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Zheng, H.; Zhu, Lilin

doi:https://doi.org/10.1155/2016/9632126

Advances in High Energy Physics

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Research Article | Open Access

Volume 2016 | Article ID 9632126 | https://doi.org/10.1155/2016/9632126

Comparing the Tsallis Distribution with and without Thermodynamical Description in Collisions

H. Zheng¹and Lilin Zhu²

Academic Editor: Adrian Buzatu

Received08 Dec 2015

Revised11 Mar 2016

Accepted24 Mar 2016

Published10 Apr 2016

Abstract

We compare two types of Tsallis distribution, that is, with and without thermodynamical description, using the experimental data from the STAR, PHENIX, ALICE, and CMS Collaborations on the rapidity and energy dependence of the transverse momentum spectra in collisions. Both of them can fit the particle spectra well. We show that the Tsallis distribution with thermodynamical description gives lower temperatures than the ones without it. The extra factor (transverse mass) in the Tsallis distribution with thermodynamical description plays an important role in the discrepancies between the two types of Tsallis distribution. But for the heavy particles, the choice to use or (transverse energy) in the Tsallis distribution becomes more crucial.

1. Introduction

The particle spectrum is a basic quantity directly measured in the experiments and it can reveal the information of particle production mechanism in heavy-ion collisions. Many physicists have devoted themselves to studying the particle spectra produced in the heavy-ion collisions using thermodynamical approaches, phenomenological methods, transport models, and so forth [1–22]. Recently, the Tsallis distribution, which was first proposed about twenty-seven years ago as a generalization of the Boltzmann-Gibbs distribution [23], has attracted many theorists’ and experimentalists’ attention in high energy collisions [5–9, 11–17, 24–34]. The excellent ability to fit the spectra of identified hadrons and charged particles in a large range of up to 200 GeV/c, which covers 15 orders of magnitude, is quite impressive. This spectacular result was first shown by Wong et al. [14–16]. In [21, 22], we have shown that Tsallis distribution can fit almost all the particle spectra produced in , , and at RHIC and LHC. From the phenomenological view, there may be real physics behind the prominently phenomenological work, for example, Regge trajectory for particle classification [35]. We also note that there are different versions of Tsallis distribution in the literature and we classify them as Type A, Type B, and Type C to clarify the comparison in [21]. Type A Tsallis distribution is obtained without resorting to thermodynamical description, but it has been adopted to analyze the particle spectra by STAR [24] and PHENIX [25] Collaborations at RHIC and ALICE [26–28] and CMS [29] Collaborations at LHC. In [21], we applied it to do the systematic analysis of identified particle spectra in collisions at RHIC and LHC and proposed a cascade particle production mechanism. On the other hand, Type B Tsallis distribution is derived by taking into account the thermodynamical consistency and is widely used by Cleymans and his collaborators to study the particle spectra in high energy collisions [8–10]. It is also used by other authors for nucleus-nucleus interactions [11]. Type A and Type B are the most popular Tsallis distributions in the literature but they give quite different temperatures while fitting the same particle spectra; for example, for pion, Type A gives GeV while Type B gives GeV. In this paper, we would like to systematically address the question regarding the discrepancies of the temperatures for the two types of Tsallis distribution, by using particle spectra in collisions. The data produced in collisions with different ranges and different rapidity cuts are collected from the experimental collaborations at RHIC and LHC [25–27, 30, 36–42].

The paper is organized as follows. In Section 2, we introduce the two types of Tsallis distribution, without and with thermodynamical description, in our comparison. In order to make our discussion clear, we also introduce another three transient distributions which are very similar to Type A and Type B Tsallis distributions. In Section 3, the results of particle spectra from the different distributions in collisions and the comparisons are shown. A brief conclusion is given in Section 4.

2. Tsallis Distributions

Type A Tsallis distribution has been widely adopted by STAR [24] and PHENIX [25] Collaborations at RHIC and ALICE [26–28] and CMS [29] Collaborations at LHC:where is the transverse mass, , , and are fitting parameters, and was used as a fitting parameter in [24], but it represents the rest mass of the particle studied in [25–29]. When , we can ignore in the last term in (1) and obtain . This result is well known because high energy particles come from hard scattering and they follow a power law distribution with . When which is the nonrelativistic limit, we obtain and , that is, a thermal distribution. The parameter in (1) plays the same role as temperature . In [21, 22], we have obtained the simpler form of (1):where the transverse energy . , , and are free fitting parameters in (2). We note that it has been used by CMS Collaboration [31, 32, 43] and by Wong et al. in their recent paper [16]. The STAR Collaboration also applied a formula which is very close to (2) [44]. We adopt (2) in the following study.

In the framework of Tsallis statistics, the distribution function isTaking into account the self-consistent thermodynamical description, one has to use the effective distribution . Therefore, the Tsallis distribution is obtained [8, 9, 11, 12]:where is the degeneracy of the particle state, is the volume, is the rapidity, is the chemical potential, is the temperature, and is the entropic factor, which measures the nonadditivity of the entropy. We dubbed it as the Type B Tsallis distribution [21]. In (4), there are four parameters, namely, , , , and . was assumed to be 0 in [8, 9, 11] which is a reasonable assumption because the energy is high enough and the chemical potential is much smaller than the temperature. In the midrapidity region, (4) is reduced toIt becomes very similar to (2), but there are some differences; for example, replaces in the bracket and there is an extra term in front of the bracket. It should be pointed out that there is no direct match between and in (2) and (5). But we could find a connection between and in the limit at large . When , from (5), we can obtainRecalling that when from (1), the relation between and isAnother treatment to find the relation between and can be found in [10].

For the other Tsallis distributions in the literature, we refer them to [21, 22]. We noted that Type A and Type B Tsallis distributions can reproduce the particle spectra in collisions very well, but Type B gives lower temperatures than the ones given by Type A. In this paper, we would like to address this discrepancies between the two types of Tsallis distribution. To make our discussion clear, another three transient distributions are used to bridge Type A and Type B distributions. In [45], a Tsallis-like distribution is obtained in the framework of nonextensive statistics for the particle invariant yield at midrapidity:where , , and are fitting parameters. Comparing (8) and (5), the only difference is the power of the distribution function, that is, for (5) and for (8). We also introduce another two forms of distribution. One iswhere we neglect the term outside of the bracket in (5) and the constants are absorbed into the parameter . The other one isSimilar to (9), we neglect the term in front of the bracket in (8) but keep the rest. Using the relation equation (7), we find that (10) becomes (2) in the limit of massless particle.

Let us put the four distributions in order of (5), (8), (10), and (2) and (5), (9), (10), and (2); one can see that any adjacent distributions have only one term different. We successfully bridge Type A and Type B Tsallis distributions and are able to conduct our investigation. We need to point out that even though we use the same symbols for the parameters in all five distributions, they may have different values when fitting the experimental data. In the next section, we will systematically apply the five distributions to the particle spectra in collisions, similar to our previous work [21]. But we update some experimental data which have larger ranges and will focus on the temperature differences among the five distributions.

3. Results

We fit the particle spectra with different ranges and different rapidity cuts from collisions at , , , , and GeV with the five distributions discussed in the last section. The fitting process is the same as that in [21, 22]. Compared with [21], the identified particle spectra data at and GeV have been updated.

In this work, we are interested in the differences of the parameter from the five distributions. As we argued in [21, 22], is one free fitting parameter and can be different for different particles even though they are produced in the same colliding system. In order to distinguish the parameters and in the distributions and make our discussion clear, we assign , , , and to (5), (8), (9), and (10), respectively, while we assign and for (2). All the five distributions can fit the particle transverse momentum spectra very well. The values of (), (), , and corresponding for , , , and charged particles are shown in Tables 1 and 2. Furthermore, the errors of , and are also provided in the tables. Here, we only show the fitting results with the five distributions for four cases: (1) at GeV, (2) at GeV, (3) at GeV, and (4) at GeV.

In Figures 1 and 2, we show the fitting results for at GeV from STAR Collaboration and GeV from ALICE Collaboration using the five distributions: (5) (solid line), (8) (dashed line), (9) (dotted line), (10) (dash-dotted line), and (2) (bold dash-dotted line), respectively. Since the lines are so close to each other, they are almost indistinguishable in the figures. To visualize the fitting quality better, we also plot the ratios of experimental data and fitting results at the bottom of the figures. We can see that the five distributions can describe the experimentally measured transverse momentum spectra very well. The errors are within 20%.

To show the fitting results other than pions, we select at GeV and at GeV at LHC in Figures 3 and 4, respectively. For the five distributions, we can not distinguish them at all in the two cases. Similar to pions, the fitting qualities are also very good.

As we can see from Tables 1 and 2, the values of parameter are different. It is the main purpose of this work to target what causes the temperature discrepancies among different Tsallis distributions, with and without thermodynamical description. In order to avoid confusion, we emphasis the meaning of each . Using the Tsallis distribution classification in [21], refers to Type B Tsallis distribution and refers to Type A Tsallis distribution. , , and are from the transient distributions to bridge Type A and Type B Tsallis distributions. To have a clear picture for the parameter from the five distributions, we plot versus the colliding energy for , , and , as shown in Figure 5. We can clearly see that Type B Tsallis distribution gives lower than Type A Tsallis distribution, as we mentioned in [21]. For the light particles, that is, pions (Figure 5(a)), and which are from the distributions ((5) and (8)) with extra term are lower than , , and which are from the distributions ((9), (10), and (2)) without it. As we see that is larger than , since (5) and (8) are similar except the power in the distributions, we conclude that the power in (5) causes larger . This can be verified by comparing with . With the same argument, since is smaller than , in (10) causes smaller . To see the effects of and in the distribution, we can compare with , which are similar. The effects of the power causing larger and causing smaller cancel each other. For the heavier particles, that is, kaons and protons (Figures 5(b) and 5(c)), the effect of in (9) wins and is smaller than . We note that the effect of the extra term is crucial for the temperature difference between Type A and Type B Tsallis distributions, especially for the light particles, while for the heavier particles, the effect of the choice of or in the Tsallis distribution becomes more important as we can see that is larger than the other for kaons and protons in Figure 5.

(a)

(b)

(c)

4. Summary

In this paper, we have presented a detailed investigation of two types of Tsallis distribution, with and without the thermodynamical description, by the spectra measured from STAR and PHENIX Collaborations at RHIC and ALICE and CMS Collaborations at LHC. The power in the Tsallis distribution with thermodynamical description is responsible for the thermodynamical consistency. To show a clear and complete comparison, another three transient distributions to bridge the two types of Tsallis distribution are also given. Good agreements are obtained, but they give different temperatures. Agreed with our previous work [21], the Tsallis distribution with thermodynamical description gives lower than the ones given by the distribution without it. The extra term in the Tsallis distribution with thermodynamical description is responsible for the discrepancies of the temperatures. But for the heavier particles, the effect of the choice of or in the Tsallis distribution beats the effect of the extra term . The data for collisions at 8 and 13 TeV are expected to provide further support for the conclusions presented here.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

This work is supported by the NSFC of China under Grant no. 11205106.

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Copyright

Copyright © 2016 H. Zheng and Lilin Zhu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP³.

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