Thermodynamic properties at the kinetic freeze‑out in the Au + Au and Cu + Cu collisions at the RHIC using the Tsallis distribution

Wei‑Hao Wu · Jun‑Qi Tao · Hua Zheng · Wen‑Chao Zhang · Xing‑Quan Liu · Li‑Lin Zhu · Aldo Bonasera

Abstract The thermodynamic properties of charged particles, such as the energy density, pressure, entropy density, particle density,and squared speed of sound at the kinetic freeze-out in the Au+Au collisions from the relativistic heavy ion collider (RHIC)beam energy scan program (and in the Cu+Cu collisions at200 GeV are studied using the thermodynamically consistent Tsallis distribution.The energy density, pressure, and particle density decrease monotonically with the collision energy for the same collision centrality; These properties also decrease monotonically from the central to peripheral collisions at the same collision energy.While the scaled energy density ε∕T4 and scaled entropy density s∕T3 demonstrate the opposite trend with the collision energy for the same collision centrality.There is a correlation between ε∕T4 and s∕T3 at the same centrality.In addition, the squared speed of sound was calculated to determine that all the collision energies share nearly the same value at different collision centralities.

Keywords Heavy-ion collision · Tsallis distribution · Kinetic freeze-out · Energy density · Entropy density · Particle density · Squared speed of sound · Pressure

1 Introduction

In relativistic heavy-ion collisions, an extremely hot and dense mixture of quarks and gluons is created, which is called the quark gluon plasma (QGP) [1–5].The QGP can only exist for a significantly short time and hadronizes into mesons and baryons owing to its color confinement.These particles interact with one another or form light nuclei [6]and continue expanding.The system cools and reaches the chemical freeze-out point when the abundances of all the particle species are unchanged.The system continues evolving to reach a kinetic freeze-out, where the distributions of all the particles do not change.Subsequently, the information of particles are recorded by detectors set around the collision region.With the measured information, such as the multiplicities of the particles and particle transverse momentum (pT) spectra, the properties of the QGP and the system can be studied at different evolution stages [7–9].

In previous experimental and theoretical studies, several statistical distributions or models based on different assumptions have been used to describe the particle transverse momentum spectra and to extract relevant information about the collision system.These include the Boltzmann-Gibbs(BG) distribution, Fermi-Dirac distribution, Bose-Einstein distribution, double exponential distribution,mT-exponential distribution [10, 11], Erlang distribution [12], multisource model [13], blast-wave model [14], Tsallis distribution [15–27], and the Generalized Fokker-Planck Solution(GFPS) [28, 29], etc.As a generalization of the BG distribution, the Tsallis distribution has been recently highly valued[18–24, 27].This is ascribed to its successful application in describing the particlepTspectra in the p+p collisions (the transverse momentum spans two orders of magnitude and the yield spans 15 orders of magnitude) presented by Wong et al.[15, 16] and in several other studies [17, 20, 25, 26]dedicated to describing the particle transverse momentum spectra produced in the pp, pA, and AA collisions.Cleymans et al.demonstrated the thermodynamic consistency of the Tsallis distribution.Utilizing the Tsallis distribution,Azmi et al.[25] described the transverse momentum spectra of charged particles produced in the Pb+Pb collisions at the Large Hadron Collider (LHC) and deduced the thermodynamic properties of the collision system at the kinetic freeze-out.Combined with the thermodynamic properties of the system at the chemical freeze-out point obtained by fitting the particle yields using the statistical model, this can provide an evolutionary picture of the thermodynamic quantities for the hadronic phase from the chemical to kinetic freeze-out point [25].

The remainder of this paper is organized as follows.The Tsallis distribution for the transverse momentum spectrum of the charged particles as well as the formulas for the thermodynamic quantities are briefly introduced in Sect.2, along with the fitting results of the experimental transverse momentum spectra of the charged particles.The thermodynamic quantities of the Au + Au and Cu + Cu collisions were calculated at different collision energies and centralities, the results of which are discussed in Sect.3.A brief summary is given in Sect.4.

2 Tsallis distribution

The Tsallis distribution is a generalization of the Boltzmann-Gibbs distribution in classical thermodynamics, which was proposed by Tsallis [35].Within the framework of the thermodynamically consistent Tsallis distribution, the momentum distribution of the final particles produced in relativistic heavy-ion collisions can be expressed as follows:

The majority of the charged particles areπ+(π-) mesons,and the number of positive and negativeπmesons are equal in the heavy-ion collisions at the RHIC and LHC, which implies despite the collision energy being as low as 7.7 GeV, which is the lowest collision energy in the BES.However, the numbers ofpand¯pare different, which leads to a nonzero chemical potential of the baryons.Considering that only a small portion of the charged particles are baryons, it is a sufficient approximation for assuming that the chemical potential of the particles is zero.The variations owing to the approximation of the zero chemical potential were determined to be small; our conclusions do not depend on the approximation.When only the particles in mid-rapidity(y≈0 ) are considered, Eq.(2) is reduced to the following:

In the experimental distribution of the charged particles in relativistic heavy-ion collisions, pseudorapidityηis occasionally used instead of rapidityy.The conversion from rapidity to pseudorapidity is given by the following:

wherei=π+,K+,p.mT,iis the transverse mass of particleiin the sum of Eq.(6).Factor 2 on the right-hand side considers the contributions from the antiparticles, which is reasonable at the LHC because the multiplicities of the particles and antiparticles are equal [25].The degeneracy factorsgof the particles aregπ+=gK+=1,gp=2.However, the experimental data demonstrate significant differences between the multiplicities of the particles and antiparticles for kaons and protons at the RHIC, particularly at lower collision energies.By considering the aforementioned, we determined the effective degeneracy factor of the particles.This factor is determined by taking half the sum of one and the multiplic-ity ratio between the antiparticles and particles for each type of particle from the experimental data of the RHIC [10, 11,34, 38].These values are listed in Table 1.

Table 1 The effective values of 8n+，gK+，and g, are used to fit thecharged particle transverse momentum spectra in the Au + Au col-lisions at sNN = 7.7 一200 GeV and in the Cu +Cu collisions asN= 62.4,200 GeV

The formulas for the thermodynamic quantities at the kinetic freeze-out in the thermodynamically consistent Tsallis statistics are as follows [25, 39]:

wherei=π+,K+,p.

To understand the behavior of the thermodynamic quantities, the analytical formulas derived for the massless particles and zero chemical potential in the Tsallis statistics are utilized for an estimation.They are provided in Ref.[40]:

wheregis the particle degeneracy factor.

Prior to calculating the thermodynamic quantities for the Au + Au and Cu + Cu collisions at the kinetic freezeout at the RHIC using Eqs.(7, 8, 9, 10, and 11 ), the Tsallis parameterqand temperature parameterTneed to be obtained.To achieve these parameters, we fitted the transverse momentum spectra of the charged particles for the Au + Au and Cu + Cu collisions atsNN=200 GeV for different collision centralities using Eq.(6).The results are presented in Fig.1.The Tsallis distribution describes the transverse momentum spectra of the charged particles with momentum values lower than 8 GeV/c.The fit/data were obtained to characterize the fit quality, as shown in the bottom panels of Fig.1, which demonstrates that most of the fit/data points fluctuated within 20%, and only a few data points wherepTwas either close to 0 GeV/cor close to 8 GeV/cfluctuated within 30%.The correspondingχ2/NDF for the fit are also listed in Tables 2 and 3, respectively.The fit quality of the peripheral collisions was better than that of the central collisions, which is consistent with our previous results [28, 29, 41–43].The transverse momentum spectra of the charged particles from the Au + Au collisions in the BES program atsNN=7.7-130 GeV and the Cu + Cu collisions atsNN=62.4 GeV were also fitted with Eq.(6)and similar results were obtained as shown in Fig.1.

Tables 2 and 3 list the temperature parameterTand Tsallis parameterqobtained by fitting the transverse momentum spectra of the charged particles from the Au + Au collissiNoNn s =a7t.7-200 GeV and Cu + Cu collisions atsNN=62.4,200 GeV.As reported in Ref.[26], for a given collision energy, as the collision centrality increases, that is,from the most central to peripheral collisions, the temperature parameterTdemonstrates a significant decreasing trend while the Tsallis parameterqdemonstrates an increasing trend; however, the absolute magnitude of the increase is significantly small.Similarly, for a given collision centrality,the temperature parameterTexhibits a decreasing trend as the collision energy increases, whereas the Tsallis parameterqdisplays the opposite trend.

3 Thermodynamic variables

In this section, the temperature parameterTand Tsallis parameterqlisted in Tables 2 and 3 are used to calculate the thermodynamic quantities for relativistic heavy-ion collisions within the framework of the thermodynamically consistent Tsallis statistics.The errors propagated by the uncertainties of the fitting parameters are also considered.Note, the thermodynamic quantities are calculated for the charged particles at the kinetic freeze-out hereafter.

3.1 Energy density

Table 2 The values of q, T,and χ2/NDF are obtained by using Eq.(6) to fit the transverse momentum spectra of the charged particles from the Au+Au collisions at sNN =7.7-200 GeV

reducuction because the kinetic freeze-out temperatureTkinstrongly depends on the centrality [44].The system size dependence of the scaled energy density nearly disappeared in the collision system when the results for the Au + Au and Cu + Cu collisions at the same collision energy were compared.In addition, the values ofε∕T4demonstrate an increasing trend as a function of the collision energy.

3.2 Pressure and squared speed of sound

In the current analysis, the pressure at the kinetic freezeout can be obtained from Eq.(10).In Fig.3a, the pressure,which is in units of GeV/fm3, demonstrates a significant and expected increase from the peripheral to the central collisions for a given collision energy.The pressure results for the Pb+Pb collisions atsNN=2760, 5020 GeV obtained from Ref.[25] are also shown in the figure.The pressure exhibited the same pattern as the particle density, as shown in Fig.6.See the explanation in Sect.3.1 for further details.

Fig.1 (Color online) Transverse momentum spectra of the charged particles in the Au+Au (left panel) and Cu+Cu (right panel) collisions at sNN =200 GeV measured by the STAR Collaboration and the PHOBOS Collaboration, respectively.The curves indicate fits using the Tsallis distribution Eq.(6), and the values of the degeneracy factor are obtained from Table 1.The lower panels of the figure demonstrate the values of the fit over the data.The experimental data are obtained from Refs.[32, 33]

The squared speed of sound can be calculated using Eqs.(11, 12); the results are shown in Fig.3b.The parameters used to calculate the squared speed of sound for the Pb+Pb collisions were obtained from Ref.[25].The values of the squared speed of sound are approximately between 0.26 and 0.275 for all the collision energies and centralities.The value for massless ideal gas is 1/3, which is the upper limit.The values of the squared speed of sound demonstrate a significantly small decreasing trend, with the collision centrality varying from the central to peripheral collisions at the same collision energy.

Fig.2 (Color online) Energy density and the scaled energy density ε∕T4 in the Au + Au collisions at sNN =7.7-200 GeV, and in the Cu + Cu collisions at sNN =62.4,200 GeV, as a function of centrality.The values are calculated by Eq.(7).The results for the Pb+Pb collisions are obtained from Ref.[25]

Fig.3 (Color online) The pressure and squared speed of sound in the Au + Au collisions at sNN =7.7-200 GeV, and in the Cu + Cu collisions at sNN =62.4,200 GeV, as a function of centrality.The pressure values are calculated using Eq.(10) and those of the squared speed of sound are calculated using Eqs.(11, 12).The pressure results for the Pb+Pb collisions are obtained from Ref.[25]

3.3 Entropy density

Entropy is a particularly important quantity in statistics.The values calculated using Eq.(9) are presented in Fig.4, where the entropy density is scaled byT3.Thes∕T3values for the Pb+Pb collisions atsNN=2760, 5020 GeV, obtained from Ref.[25], are shown in the inset.Similar to the scaled energy densities shown in Fig.2b, the scaled entropy density presents a significantly weak centrality dependence for a given collision energy.No system size effect was observed for the Cu + Cu and Au + Au collisions.Furthermore, the values ofs∕T3generally increased as the collision energy increased.

The thermodynamic relationship was also verified explicitly:

which holds.

As illustrated in Fig.5, the scaledε∕T4ands∕T3are plotted for the most central collisions (0–5% or 0–6% ) and for the most peripheral collisions (60–80% ) from 7.7 to 5020 GeV as a function of ln(sNN).The scaledε∕T4ands∕T3as a function of the centrality demonstrate the same trend.The data points were fit with power-law functions, as indicated by the lines in the figure.The curves were similar and the fitting parameters were approximately the same when the collision centrality was the same.This can be indicated by the massless particle limit; the analytical formulas (Eqs.(13, 15)) ofε∕T4ands∕T3for the massless particles are proportional.Furthermore, the figure demonstrates that the difference in the values between the most central and peripheral collisions is subtle at high collision energies.This is reasonable because similar nuclear reaction environments are created at different centralities at higher collision energies.

Fig.4 (Color online) The scaled entropy density in the Au + Au collisions at sNN =7.7-200 GeV, and in the Cu + Cu collisions at sNN =62.4,200 GeV, as a function of the centrality.The values are calculated using Eq.(9).The results for the Pb+Pb collisions are obtained from Ref.[25] and shown in the insert

Fig.6 (Color online) Particle density in the Au + Au collisions at sNN =7.7-200 GeV, and in the Cu + Cu collisions at sNN =62.4,200 GeV, as a function of centrality.The values are calculated using Eq.(8).The results for the Pb+Pb collisions are obtained from Ref.[25]

Fig.5 (Color online) The scaled ε∕T4 and s∕T3 for the most central(black) collision and most peripheral (red) collision in the Au + Au collisions at sNN =7.7-200 GeV, and in the Cu + Cu collisions at sNN =62.4,200 GeV, as a function of ln(sNN).The lines are fitted with the expressions shown at the bottom of the figure.The parameters used to calculate the thermodynamic quantities for the Pb+Pb collisions are obtained from Ref.[25]

3.4 Particle density

4 Conclusion

Author contributionsAll authors contributed to the study conception and design.Material preparation, data collection and analysis were performed by Wei-Hao Wu, Jun-Qi Tao, Hua Zheng, Wen-Chao Zhang,Xing-Quan Liu, Li-Lin Zhu and Aldo Bonasera.The first draft of the manuscript was written by Wei-Hao Wu and Hua Zheng and all authors commented on previous versions of the manuscript.All authors read and approved the final manuscript.

Data availabilityThe data that support the findings of this study are openly available in Science Data Bank at https:// www.doi.org/ 10.57760/ scien cedb.j00186.00239 and https:// cstr.cn/ 31253.11.scien cedb.j00186.00239.

Declarations

Conflict of interestsThe authors declare that they have no competing interests.