The Tsallis distribution in proton–proton collisions at $\sqrt{s}$ = 0.9 TeV at the LHC

The Tsallis distribution has gained prominence recently in high-energy physics with very high-quality fits of the transverse momentum distributions made by the STAR [1] and PHENIX [2] collaborations at the Relativistic Heavy Ion Collider and by the ALICE [3] and CMS [4] collaborations at the Large Hadron Collider.

In the literature, there exists more than one version of the Tsallis distribution [5, 6], and we investigate in this paper a version which we consider suited for describing results in high-energy particle physics. Our main guiding criterion will be thermodynamic consistency which has not always been implemented correctly (see, e.g., [7–9]). The explicit form which we use is

$$\begin{eqnarray} &&\frac{{\rm d}N}{{\rm d}p_T {\rm d}y} = gV\frac{p_Tm_T\cosh y}{(2\pi )^2} \left[1+(q-1)\frac{m_T\cosh y -\mu }{T}\right]^{q/(1-q)}, \end{eqnarray} \tag{ 1 }$$

where p_T and m_T are the transverse momentum and mass, respectively, y is the rapidity, T and μ are the temperature and the chemical potential, V is the volume and g is the degeneracy factor. In the limit where the parameter q tends to 1, this reproduces the standard Boltzmann distribution

$$\begin{eqnarray} &&\lim _{q\rightarrow 1}\frac{{\rm d}N}{{\rm d}p_T\, {\rm d}y} = gV\frac{p_Tm_T\cosh y}{(2\pi )^2} \exp \left(-\frac{m_T\cosh y -\mu }{T}\right). \end{eqnarray} \tag{ 2 }$$

In order to distinguish equation (1) from the form used by the STAR, PHENIX, ALICE and CMS collaborations [1–4], the motivation for preferring this form is presented in detail in the rest of this paper. The parameterization given in equation (1) is close (but different) to the one used by the STAR, PHENIX, ALICE and CMS collaborations:

$$\begin{equation} \frac{{\rm d}^2N}{{\rm d}p_{\rm T}\,{\rm d}y} = p_{\rm T} \frac{{\rm d}N}{{\rm d}y} \frac{(n-1)(n-2)}{nC(nC + m_{0} (n-2))} \left( 1 + \frac{m_{\rm T} - m_{0}}{nC} \right)^{-n}, \end{equation} \tag{ 3 }$$

where n, C and m₀ are the fit parameters. The analytic expression used in [1–4] corresponds to identifying

$$\begin{equation} n\rightarrow \frac{q}{q-1} \end{equation} \tag{ 4 }$$

and

$$\begin{equation} nC \rightarrow \frac{T}{q-1}. \end{equation} \tag{ 5 }$$

But differences do not allow for the above identification to be made complete due to an additional factor of the transverse mass on the right-hand side and a shift in the transverse mass. They are close but not the same. In particular, no clear pattern emerges for the values of n and C, while an interesting regularity is obtained for q and T as shown in figures 6 and 7.

In the next section, we review the derivation of the Tsallis distribution by emphasizing the quantum statistical form and the thermodynamic consistency.

In the following, we discuss the Tsallis form of the Fermi–Dirac distribution proposed in [9–13], which uses

$$\begin{equation} n^{{\rm FD}}_T(E) \equiv \frac{1}{1+\exp _q\left(\frac{E-\mu }{T}\right)} , \end{equation} \tag{ 6 }$$

where the function exp _q(x) is defined as

$$\begin{equation} \exp _q(x) \equiv \left\lbrace \begin{array}{@{}ll@{}} \left[1+(q-1)x\right]^{1/(q-1)}& \mathrm{if}\quad x > 0, \\ \left[1+(1-q)x\right]^{1/(1-q)}& \mathrm{if}\quad x \le 0, \\ \end{array} \right. \end{equation} \tag{ 7 }$$

and in the limit where q → 1 reduces to the standard exponential:

$$\begin{eqnarray*} &&\lim _{q\rightarrow 1}\exp _q(x)\rightarrow \exp (x) . \end{eqnarray*}$$

The form given in equation (6) will be referred to as the Tsallis–FD distribution. The Bose–Einstein version (given below) will be referred to as the Tsallis–BE distribution [14].

All forms of the Tsallis distribution introduce a new parameter q. In practice, this parameter is always close to 1, e.g., in the results obtained by the ALICE and CMS collaborations, typical values for the parameter q can be obtained from fits to the transverse momentum distribution for identified charged particles [3] and are in the range from 1.1 to 1.2 (see below). The value of q should thus be considered as never being far from 1, deviating from it by 20% at most. An analysis of the composition of final-state particles leads to a similar result [15] for the parameter q.

The classical limit will be referred to as Tsallis–B distribution (the 'B' stands for the fact that it reduces to the Boltzmann distribution in the limit where q → 1) and is given by the result [5, 6]

$$\begin{equation} n_T^B(E) = \left[ 1 + (q-1) \frac{E-\mu }{T}\right]^{-\frac{1}{q-1}} . \end{equation} \tag{ 8 }$$

Note that we do not use the normalized q-probabilities which have been proposed in [6] since we use here mean occupation numbers which do not need to be normalized. In the limit where q → 1, all distributions coincide with the standard statistical distributions:

$$\begin{eqnarray} \lim _{q\rightarrow 1} n_T^B(E) &=& n^B(E), \end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray} \lim _{q\rightarrow 1} n_T^{{\rm FD}}(E) &=& n^{{\rm FD}}(E), \end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray} \lim _{q\rightarrow 1} n_T^{{\rm BE}}(E) &=& n^{{\rm BE}}(E) . \end{eqnarray} \tag{ 11 }$$

A derivation of the Tsallis distribution, based on the Boltzmann equation, has been given in [16, 17]. Numerically, the difference between equation (6) and the Fermi–Dirac distribution is small, as shown in figure 1 for a value of q = 1.1.

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The Tsallis–B distribution is always larger than the Boltzmann one if q > 1. Taking into account the large p_T results for particle production, we will only consider this possibility in this paper. As a consequence, in order to keep the particle yields the same, the Tsallis distribution always leads to the smaller values of the freeze-out temperature for the same set of particle yields [15].

The first and second laws of thermodynamics lead to the following two differential relations [18]:

$$\begin{eqnarray} &&{\rm d}\epsilon = T\,{\rm d}s + \mu\, {\rm d}n, \end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray} &&{\rm d}P = s\,{\rm d}T + n\,{\rm d}\mu , \end{eqnarray} \tag{ 13 }$$

where = E/V, s = S/V and n = N/V are the energy, entropy and particle densities, respectively. Thermodynamic consistency requires the following relations to be satisfied:

$$\begin{eqnarray} T &=& \left.\frac{\partial \epsilon }{\partial s}\right|_{n}, \end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray} \mu &=&\left.\frac{\partial \epsilon }{\partial n}\right|_{s}, \end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray} n &=& \left.\frac{\partial P}{\partial \mu }\right|_{T}, \end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray} s &=& \left.\frac{\partial P}{\partial T}\right|_{\mu }. \end{eqnarray} \tag{ 17 }$$

The pressure, energy density and entropy density are all given by the corresponding integrals over Tsallis distributions and the derivatives have to reproduce the corresponding physical quantities. For completeness, in the next section, we derive Tsallis thermodynamics using the maximal entropy principle and discuss quantum q-statistics in particular the Bose–Einstein and the Fermi–Dirac distributions by maximizing the entropy of the system for quantum distributions. This extends the derivation of [9]. We will show that the consistency conditions given above are indeed obeyed by the Tsallis–FD distribution.

The entropy in standard statistical mechanics for fermions is given in the large volume limit by

$$\begin{eqnarray} &&S^{{\rm FD}}=-gV\displaystyle \int \frac{{\rm d}^3p}{(2\pi )^3} [ n^{{\rm FD}}\ln n^{{\rm FD}}+(1-n^{{\rm FD}}) \ln (1-n^{{\rm FD}}) ], \end{eqnarray} \tag{ 18 }$$

where g is the degeneracy factor and V is the volume of the system. For simplicity, equation (18) refers to one particle species but can be easily generalized to many. In the limit, where momenta are quantized, the entropy is given by

$$\begin{equation} S^{{\rm FD}}=-g\displaystyle \sum _{i}\left[ n_i\ln n_i+(1-n_i) \ln (1-n_i) \right]. \end{equation} \tag{ 19 }$$

For convenience, we will work with the discrete form in the rest of this section. The large volume limit can be recovered with the standard replacement

$$\begin{equation} \displaystyle \sum _i \rightarrow V\displaystyle \int \frac{{\rm d}^3p}{(2\pi )^3}. \end{equation} \tag{ 20 }$$

The generalization, using the Tsallis prescription, leads to [10–12]

$$\begin{equation} S^{{\rm FD}}_T=-g\displaystyle \sum _{i}\left[ n_{i}^{q}\ln _{q} n_{i}+(1-n_{i})^{q} \ln _{q}(1-n_{i}) \right], \end{equation} \tag{ 21 }$$

where use has been made of the function

$$\begin{equation} \ln _q (x)\equiv \frac{x^{1-q}-1}{1-q} , \end{equation} \tag{ 22 }$$

often referred to as the q-logarithm. It can be easily shown that in the limit where the Tsallis parameter q tends to 1, one has

$$\begin{equation} \lim _{q\rightarrow 1}\ln _q (x) = \ln (x) . \end{equation} \tag{ 23 }$$

In a similar vein, the generalized form of the entropy for bosons is given by

$$\begin{equation} S^{{\rm BE}}_T=- g\displaystyle \sum _i\left[ n_i^q\ln _{q} n_i-(1+n_i)^q \ln _{q}(1+n_i) \right]. \end{equation} \tag{ 24 }$$

In the limit q → 1, equations (21) and (24) reduce to the standard Fermi–Dirac and Bose–Einstein distributions. Furthermore, as we currently explain, the formulation of a variational principle in terms of the above equations allows us to prove the validity of the general relations of thermodynamics. One of the relevant constraints is given by the average number of particles:

$$\begin{equation} \displaystyle \sum _{i} n_{i}^{q} = N . \end{equation} \tag{ 25 }$$

Likewise, the energy of the system gives a constraint

$$\begin{equation} \displaystyle \sum _{i} n_{i}^{q}E_{i} = E . \end{equation} \tag{ 26 }$$

It is necessary to have the power q on the left-hand side as no thermodynamic consistency would be achieved without it. The maximization of the entropic measure under constraints (25) and (26) leads to the variational equation

$$\begin{equation} \frac{\delta }{\delta n_{i}} \bigg[ S^{{\rm FD}}_T+\alpha \bigg(N - \sum _i n_i^q\bigg) +\beta \bigg(E - \sum _i n_i^qE_i\bigg)\bigg] =0, \end{equation} \tag{ 27 }$$

where α and β are the Lagrange multipliers associated, respectively, with the total number of particles and the total energy. Differentiating each expression in equation (27),

$$\begin{equation} \frac{\delta }{\delta n_{i}}\big( S^{{\rm FD}}_T\big) =\frac{q}{q-1}\left[\left( \frac{1-n_{i}}{n_{i}}\right)^{q-1}-1 \right] n_{i}^{q-1}, \end{equation} \tag{ 28 }$$

$$\begin{equation} \frac{\delta }{\delta n_{i}}\left( N - \displaystyle \sum _in_i^q\right) = -qn_i^{q-1} \end{equation} \tag{ 29 }$$

and

$$\begin{equation} \frac{\delta }{\delta n_{i}}\left(E - \displaystyle \sum _in_i^qE_i\right) =-qE_in_i^{q-1}. \end{equation} \tag{ 30 }$$

By substituting equations (28), (29) and (30) into (27), we obtain

$$\begin{equation} qn_{i}^{q-1} \left\lbrace \frac{1}{q-1}\left[ -1+\left(\frac{1-n_{i}}{n_{i}} \right)^{q-1} \right] -\beta E_{i}-\alpha \right\rbrace =0, \end{equation} \tag{ 31 }$$

which can be rewritten as

$$\begin{equation} \frac{1}{q-1}\left[ -1+\left(\frac{1-n_{i}}{n_{i}} \right)^{q-1} \right]=\beta E_{i}+\alpha , \end{equation} \tag{ 32 }$$

and by rearranging equation (32), we obtain

$$\begin{equation} \frac{1-n_{i}}{n_{i}}=\left[ 1+(q-1)(\beta E_{i}+\alpha )\right]^{\frac{1}{q-1}}, \end{equation}$$

which gives the Tsallis–FD form referred to earlier in this paper as [10–12]

$$\begin{eqnarray} n_{i}&=&\frac{1}{\left[ 1+(q-1)(\beta E_{i}+\alpha )\right]^{\frac{1}{q-1}}+1},\nonumber \\ &=&\frac{1}{\left[\exp _{q}(\alpha +\beta E_{i} )\right] +1}. \end{eqnarray} \tag{ 33 }$$

Using a similar approach, one can also determine the Tsallis–BE distribution by starting from the extremum of the entropy subject to the same two conditions:

$$\begin{equation} \frac{\delta }{\delta n_{i}}\left[ S^{{\rm BE}}_T+\alpha \bigg(N - \sum _{i} n_i^{q}\bigg) +\beta \bigg(E - \sum _{i} n_{i}^{q}E_{i}\bigg)\right] =0, \end{equation} \tag{ 34 }$$

which leads to

$$\begin{eqnarray} n_{i}&=&\frac{1}{[ 1+(q-1)(\beta E_{i}+\alpha )]^{\frac{1}{q-1}}-1},\nonumber \\ &=&\frac{1}{[\exp _{q}((E_{i} -\mu )/T)]-1} , \end{eqnarray} \tag{ 35 }$$

where the usual identifications α = −μ/T and β = 1/T have been made.

In order to use the above expressions, it has to be shown that they satisfy the thermodynamic consistency conditions. To show this in detail, we use the first law of thermodynamics [18]

$$\begin{equation} P = \frac{-E + TS + \mu N}{V}, \end{equation} \tag{ 36 }$$

and take the partial derivative with respect to μ in order to check for thermodynamic consistency; it leads to

$$\begin{eqnarray} &&\fl\left.\frac{\partial P}{\partial \mu }\right|_T = \frac{1}{V} \left[-\frac{\partial E}{\partial \mu } +T\frac{\partial S}{\partial \mu } + N + \mu \frac{\partial N}{\partial \mu }\right],\nonumber \\ &&\hphantom{\fl\left.\frac{\partial P}{\partial \mu }\right|_T } = \frac{1}{V}\left[N + \displaystyle \sum _{i}-\frac{T}{q-1}\left(1 + (q-1)\frac{E_{i} -\mu }{T}\right)\frac{\partial n_{i}^{q}}{\partial \mu } +\frac{Tq(1-n_{i})^{q-1}}{q-1}\frac{\partial n_{i}}{\partial \mu } \right], \end{eqnarray} \tag{ 37 }$$

and then, by explicit calculation,

$$\begin{equation} \frac{\partial n_{i}^{q}}{\partial \mu } = \frac{qn_{i}^{q+1}}{T}\left[1+(q-1)\frac{E_{i}-\mu }{T}\right]^{-1+\frac{1}{1-q}}, \end{equation}$$

$$\begin{equation} \frac{\partial n_{i}}{\partial \mu } = \frac{n_{i}^{2}}{T}\left[1+(q-1)\frac{E_{i}-\mu }{T}\right]^{-1+\frac{1}{1-q}} \end{equation}$$

and

$$\begin{equation} \left(1-n_{i}\right)^{q-1} = n_{i}^{q-1}\left[1+\frac{(q-1)(E_{i}-\mu )}{T}\right]. \end{equation}$$

Introducing this into equation (37) yields

$$\begin{equation} \left. \frac{\partial P}{\partial \mu }\right|_T = n, \end{equation} \tag{ 38 }$$

which proves the thermodynamical consistency (16).

We also calculate explicitly the relation in equation (14) that can be rewritten as

$$\begin{eqnarray} \left. \frac{\partial E}{\partial S}\right|_n& = &\frac{\frac{\partial E}{\partial T}\,{\rm d}T + \frac{\partial E}{\partial \mu }\,{\rm d}\mu }{\frac{\partial S}{\partial T}\,{\rm d}T + \frac{\partial S}{\partial \mu }\,{\rm d}\mu },\nonumber \\ & = &\frac{\frac{\partial E}{\partial T} + \frac{\partial E}{\partial \mu }\frac{{\rm d}\mu }{{\rm d}T}}{\frac{\partial S}{\partial T} + \frac{\partial S}{\partial \mu }\frac{{\rm d}\mu }{{\rm d}T}}. \end{eqnarray} \tag{ 39 }$$

Since n is kept fixed, one has the additional constraint

$$\begin{eqnarray*} &&{\rm d}n = \frac{\partial n}{\partial T}\,{\rm d}T + \frac{\partial n}{\partial \mu }\,{\rm d}\mu = 0 \end{eqnarray*}$$

leading to

$$\begin{equation} \frac{{\rm d}\mu }{{\rm d}T} = -\frac{\frac{\partial n}{\partial T}}{\frac{\partial n}{\partial \mu }}. \end{equation} \tag{ 40 }$$

Now, we rewrite (39) and (40) in terms of the following expressions:

$$\begin{equation} \frac{\partial E}{\partial T} = \displaystyle \sum _{i} qE_{i}n_{i}^{q-1}\frac{\partial n_{i}}{\partial T}, \end{equation}$$

$$\begin{equation} \frac{\partial E}{\partial \mu } = \displaystyle \sum _{i} qE_{i}n_{i}^{q-1}\frac{\partial n_{i}}{\partial \mu }, \end{equation}$$

$$\begin{equation} \frac{\partial S}{\partial T} = \displaystyle \sum _{i} q\left[\frac{-n_{i}^{q-1}+(1-n_{i})^{q-1}}{q-1}\right]\frac{\partial n_{i}}{\partial T}, \end{equation}$$

$$\begin{equation} \frac{\partial S}{\partial \mu } = \displaystyle \sum _{i}q\left[\frac{-n_{i}^{q-1}+(1-n_{i})^{q-1}}{q-1}\right]\frac{\partial n_{i}}{\partial \mu }, \end{equation}$$

$$\begin{equation} \frac{\partial n}{\partial T} = \frac{1}{V}\left[\displaystyle \sum _{i} qn_{i}^{q-1}\frac{\partial n_{i}}{\partial T}\right] \end{equation}$$

and

$$\begin{equation} \frac{\partial n}{\partial \mu } = \frac{1}{V}\left[\displaystyle \sum _{i} qn_{i}^{q-1}\frac{\partial n_{i}}{\partial \mu }\right]. \end{equation}$$

By introducing the above relations into equation (39), the numerator of the equation becomes

$$\begin{eqnarray} &&\fl \frac{\partial E}{\partial T} + \frac{\partial E}{\partial \mu }\frac{{\rm d}\mu }{{\rm d}T} = \sum _{i} qE_{i}n_{i}^{q-1}\frac{\partial n_i}{\partial T}-\frac{ \sum _{i,j} q^{2}E_{j}(n_{i}n_{j})^{q-1}\frac{\partial n_{j}}{\partial \mu }\frac{\partial n_{i}}{\partial T}}{ \sum _{j} qn_{j}^{q-1}\frac{\partial n_{j}}{\partial \mu }},\nonumber \\ &&\hphantom{\fl \frac{\partial E}{\partial T} + \frac{\partial E}{\partial \mu }\frac{{\rm d}\mu }{{\rm d}T} }= \frac{ \sum _{i,j} qE_{i}(n_{i}n_{j})^{q-1}C_{ij}}{ \sum _{j} n_{j}^{q-1}\frac{\partial n_{j}}{\partial \mu }}, \end{eqnarray} \tag{ 41 }$$

where the abbreviation

$$\begin{equation} C_{ij}\equiv (n_{i}n_{j})^{q-1}\left[\frac{\partial n_{i}}{\partial T}\frac{\partial n_{j}}{\partial \mu }- \frac{\partial n_{j}}{\partial T}\frac{\partial n_{i}}{\partial \mu }\right] \end{equation} \tag{ 42 }$$

has been introduced. One can rewrite the denominator of equation (39) as

$$\begin{eqnarray} \frac{\partial S}{\partial T} + \frac{\partial S}{\partial \mu }\frac{{\rm d}\mu }{{\rm d}T} & = \frac{ q\sum _{i,j}\big[{{-}n_{i}^{q-1}+(1-n_{i})^{q-1}}\big]n_{j}^{q-1} C_{i,j}}{(q-1) \sum _{j} n_{j}^{q-1}\frac{\partial n_{j}}{\partial \mu }},\nonumber \\ & = \frac{ q\sum _{i,j}(E_{i}-\mu )(n_{i}n_{j})^{q-1} C_{i,j}}{T \sum _{j} n_{j}^{q-1}\frac{\partial n_{j}}{\partial \mu }}, \end{eqnarray} \tag{ 43 }$$

where

$$\begin{equation} \frac{-n_{i}^{q-1}+(1-n_{i})^{q-1}}{q-1} = \frac{(E_{i}-\mu )}{T}n_{i}^{q-1}. \end{equation}$$

Hence, by substituting equations (41) and (43) into (39), we find

$$\begin{equation} \left. \frac{\partial E}{\partial S}\right|_n = T\frac{ \sum _{i,j}E_{i}C_{ij}}{ \sum _{i,j}(E_{i}-\mu )C_{ij}}. \end{equation} \tag{ 44 }$$

Since ∑_{i, j}C_ij = 0, this finally leads to the desired result

$$\begin{equation} \left.\frac{\partial E}{\partial S}\right|_n = T. \end{equation} \tag{ 45 }$$

Hence, the thermodynamic consistency is satisfied.

It has thus been shown that the definitions of temperature and pressure within the Tsallis formalism for non-extensive thermostatistics lead to expressions which satisfy consistency with the first and second laws of thermodynamics.

The total number of particles is given by the integral version of equation (25):

$$\begin{equation} N = gV\displaystyle \int \frac{{\rm d}^3p}{(2\pi )^3} \left[1+(q-1)\frac{E-\mu }{T}\right]^{q/(1-q)} . \end{equation} \tag{ 46 }$$

The corresponding (invariant) momentum distribution deduced from the above equation is given by

$$\begin{equation} E\frac{{\rm d}N}{{\rm d}^3p} = gVE\frac{1}{(2\pi )^3} \left[1+(q-1)\frac{E-\mu }{T}\right]^{q/(1-q)}, \end{equation} \tag{ 47 }$$

which, in terms of the rapidity and transverse mass variables, becomes

$$\begin{eqnarray} &&\frac{{\rm d}N}{{\rm d}y\, p_T{\rm d}p_T} = gV\frac{m_T\cosh y}{(2\pi )^2}\left[1+(q-1)\frac{m_T\cosh y -\mu }{T}\right]^{q/(1-q)}. \end{eqnarray} \tag{ 48 }$$

At mid-rapidity y = 0 and for zero chemical potential, this reduces to the following expression:

$$\begin{equation} \left.\frac{{\rm d}N}{{\rm d}p_T {\rm d}y}\right|_{y=0} = gV\frac{p_Tm_T}{(2\pi )^2} \left[1+(q-1)\frac{m_T}{T}\right]^{q/(1-q)}. \end{equation} \tag{ 49 }$$

Fits using the above expressions based on the Tsallis–B distribution to experimental measurements published by the CMS collaboration [4] are shown in figures 2–4, which are comparable with those shown by the CMS collaboration. Fits to the results obtained by the ALICE collaboration [3] are shown in figure 5 for $\pi ^-, K^-\ {\rm and }\ \bar{p}$. The resulting parameters are considerably different from those obtained from equation (3) and are tabulated in table 1. The most striking feature is that the values of the parameter q are fairly stable in the range from 1.1 to 1.2 for all particles considered at 0.9 TeV. The temperature T cannot be determined very accurately for all hadrons but they are consistent with a value around 70 MeV.

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Table 1. Fitted values of the T and q parameters for strange particles measured by the ALICE [3] and CMS collaborations [4] using the Tsallis–B form for the momentum distribution.

Particle	q	T (GeV)
π+	1.154 ± 0.036	0.0682 ± 0.0026
π−	1.146 ± 0.036	0.0704 ± 0.0027
K+	1.158 ± 0.142	0.0690 ± 0.0223
K−	1.157 ± 0.139	0.0681 ± 0.0217
K0S	1.134 ± 0.079	0.0923 ± 0.0139
p	1.107 ± 0.147	0.0730 ± 0.0425
$\bar{p}$	1.106 ± 0.158	0.0764 ± 0.0464
Λ	1.114 ± 0.047	0.0698 ± 0.0148
Ξ−	1.110 ± 0.218	0.0440 ± 0.0752

For clarity, we show these results also in figure 6 for the values of the parameter q and in figure 7 for the values of the Tsallis parameter T. The striking feature is that the values of q are consistently between 1.1 and 1.2 for all species of hadrons.

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In this paper, we have presented a detailed derivation of the quantum form of the Tsallis distribution and considered in detail the thermodynamic consistency of the resulting distribution. It was emphasized that an additional power of q is needed to achieve consistency with the laws of thermodynamics [9]. The resulting distribution, called Tsallis–B, was compared with recent measurements from the ALICE [3] and CMS [4] collaborations and good agreement was obtained. The resulting parameter q which is a measure for the deviation from a standard Boltzmann distribution was found to be around 1.11. The resulting values of the temperature are also consistent within the errors and lead to a value of around 70 MeV.