Thermodynamics of partonic matter in relativistic heavy-ion collisions from a multiphase transport model
aa r X i v : . [ nu c l - t h ] F e b Thermodynamics of partonic matter in relativistic heavy-ion collisions from amultiphase transport model
Han-Sheng Wang, Guo-Liang Ma, ∗ Zi-Wei Lin, and Wei-jie Fu Key Laboratory of Nuclear Physics and Ion-beam Application (MOE),Institute of Modern Physics, Fudan University, Shanghai 200433, China Department of Physics, East Carolina University,C-209 Howell Science Complex, Greenville, NC 27858 School of Physics, Dalian University of Technology, Dalian, 116024, China
Using the string melting version of a multiphase transport (AMPT) model, we focus on theevolution of thermodynamic properties of the central cell of parton matter produced in Au+Aucollisions ranging from 200 GeV down to 2.7 GeV. The temperature and baryon chemical potentialare calculated for Au+Au collisions at different energies to locate their evolution trajectories in theQCD phase diagram. The evolution of pressure anisotropy indicates that only partial thermalizationcan be achieved, especially at lower energies. Through event-by-event temperature fluctuations, wepresent the specific heat of the partonic matter as a function of temperature and baryon chemicalpotential that is related to the partonic matter’s approach to equilibrium.
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I. INTRODUCTION
The ultrarelativistic heavy ion collisions at the BNLRelativistic Heavy Ion Collider (RHIC) and the CERNLarge Hadron Collider (LHC) have created the partonicmatter at extreme conditions of temperature and energydensities, the Quark-Gluon Plasma (QGP), which is gov-erned by the Quantum Chromodynamics (QCD) theory.The first principle Lattice QCD calculation shows thatthe transition from hadronic to the partonic matter atzero baryon chemical potential µ B is a smooth crossover[1–3]. But the calculation of phase transition in theQCD phase diagram at finite baryon chemical potentialstill has large uncertainties [4–6], especially regarding theconjectured end point of the first order phase transitionboundary that is the so-called QCD critical end point(CEP) [7–9], due to the famous sign problem [10–12].To explore the nature of the QCD phase diagram, theBeam Energy Scan (BES) program at the RelativisticHeavy-Ion Collider (RHIC) is searching for the QCD crit-ical point with Au+Au collisions at a large range of colli-sion energies [13–18]. The fireballs created in Au+Au col-lisions at different energies freeze out at different pointsof the QCD phase diagram. Because certain singularitieswill appear at the CEP in the thermodynamic limit [19],we expect to observe certain non-monotonic behaviors ifthe evolution trajectory of the colliding system is closeenough to the CEP. For example, event-by-event fluc-tuations of various conserved quantities are proposed aspossible signatures of the existence of the CEP [20–22]because they are proportional to the corresponding sus-ceptibilities and correlation lengths. Many recent experi-mental results on net-proton fluctuations hint that a crit-ical point might have been reached during the evolution ∗ [email protected] of Au+Au collisions at a low collision energy [14, 18, 23],which serves as a main motivation for the upcoming re-search projects such as those at FAIR in Germany, NICAin Russia, and HIAF in China.Temperature and baryon chemical potential specify athermodynamic state of the QCD matter. Meanwhile,their fluctuations are deeply related to the phase proper-ties and phase transitions. The fluctuations of thermody-namic quantities are affected by the interactions amongthe constituent particles. For example, fluctuations ofthe temperature of thermal system provide an measureof the specific heat C v , and the irregular behavior of spe-cific heat is related to phase transitions. There have beenmany investigations on the specific heat in relativisticheavy ion collisions [24–34].On the other hand, it is difficult to connect thermalproperties of static QCD matter with the experimen-tal measurements, since relativistic heavy-ion collisionsinvolve many different dynamical evolution stages. Tostudy the full evolution history of the thermodynamicproperties of the QCD matter with a dynamical transportmodel may serve as a bridge between the gap [35, 36]. Inthis work, we investigate the space-time evolution of theparton matter created in Au+Au collisions at differentenergies, including transverse flow, effective temperatureand baryon chemical potential, pressure anisotropy andspecific heat by using the string melting version of a mul-tiphase transport (AMPT) model [37].The paper is organized as follows. Section II brieflyintroduces the string melting version of AMPT modeland the parameters that we use. Comparison of thespace-time evolution of transverse flow at different colli-sion energies are presented in Sec. III A. We then discussthe space-time evolution of the effective temperature andbaryon chemical potential in Sec. III B. We show the tra-jectories of Au+Au collisions at different energies in theQCD phase diagram in Sec. III C. We discuss the timeevolution of pressure anisotropy in Sec. III D. The evo-lution of event-by-event fluctuations of the temperatureand specific heat of the partonic matter are discussed inSec. III E. Finally, a summary is given in Sec. IV. II. THE STRING MELTING VERSION OF AMULTIPHASE TRANSPORT MODEL
The string melting version of AMPT model consistsof fluctuating initial conditions from the heavy ion jetinteraction generator (HIJING) model. In this model,minijet partons and strings are produced from hard pro-cesses and soft processes, respectively. Meanwhile, all ex-cited hadronic strings in the overlap volume are convertedinto partons. The interactions among these partons aredescribed by Zhang’s parton cascade (ZPC) model [38],which includes parton two-body scatterings based on theleading order pQCD gg → gg cross section: dσdt = 9 πα s µ s ) 1( t − µ ) . (1)In the above, α s is the strong coupling constant (taken as0.33), while s and t are the usual Mandelstam variables.The effective screening mass µ is taken as a parameterin ZPC for the parton scattering cross section, and weset µ as 2.265 fm − leading to a total cross section ofabout 3 mb for elastic scatterings. The AMPT model im-plements a simple quark coalescence model, which com-bines nearby freeze-out partons into mesons or baryons,to simulate the transition from the partonic matter to thehadronic matter. The final-stage hadronic evolutions aremodeled by an extension of a relativistic transport model(ART) including both elastic and inelastic scatterings forbaryon-baryon, baryon-meson and meson-meson interac-tions [39]. Our other parameters are taken as same asthose from Ref. [36, 41], which can reasonably repro-duce many experimental observerbles such as rapiditydistributions, p T spectra, and anisotropic flows [40–42]for both Au+Au collisions at RHIC and Pb+Pb colli-sions at LHC energies.Using the string-melting version of the AMPT model,10 ,
000 events of Au+Au central collisions (0 − ≤ √ s NN =200, 62.4, 39, 27, 19.6, 11.5, 7.7, 4.9, and 2.7GeV) which can be provided by RHIC, FAIR and NICAfacilities. We only consider the space-time evolution ofthe partonic matter in the parton cascade, for which nec-essary information is recorded and analyzed for its ther-modynamics properties. In our convention, the x axis ischosen along the direction of impact parameter b from thetarget center to the projectile center, the z axis is alongthe beam direction, and the y axis is perpendicular toboth x and z directions. The time t starts when the twogold nuclei are fully overlapped in the longitudinal direc-tion, while the proper time τ is defined as ( t − z ) / . III. RESULTS AND DISCUSSIONSA. Space-time evolution of transverse flow
We first calculate the radial flow employing ~β =( P i ~p i / P i E i ), where the sum over index i takes intoaccount all partons in the cell from all events of a givencollision system.Flow component β x along the x direction as functionsof coordinate x and space-time rapidity η s at differenttimes in cells within 0 . < y < . √ s NN =200 GeV and 7.7 GeV are shown inFig. 1. We can see the antisymmetry of the transverseflow along x axis at all space-time rapidity, after aver-aging over many events of central collisions. The flowis very small at the early time τ = 0 . x [36].Figure 2 shows the transverse flows of partons in thetwo selected cells at ( x = 3 fm and y = 0 fm) and ( x = 7fm and y = 0 fm) within space-time rapidity | η s | < . x = 3 fm), butsimilar for the outer cell (at x = 7 fm). B. Space-time evolution of temperature andbaryon chemical potential
In the AMPT model, the energy-momentum tensor canbe calculated by averaging over particles and events in avolume V [35], i.e., T µν = 1 V X i p µi p νi E i . (2)The energy density can be given by ǫ = T , whilethe pressure components are related to the energy-momentum tensor by P x = T , P y = T , P z = T .For a massless quark-gluon plasma in full chemical andthermal equilibrium as given by the Bose-Einstein andFermi-Dirac distributions, the energy density and tem-perature T obey the following relation: ǫ = d QGP π T , (3)where d QGP = d g + 7 d q / . N f is the totaldegeneracy factor of the QGP, and N f represents thenumber of relevant quark flavors ( N f = 3 is consideredin our investigation). With the above assumption, wecan calculate the temperature [36] and baryon chemicalpotential [44] according to T = s d QGP π ǫ, (4) Au+Au200GeVb 3fmAu+Au7.7GeVb 3fm
FIG. 1: (Color online) Transverse flow component β x along the x axis (0 . < y < . x and η s at differentproper times in central Au+Au collisions at √ s NN =200 GeV (first row) and 7.7 GeV (second row). x=3 fm 200 GeV 62.4 GeV 39 GeV 27 GeV 19.6 GeV 11.5 GeV 7.7 GeV 4.9 GeV 2.7 GeV b x t (fm/c)AMPT: Au+Au b 3fm |h s |<0.5 |y|<0.5fmx=7 fm FIG. 2: (Color online) Proper time evolution of transverseflow component β x of partons within space-time rapidity | η s | < . µ B = (5 n B − n Q + 2 n S ) /T , (5)where n B,Q,S are the density of net baryon, net elec-tric charge, and net strangeness, respectively. Note thatEq. (5) holds for a non-interacting massless parton sys-tem with a small chemical potential in principle. Theenergy density in the rest frame of a cell should be usedto obtained the local effective temperature in order toremove the effect of radial flow. However, the central cellthat we consider here is special because it is always atrest. Also note that, since partons in the cell may not bein full thermal and chemical equilibrium [36] (which willbe discussed in Sec. III D), these extracted temperature -3 -2 -1 r t (fm/c) 200 GeV 62.4 GeV 39 GeV 27 GeV 19.6 GeV 11.5 GeV 7.7 GeV 4.9 GeV 2.7 GeVAMPT: Au+Au b 3fm|h s |<0.5 r <3fm FIG. 3: (Color online) Proper time evolution of the partondensity in the central cell ( r ⊥ < | η s | < . and baryon chemical potential are the effective values.Figure 3 shows the densities of partons at the center offireball ( r ⊥ < | η s | < . x and space-time rapidity atdifferent proper times in central Au+Au collisions at √ s NN =200 and 7.7 GeV are shown in Fig. 4. We cansee that the highest temperature is reached in the earlytime at the center of the overlap region, and the temper-ature decreases with the expansion and evolution of the -404 s t=0.2fm/c t=2fm/c t=4fm/c t=6fm/c T e (MeV)t=8fm/c Au+Au200GeVb 3fmAu+Au7.7GeVb 3fm -8 0 8-404 s x (fm)t=0.2fm/c -8 0 8 x (fm)t=2fm/c -8 0 8 x (fm)t=4fm/c -8 0 8 x (fm)t=6fm/c -8 0 8 x (fm) T e (MeV)t=8fm/c FIG. 4: (Color online) Contour plots of effective temperatureas a function of x coordinate and space-time rapidity η s atdifferent proper times in central Au+Au collisions at √ s NN =200 GeV (first row) and 7.7 GeV (second row), where y coor-dinate is chosen within | y | < system. m B ( M e V ) AMPT, Au+Au, b 3 fm, |h s |<0.5 (Central cell)(a)(b) T ( M e V ) t (fm/c) 200 GeV 62.4 GeV 39 GeV 27 GeV 19.6 GeV 11.5 GeV 7.7 GeV 4.9 GeV 2.7 GeV FIG. 5: (Color online) Proper time evolution of baryon chemi-cal potential (a) and temperature (b) of the central cell withinspace-time rapidity | η s | < . The proper time evolutions of the baryon chemical po-tential and temperature for the central cell within thespace-time rapidity range of − . < η s < . s |<0.5 (Central cell) 200GeV (b) m B o r m B c o m ponen t ( M e V ) t (fm/c) m B =(5n B -n Q +2n S )/T B /T -n Q /T S /T (a) FIG. 6: (Color online) Proper time evolution of componentsof baryon chemical potential of the central cell within space-time rapidity | η s | < . proper time evolution of the three components of baryonchemical potential [(5 n B ) /T , ( − n Q ) /T , (2 n S ) /T ] ofthe central cell in central Au+Au collisions at 7.7 and 200GeV, respectively. We can see that the baryon chemicalpotential is dominated by the net baryon number density.We also find that the total baryon chemical potential isslightly lower than the component from the net baryonnumber density in Au+Au collisions at 7.7 GeV whileit is often the opposite at 200 GeV; this is mainly dueto the increase of the relative contribution from the netstrangeness number density at the higher colliding en-ergy. C. Trajectories in the QCD phase diagram
In Fig. 7 we present the event-averaged evolution tra-jectory of the central cell of the partonic matter producedin central Au+Au collisions at different beam energiesfrom the time of 1 fm/c to the moment when it reachesthe crossover curve in the temperature and baryon chem-ical potential diagram. Note that the crossover phaseboundary is obtained from the functional renormalisationgroup (FRG) method with N f = 2 + 1, which well agreeswith the phase boundary from the Lattice QCD[6]. It isinteresting to find that the durations of the partonic stageare around 4 fm/c, which is consistent with the previousAMPT results for mid-central Au+Au collisions [45]. Ifwe take the location of the critical end point at ( T CEP , µ B CEP ) = (107, 635) MeV from the FRG calculation, thebeam energies between 2.7 and 4.9 GeV [6, 48] seem tobe the most promising to reach the CEP, which couldbe accessed at fixed target experiments at RHIC or in t fo (fm/c) The phase boundary T ( M e V ) m B (MeV) 200 GeV 62.4 GeV 39 GeV 27 GeV 19.6 GeV 11.5 GeV 7.7 GeV 4.9 GeV 2.7 GeVAMPT, Au+Au, b 3 fm, |h s |<0.5 (Central cell) CEP
FIG. 7: (Color online) AMPT results on the average trajec-tory of the central cell in central Au+Au collisions at dif-ferent energies in the QCD phase diagram of temperatureversus baryon chemical potential. The black curve showsthe crossover phase boundary with the critical end point ob-tained from the functional renormalisation group approachwith N f = 2 + 1 [6]. The time values at which each trajectoryintersects the phase boundary are also shown. the future. Note that it has been found that the chemi-cal and kinetic freeze-out parameters extracted from theAMPT model agree with the RHIC experimental mea-surements [49]. T ( M e V ) B (MeV) s = 0 mb s = 1 mb s = 2 mb s = 3 mb The phase boundaryAMPT:Au+Au @ (cid:214)s NN = 7.7GeVb 3fm |h s |<0.5(Central cell) CEP
FIG. 8: (Color online) AMPT results on the average trajec-tory of the central cell in central Au+Au collisions at 7.7 GeVat different parton cross sections in the QCD phase diagramof temperature versus baryon chemical potential.
We also investigate the effect of parton cross sectionon the event-averaged trajectory in the QCD phase di-agram for Au+Au collisions at 7.7 GeV, since differentinteraction strength can affect the evolution of the col-lision system. In Fig. 8, we can see that a larger crosssection pushes the evolution trajectory downward; conse-quently the intersection point is closer to the CEP with T ( s ) / T ( s = m b ) (a) B ( s ) / B ( s = m b ) t (fm/c) s = 0 mb s = 1 mb s = 2 mb s = 3 mb (b)Au+Au @ (cid:214)s NN = 7.7GeV b 3fm|h s |<0.5 FIG. 9: (Color online) Time evolution of the ratios of temper-ature (a) and baryon chemical potential (b) at a given partoncross section over that at 3mb cross section for central Au+Aucollisions at 7.7 GeV. a larger parton cross section for Au+Au collisions at7.7GeV. More details can be seen in Fig. 9, which showsthe time evolution of the ratios of temperature or baryonchemical potential at each parton cross section over thatat 3mb. We can see that the decrease of temperature isslower with a smaller cross section, but the opposite istrue for the baryon chemical potential.We note that the string melting version of the AMPTmodel that we use in this study does not include the finitenuclear thickness, which effect could be important forheavy ion collisions at low energies. For example, it hasbeen found [46, 47] that at low energies the finite nuclearthickness along the beam direction leads to a lower peakvalue but a longer evolution time for the produced initialenergy density. This would affect the time evolution ofthe effective temperature and consequently the evolutiontrajectory of heavy ion collisions at the BES energies andbelow. Further studies that includes the finite nuclearthickness are warranted.We should point out that the above results come fromthe average of 10,000 central Au+Au events. However,event-by-event fluctuations can not be neglected, andthese fluctuations could affect the search for the CEP ofthe QCD phase diagram. Figure 10 shows the event-by-event trajectories of central Au+Au collisions at differ-ent beam energies from the AMPT-SM model, where theblack curve in each panel represents the phase transitioncrossover boundary from FRG [6]. To suppress the effectfrom multiplicity or volume fluctuation, a multiplicity cutis further applied, where we divide the total events into100 bins by multiplicity and only use the events in onemiddle bin around the average. Even so, we can see thatthe fluctuation is quite large, especially at high energies.From the initial state to the late stage, these trajectoriesare distributed in a wide area around the averaged tra-jectory; this is consistent with the recent hydrodynamicalresult [50]. It is interesting to see that some trajectories T ( M e V ) AMPT Au+Au, |h s |<0.5(Central cell) m B (MeV) FIG. 10: (Color online) AMPT results on event-by-event tra-jectories of the central cell in central Au+Au collisions atdifferent beam energies in the QCD phase diagram of tem-perature versus baryon chemical potential. can pass the CEP because of event-by-event fluctuations,which could further lead to possible critical behaviors.
D. Pressure anisotropy
In the central cell of central Au+Au collisions, dueto the cylindrical symmetry around the beam axis, thetwo transverse pressure components P x and P y are equal.Therefore, the transverse pressure can be defined to be P T = ( P x + P y ) / P L is just P z . For a system in thermal equilibrium, itspressure must be isotropic, which satisfies the relation of P T = P L = P . Therefore, a pressure anisotropy parame-ter, P L /P T , is defined to describe the degree of pressureanisotropy of the system in thermal equilibrium. Thecloser the value of P L /P T is to unity, the closer the sys-tem to the state of thermal equilibrium.Figure 11 shows how the pressure anisotropy param-eter of partons in the central cell evolves with propertime in central Au+Au collisions at different beam ener-gies. We see that P L /P T keeps increasing up to 5 fm/cin Au+Au collisions at 200 GeV; however, it still doesnot reach unity. It indicates that even for the top RHICenergy the central cell of the system actually does notreach thermal equilibrium when it arrives at the phaseboundary in the AMPT model, consistent with the pre-vious result [36]. For lower energies, P L /P T increases atearly times but saturates around 0.7 at late times. Forthe lowest energy of 2.7 GeV, P L /P T first increases andthen gradually decreases with time.In order to understand why the AMPT model showsthe above time evolution of the pressure anisotropy pa-rameter, Fig. 12 presents the average number of colli-sions of each parton per unit proper time for the centralcell in central Au+Au collisions at different beam en- P L / P T t (fm/c) 200 GeV 62.4 GeV 39 GeV 27 GeV 19.6 GeV 11.5 GeV 7.7 GeV 4.9 GeV 2.7 GeVAMPT: Au+Au b 3 fm |h s |<0.5(Central cell) FIG. 11: (Color online) AMPT results on the time evolutionof the pressure anisotropy parameter of partons in the centralcell in central Au+Au collisions at different beam energies. / N ( t ) d N c o ll / d t t (fm/c) 200 GeV 62.4 GeV 39 GeV 27 GeV 19.6 GeV 11.5 GeV 7.7 GeV 4.9 GeV 2.7 GeV AMPT: Au+Au b 3 fm |h s |<0.5 (Central cell) FIG. 12: (Color online) AMPT results on the average numberof collisions of each parton per unit proper time as a functionof the proper time for the central cell in central Au+Au col-lisions at different beam energies. ergies. We see that the partons have few collisions atearly times since most partons have not been formed,and then they collide most frequently around 0.5 fm/c.For higher beam energies, they can keep colliding untilthe phase boundary ( ∼ E. Temperature fluctuations and specific heat
Thermodynamic quantities always fluctuate aroundtheir thermal averages, because the entropy is requiredto reach its maximum in equilibrium. By taking advan-tage of thermodynamic fluctuations, we can learn moreabout thermal properties of an equilibrium system. Forinstance, the event-by-event temperature fluctuation ofan equilibrium system obeys the probability distribu-tion [24]: P ( T ) ∼ exp [ − C v T ) h T i ] , (6)where h T i is the mean temperature and (∆ T ) = ( T −h T i ) is the variance in temperature [19]. For a Gaus-sian fluctuation, the specific heat can be easily obtainedas [27] 1 C v = (∆ T ) h T i . (7) T (MeV) 7.7GeV 4.9GeV 2.7GeV 200GeV 62.4GeV 39GeV N u m be r o f e v en t s AMPT:t=0.2fm/cAu+Aub 3 fm| s |<0.5 (c)(b)(a) (d)t=2fm/c (c)t=4fm/c t=6fm/c 27GeV 19.6GeV 11.5GeV FIG. 13: (Color online) AMPT results on the event-by-eventeffective temperature distributions of partons for the centralcell within | η s | < τ = 0.2 fm/c, (b) τ = 2 fm/c, (c) τ = 4 fm/c, and (d) τ = 6 fm/c in central Au+Au collisions atdifferent beam energies, where symbols and curves representthe AMPT results and Gaussian fits, respectively. The AMPT results on the event-by-event effective tem-perature distributions for the partons in the central cell(within | η s | < s T t (fm/c) 200 GeV 62.4 GeV 39 GeV 27 GeV 19.6 GeV 11.5 GeV 7.7 GeV 4.9 GeV 2.7 GeVAMPT, Au+Au, b 3 fm, |h s |<0.5, (Central cell) FIG. 14: (Color online) AMPT results on the time evolutionof the width of the temperature distribution of partons in thecentral cell in central Au+Au collisions at different energies. cascade. Although collisions at higher collisions energieshave a larger temperature fluctuation at the beginning,they reach similar fluctuations at late times. b 3fm |h s |<0.5 (Central cell) T ( M e V ) m B (MeV) C v /N AMPT: Au+AuCEP T ( M e V )
200 GeV 62.4 GeV 39 GeV 27 GeV 19.6 GeV 11.5 GeV 7.7 GeV 4.9 GeV 2.7 GeV The phase boundary
FIG. 15: (Color online) AMPT results on the specific heat perparton, C v /N , of the central cell in central Au+Au collisionsas a function of baryon chemical potential and temperature,where the average evolution trajectories of the central cell andthe crossover phase boundary with the critical end point arealso shown. Based on Eq. (7), the specific heat of the central cell ofthe partonic matter can be extracted. Scine the specificheat, as an extensive quantity, reflects the total energyper unit of temperature change needed by all partons inthe cell, we introduce a normalized specific heat C v /N ,where N is the parton multiplicity in the central cell. Fig-ure 15 shows the normalized specific heat as functions ofbaryon chemical potential and temperature for partonsin the central cell in central Au+Au collisions from theAMPT model. We find that the normalized specific heatincreases with the decrease of temperature and the in- AMPT: Au+Au15.6 b 3fm |h s |<0.5 (Central cell) T ( M e V ) m B (MeV) C v /(NT ) CEP T ( M e V )
200 GeV 62.4 GeV 39 GeV 27 GeV 19.6 GeV 11.5 GeV 7.7 GeV 4.9 GeV 2.7 GeV The phase boundary
FIG. 16: (Color online) AMPT results on the dimensionlessspecific heat per parton, C v / ( NT ), of the central cell in cen-tral Au+Au collisions as a function of baryon chemical poten-tial and temperature, where the average evolution trajectoriesof the central cell and the crossover phase boundary with thecritical end point and the line for 15 . crease of baryon chemical potential in general. It alsoreaches a maximum around the CEP within the energyranges shown in the figure. Compared to preliminaryresults from the RHIC-STAR experiment [34], however,our result is systematically lower than the experimentaldata for lower energies. But it should be noticed that ourspecific heat is aimed at the evolution of partonic matter,however the experimental results are obtained based onthe information from final freeze out hadrons.Since we assume that the partonic matter is a masslessquark-gluon plasma in full chemical and thermal equilib-rium in extracting the effective temperature, C v / ( N T )should be a dimensionless constant of d QGP π / ∼ . C v / ( N T ), of the central cell in centralAu+Au collisions as a function of baryon chemical poten-tial and temperature. We can see that it increases froma small value in the early time of Au+Au collisions toa large value along the trajectory, close to the expectedvalue of ∼ . C v / ( N T ) value could re-sult from dynamical fluctuations or the non-equilibriumevolution of partonic matter. IV. SUMMARY
We have studied the space-time evolution of the partonmatter produced in 0-5% most central Au+Au collisionsat different collision energies using the AMPT modelwith string melting. We extract the effective tempera- ture and baryon chemical potential of the partons in thecentral cell. The temperature and baryon chemical po-tential both decrease with time, but their dependenceson the collision energy are opposite. By investigating theevolution of the partonic matter created in Au+Au colli-sions at different energies, we obtain their trajectories inthe QCD phase diagram. The results indicate that thepartonic state exists until about 4 fm/c, and the CEPproposed by the FRG approach is located between thetrajectories at the collision energies of 2.7 and 4.9 GeV.We find that the parton interaction cross section influ-ences the evolution trajectory of the collision system andthe trajectory also fluctuates event-by-event. These fea-tures could be important for the experimental search forthe CEP. By studying pressure anisotropies in 0-5% mostcentral Au+Au collisions at different collision energies,we find that the parton matter from the AMPT model isstill in incomplete thermal equilibrium in the region nearthe QCD phase boundary. We also use temperature fluc-tuations to investigate the evolution of the specific heatof the partonic matter in the AMPT model. We find thatthe dimensionless specific heat per parton increases withtime and gets closer to the expected value near the QCDphase boundary, indicating that the partonic system getscloser to thermal equilibrium.It should be noted that the AMPT model currentlyonly includes two-body elastic parton collisions with aleading-order pQCD gluon elastic scattering cross sec-tion, which is controlled by a Debye screening mass.However, it has been demonstrated that the pQCD gg ↔ ggg bremsstrahlung processes can isotropize momen-tum more efficiently than elastic scatterings [52]. In ad-dition, the QGP created in high energy heavy-ion col-lisions, which may consist of gluons and quarks in ornear chemical and thermal equilibrium, should be gov-erned by non-perturbative QCD interactions. Further-more, the method that we used to extract temperatureand baryon chemical potential works for non-interactingmassless parton system with a small chemical potentialin principle, and we have not include the effect of finitenuclear thickness that has been shown to be important atlow energies. Due the these limitations, we expect thatour calculations can only provide qualitative features ofthe non-equilibrium dynamical evolution. In addition,we have focused on the central space-time rapidity andonly studied the partonic matter without the subsequentphase transition and hadronic evolution. Further stud-ies of the evolution and the thermodynamic properties ofthe matter in heavy-ion collisions are indispensable forstudying the QCD phase structure and the experimentalsearch for the critical point. Acknowledgments
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