid:1) # p [GeV] ωα = ωmce + α2 (ωgce − ωmce) , (10) p 1.82 Da w 1.6 where for our system ω =1 and ω =1/4. As it gce mce 1.4 canbe expected,ω =ω forα=0,andω =ω α mce α gce 1.2 for α = 1. It also follows, ω < ω < ω for mce α gce 1 0<α<1, and ω >ω for α>1. The α-ensemble α gce 0.8 definedbyEqs.(2,7)presentsanextensionoftheGCE 0.6 (α = 1) and MCE (α = 0) to a more general energy 0.4 distribution. Differentvaluesofαcorrespond,bycon- 0.2 struction,tothesameexpectationvaluesofenergyE, 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 and multiplicity N. Hence in the TL all ensembles p [GeV] defined by Eq.(7) are thermodynamically equivalent. Energy and multiplicity fluctuations are however dif- FIG.1: Momentumdependenceofthescaledvarianceω∆p ferent. α for a classical ultra-relativistic gas at T = 160 MeV. Mo- mentum bins are constructed in a way that each bin con- As a next step we want to discuss the ef- tains the same fraction q = 0.1 (upper panel), or q = 0.2 fect of energy fluctuations, Eq.(7), on multiplic- (lowerpanel)oftheaverageparticleyield. Thehorizontal ity fluctuations in limited segments of momentum barsindicatethewidthofbins,whilethemarkerindicates space for a classical ultra-relativistic gas in the the center of gravity of the corresponding bin. Calcula- TL. The results for the scaled variances, ω∆p ≡ tions are done for different values of α. The total par- α hN2 i −hN i2 /hN i ,indifferentmomentum ticle yield would correspond to q = 1, and according to ∆p α ∆p α ∆p α bins ∆p are shown in Fig. 1 for several values of Eq. (10) the scaled variance ωα for the total yield fluctu- (cid:0) (cid:1) ations equals to 1/4, 7/16, 1, 31/16 at α=0, 0.5, 1, 1.5, parameter α. As in Ref.[12], each momentum bin respectively. ∆p = [p ,p ] contains the same fraction q of the to- 1 2 tal average multiplicity, q ≡ hN i /hNi . In the ∆p α α MCE(α=0),multiplicity fluctuationsareessentially eralized to more complicated cases of non-zero parti- suppressed with respect to the GCE (α = 1). Global cle masses, quantum statistics, many particle species, energy conservation in the MCE introduces correla- several conserved charges, and multiplicity fluctuations in momentum space [12]. The larger the frac- tions in limited segments of momentum space. In tion of the total energy in a given momentum bin, all these cases the expressions for P (N;A~) and mce the stronger is the MCE suppression effect (α = 0 in P (A~) are already obtained [11, 12]. The MCE Fig. 1). For 0 < α < 1 the suppression effects be- gce multiplicity distributionsinsystemswithfullhadron- come weaker. α = 1 corresponds to the GCE results resonance spectrum and quantum statistics effects for multiplicity fluctuations: ω =1, both in full mo- α were presented in Ref. [11]. The role of momentum mentum space, and in different momentum bins. For conservationwasdiscussedinRef.[12]. Thenextstep ‘strong’ energy fluctuations, α > 1, we find an in- wouldbeanextensionofthisformulationtoanexter- crease of multiplicity fluctuations with increasing bin nal distributions P (A~) = P (V,E,P~,B,S,Q), e.g., momentum, as shown in Fig. 1. α α taking into accountfluctuations ofenergyE, momen- In order to make a quantitative comparison with tum P~, and three Abelian charges - baryon number present and future data on multiplicity fluctuations B, strangeness S, and electric charge Q. This is to in relativistic nucleus-nucleus collisions one needs to be done according to Eq. (1). Fluctuations and cor- extend essentially the formulation considered in this relations of the extensive quantities presented by the letter. PleasenotethatEqs.(2,7-8)canbereadilygen- distribution P (V,E,P~,B,S,Q) could then be con- α 4 nected to measured fluctuations of hadron multiplic- Additionally, in Boltzmann approximation, κE,N = 2 ities in limited segments of momentum space. This κE andκN,N =κN. Thecumulantsinthemomentum 1 2 1 will be a subject of future studies. segments ∆p=[p ,p ] are: 2 1 In this letter we suggested to extend the concepts ofstatisticalensemblesandintroducedtheα-ensemble p2 g defined by anexternaldistribution ofextensive quan- κN = dp p2 f(p) tities. Thekeyassumptionusedforcalculationofmul- (cid:16) 1 (cid:17)∆p 2π2pZ1 tiplicitydistributionsisthe‘equiprobability’ofallmi- p1 g crostateswiththesamesetofextensivequantities. We = 2T2 + 2pT + p2 f(p) , havediscussedasimple exampleofagascomposedof 2π2 (cid:12) h i (cid:12)p2 classical massless particles to demonstrate the effects p2 (cid:12)(cid:12) of energy fluctuations on multiplicity fluctuations in κE,N = g dp p3 f(p) (cid:12) full momentum space and in limited segments of mo- 2 ∆p 2π2 mentumspace. Measurementofevent-by-eventmulti- (cid:16) (cid:17) pZ1 plicity fluctuationsindifferentmomentumbins corre- g p1 spond to current experimental studies of hadron pro- = 2π2 6T3 + 6T2p + 3Tp2 + p3 f(p) (cid:12) . duction in relativistic nucleus-nucleus collisions. We h i (cid:12)p2 (cid:12) believe also that the concept of statistical ensembles (cid:12) (cid:12) withfluctuatingextensivequantitiesmaybeappropri- Additionally, in the Boltzmann approximation ate in other situations too. In fact, in all cases when (κN,N) =(κN) . 2 ∆p 1 ∆p fluctuations of extensive quantities are a subject of interest and can be measured experimentally. Acknowledgments. We would like to thank V.V. Begun, W. Broniowski, M. Ga´zdzicki, P. Stein- [1] J. Cleymans, H. Oeschler, K. Redlich, and berg, and G. Torrieri for fruitful discussions. One S. Wheaton, Phys. Rev. C 73, 034905 (2006); F. Be- of the authors, M.I.G., would like to thank for the cattini, J. Manninen, and M. Ga´zdzicki, ibid. 73, 044905 (2006); A. 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