Some Intensive and Extensive Quantities in High-Energy Collisions
aa r X i v : . [ h e p - ph ] O c t Some Intensive and Extensive Quantities in High-EnergyCollisions ∗ A. Tawfik † Egyptian Center for Theoretical Physics (ECTP), MTI University, 11571 Cairo,Egypt andWorld Laboratory for Cosmology And Particle Physics (WLCAPP), Cairo, EgyptWe review the evolution of some statistical and thermodynamical quan-tities measured in difference sizes of high-energy collisions at different en-ergies. We differentiate between intensive and extensive quantities anddiscuss the importance of their distinguishability in characterizing possiblecritical phenomena of nuclear collisions at various energies with differentinitial conditions.PACS numbers: 25.75.Gz,25.75.Dw,05.40.-a,05.70.Fh,
1. Introduction
The terminology ”intensive and extensive quantity” was introduced byRichard C. Tolman [1] in order to distinguish between different thermody-namical parameters, properties, variables, etc. Therefore, the defining ofsuch quantities as intensive or extensive may depend on the way in whichsubsystems are arranged [1]. In order to characterize possible critical phe-nomena of the nuclear collisions, which likely become complex at ultra highenergy, various signatures have been proposed [2]. It is obvious that thecritical phenomena of intensive or extensive variables [3] should be differen-tiated. The extensive variables, like total charge multiplicity, obtain aboutequal contributions from the initial (due to fluctuations in spectators) andfinal stage (resonances). The intensive variables, like particle ratios, arewell described by resonances at the freeze-out [4, 5, 6, 7]. In the presentwork, we show how the distinguishability between extensive and intensivequantities behaves at various energies and with different initial conditions. ∗ Invited Talk at XXXI Max Born Symposium and HIC for FAIR Workshop, ThreeDays of Critical Behaviour in hot and dense QCD, Wroclaw-Poland, 14 - 16 JUNE2013 † http://atawfik.net/; a.tawfi[email protected] (1) The implication of statistical-thermal models on high-energy physicsdates back to about six decades [8]. Koppe introduced an almost-completerecipe for the statistical description of particle production [9]. The par-ticle abundances in Fermi model [10] are treated by means of statisticalweights. Furthermore, Fermi model [10] gives a generalization of the ”sta-tistical model” , in which one starts with a general cross-section formula andinserts into it a simplifying assumption about the matrix element of the pro-cess, which reflects many features of the high-energy reactions dominatedby the density in phase space of the final states. In 1951, Pomeranchuk [11]came up with the conjecture that a finite hadron size would imply a crit-ical density above which the hadronic matter cannot be in the compoundstate, known as hadrons. Using all tools of statistical physics, Hagedornintroduced in 1965 the mass spectrum to describe the abundant formationof resonances with increasing masses and rotational degrees of freedom [12]which relate the number of hadronic resonances to their masses as an expo-nential. Accordingly, Hagedorn formulated the concept of limiting temper-ature based on the statistical bootstrap model.The statistical and thermodynamical variables, properties and parame-ters can be classified into intensive, extensive, normalized intensive and ex-tensive, process and conjugate. There are physical properties which neitherintensive nor extensive, e.g. electric resistance, invariant mass and specialrelativity. The intensivity is apparently additive and therefore a state vari-able. The intensive (bulk) properties do not depend on the system size orthe amount of existing material. Therefore, it is scale invariant. The exten-sivity is field and point variable but not additive. The extensive propertiesare additive for independent and non-interacting subsystems. They are di-rectly proportional to the amount of existing material. Normalized intensiveand extensive quantities are densities. They are not additive. The processdepends on past history of the system. Therefore, they are differentiable, in-exactly. The conjugates are intensive and extensive pairs, like temperatureand entropy. For example, in grand canonical ensemble, strongly inten-sive quantities have been suggested as fluctuation measures not dependingon the system volume and its fluctuations [13]. The charge distributionis inclusive, while isotropically resolved particle observation is an exclusiveproperty. We review the evolution of some statistical and thermodynam-ical quantities measured in difference sizes of the high-energy collisions atdifferent energies.The present paper is organized as follows. The intensivity and extensiv-ity of statistical properties are shortly reviewed in section 2. The dissipativeproperties are elaborated in section 3. The energy dependence of tempera-ture shall be estimated in section 4. Section 5 is devoted to the conclusionsand outlook.
2. Statistical properties: multiplicities and particle ratios (GeV) NN s æ pa r t N Æ ) / h / d c h ( d N Central AA ALICEATLASCMS RHIC SPS AGS FOPI ) NSD p pp (pALICECDFCMS RHIC UA1 UA5 ) INEL p pp (pISR UA5 PHOBOS ALICE pPb ALICE dAu PHOBOSpAu NA35 s (cid:181) s (cid:181) s (cid:181) Fig. 1. A comparison of dN ch /dη per participating nucleon at mid-rapidity incentral heavy-ion collisions to corresponding results from p + p (¯ p ) and p ( d )+A col-lisions. The quantities are given in physical units. Graph taken from Ref. [36]. Only two independent intensive variables are needed in order to fullyspecify the entire state of the system of interest. Other intensive propertiescan be derived from these known ones. An exclusive property implies thatenergy and momentum, for instance, of all products are measured. Theintensivity means that some quantities of the products are left unmeasured.An extensive comparison between the particle multiplicity dN ch /dη perparticipating nucleon at mid-rapidity in central heavy-ion collisions [14,15, 16, 17, 18, 19, 20, 21, 22, 23, 24] and the corresponding results from p + p (¯ p ) [25, 26, 27, 28, 29, 30, 31, 32, 33] and p ( d )+A collisions [34, 35, 14]is presented in Fig. 1. It is obvious that the energy dependence of the totalmultiplicity is distinguishable. In order words, the initial state plays anessential role. The extenstivity can be related to canonical ensemble, Z ( N, T, V ) = Tr N exp (cid:18) − HT (cid:19) , (1)where H is the Hamiltonian, while grand canonical ensemble is related tointensivity, Z ( µ, T, V ) = Tr N exp (cid:18) − H − µ NT (cid:19) , (2) where N stands for the degrees of freedom. With Dirac delta function andwhen the chemical potential µ is Wick rotated, then extenstivity can berelated to intensivity Z ( N, T, V ) = 12 π Z π Z ( iT θ, T, V ) exp( − iN θ ) dθ. (3) n a n ti p / n p (cid:214) s [GeV]ISRNA49 PHOBOSSTARBRAHMSPHENIX A L I C EReggeHRG Model Fig. 2. n ¯ p /n p ratios depicted in whole available range of √ s . Open symbols standfor the results from various pp experiments (labeled). The solid symbols give theheavy-ion results from AGS, SPS and RHIC, respectively. The fitting of pp resultsaccording to Regge model is given by the dashed curve [37]. The solid curve isthe HRG results. Contrary to the dashed curve, the solid line is not a fitting toexperimental data. The graph taken from Ref. [38]. In Fig. 2, the results of ¯ p/p calculated in HRG are represented bysolid line, which seems to be a kind of a universal curve. In heavy-ioncollisions, the proton ratio varies strongly with the center-of-mass energy √ s . The HRG models describes very well the heavy-ion results. Also, AL-ICE pp results are reproduced by means of HRG model. The ratios from pp - and AA -collisions runs very close to unity implying almost vanishingmatter-antimatter asymmetry. On the other hand, it can also be concludedthat the statistical-thermal models including HRG seem to excellently de-scribe the hadronization at very large energies and the condition derivingthe chemical freeze-out at the final state of hadronization, the constantdegrees of freedom or S ( √ s, T ) = 7(4 /π ) V T , seems to be valid at allcenter-of-mass-energies spanning between AGS and LHC. So far, we con-clude that the distinguishability between proton ratios in pp-collisions andthat in AA-collisions disappears with increasing √ s .
3. Dissipative properties: elliptic flow
The azimuthal distribution with respect to the reaction plane reads d Nd ( φ i − Ψ n ) ∼ X n =1 v n cos [ n ( φ i − Ψ n )] . (4)The reaction plane angle Ψ n is not directly measurable, but can be de-termined from particle azimuthal distributions. There are various possiblesources of azimuthal correlations like, jet formation, resonances exist, whichdo not depend on the reaction plane (non-flow correlations). The Fouriercoefficient v n , which refers to the correlation in n particle emission withrespect to the reaction plane, is given by v n = h cos [ n ( φ i − Ψ n )] i . (5) Fig. 3. Integrated elliptic flow measured in central heavy-ion collisions (20 − Fig. 3 shows data collected over about four decades spanning from GSI,AGS, SPS, RHIC to LHC facilities. The integrated elliptic flow measuredin relative central heavy-ion collisions (20 − p t cutoff of 0 . c . Theestimated magnitude of this correction is 12 ±
5% based on calculationswith Therminator. The figure shows that there is a continuous increasein the magnitude of elliptic flow for this centrality region from RHIC to
LHC energies. In comparison to the elliptic flow measurements in Au-Aucollisions at √ s NN = 200 GeV we observe about a 30% increase in themagnitude of v at √ s NN = 2 .
76 TeV. The rapid decrease of v at very lowenergy, FOPI data, refers to bounce-off . Increasing √ s NN , a squeeze-out will set on. At larger energies, the behavior can be described by in-planeelliptic flow due to pressure gradient.the elliptic flow shows a rich structure; a transition from in-plane toout-of-plane and back to in-plane emission. Apparently, it is sensitive tothe properties of the medium created in heavy-ion collisions. There areevidences that the elliptic flow of charged and identified particles indicatesa strong rise of the expansion velocity of the medium (radial fow) at RHICvs LHC.On the other hand, it was assumed that there are no correlations due toelliptic flow in pp collisions at RHIC energy [40]. The methods of measuringelliptic flow can hardly be employed with the currently available number ofrecorded pp interactions of ALICE at the LHC. Furthermore, none of avail-able microscopic Monte Carlo (MC) models describes the development ofanisotropic flow in elementary hadron-hadron interactions yet [40]. Particu-lar non-perturbative approach was suggested as a mechanism of anisotropicflow might be a leading one in hadron collisions, since those have smallergeometrical extension and the probability of hydrodynamical generation ofelliptic flow is lower compared to the collisions of nuclei [41]. pp collisions simulated by PYTHIA, PHOJET and EPOS at 900 and7000 GeV are analyzed by two-particle correlation methods. The integrated v coefficients reconstructed by the methods are found to vary from 10% −
4. Hagedorn temperature: energy and system size dependence
The transverse mass spectra of well-identified particles have been stud-ied at various energies, for instance [45]. Accordingly, Stefan-Boltzmannapproximation results is1 m T dNdm T dy = a exp (cid:18) − m T T (cid:19) , (6)where m T = q p T + m is the dispersion relation and a is a fitting param-eter. Fig. 4 presents the energy dependence of the inverse slope parameter T of the transverse mass spectra of K + (left panel) and K − mesons (rightpanel) produced in central Pb+Pb and Au+Au collisions. There is a plateauat SPS energies [45] which is preceded by a steep rise of T measured at theAGS [42] and followed by an indication of a further increase of the RHICdata [43]. Although the scatter of data points is large, T appears to increase (GeV) NN s1 10 T ( M e V ) + K AGSNA49RHICp+p (GeV) NN s1 10 T ( M e V ) - K HSDHSD + ISSUrQMDHydro + PT
Fig. 4. Energy dependence of T related to the transverse mass spectra of K + (left panel) and K − mesons (right panel) produced in central Pb+Pb and Au+Aucollisions. The graphs taken from Ref. [45]. smoothly in p + p (¯ p ) collisions [44], left panel of Fig. 4. The dependence of T on the system size is obvious. For completeness, we recall that the directthermal photons have been used to estimate the Hagedorn temperature, T = ± MeV ALICE [46]221 ± MeV PHENIX [47] (7)The dependence on system size is illustrated in left panel of Fig. 4. TheHagedorn temperature in pp collisions seems to be smaller than that in AA collisions. Its variation with the center-of-mass energy is apparently weakerthan the variation in AA collisions. A much more systematic measurementwould help in proving or disproving such a conclusion.
5. Conclusions and outlook
The ultimate goal of the physics program of high-energy collisions is thestudy of properties of strongly interacting matter under extreme conditionsof temperature and/or compression. The particle multiplicities and theirfluctuations and correlations are experimental tools to analyse the nature,composition, and size of the medium, from which they are originating. Ofparticular interest is the extent to which the measured particle yields areshowing equilibration. Based on analysing the particle abundances or mo-mentum spectra, the degree of equilibrium of the produced particles canbe estimated. The particle abundances can help to establish the chemical composition of the system. The momentum spectra can give additionalinformation on the dynamical evolution and the collective flow.In order to characterize possible critical phenomena, signatures basedon particle multiplicities and their fluctuations and correlations have beenproposed. Intensive or extensive quantities should be separated, systemat-ically. Extensivity obtains about equal contributions from the initial andfinal stage. Intensivity is well described by produced particles in final state.The present work introduces the importance of distinguishability betweenextensive and intensive quantities at various energies and in different systemsizes.
Acknowledgement
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