We analyze transverse momentum spectra of KS0 and K*0 at midrapidity in d + Au, Cu + Cu, and p+p collisions at sNN=200 GeV in the formworks of Tsallis statistics and Boltzmann statistics, respectively. Both of them can describe the transverse momentum spectra and extract the thermodynamics parameters of matter evolution in the collisions. The parameters are helpful for us to understand the thermodynamics factors of the particle production.
1. Introduction
High-energy collisions provide many final-state particles, which can be observed in experiments [1–3]. By an investigation of the particle distribution produced in different kinds of collisions, we may speculate the collision process in some ways. Among the properties of the observed particles, the transverse momentum plays a significant role in the colliding experiment. Transverse momentum spectra of hadrons produced in proton and heavy-ion collisions at RHIC and LHC energies have been described successfully through nonextensive statistical mechanics [4]. In our previous work [5], we have systematically investigated the pseudorapidity distributions of charged particles produced in high-energy nucleon-nucleon (pp or pp¯) collisions and high-energy nucleus-nucleus (AA) collisions with different centralities by combining Tsallis statistics with a multisource thermal model.
Recently, Tsallis statistics [6–8] and Boltzmann statistics [9] have been used to analyze the transverse momentum spectra in heavy-ion collisions at high energy. They can both extract the thermodynamics parameters of matter produced in the collisions. What is a parameter difference between the different models? What does the difference mean? In order to concretely understand the thermodynamics properties [10], we implant the Tsallis distribution and Boltzmann distribution in the multisource thermal model. In this paper, we compare the two model descriptions of the transverse momentum spectra of KS0 and K*0 produced in d + Au, Cu + Cu, and p+p collisions at sNN=200 GeV. Two forms of the Tsallis distribution will be taken in Tsallis statistics. One is a conventional choice and the other has been improved to satisfy the thermodynamical consistency [8].
2. Tsallis Distribution and Boltzmann Distribution
In the Tsallis statistics [6–8], the momentum distribution is given by
(1)d3Nd3P=gV2π31+q-1E-μT-q/q-1,
where T is the Tsallis temperature, q is called the “nonequilibrium degree” of the collision system, and g is the degeneracy degree. The parameters p, V, E, and μ are the particle momentum, the system volume, the energy, and the chemical potential, respectively. In terms of the transverse mass mT and the rapidity y, the transverse momentum spectra of the particles can be written as
(2)d2NdPTdy=gVPTmTcoshy2π21+q-1mTcoshy-μT-q/q-1.
When μ=0 and y=0, the distribution function is
(3)d2NdPTdyy=0=gVPTmT2π21+q-1mTT-q/q-1.
Considering a fixed rapidity interval [5], the distribution function should be
(4)dNPTdPT=C∫yminymaxcoshymT1+q-1mTcoshyT-q/q-1,
where the interval of the integral represents the rapidity range observed in the experiment. C is a normalization constant. The function is an improved form to satisfy the thermodynamical consistency [8]. Under the limit q→1, the pT spectrum becomes a conventional form,
(5)dNPTdPT=C∫yminymaxcoshymT1+q′-1mTcoshyT′-1/q′-1.
Next, we review the Boltzmann distribution. For Maxwell’s ideal gas, the momentum distribution function is
(6)dNNd3P=P2m0kT3/22πexp-P22m0kT,
where k is a Boltzmann constant and m0 is the particle mass. If the relativistic effect is taken into account, the momentum distribution function is
(7)dNNd3P=P2m02kTK2m0/kTexp-P2+m02kT,
where K2(m0/kT) is the second-order modified Bessel function. For the isotropic emission in the collision, the transverse momentum distribution is
(8)dNNdPT=C′PTexp-PT2+m02kT=gC′,T,
where C′ is a normalization constant. The two-component distribution of the transverse momentum is
(9)dNNdPT=wgC1′,T1+1-wgC2′,T2,
where w indicates the contribution percentage of the first component.
3. Discussion and Conclusion
Figure 1 shows the transverse momentum spectra of KS0 meson at midrapidity in d + Au, Cu + Cu, and p+p collisions at sNN=200 GeV. The experimental points measured by STAR and PHENIX collaborations [11–15] are shown with different symbols. For Cu + Cu and d + Au, different centrality bins are marked by the different shapes. At the bottom of the figure, we show the p+p data as a reference. The dashed lines and solid lines are numerical results from the thermodynamically consistent Tsallis distribution equation 4 and the conventional Tsallis distribution equation 5, respectively. It is seen that the two forms of Tsallis distribution can both agree with the data. The difference of the numerical results is very small. The parameters T, q, T′, and q′ in the calculations are listed in Table 1 with χ2 per degree of freedom (χ2/dof). Their values do not change obviously due to a PT scaling behavior. In Figure 2, we also give a comparison between the numerical results and the experimental points of K*0 (or K*0¯). The parameters T, q, T′, and q′ are listed in Table 2 with χ2/dof. Similarly, the values have no significant or no regular changes. With Tsallis statistics’ success in dealing with nonequilibrated complex systems in condensed matter research, it is used to study the particle production in high-energy physics. The Tsallis statistics is widely applied in the description of the experimental data in RHIC [12, 16, 17] and LHC [18–20]. It is an advantage that the Tsallis statistics is connected to thermodynamics by the entropy; for example, see [21] for more detailed discussions and its references.
Values of T, q, T′, and q′ taken in Figure 1. T and T′ units are GeV.
Figure
Centrality
T
q
χ2/dof
Figure
Centrality
T′
q′
χ2/dof
Figure 1(a)
MinBias
0.122
1.085
0.852
Figure 1(c)
MinBias
0.120
1.077
0.755
0–20%
0.124
1.084
0.621
0–20%
0.120
1.077
0.782
20–40%
0.124
1.084
0.566
20–40%
0.121
1.077
0.528
40–60%
0.126
1.084
0.589
40–60%
0.122
1.077
0.495
60–88%
0.129
1.084
0.631
60–88%
0.122
1.077
0.670
pp data
0.134
1.083
0.952
pp data
0.122
1.077
1.026
Figure 1(b)
MinBias
0.122
1.086
0.426
Figure 1(d)
MinBias
0.120
1.079
0.383
0–20%
0.124
1.086
0.215
0–20%
0.122
1.079
0.402
20–60%
0.124
1.086
0.270
20–60%
0.122
1.079
0.240
60–94%
0.125
1.085
0.354
60–94%
0.122
1.079
0.319
Values of T, q, T′, and q′ taken in Figure 2. T and T′ units are GeV.
Figure
Centrality
T
q
χ2/dof
Figure
Centrality
T′
q′
χ2/dof
Figure 2(a)
MinBias
0.120
1.084
0.372
Figure 2(c)
MinBias
0.122
1.075
0.391
0–20%
0.120
1.084
0.446
0–20%
0.122
1.075
0.426
20–40%
0.130
1.082
0.515
20–40%
0.124
1.075
0.510
40–60%
0.130
1.082
0.658
40–60%
0.124
1.073
0.624
60–88%
0.140
1.081
0.675
60–88%
0.125
1.073
0.759
pp
data
0.150
1.077
0.788
pp data
0.127
1.073
0.806
Figure 2(b)
MinBias
0.120
1.080
0.229
Figure 2(d)
MinBias
0.118
1.073
0.255
0–20%
0.120
1.080
0.275
0–20%
0.118
1.073
0.279
20–40%
0.122
1.080
0.210
20–40%
0.118
1.073
0.246
40–60%
0.124
1.080
0.462
40–60%
0.122
1.074
0.503
60–94%
0.126
1.080
0.404
60–94%
0.122
1.074
0.429
KS0 transverse momentum spectra in d + Au and Cu + Cu collisions for different centrality bins and p+p collisions at sNN=200 GeV. Experimental data [11–15] are shown with different symbols. The dashed lines and solid lines are the results of 4 and 5, respectively.
K*0 and K*0¯ transverse momentum spectra in d + Au and Cu + Cu collisions for different centrality bins and p+p collisions at sNN=200 GeV. Experimental data [11–15] are shown with different symbols. The dashed lines and solid lines are the results of 4 and 5, respectively.
In Figures 3 and 4, we present a comparison between the two-component Boltzmann distribution and the experimental data measured in d + Au, Cu + Cu, and p+p collisions at sNN=200 GeV. The solid lines denote the results of the two-component Boltzmann distribution equation 9. The two-component Boltzmann distribution also can agree with the experimental points. The dashed lines and the dotted lines denote the contributions of the first component and the second component, respectively. It is seen clearly that the soft and hard interactions behave in the low and high transverse momentum of the identified particles. The parameters T1, T2, and w used in the calculations are given in Table 3 with χ2/dof. The values of T1 are two to four times the values of T or T′. The values of T2 are about twice the values of T1 because of the hard interaction. In 9, the first component is the contribution of soft process and the second component is the contribution of hard process. The distribution in the low transverse momentum region is mainly contributed by the soft processes. The hard processes contribute high transverse momentums in the pT spectra. For the two-component distribution of the Boltzmann distribution, the parameter w is used to denote the contribution of the soft process and 1-w is used to denote the contribution of the hard process.
Values of T1, T2, and w taken in Figures 3 and 4. T1 and T2 units are GeV.
Figure
Centrality
T1
T2
w
χ2/dof
Figure
Centrality
T1
T2
w
χ2/dof
Figure 3(a)
MinBias
0.380
0.900
0.996
1.057
Figure 4(a)
MinBias
0.300
0.600
0.977
1.248
0–20%
0.380
0.900
0.998
0.932
0–20%
0.310
0.580
0.974
0.890
20–40%
0.390
0.900
0.996
0.811
20–40%
0.310
0.580
0.950
0.780
40–60%
0.400
0.910
0.996
0.670
40–60%
0.310
0.580
0.974
0.605
60–88%
0.410
0.920
0.996
0.715
60–88%
0.320
0.580
0.940
0.643
pp data
0.420
0.930
0.995
0.600
pp data
0.300
0.600
0.979
0.580
Figure 3(c)
MinBias
0.450
0.900
0.936
0.858
Figure 4(c)
MinBias
0.280
0.580
0.979
1.150
0–20%
0.470
0.920
0.989
0.705
0–20%
0.280
0.580
0.984
0.700
20–60%
0.470
0.920
0.990
0.426
20–40%
0.280
0.580
0.982
0.570
60–94%
0.470
0.920
0.987
0.395
40–60%
0.280
0.600
0.979
0.352
—
—
—
—
—
60–94%
0.280
0.600
0.980
0.320
The same as Figure 1, but the solid lines are the results of 9. The dashed lines and the dotted lines denote the contributions of the first component and the second component, respectively.
The same as Figure 2, but the solid lines are the results of 9. The dashed lines and the dotted lines denote the contributions of the first component and the second component, respectively.
In summary, we have compared Tsallis statistics and the Boltzmann distribution in the analysis of the transverse momentum spectra of KS0 and K*0 at midrapidity in d + Au, Cu + Cu, and p+p collisions at sNN=200 GeV. The two methods can both describe the distribution of the final-state particles. They have their own advantage and proper scope. The two forms of Tsallis distribution can consistently agree with the experimental points in the low and high pT region. Tsallis statistics is nonextensive statistics [4]. The parameter T is temperature and the parameter q summarily describes all features causing a departure from the Boltzmann-Gibbs statistics. In [6], Var(T)/T2=q-1 directly reflects intrinsic fluctuations of the temperature. However, the Tsallis distribution also emerges from a number of other dynamical mechanisms [22]. The two-component Boltzmann distribution can directly show the contribution of the soft interaction and the hard interaction in the observed spectra by the weight parameter w.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work is supported by the National Natural Science Foundation of China under Grant no. 11247250, no. 11005071, and no. 10975095 and the Shanxi Provincial Natural Science Foundation under Grant no. 2013021006.
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