Research ArticleTransverse Momentum Spectra of 1198700

119878and 119870lowast0 at Midrapidity in

119889 + Au Cu + Cu and 119901 + 119901 Collisions at radic119904119873119873= 200GeV

Bao-Chun Li Guo-Xing Zhang and Yuan-Yuan Guo

Department of Physics Shanxi University Taiyuan Shanxi 030006 China

Correspondence should be addressed to Bao-Chun Li libc2010163com

Received 30 October 2014 Revised 31 December 2014 Accepted 31 December 2014

Academic Editor Bhartendu K Singh

Copyright copy 2015 Bao-Chun Li et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited Thepublication of this article was funded by SCOAP3

We analyze transversemomentum spectra of1198700119878and119870lowast0 at midrapidity in 119889 +Au Cu +Cu and 119901+119901 collisions atradic119904

119873119873= 200GeV

in the formworks of Tsallis statistics and Boltzmann statistics respectively Both of them can describe the transverse momentumspectra and extract the thermodynamics parameters of matter evolution in the collisions The parameters are helpful for us tounderstand the thermodynamics factors of the particle production

1 Introduction

High-energy collisions provide many final-state particleswhich can be observed in experiments [1ndash3] By an investi-gation of the particle distribution produced in different kindsof collisions we may speculate the collision process in someways Among the properties of the observed particles thetransverse momentum plays a significant role in the collid-ing experiment Transverse momentum spectra of hadronsproduced in proton and heavy-ion collisions at RHIC andLHC energies have been described successfully throughnonextensive statistical mechanics [4] In our previous work[5] we have systematically investigated the pseudorapiditydistributions of charged particles produced in high-energynucleon-nucleon (119901119901 or 119901119901) collisions and high-energynucleus-nucleus (AA) collisions with different centralitiesby combining Tsallis statistics with a multisource thermalmodel

Recently Tsallis statistics [6ndash8] and Boltzmann statistics[9] have been used to analyze the transverse momentumspectra in heavy-ion collisions at high energy They can bothextract the thermodynamics parameters of matter producedin the collisions What is a parameter difference between thedifferent models What does the difference mean In orderto concretely understand the thermodynamics properties

[10] we implant the Tsallis distribution and Boltzmanndistribution in the multisource thermal model In this paperwe compare the two model descriptions of the transversemomentum spectra of 1198700

119878and 119870

lowast0 produced in 119889 + Au Cu+ Cu and 119901 + 119901 collisions atradic119904

119873119873= 200GeV Two forms of

the Tsallis distribution will be taken in Tsallis statistics Oneis a conventional choice and the other has been improved tosatisfy the thermodynamical consistency [8]

2 Tsallis Distribution andBoltzmann Distribution

In the Tsallis statistics [6ndash8] the momentum distribution isgiven by

1198893119873

1198893119875=

119892119881

(2120587)3[1 + (119902 minus 1)

119864 minus 120583

119879]

minus119902(119902minus1)

(1)

where119879 is the Tsallis temperature 119902 is called the ldquononequilib-rium degreerdquo of the collision system and 119892 is the degeneracydegree The parameters 119901 119881 119864 and 120583 are the particlemomentum the system volume the energy and the chemical

Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2015 Article ID 684950 8 pageshttpdxdoiorg1011552015684950

2 Advances in High Energy Physics

10minus17

10minus15

10minus13

10minus11

10minus9

10minus7

10minus5

10minus3

10minus1

101

2 4 6 8 10 12

K0S rarr 12058701205870

d + Au 200GeV

PT (GeVc)

d2N(2120587PTdPTdy)(G

eVc)minus2

(a)

2 4 6 8 10 12

10minus17

10minus15

10minus13

10minus11

10minus9

10minus7

10minus5

10minus3

10minus1

101

PT (GeVc)

K0S rarr 12058701205870

Cu + Cu 200GeV

d2N(2120587PTdPTdy)(G

eVc)minus2

(b)

2 4 6 8 10 12

10minus5

10minus3

10minus1

10minus17

10minus19

10minus15

10minus13

10minus11

10minus9

10minus7

PT (GeVc)

MinBias times 102

0ndash20 times 10120ndash40

40ndash60 times 10minus1

times 10minus3

d2N(2120587PTdPTdy)(G

eVc)minus2

60ndash88 times 10minus2

pp data

(c)

2 4 6 8 10 12

10minus5

10minus3

10minus1

10minus17

10minus19

10minus15

10minus13

10minus11

10minus9

10minus7

PT (GeVc)

MinBias times 102

times 10minus20ndash20 times 220ndash60 times 02

60ndash94 times 005

d2N(2120587PTdPTdy)(G

eVc)minus2

pp data

(d)

Figure 1 1198700119878transverse momentum spectra in 119889 + Au and Cu + Cu collisions for different centrality bins and 119901 + 119901 collisions at radic119904

119873119873=

200GeV Experimental data [11ndash15] are shownwith different symbolsThedashed lines and solid lines are the results of (4) and (5) respectively

potential respectively In terms of the transverse mass 119898119879

and the rapidity 119910 the transverse momentum spectra of theparticles can be written as

1198892119873

119889119875119879119889119910

= 119892119881119875119879119898119879cosh119910

(2120587)2

[1 + (119902 minus 1)119898119879cosh119910 minus 120583

119879]

minus119902(119902minus1)

(2)

When 120583 = 0 and 119910 = 0 the distribution function is

1198892119873

119889119875119879119889119910

100381610038161003816100381610038161003816100381610038161003816119910=0

= 119892119881119875119879119898119879

(2120587)2[1 + (119902 minus 1)

119898119879

119879]

minus119902(119902minus1)

(3)

Considering a fixed rapidity interval [5] the distributionfunction should be

119889119873

119875119879119889119875119879

= 119862int

119910max

119910mincosh119910119898

119879[1 + (119902 minus 1)

119898119879cosh119910119879

]

minus119902(119902minus1)

(4)

Advances in High Energy Physics 3

10minus5

10minus3

10minus1

10minus13

10minus11

10minus9

10minus7

101d + Au 200GeV

Klowast0 rarr K+120587minus

Klowast0 rarr Kminus120587+

(Klowast0 + Klowast0)2

1 2 3 4 5 6 7 8PT (GeVc)

d2N(2120587PTdPTdy)(G

eVc)minus2

(a)

10minus5

10minus3

10minus1

10minus13

10minus11

10minus9

10minus7

101Cu + Cu 200GeV

Klowast0 rarr K+120587minus

Klowast0 rarr Kminus120587+

(Klowast0 + Klowast0)2

1 2 3 4 5 6 7 8PT (GeVc)

d2N(2120587PTdPTdy)(G

eVc)minus2

(b)

1 2 3 4 5 6 7 8

10minus5

10minus3

10minus1

10minus13

10minus11

10minus9

10minus7

101

PT (GeVc)

MinBias times 2 times 103

0ndash20 times 102

20ndash40 times 2 times 101

40ndash60 times 2 times 100

60ndash88 times 5 times 10minus1

times 2 times 10minus1

d2N(2120587PTdPTdy)(G

eVc)minus2

pp data

(c)

10minus5

10minus3

10minus1

10minus13

10minus11

10minus9

10minus7

101

1 2 3 4 5 6 7 8PT (GeVc)

MinBias times 800ndash20 times 4

40ndash60 times 05

20ndash40 times 160ndash94 times 03

times 0008

d2N(2120587PTdPTdy)(G

eVc)minus2

pp data

(d)

Figure 2 119870lowast0 and 119870lowast0 transverse momentum spectra in 119889 + Au and Cu + Cu collisions for different centrality bins and 119901 + 119901 collisions atradic119904119873119873

= 200GeV Experimental data [11ndash15] are shown with different symbols The dashed lines and solid lines are the results of (4) and (5)respectively

where the interval of the integral represents the rapidityrange observed in the experiment 119862 is a normalizationconstant The function is an improved form to satisfy thethermodynamical consistency [8] Under the limit 119902 rarr 1the 119901119879spectrum becomes a conventional form

119889119873

119875119879119889119875119879

= 119862int

119910max

119910mincosh119910119898

119879[1 + (119902

1015840minus 1)

119898119879cosh1199101198791015840

]

minus1(1199021015840minus1)

(5)

Next we review the Boltzmann distribution For Maxwellrsquosideal gas the momentum distribution function is

119889119873

1198731198893119875=

1198752

(1198980119896119879)32

radic2

120587exp(minus 119875

2

21198980119896119879

) (6)

where 119896 is a Boltzmann constant and1198980is the particle mass

If the relativistic effect is taken into account the momentumdistribution function is

119889119873

1198731198893119875=

1198752

1198982

01198961198791198702(1198980119896119879)

exp(minus

radic1198752 + 1198982

0

119896119879) (7)

4 Advances in High Energy Physics

Table 1 Values of 119879 119902 1198791015840 and 1199021015840 taken in Figure 1 119879 and 119879

1015840 units are GeV

Figure Centrality 119879 119902 1205942dof Figure Centrality 119879

10158401199021015840

1205942dof

Figure 1(a)

MinBias 0122 1085 0852

Figure 1(c)

MinBias 0120 1077 07550ndash20 0124 1084 0621 0ndash20 0120 1077 078220ndash40 0124 1084 0566 20ndash40 0121 1077 052840ndash60 0126 1084 0589 40ndash60 0122 1077 049560ndash88 0129 1084 0631 60ndash88 0122 1077 0670pp data 0134 1083 0952 pp data 0122 1077 1026

Figure 1(b)

MinBias 0122 1086 0426

Figure 1(d)

MinBias 0120 1079 03830ndash20 0124 1086 0215 0ndash20 0122 1079 040220ndash60 0124 1086 0270 20ndash60 0122 1079 024060ndash94 0125 1085 0354 60ndash94 0122 1079 0319

Table 2 Values of 119879 119902 1198791015840 and 1199021015840 taken in Figure 2 119879 and 119879

1015840 units are GeV

Figure Centrality 119879 119902 1205942dof Figure Centrality 119879

10158401199021015840

1205942dof

Figure 2(a)

MinBias 0120 1084 0372

Figure 2(c)

MinBias 0122 1075 03910ndash20 0120 1084 0446 0ndash20 0122 1075 042620ndash40 0130 1082 0515 20ndash40 0124 1075 051040ndash60 0130 1082 0658 40ndash60 0124 1073 062460ndash88 0140 1081 0675 60ndash88 0125 1073 0759pp data 0150 1077 0788 pp data 0127 1073 0806

Figure 2(b)

MinBias 0120 1080 0229

Figure 2(d)

MinBias 0118 1073 02550ndash20 0120 1080 0275 0ndash20 0118 1073 027920ndash40 0122 1080 0210 20ndash40 0118 1073 024640ndash60 0124 1080 0462 40ndash60 0122 1074 050360ndash94 0126 1080 0404 60ndash94 0122 1074 0429

where 1198702(1198980119896119879) is the second-order modified Bessel func-

tion For the isotropic emission in the collision the transversemomentum distribution is

119889119873

119873119889119875119879

= 1198621015840119875119879exp(minus

radic1198752

119879+ 1198982

0

119896119879) = 119892 (119862

1015840 119879) (8)

where 1198621015840 is a normalization constant The two-component

distribution of the transverse momentum is119889119873

119873119889119875119879

= 119908119892 (1198621015840

1 1198791) + (1 minus 119908) 119892 (119862

1015840

2 1198792) (9)

where 119908 indicates the contribution percentage of the firstcomponent

3 Discussion and Conclusion

Figure 1 shows the transverse momentum spectra of 1198700

119878

meson at midrapidity in 119889 +Au Cu + Cu and 119901+119901 collisionsat radic119904119873119873

= 200GeV The experimental points measured bySTAR and PHENIX collaborations [11ndash15] are shown withdifferent symbols ForCu+Cu and119889+Au different centralitybins are marked by the different shapes At the bottom ofthe figure we show the 119901 + 119901 data as a reference Thedashed 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 distributioncan both agree with the data The difference of the numericalresults is very small The parameters 119879 119902 1198791015840 and 119902

1015840 in thecalculations are listed inTable 1 with1205942 per degree of freedom(1205942dof) Their values do not change obviously due to a119875119879scaling behavior In Figure 2 we also give a comparison

between the numerical results and the experimental pointsof 119870lowast0 (or 119870lowast0) The parameters 119879 119902 1198791015840 and 119902

1015840 are listed inTable 2 with 120594

2dof Similarly the values have no significantor no regular changes With Tsallis statisticsrsquo success indealing with nonequilibrated complex systems in condensedmatter research it is used to study the particle production inhigh-energy physics The Tsallis statistics is widely applied inthe description of the experimental data in RHIC [12 16 17]and LHC [18ndash20] It is an advantage that the Tsallis statisticsis connected to thermodynamics by the entropy for examplesee [21] for more detailed discussions and its references

In Figures 3 and 4 we present a comparison between thetwo-component Boltzmann distribution and the experimen-tal data measured in 119889 + Au Cu + Cu and 119901 + 119901 collisions atradic119904119873119873

= 200GeV The solid lines denote the results of thetwo-component Boltzmann distribution equation (9) Thetwo-component Boltzmann distribution also can agree withthe experimental points The dashed lines and the dotted

Advances in High Energy Physics 5

MinBias times 102

0ndash20 times 10120ndash40

40ndash60 times 10minus1

times 10minus3pp data

2 4 6 8 10 12

10minus5

10minus3

10minus1

10minus13

10minus15

10minus17

10minus19

10minus11

10minus9

10minus7

101

PT (GeVc)

d + Au 200GeVK0S rarr 120587

01205870

d2N(2120587PTdPTdy)(G

eVc)minus2

60ndash88 times 10minus2

(a)

2 4 6 8 10 12

10minus5

10minus3

10minus1

10minus13

10minus15

10minus17

10minus19

10minus11

10minus9

10minus7

101

PT (GeVc)

d2N(2120587PTdPTdy)(G

eVc)minus2

(b)

2 4 6 8 10 12

10minus5

10minus3

10minus1

10minus13

10minus15

10minus17

10minus19

10minus11

10minus9

10minus7

101

PT (GeVc)

K0S rarr 12058701205870

Cu + Cu 200GeV

MinBias times 102

times 10minus20ndash20 times 220ndash60 times 02

60ndash94 times 005

d2N(2120587PTdPTdy)(G

eVc)minus2

pp data

(c)

2 4 6 8 10 12

10minus5

10minus3

10minus1

10minus13

10minus15

10minus17

10minus19

10minus11

10minus9

10minus7

101

PT (GeVc)

d2N(2120587PTdPTdy)(G

eVc)minus2

(d)

Figure 3 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 thefirst component and the second component respectively

lines denote the contributions of the first component and thesecond component respectively It is seen clearly that the softand hard interactions behave in the low and high transversemomentum of the identified particles The parameters 119879

1

1198792 and 119908 used in the calculations are given in Table 3 with

1205942dof The values of 119879

1are two to four times the values of

119879 or 1198791015840 The values of 1198792are about twice the values of 119879

1

because of the hard interaction In (9) the first componentis the contribution of soft process and the second component

is the contribution of hard processThedistribution in the lowtransverse momentum region is mainly contributed by thesoft processes The hard processes contribute high transversemomentums in the 119901

119879spectra For the two-component

distribution of the Boltzmann distribution the parameter 119908is used to denote the contribution of the soft process and 1minus 119908

is used to denote the contribution of the hard processIn summary we have compared Tsallis statistics and

the Boltzmann distribution in the analysis of the transverse

6 Advances in High Energy Physics

10minus5

10minus3

10minus1

10minus13

10minus11

10minus9

10minus7

101

1 2 3 4 5 6 7 8

MinBias times 2 times 103

0ndash20 times 102

20ndash40 times 2 times 101

40ndash60 times 2 times 100

60ndash88 times 5 times 10minus1

times 2 times 10minus1

PT (GeVc)

d + Au 200GeVKlowast0 rarr K+120587minus

Klowast0 rarr Kminus120587+

(Klowast0 + Klowast0)2

d2N(2120587PTdPTdy)(G

eVc)minus2

pp data

(a)

10minus5

10minus3

10minus1

10minus13

10minus11

10minus9

10minus7

101

1 2 3 4 5 6 7 8PT (GeVc)

d2N(2120587PTdPTdy)(G

eVc)minus2

(b)

1 2 3 4 5 6 7 8

10minus5

10minus3

10minus1

10minus13

10minus11

10minus9

10minus7

101

PT (GeVc)

MinBias times 800ndash20 times 4

40ndash60 times 05

20ndash40 times 160ndash94 times 03

times 0008

Cu + Cu 200GeVKlowast0 rarr K+120587minus

Klowast0 rarr Kminus120587+

(Klowast0 + Klowast0)2

d2N(2120587PTdPTdy)(G

eVc)minus2

pp data

(c)

1 2 3 4 5 6 7 8

10minus5

10minus3

10minus1

10minus13

10minus11

10minus9

10minus7

101

PT (GeVc)

d2N(2120587PTdPTdy)(G

eVc)minus2

(d)

Figure 4The same as Figure 2 but the solid lines are the results of (9) The dashed lines and the dotted lines denote the contributions of thefirst component and the second component respectively

momentum spectra of 1198700119878and 119870

lowast0 at midrapidity in 119889 +Au Cu + Cu and 119901 + 119901 collisions at radic119904

119873119873= 200GeV

The two methods can both describe the distribution of thefinal-state particles They have their own advantage andproper scope The two forms of Tsallis distribution canconsistently agree with the experimental points in the lowand high 119901

119879region Tsallis statistics is nonextensive statistics

[4] The parameter 119879 is temperature and the parameter 119902

summarily describes all features causing a departure fromthe Boltzmann-Gibbs statistics In [6] Var(119879)⟨119879⟩2 = 119902 minus

1 directly reflects intrinsic fluctuations of the temperatureHowever the Tsallis distribution also emerges from a numberof other dynamical mechanisms [22] The two-componentBoltzmann distribution can directly show the contribution ofthe soft interaction and the hard interaction in the observedspectra by the weight parameter 119908

Advances in High Energy Physics 7

Table 3 Values of 1198791 1198792 and 119908 taken in Figures 3 and 4 119879

1and 119879

2units are GeV

Figure Centrality 1198791

1198792

119908 1205942dof Figure Centrality 119879

11198792

119908 1205942dof

Figure 3(a)

MinBias 0380 0900 0996 1057

Figure 4(a)

MinBias 0300 0600 0977 12480ndash20 0380 0900 0998 0932 0ndash20 0310 0580 0974 089020ndash40 0390 0900 0996 0811 20ndash40 0310 0580 0950 078040ndash60 0400 0910 0996 0670 40ndash60 0310 0580 0974 060560ndash88 0410 0920 0996 0715 60ndash88 0320 0580 0940 0643pp data 0420 0930 0995 0600 pp data 0300 0600 0979 0580

Figure 3(c)

MinBias 0450 0900 0936 0858

Figure 4(c)

MinBias 0280 0580 0979 11500ndash20 0470 0920 0989 0705 0ndash20 0280 0580 0984 070020ndash60 0470 0920 0990 0426 20ndash40 0280 0580 0982 057060ndash94 0470 0920 0987 0395 40ndash60 0280 0600 0979 0352

mdash mdash mdash mdash mdash 60ndash94 0280 0600 0980 0320

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China under Grant no 11247250 no 11005071and no 10975095 and the Shanxi Provincial Natural ScienceFoundation under Grant no 2013021006

References

[1] L Adamczyk G Agakishiev M M Aggarwal et al ldquoDirectedflow of identified particles in Au+Au collisions at radic119878

119873119873=

200GeV at RHICrdquo Physical Review Letters vol 108 no 20Article ID 202301 6 pages 2012

[2] K Adcox S S Adler N N Ajitanand et al ldquoCentralitydependence of 120587+minus 119870+minus p and 119901

minus production from radic119904NN =

130GeV Au+Au collisions at RHICrdquo Physical Review Lettersvol 88 Article ID 242301 2002

[3] P S B Dev A Pilaftsis and U K Yang ldquoNew productionmechanism for heavy neutrinos at the LHCrdquo Physical ReviewLetters vol 112 no 8 Article ID 081801 5 pages 2014

[4] C Tsallis ldquoPossible generalization of Boltzmann-Gibbs statis-ticsrdquo Journal of Statistical Physics vol 52 no 1-2 pp 479ndash4871988

[5] B C Li Y ZWang FH Liu X JWen andY EDong ldquoParticleproduction in relativistic 119875119875(119875) and119860119860 collisions at RHIC andLHC energies with Tsallis statistics using the two-cylindricalmultisource thermal modelrdquo Physical Review D vol 89 ArticleID 054014 2014

[6] C Y Wong and G Wilk ldquoTsallis fits to 119901119879spectra and multiple

hard scattering in 119901119901 collisions at the LHCrdquo Physical Review Dvol 87 Article ID 114007 2013

[7] B C Li Y Z Wang and F H Liu ldquoFormulation of transversemass distributions in AundashAu collisions at radic119904

119873119873= 200

GeVnucleonrdquo Physics Letters B vol 725 no 4-5 pp 352ndash3562013

[8] M Rybczynski and Z Włodarczyk ldquoTsallis statistics approachto the transverse momentum distributions in pndashp collisionsrdquoThe European Physical Journal C vol 74 no 2 p 2785 2014

[9] F-H Liu Y-H Chen H-R Wei and B-C Li ldquoTransversemomentum distributions of final-state particles produced insoft excitation process in high energy collisionsrdquo Advances inHigh Energy Physics vol 2013 Article ID 965735 15 pages 2013

[10] B-C Li Y-Y Fu E-Q Wang L-L Wang and F-H LiuldquoTransverse momentum dependence of charged and strangehadron elliptic flows in CundashCu collisionsrdquo Journal of Physics GNuclear and Particle Physics vol 39 no 8 Article ID 0851092012

[11] A Adare S Afanasiev C Aidala et al ldquoMeasurement of1198700119878and

119870lowast0 in p+p d+Au and Cu+Cu collisions at radic119904NN = 200GeVrdquo

Physical Review C vol 90 Article ID 054905 2014[12] A Adare S Afanasiev C Aidala et al ldquoMeasurement of

neutral mesons in p+p collisions at radic119904 = 200GeV and scalingproperties of hadron productionrdquo Physical Review D vol 83Article ID 052004 2011

[13] J Adams V Eckardt J Putschke et al ldquo119870(892)lowast resonance

production in Au+Au and 119901+119901 collisions atradic119904119873119873

= 200GeVrdquoPhysical Review C vol 71 Article ID 064902 2005

[14] B I Abelev M M Aggarwal Z Ahammed et al ldquoHadronicresonance production in 119889 + Au collisions at radic119904

119873119873= 200 GeV

measured at the BNL relativistic heavy ion colliderrdquo PhysicalReview C vol 78 no 4 Article ID 044906 20 pages 2008

[15] M M Aggarwal Z Ahammed and A V AlakhverdyantsldquoKlowast0 production in Cu+Cu and Au+Au collisions at radic119904

119873119873=

624GeV and 200GeVrdquo Physical Review C vol 84 no 3 ArticleID 034909 2011

[16] A Adare S Afanasiev C Aidala et al ldquoIdentified chargedhadron production in119901+119901 collisions atradic119904 = 200 and 624GeVrdquoPhysical Review C vol 83 Article ID 064903 2011

[17] B I Abelev J Adams M M Aggarwal et al ldquoStrange particleproduction in 119901+119901 collisions atradic119904 = 200GeVrdquo Physical ReviewC vol 75 no 6 Article ID 064901 21 pages 2007

[18] G Aad B Abbott J Abdallah et al ldquoCharged-particle multi-plicities in 119901119901 interactions measured with the ATLAS detectorat the LHCrdquo New Journal of Physics vol 13 Article ID 0530332011

[19] K Aamodt N Abel U Abeysekara et al ldquoTransverse momen-tum spectra of charged particles in proton-proton collisions atradic119904 = 900GeV with ALICE at the LHCrdquo Physics Letters B vol693 no 2 pp 53ndash68 2010

[20] V Khachatryan A M Sirunyan A Tumasyan et alldquoTransverse-momentum and pseudorapidity distributions

8 Advances in High Energy Physics

of charged hadrons in pp collisions at radic119904 = 7 TeVrdquo PhysicalReview Letters vol 105 Article ID 022002 2010

[21] G Wilk and Z Włodarczyk ldquoInterpretation of the nonexten-sivity parameter q in some applications of Tsallis statistics andLevy distributionsrdquo Physical Review Letters vol 84 no 13 pp2770ndash2773 2000

[22] G Wilk and Z Wlodarczyk ldquoConsequences of temperaturefluctuations in observablesmeasured in high-energy collisionsrdquoThe European Physical Journal A vol 48 article 161 2012

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2 Advances in High Energy Physics

10minus17

10minus15

10minus13

10minus11

10minus9

10minus7

10minus5

10minus3

10minus1

101

2 4 6 8 10 12

K0S rarr 12058701205870

d + Au 200GeV

PT (GeVc)

d2N(2120587PTdPTdy)(G

eVc)minus2

(a)

2 4 6 8 10 12

10minus17

10minus15

10minus13

10minus11

10minus9

10minus7

10minus5

10minus3

10minus1

101

PT (GeVc)

K0S rarr 12058701205870

Cu + Cu 200GeV

d2N(2120587PTdPTdy)(G

eVc)minus2

(b)

2 4 6 8 10 12

10minus5

10minus3

10minus1

10minus17

10minus19

10minus15

10minus13

10minus11

10minus9

10minus7

PT (GeVc)

MinBias times 102

0ndash20 times 10120ndash40

40ndash60 times 10minus1

times 10minus3

d2N(2120587PTdPTdy)(G

eVc)minus2

60ndash88 times 10minus2

pp data

(c)

2 4 6 8 10 12

10minus5

10minus3

10minus1

10minus17

10minus19

10minus15

10minus13

10minus11

10minus9

10minus7

PT (GeVc)

MinBias times 102

times 10minus20ndash20 times 220ndash60 times 02

60ndash94 times 005

d2N(2120587PTdPTdy)(G

eVc)minus2

pp data

(d)

Figure 1 1198700119878transverse momentum spectra in 119889 + Au and Cu + Cu collisions for different centrality bins and 119901 + 119901 collisions at radic119904

119873119873=

200GeV Experimental data [11ndash15] are shownwith different symbolsThedashed lines and solid lines are the results of (4) and (5) respectively

potential respectively In terms of the transverse mass 119898119879

and the rapidity 119910 the transverse momentum spectra of theparticles can be written as

1198892119873

119889119875119879119889119910

= 119892119881119875119879119898119879cosh119910

(2120587)2

[1 + (119902 minus 1)119898119879cosh119910 minus 120583

119879]

minus119902(119902minus1)

(2)

When 120583 = 0 and 119910 = 0 the distribution function is

1198892119873

119889119875119879119889119910

100381610038161003816100381610038161003816100381610038161003816119910=0

= 119892119881119875119879119898119879

(2120587)2[1 + (119902 minus 1)

119898119879

119879]

minus119902(119902minus1)

(3)

Considering a fixed rapidity interval [5] the distributionfunction should be

119889119873

119875119879119889119875119879

= 119862int

119910max

119910mincosh119910119898

119879[1 + (119902 minus 1)

119898119879cosh119910119879

]

minus119902(119902minus1)

(4)

Advances in High Energy Physics 3

10minus5

10minus3

10minus1

10minus13

10minus11

10minus9

10minus7

101d + Au 200GeV

Klowast0 rarr K+120587minus

Klowast0 rarr Kminus120587+

(Klowast0 + Klowast0)2

1 2 3 4 5 6 7 8PT (GeVc)

d2N(2120587PTdPTdy)(G

eVc)minus2

(a)

10minus5

10minus3

10minus1

10minus13

10minus11

10minus9

10minus7

101Cu + Cu 200GeV

Klowast0 rarr K+120587minus

Klowast0 rarr Kminus120587+

(Klowast0 + Klowast0)2

1 2 3 4 5 6 7 8PT (GeVc)

d2N(2120587PTdPTdy)(G

eVc)minus2

(b)

1 2 3 4 5 6 7 8

10minus5

10minus3

10minus1

10minus13

10minus11

10minus9

10minus7

101

PT (GeVc)

MinBias times 2 times 103

0ndash20 times 102

20ndash40 times 2 times 101

40ndash60 times 2 times 100

60ndash88 times 5 times 10minus1

times 2 times 10minus1

d2N(2120587PTdPTdy)(G

eVc)minus2

pp data

(c)

10minus5

10minus3

10minus1

10minus13

10minus11

10minus9

10minus7

101

1 2 3 4 5 6 7 8PT (GeVc)

MinBias times 800ndash20 times 4

40ndash60 times 05

20ndash40 times 160ndash94 times 03

times 0008

d2N(2120587PTdPTdy)(G

eVc)minus2

pp data

(d)

Figure 2 119870lowast0 and 119870lowast0 transverse momentum spectra in 119889 + Au and Cu + Cu collisions for different centrality bins and 119901 + 119901 collisions atradic119904119873119873

= 200GeV Experimental data [11ndash15] are shown with different symbols The dashed lines and solid lines are the results of (4) and (5)respectively

where the interval of the integral represents the rapidityrange observed in the experiment 119862 is a normalizationconstant The function is an improved form to satisfy thethermodynamical consistency [8] Under the limit 119902 rarr 1the 119901119879spectrum becomes a conventional form

119889119873

119875119879119889119875119879

= 119862int

119910max

119910mincosh119910119898

119879[1 + (119902

1015840minus 1)

119898119879cosh1199101198791015840

]

minus1(1199021015840minus1)

(5)

Next we review the Boltzmann distribution For Maxwellrsquosideal gas the momentum distribution function is

119889119873

1198731198893119875=

1198752

(1198980119896119879)32

radic2

120587exp(minus 119875

2

21198980119896119879

) (6)

where 119896 is a Boltzmann constant and1198980is the particle mass

If the relativistic effect is taken into account the momentumdistribution function is

119889119873

1198731198893119875=

1198752

1198982

01198961198791198702(1198980119896119879)

exp(minus

radic1198752 + 1198982

0

119896119879) (7)

4 Advances in High Energy Physics

Table 1 Values of 119879 119902 1198791015840 and 1199021015840 taken in Figure 1 119879 and 119879

1015840 units are GeV

Figure Centrality 119879 119902 1205942dof Figure Centrality 119879

10158401199021015840

1205942dof

Figure 1(a)

MinBias 0122 1085 0852

Figure 1(c)

MinBias 0120 1077 07550ndash20 0124 1084 0621 0ndash20 0120 1077 078220ndash40 0124 1084 0566 20ndash40 0121 1077 052840ndash60 0126 1084 0589 40ndash60 0122 1077 049560ndash88 0129 1084 0631 60ndash88 0122 1077 0670pp data 0134 1083 0952 pp data 0122 1077 1026

Figure 1(b)

MinBias 0122 1086 0426

Figure 1(d)

MinBias 0120 1079 03830ndash20 0124 1086 0215 0ndash20 0122 1079 040220ndash60 0124 1086 0270 20ndash60 0122 1079 024060ndash94 0125 1085 0354 60ndash94 0122 1079 0319

Table 2 Values of 119879 119902 1198791015840 and 1199021015840 taken in Figure 2 119879 and 119879

1015840 units are GeV

Figure Centrality 119879 119902 1205942dof Figure Centrality 119879

10158401199021015840

1205942dof

Figure 2(a)

MinBias 0120 1084 0372

Figure 2(c)

MinBias 0122 1075 03910ndash20 0120 1084 0446 0ndash20 0122 1075 042620ndash40 0130 1082 0515 20ndash40 0124 1075 051040ndash60 0130 1082 0658 40ndash60 0124 1073 062460ndash88 0140 1081 0675 60ndash88 0125 1073 0759pp data 0150 1077 0788 pp data 0127 1073 0806

Figure 2(b)

MinBias 0120 1080 0229

Figure 2(d)

MinBias 0118 1073 02550ndash20 0120 1080 0275 0ndash20 0118 1073 027920ndash40 0122 1080 0210 20ndash40 0118 1073 024640ndash60 0124 1080 0462 40ndash60 0122 1074 050360ndash94 0126 1080 0404 60ndash94 0122 1074 0429

where 1198702(1198980119896119879) is the second-order modified Bessel func-

tion For the isotropic emission in the collision the transversemomentum distribution is

119889119873

119873119889119875119879

= 1198621015840119875119879exp(minus

radic1198752

119879+ 1198982

0

119896119879) = 119892 (119862

1015840 119879) (8)

where 1198621015840 is a normalization constant The two-component

distribution of the transverse momentum is119889119873

119873119889119875119879

= 119908119892 (1198621015840

1 1198791) + (1 minus 119908) 119892 (119862

1015840

2 1198792) (9)

where 119908 indicates the contribution percentage of the firstcomponent

3 Discussion and Conclusion

Figure 1 shows the transverse momentum spectra of 1198700

119878

meson at midrapidity in 119889 +Au Cu + Cu and 119901+119901 collisionsat radic119904119873119873

= 200GeV The experimental points measured bySTAR and PHENIX collaborations [11ndash15] are shown withdifferent symbols ForCu+Cu and119889+Au different centralitybins are marked by the different shapes At the bottom ofthe figure we show the 119901 + 119901 data as a reference Thedashed 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 distributioncan both agree with the data The difference of the numericalresults is very small The parameters 119879 119902 1198791015840 and 119902

1015840 in thecalculations are listed inTable 1 with1205942 per degree of freedom(1205942dof) Their values do not change obviously due to a119875119879scaling behavior In Figure 2 we also give a comparison

between the numerical results and the experimental pointsof 119870lowast0 (or 119870lowast0) The parameters 119879 119902 1198791015840 and 119902

1015840 are listed inTable 2 with 120594

2dof Similarly the values have no significantor no regular changes With Tsallis statisticsrsquo success indealing with nonequilibrated complex systems in condensedmatter research it is used to study the particle production inhigh-energy physics The Tsallis statistics is widely applied inthe description of the experimental data in RHIC [12 16 17]and LHC [18ndash20] It is an advantage that the Tsallis statisticsis connected to thermodynamics by the entropy for examplesee [21] for more detailed discussions and its references

In Figures 3 and 4 we present a comparison between thetwo-component Boltzmann distribution and the experimen-tal data measured in 119889 + Au Cu + Cu and 119901 + 119901 collisions atradic119904119873119873

= 200GeV The solid lines denote the results of thetwo-component Boltzmann distribution equation (9) Thetwo-component Boltzmann distribution also can agree withthe experimental points The dashed lines and the dotted

Advances in High Energy Physics 5

MinBias times 102

0ndash20 times 10120ndash40

40ndash60 times 10minus1

times 10minus3pp data

2 4 6 8 10 12

10minus5

10minus3

10minus1

10minus13

10minus15

10minus17

10minus19

10minus11

10minus9

10minus7

101

PT (GeVc)

d + Au 200GeVK0S rarr 120587

01205870

d2N(2120587PTdPTdy)(G

eVc)minus2

60ndash88 times 10minus2

(a)

2 4 6 8 10 12

10minus5

10minus3

10minus1

10minus13

10minus15

10minus17

10minus19

10minus11

10minus9

10minus7

101

PT (GeVc)

d2N(2120587PTdPTdy)(G

eVc)minus2

(b)

2 4 6 8 10 12

10minus5

10minus3

10minus1

10minus13

10minus15

10minus17

10minus19

10minus11

10minus9

10minus7

101

PT (GeVc)

K0S rarr 12058701205870

Cu + Cu 200GeV

MinBias times 102

times 10minus20ndash20 times 220ndash60 times 02

60ndash94 times 005

d2N(2120587PTdPTdy)(G

eVc)minus2

pp data

(c)

2 4 6 8 10 12

10minus5

10minus3

10minus1

10minus13

10minus15

10minus17

10minus19

10minus11

10minus9

10minus7

101

PT (GeVc)

d2N(2120587PTdPTdy)(G

eVc)minus2

(d)

Figure 3 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 thefirst component and the second component respectively

lines denote the contributions of the first component and thesecond component respectively It is seen clearly that the softand hard interactions behave in the low and high transversemomentum of the identified particles The parameters 119879

1

1198792 and 119908 used in the calculations are given in Table 3 with

1205942dof The values of 119879

1are two to four times the values of

119879 or 1198791015840 The values of 1198792are about twice the values of 119879

1

because of the hard interaction In (9) the first componentis the contribution of soft process and the second component

is the contribution of hard processThedistribution in the lowtransverse momentum region is mainly contributed by thesoft processes The hard processes contribute high transversemomentums in the 119901

119879spectra For the two-component

distribution of the Boltzmann distribution the parameter 119908is used to denote the contribution of the soft process and 1minus 119908

is used to denote the contribution of the hard processIn summary we have compared Tsallis statistics and

the Boltzmann distribution in the analysis of the transverse

6 Advances in High Energy Physics

10minus5

10minus3

10minus1

10minus13

10minus11

10minus9

10minus7

101

1 2 3 4 5 6 7 8

MinBias times 2 times 103

0ndash20 times 102

20ndash40 times 2 times 101

40ndash60 times 2 times 100

60ndash88 times 5 times 10minus1

times 2 times 10minus1

PT (GeVc)

d + Au 200GeVKlowast0 rarr K+120587minus

Klowast0 rarr Kminus120587+

(Klowast0 + Klowast0)2

d2N(2120587PTdPTdy)(G

eVc)minus2

pp data

(a)

10minus5

10minus3

10minus1

10minus13

10minus11

10minus9

10minus7

101

1 2 3 4 5 6 7 8PT (GeVc)

d2N(2120587PTdPTdy)(G

eVc)minus2

(b)

1 2 3 4 5 6 7 8

10minus5

10minus3

10minus1

10minus13

10minus11

10minus9

10minus7

101

PT (GeVc)

MinBias times 800ndash20 times 4

40ndash60 times 05

20ndash40 times 160ndash94 times 03

times 0008

Cu + Cu 200GeVKlowast0 rarr K+120587minus

Klowast0 rarr Kminus120587+

(Klowast0 + Klowast0)2

d2N(2120587PTdPTdy)(G

eVc)minus2

pp data

(c)

1 2 3 4 5 6 7 8

10minus5

10minus3

10minus1

10minus13

10minus11

10minus9

10minus7

101

PT (GeVc)

d2N(2120587PTdPTdy)(G

eVc)minus2

(d)

Figure 4The same as Figure 2 but the solid lines are the results of (9) The dashed lines and the dotted lines denote the contributions of thefirst component and the second component respectively

momentum spectra of 1198700119878and 119870

lowast0 at midrapidity in 119889 +Au Cu + Cu and 119901 + 119901 collisions at radic119904

119873119873= 200GeV

The two methods can both describe the distribution of thefinal-state particles They have their own advantage andproper scope The two forms of Tsallis distribution canconsistently agree with the experimental points in the lowand high 119901

119879region Tsallis statistics is nonextensive statistics

[4] The parameter 119879 is temperature and the parameter 119902

summarily describes all features causing a departure fromthe Boltzmann-Gibbs statistics In [6] Var(119879)⟨119879⟩2 = 119902 minus

1 directly reflects intrinsic fluctuations of the temperatureHowever the Tsallis distribution also emerges from a numberof other dynamical mechanisms [22] The two-componentBoltzmann distribution can directly show the contribution ofthe soft interaction and the hard interaction in the observedspectra by the weight parameter 119908

Advances in High Energy Physics 7

Table 3 Values of 1198791 1198792 and 119908 taken in Figures 3 and 4 119879

1and 119879

2units are GeV

Figure Centrality 1198791

1198792

119908 1205942dof Figure Centrality 119879

11198792

119908 1205942dof

Figure 3(a)

MinBias 0380 0900 0996 1057

Figure 4(a)

MinBias 0300 0600 0977 12480ndash20 0380 0900 0998 0932 0ndash20 0310 0580 0974 089020ndash40 0390 0900 0996 0811 20ndash40 0310 0580 0950 078040ndash60 0400 0910 0996 0670 40ndash60 0310 0580 0974 060560ndash88 0410 0920 0996 0715 60ndash88 0320 0580 0940 0643pp data 0420 0930 0995 0600 pp data 0300 0600 0979 0580

Figure 3(c)

MinBias 0450 0900 0936 0858

Figure 4(c)

MinBias 0280 0580 0979 11500ndash20 0470 0920 0989 0705 0ndash20 0280 0580 0984 070020ndash60 0470 0920 0990 0426 20ndash40 0280 0580 0982 057060ndash94 0470 0920 0987 0395 40ndash60 0280 0600 0979 0352

mdash mdash mdash mdash mdash 60ndash94 0280 0600 0980 0320

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China under Grant no 11247250 no 11005071and no 10975095 and the Shanxi Provincial Natural ScienceFoundation under Grant no 2013021006

References

[1] L Adamczyk G Agakishiev M M Aggarwal et al ldquoDirectedflow of identified particles in Au+Au collisions at radic119878

119873119873=

200GeV at RHICrdquo Physical Review Letters vol 108 no 20Article ID 202301 6 pages 2012

[2] K Adcox S S Adler N N Ajitanand et al ldquoCentralitydependence of 120587+minus 119870+minus p and 119901

minus production from radic119904NN =

130GeV Au+Au collisions at RHICrdquo Physical Review Lettersvol 88 Article ID 242301 2002

[3] P S B Dev A Pilaftsis and U K Yang ldquoNew productionmechanism for heavy neutrinos at the LHCrdquo Physical ReviewLetters vol 112 no 8 Article ID 081801 5 pages 2014

[4] C Tsallis ldquoPossible generalization of Boltzmann-Gibbs statis-ticsrdquo Journal of Statistical Physics vol 52 no 1-2 pp 479ndash4871988

[5] B C Li Y ZWang FH Liu X JWen andY EDong ldquoParticleproduction in relativistic 119875119875(119875) and119860119860 collisions at RHIC andLHC energies with Tsallis statistics using the two-cylindricalmultisource thermal modelrdquo Physical Review D vol 89 ArticleID 054014 2014

[6] C Y Wong and G Wilk ldquoTsallis fits to 119901119879spectra and multiple

hard scattering in 119901119901 collisions at the LHCrdquo Physical Review Dvol 87 Article ID 114007 2013

[7] B C Li Y Z Wang and F H Liu ldquoFormulation of transversemass distributions in AundashAu collisions at radic119904

119873119873= 200

GeVnucleonrdquo Physics Letters B vol 725 no 4-5 pp 352ndash3562013

[8] M Rybczynski and Z Włodarczyk ldquoTsallis statistics approachto the transverse momentum distributions in pndashp collisionsrdquoThe European Physical Journal C vol 74 no 2 p 2785 2014

[9] F-H Liu Y-H Chen H-R Wei and B-C Li ldquoTransversemomentum distributions of final-state particles produced insoft excitation process in high energy collisionsrdquo Advances inHigh Energy Physics vol 2013 Article ID 965735 15 pages 2013

[10] B-C Li Y-Y Fu E-Q Wang L-L Wang and F-H LiuldquoTransverse momentum dependence of charged and strangehadron elliptic flows in CundashCu collisionsrdquo Journal of Physics GNuclear and Particle Physics vol 39 no 8 Article ID 0851092012

[11] A Adare S Afanasiev C Aidala et al ldquoMeasurement of1198700119878and

119870lowast0 in p+p d+Au and Cu+Cu collisions at radic119904NN = 200GeVrdquo

Physical Review C vol 90 Article ID 054905 2014[12] A Adare S Afanasiev C Aidala et al ldquoMeasurement of

neutral mesons in p+p collisions at radic119904 = 200GeV and scalingproperties of hadron productionrdquo Physical Review D vol 83Article ID 052004 2011

[13] J Adams V Eckardt J Putschke et al ldquo119870(892)lowast resonance

production in Au+Au and 119901+119901 collisions atradic119904119873119873

= 200GeVrdquoPhysical Review C vol 71 Article ID 064902 2005

[14] B I Abelev M M Aggarwal Z Ahammed et al ldquoHadronicresonance production in 119889 + Au collisions at radic119904

119873119873= 200 GeV

measured at the BNL relativistic heavy ion colliderrdquo PhysicalReview C vol 78 no 4 Article ID 044906 20 pages 2008

[15] M M Aggarwal Z Ahammed and A V AlakhverdyantsldquoKlowast0 production in Cu+Cu and Au+Au collisions at radic119904

119873119873=

624GeV and 200GeVrdquo Physical Review C vol 84 no 3 ArticleID 034909 2011

[16] A Adare S Afanasiev C Aidala et al ldquoIdentified chargedhadron production in119901+119901 collisions atradic119904 = 200 and 624GeVrdquoPhysical Review C vol 83 Article ID 064903 2011

[17] B I Abelev J Adams M M Aggarwal et al ldquoStrange particleproduction in 119901+119901 collisions atradic119904 = 200GeVrdquo Physical ReviewC vol 75 no 6 Article ID 064901 21 pages 2007

[18] G Aad B Abbott J Abdallah et al ldquoCharged-particle multi-plicities in 119901119901 interactions measured with the ATLAS detectorat the LHCrdquo New Journal of Physics vol 13 Article ID 0530332011

[19] K Aamodt N Abel U Abeysekara et al ldquoTransverse momen-tum spectra of charged particles in proton-proton collisions atradic119904 = 900GeV with ALICE at the LHCrdquo Physics Letters B vol693 no 2 pp 53ndash68 2010

[20] V Khachatryan A M Sirunyan A Tumasyan et alldquoTransverse-momentum and pseudorapidity distributions

8 Advances in High Energy Physics

of charged hadrons in pp collisions at radic119904 = 7 TeVrdquo PhysicalReview Letters vol 105 Article ID 022002 2010

[21] G Wilk and Z Włodarczyk ldquoInterpretation of the nonexten-sivity parameter q in some applications of Tsallis statistics andLevy distributionsrdquo Physical Review Letters vol 84 no 13 pp2770ndash2773 2000

[22] G Wilk and Z Wlodarczyk ldquoConsequences of temperaturefluctuations in observablesmeasured in high-energy collisionsrdquoThe European Physical Journal A vol 48 article 161 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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FluidsJournal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

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AstronomyAdvances in

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Superconductivity

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Statistical MechanicsInternational Journal of

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GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

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Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

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AerodynamicsJournal of

Volume 2014

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PhotonicsJournal of

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Journal of

Biophysics

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ThermodynamicsJournal of

Advances in High Energy Physics 3

10minus5

10minus3

10minus1

10minus13

10minus11

10minus9

10minus7

101d + Au 200GeV

Klowast0 rarr K+120587minus

Klowast0 rarr Kminus120587+

(Klowast0 + Klowast0)2

1 2 3 4 5 6 7 8PT (GeVc)

d2N(2120587PTdPTdy)(G

eVc)minus2

(a)

10minus5

10minus3

10minus1

10minus13

10minus11

10minus9

10minus7

101Cu + Cu 200GeV

Klowast0 rarr K+120587minus

Klowast0 rarr Kminus120587+

(Klowast0 + Klowast0)2

1 2 3 4 5 6 7 8PT (GeVc)

d2N(2120587PTdPTdy)(G

eVc)minus2

(b)

1 2 3 4 5 6 7 8

10minus5

10minus3

10minus1

10minus13

10minus11

10minus9

10minus7

101

PT (GeVc)

MinBias times 2 times 103

0ndash20 times 102

20ndash40 times 2 times 101

40ndash60 times 2 times 100

60ndash88 times 5 times 10minus1

times 2 times 10minus1

d2N(2120587PTdPTdy)(G

eVc)minus2

pp data

(c)

10minus5

10minus3

10minus1

10minus13

10minus11

10minus9

10minus7

101

1 2 3 4 5 6 7 8PT (GeVc)

MinBias times 800ndash20 times 4

40ndash60 times 05

20ndash40 times 160ndash94 times 03

times 0008

d2N(2120587PTdPTdy)(G

eVc)minus2

pp data

(d)

Figure 2 119870lowast0 and 119870lowast0 transverse momentum spectra in 119889 + Au and Cu + Cu collisions for different centrality bins and 119901 + 119901 collisions atradic119904119873119873

= 200GeV Experimental data [11ndash15] are shown with different symbols The dashed lines and solid lines are the results of (4) and (5)respectively

where the interval of the integral represents the rapidityrange observed in the experiment 119862 is a normalizationconstant The function is an improved form to satisfy thethermodynamical consistency [8] Under the limit 119902 rarr 1the 119901119879spectrum becomes a conventional form

119889119873

119875119879119889119875119879

= 119862int

119910max

119910mincosh119910119898

119879[1 + (119902

1015840minus 1)

119898119879cosh1199101198791015840

]

minus1(1199021015840minus1)

(5)

Next we review the Boltzmann distribution For Maxwellrsquosideal gas the momentum distribution function is

119889119873

1198731198893119875=

1198752

(1198980119896119879)32

radic2

120587exp(minus 119875

2

21198980119896119879

) (6)

where 119896 is a Boltzmann constant and1198980is the particle mass

If the relativistic effect is taken into account the momentumdistribution function is

119889119873

1198731198893119875=

1198752

1198982

01198961198791198702(1198980119896119879)

exp(minus

radic1198752 + 1198982

0

119896119879) (7)

4 Advances in High Energy Physics

Table 1 Values of 119879 119902 1198791015840 and 1199021015840 taken in Figure 1 119879 and 119879

1015840 units are GeV

Figure Centrality 119879 119902 1205942dof Figure Centrality 119879

10158401199021015840

1205942dof

Figure 1(a)

MinBias 0122 1085 0852

Figure 1(c)

MinBias 0120 1077 07550ndash20 0124 1084 0621 0ndash20 0120 1077 078220ndash40 0124 1084 0566 20ndash40 0121 1077 052840ndash60 0126 1084 0589 40ndash60 0122 1077 049560ndash88 0129 1084 0631 60ndash88 0122 1077 0670pp data 0134 1083 0952 pp data 0122 1077 1026

Figure 1(b)

MinBias 0122 1086 0426

Figure 1(d)

MinBias 0120 1079 03830ndash20 0124 1086 0215 0ndash20 0122 1079 040220ndash60 0124 1086 0270 20ndash60 0122 1079 024060ndash94 0125 1085 0354 60ndash94 0122 1079 0319

Table 2 Values of 119879 119902 1198791015840 and 1199021015840 taken in Figure 2 119879 and 119879

1015840 units are GeV

Figure Centrality 119879 119902 1205942dof Figure Centrality 119879

10158401199021015840

1205942dof

Figure 2(a)

MinBias 0120 1084 0372

Figure 2(c)

MinBias 0122 1075 03910ndash20 0120 1084 0446 0ndash20 0122 1075 042620ndash40 0130 1082 0515 20ndash40 0124 1075 051040ndash60 0130 1082 0658 40ndash60 0124 1073 062460ndash88 0140 1081 0675 60ndash88 0125 1073 0759pp data 0150 1077 0788 pp data 0127 1073 0806

Figure 2(b)

MinBias 0120 1080 0229

Figure 2(d)

MinBias 0118 1073 02550ndash20 0120 1080 0275 0ndash20 0118 1073 027920ndash40 0122 1080 0210 20ndash40 0118 1073 024640ndash60 0124 1080 0462 40ndash60 0122 1074 050360ndash94 0126 1080 0404 60ndash94 0122 1074 0429

where 1198702(1198980119896119879) is the second-order modified Bessel func-

tion For the isotropic emission in the collision the transversemomentum distribution is

119889119873

119873119889119875119879

= 1198621015840119875119879exp(minus

radic1198752

119879+ 1198982

0

119896119879) = 119892 (119862

1015840 119879) (8)

where 1198621015840 is a normalization constant The two-component

distribution of the transverse momentum is119889119873

119873119889119875119879

= 119908119892 (1198621015840

1 1198791) + (1 minus 119908) 119892 (119862

1015840

2 1198792) (9)

where 119908 indicates the contribution percentage of the firstcomponent

3 Discussion and Conclusion

Figure 1 shows the transverse momentum spectra of 1198700

119878

meson at midrapidity in 119889 +Au Cu + Cu and 119901+119901 collisionsat radic119904119873119873

= 200GeV The experimental points measured bySTAR and PHENIX collaborations [11ndash15] are shown withdifferent symbols ForCu+Cu and119889+Au different centralitybins are marked by the different shapes At the bottom ofthe figure we show the 119901 + 119901 data as a reference Thedashed 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 distributioncan both agree with the data The difference of the numericalresults is very small The parameters 119879 119902 1198791015840 and 119902

1015840 in thecalculations are listed inTable 1 with1205942 per degree of freedom(1205942dof) Their values do not change obviously due to a119875119879scaling behavior In Figure 2 we also give a comparison

between the numerical results and the experimental pointsof 119870lowast0 (or 119870lowast0) The parameters 119879 119902 1198791015840 and 119902

1015840 are listed inTable 2 with 120594

2dof Similarly the values have no significantor no regular changes With Tsallis statisticsrsquo success indealing with nonequilibrated complex systems in condensedmatter research it is used to study the particle production inhigh-energy physics The Tsallis statistics is widely applied inthe description of the experimental data in RHIC [12 16 17]and LHC [18ndash20] It is an advantage that the Tsallis statisticsis connected to thermodynamics by the entropy for examplesee [21] for more detailed discussions and its references

In Figures 3 and 4 we present a comparison between thetwo-component Boltzmann distribution and the experimen-tal data measured in 119889 + Au Cu + Cu and 119901 + 119901 collisions atradic119904119873119873

= 200GeV The solid lines denote the results of thetwo-component Boltzmann distribution equation (9) Thetwo-component Boltzmann distribution also can agree withthe experimental points The dashed lines and the dotted

Advances in High Energy Physics 5

MinBias times 102

0ndash20 times 10120ndash40

40ndash60 times 10minus1

times 10minus3pp data

2 4 6 8 10 12

10minus5

10minus3

10minus1

10minus13

10minus15

10minus17

10minus19

10minus11

10minus9

10minus7

101

PT (GeVc)

d + Au 200GeVK0S rarr 120587

01205870

d2N(2120587PTdPTdy)(G

eVc)minus2

60ndash88 times 10minus2

(a)

2 4 6 8 10 12

10minus5

10minus3

10minus1

10minus13

10minus15

10minus17

10minus19

10minus11

10minus9

10minus7

101

PT (GeVc)

d2N(2120587PTdPTdy)(G

eVc)minus2

(b)

2 4 6 8 10 12

10minus5

10minus3

10minus1

10minus13

10minus15

10minus17

10minus19

10minus11

10minus9

10minus7

101

PT (GeVc)

K0S rarr 12058701205870

Cu + Cu 200GeV

MinBias times 102

times 10minus20ndash20 times 220ndash60 times 02

60ndash94 times 005

d2N(2120587PTdPTdy)(G

eVc)minus2

pp data

(c)

2 4 6 8 10 12

10minus5

10minus3

10minus1

10minus13

10minus15

10minus17

10minus19

10minus11

10minus9

10minus7

101

PT (GeVc)

d2N(2120587PTdPTdy)(G

eVc)minus2

(d)

Figure 3 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 thefirst component and the second component respectively

lines denote the contributions of the first component and thesecond component respectively It is seen clearly that the softand hard interactions behave in the low and high transversemomentum of the identified particles The parameters 119879

1

1198792 and 119908 used in the calculations are given in Table 3 with

1205942dof The values of 119879

1are two to four times the values of

119879 or 1198791015840 The values of 1198792are about twice the values of 119879

1

because of the hard interaction In (9) the first componentis the contribution of soft process and the second component

is the contribution of hard processThedistribution in the lowtransverse momentum region is mainly contributed by thesoft processes The hard processes contribute high transversemomentums in the 119901

119879spectra For the two-component

distribution of the Boltzmann distribution the parameter 119908is used to denote the contribution of the soft process and 1minus 119908

is used to denote the contribution of the hard processIn summary we have compared Tsallis statistics and

the Boltzmann distribution in the analysis of the transverse

6 Advances in High Energy Physics

10minus5

10minus3

10minus1

10minus13

10minus11

10minus9

10minus7

101

1 2 3 4 5 6 7 8

MinBias times 2 times 103

0ndash20 times 102

20ndash40 times 2 times 101

40ndash60 times 2 times 100

60ndash88 times 5 times 10minus1

times 2 times 10minus1

PT (GeVc)

d + Au 200GeVKlowast0 rarr K+120587minus

Klowast0 rarr Kminus120587+

(Klowast0 + Klowast0)2

d2N(2120587PTdPTdy)(G

eVc)minus2

pp data

(a)

10minus5

10minus3

10minus1

10minus13

10minus11

10minus9

10minus7

101

1 2 3 4 5 6 7 8PT (GeVc)

d2N(2120587PTdPTdy)(G

eVc)minus2

(b)

1 2 3 4 5 6 7 8

10minus5

10minus3

10minus1

10minus13

10minus11

10minus9

10minus7

101

PT (GeVc)

MinBias times 800ndash20 times 4

40ndash60 times 05

20ndash40 times 160ndash94 times 03

times 0008

Cu + Cu 200GeVKlowast0 rarr K+120587minus

Klowast0 rarr Kminus120587+

(Klowast0 + Klowast0)2

d2N(2120587PTdPTdy)(G

eVc)minus2

pp data

(c)

1 2 3 4 5 6 7 8

10minus5

10minus3

10minus1

10minus13

10minus11

10minus9

10minus7

101

PT (GeVc)

d2N(2120587PTdPTdy)(G

eVc)minus2

(d)

Figure 4The same as Figure 2 but the solid lines are the results of (9) The dashed lines and the dotted lines denote the contributions of thefirst component and the second component respectively

momentum spectra of 1198700119878and 119870

lowast0 at midrapidity in 119889 +Au Cu + Cu and 119901 + 119901 collisions at radic119904

119873119873= 200GeV

The two methods can both describe the distribution of thefinal-state particles They have their own advantage andproper scope The two forms of Tsallis distribution canconsistently agree with the experimental points in the lowand high 119901

119879region Tsallis statistics is nonextensive statistics

[4] The parameter 119879 is temperature and the parameter 119902

summarily describes all features causing a departure fromthe Boltzmann-Gibbs statistics In [6] Var(119879)⟨119879⟩2 = 119902 minus

1 directly reflects intrinsic fluctuations of the temperatureHowever the Tsallis distribution also emerges from a numberof other dynamical mechanisms [22] The two-componentBoltzmann distribution can directly show the contribution ofthe soft interaction and the hard interaction in the observedspectra by the weight parameter 119908

Advances in High Energy Physics 7

Table 3 Values of 1198791 1198792 and 119908 taken in Figures 3 and 4 119879

1and 119879

2units are GeV

Figure Centrality 1198791

1198792

119908 1205942dof Figure Centrality 119879

11198792

119908 1205942dof

Figure 3(a)

MinBias 0380 0900 0996 1057

Figure 4(a)

MinBias 0300 0600 0977 12480ndash20 0380 0900 0998 0932 0ndash20 0310 0580 0974 089020ndash40 0390 0900 0996 0811 20ndash40 0310 0580 0950 078040ndash60 0400 0910 0996 0670 40ndash60 0310 0580 0974 060560ndash88 0410 0920 0996 0715 60ndash88 0320 0580 0940 0643pp data 0420 0930 0995 0600 pp data 0300 0600 0979 0580

Figure 3(c)

MinBias 0450 0900 0936 0858

Figure 4(c)

MinBias 0280 0580 0979 11500ndash20 0470 0920 0989 0705 0ndash20 0280 0580 0984 070020ndash60 0470 0920 0990 0426 20ndash40 0280 0580 0982 057060ndash94 0470 0920 0987 0395 40ndash60 0280 0600 0979 0352

mdash mdash mdash mdash mdash 60ndash94 0280 0600 0980 0320

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China under Grant no 11247250 no 11005071and no 10975095 and the Shanxi Provincial Natural ScienceFoundation under Grant no 2013021006

References

[1] L Adamczyk G Agakishiev M M Aggarwal et al ldquoDirectedflow of identified particles in Au+Au collisions at radic119878

119873119873=

200GeV at RHICrdquo Physical Review Letters vol 108 no 20Article ID 202301 6 pages 2012

[2] K Adcox S S Adler N N Ajitanand et al ldquoCentralitydependence of 120587+minus 119870+minus p and 119901

minus production from radic119904NN =

130GeV Au+Au collisions at RHICrdquo Physical Review Lettersvol 88 Article ID 242301 2002

[3] P S B Dev A Pilaftsis and U K Yang ldquoNew productionmechanism for heavy neutrinos at the LHCrdquo Physical ReviewLetters vol 112 no 8 Article ID 081801 5 pages 2014

[4] C Tsallis ldquoPossible generalization of Boltzmann-Gibbs statis-ticsrdquo Journal of Statistical Physics vol 52 no 1-2 pp 479ndash4871988

[5] B C Li Y ZWang FH Liu X JWen andY EDong ldquoParticleproduction in relativistic 119875119875(119875) and119860119860 collisions at RHIC andLHC energies with Tsallis statistics using the two-cylindricalmultisource thermal modelrdquo Physical Review D vol 89 ArticleID 054014 2014

[6] C Y Wong and G Wilk ldquoTsallis fits to 119901119879spectra and multiple

hard scattering in 119901119901 collisions at the LHCrdquo Physical Review Dvol 87 Article ID 114007 2013

[7] B C Li Y Z Wang and F H Liu ldquoFormulation of transversemass distributions in AundashAu collisions at radic119904

119873119873= 200

GeVnucleonrdquo Physics Letters B vol 725 no 4-5 pp 352ndash3562013

[8] M Rybczynski and Z Włodarczyk ldquoTsallis statistics approachto the transverse momentum distributions in pndashp collisionsrdquoThe European Physical Journal C vol 74 no 2 p 2785 2014

[9] F-H Liu Y-H Chen H-R Wei and B-C Li ldquoTransversemomentum distributions of final-state particles produced insoft excitation process in high energy collisionsrdquo Advances inHigh Energy Physics vol 2013 Article ID 965735 15 pages 2013

[10] B-C Li Y-Y Fu E-Q Wang L-L Wang and F-H LiuldquoTransverse momentum dependence of charged and strangehadron elliptic flows in CundashCu collisionsrdquo Journal of Physics GNuclear and Particle Physics vol 39 no 8 Article ID 0851092012

[11] A Adare S Afanasiev C Aidala et al ldquoMeasurement of1198700119878and

119870lowast0 in p+p d+Au and Cu+Cu collisions at radic119904NN = 200GeVrdquo

Physical Review C vol 90 Article ID 054905 2014[12] A Adare S Afanasiev C Aidala et al ldquoMeasurement of

neutral mesons in p+p collisions at radic119904 = 200GeV and scalingproperties of hadron productionrdquo Physical Review D vol 83Article ID 052004 2011

[13] J Adams V Eckardt J Putschke et al ldquo119870(892)lowast resonance

production in Au+Au and 119901+119901 collisions atradic119904119873119873

= 200GeVrdquoPhysical Review C vol 71 Article ID 064902 2005

[14] B I Abelev M M Aggarwal Z Ahammed et al ldquoHadronicresonance production in 119889 + Au collisions at radic119904

119873119873= 200 GeV

measured at the BNL relativistic heavy ion colliderrdquo PhysicalReview C vol 78 no 4 Article ID 044906 20 pages 2008

[15] M M Aggarwal Z Ahammed and A V AlakhverdyantsldquoKlowast0 production in Cu+Cu and Au+Au collisions at radic119904

119873119873=

624GeV and 200GeVrdquo Physical Review C vol 84 no 3 ArticleID 034909 2011

[16] A Adare S Afanasiev C Aidala et al ldquoIdentified chargedhadron production in119901+119901 collisions atradic119904 = 200 and 624GeVrdquoPhysical Review C vol 83 Article ID 064903 2011

[17] B I Abelev J Adams M M Aggarwal et al ldquoStrange particleproduction in 119901+119901 collisions atradic119904 = 200GeVrdquo Physical ReviewC vol 75 no 6 Article ID 064901 21 pages 2007

[18] G Aad B Abbott J Abdallah et al ldquoCharged-particle multi-plicities in 119901119901 interactions measured with the ATLAS detectorat the LHCrdquo New Journal of Physics vol 13 Article ID 0530332011

[19] K Aamodt N Abel U Abeysekara et al ldquoTransverse momen-tum spectra of charged particles in proton-proton collisions atradic119904 = 900GeV with ALICE at the LHCrdquo Physics Letters B vol693 no 2 pp 53ndash68 2010

[20] V Khachatryan A M Sirunyan A Tumasyan et alldquoTransverse-momentum and pseudorapidity distributions

8 Advances in High Energy Physics

of charged hadrons in pp collisions at radic119904 = 7 TeVrdquo PhysicalReview Letters vol 105 Article ID 022002 2010

[21] G Wilk and Z Włodarczyk ldquoInterpretation of the nonexten-sivity parameter q in some applications of Tsallis statistics andLevy distributionsrdquo Physical Review Letters vol 84 no 13 pp2770ndash2773 2000

[22] G Wilk and Z Wlodarczyk ldquoConsequences of temperaturefluctuations in observablesmeasured in high-energy collisionsrdquoThe European Physical Journal A vol 48 article 161 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

4 Advances in High Energy Physics

Table 1 Values of 119879 119902 1198791015840 and 1199021015840 taken in Figure 1 119879 and 119879

1015840 units are GeV

Figure Centrality 119879 119902 1205942dof Figure Centrality 119879

10158401199021015840

1205942dof

Figure 1(a)

MinBias 0122 1085 0852

Figure 1(c)

MinBias 0120 1077 07550ndash20 0124 1084 0621 0ndash20 0120 1077 078220ndash40 0124 1084 0566 20ndash40 0121 1077 052840ndash60 0126 1084 0589 40ndash60 0122 1077 049560ndash88 0129 1084 0631 60ndash88 0122 1077 0670pp data 0134 1083 0952 pp data 0122 1077 1026

Figure 1(b)

MinBias 0122 1086 0426

Figure 1(d)

MinBias 0120 1079 03830ndash20 0124 1086 0215 0ndash20 0122 1079 040220ndash60 0124 1086 0270 20ndash60 0122 1079 024060ndash94 0125 1085 0354 60ndash94 0122 1079 0319

Table 2 Values of 119879 119902 1198791015840 and 1199021015840 taken in Figure 2 119879 and 119879

1015840 units are GeV

Figure Centrality 119879 119902 1205942dof Figure Centrality 119879

10158401199021015840

1205942dof

Figure 2(a)

MinBias 0120 1084 0372

Figure 2(c)

MinBias 0122 1075 03910ndash20 0120 1084 0446 0ndash20 0122 1075 042620ndash40 0130 1082 0515 20ndash40 0124 1075 051040ndash60 0130 1082 0658 40ndash60 0124 1073 062460ndash88 0140 1081 0675 60ndash88 0125 1073 0759pp data 0150 1077 0788 pp data 0127 1073 0806

Figure 2(b)

MinBias 0120 1080 0229

Figure 2(d)

MinBias 0118 1073 02550ndash20 0120 1080 0275 0ndash20 0118 1073 027920ndash40 0122 1080 0210 20ndash40 0118 1073 024640ndash60 0124 1080 0462 40ndash60 0122 1074 050360ndash94 0126 1080 0404 60ndash94 0122 1074 0429

where 1198702(1198980119896119879) is the second-order modified Bessel func-

tion For the isotropic emission in the collision the transversemomentum distribution is

119889119873

119873119889119875119879

= 1198621015840119875119879exp(minus

radic1198752

119879+ 1198982

0

119896119879) = 119892 (119862

1015840 119879) (8)

where 1198621015840 is a normalization constant The two-component

distribution of the transverse momentum is119889119873

119873119889119875119879

= 119908119892 (1198621015840

1 1198791) + (1 minus 119908) 119892 (119862

1015840

2 1198792) (9)

where 119908 indicates the contribution percentage of the firstcomponent

3 Discussion and Conclusion

Figure 1 shows the transverse momentum spectra of 1198700

119878

meson at midrapidity in 119889 +Au Cu + Cu and 119901+119901 collisionsat radic119904119873119873

= 200GeV The experimental points measured bySTAR and PHENIX collaborations [11ndash15] are shown withdifferent symbols ForCu+Cu and119889+Au different centralitybins are marked by the different shapes At the bottom ofthe figure we show the 119901 + 119901 data as a reference Thedashed 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 distributioncan both agree with the data The difference of the numericalresults is very small The parameters 119879 119902 1198791015840 and 119902

1015840 in thecalculations are listed inTable 1 with1205942 per degree of freedom(1205942dof) Their values do not change obviously due to a119875119879scaling behavior In Figure 2 we also give a comparison

between the numerical results and the experimental pointsof 119870lowast0 (or 119870lowast0) The parameters 119879 119902 1198791015840 and 119902

1015840 are listed inTable 2 with 120594

2dof Similarly the values have no significantor no regular changes With Tsallis statisticsrsquo success indealing with nonequilibrated complex systems in condensedmatter research it is used to study the particle production inhigh-energy physics The Tsallis statistics is widely applied inthe description of the experimental data in RHIC [12 16 17]and LHC [18ndash20] It is an advantage that the Tsallis statisticsis connected to thermodynamics by the entropy for examplesee [21] for more detailed discussions and its references

In Figures 3 and 4 we present a comparison between thetwo-component Boltzmann distribution and the experimen-tal data measured in 119889 + Au Cu + Cu and 119901 + 119901 collisions atradic119904119873119873

= 200GeV The solid lines denote the results of thetwo-component Boltzmann distribution equation (9) Thetwo-component Boltzmann distribution also can agree withthe experimental points The dashed lines and the dotted

Advances in High Energy Physics 5

MinBias times 102

0ndash20 times 10120ndash40

40ndash60 times 10minus1

times 10minus3pp data

2 4 6 8 10 12

10minus5

10minus3

10minus1

10minus13

10minus15

10minus17

10minus19

10minus11

10minus9

10minus7

101

PT (GeVc)

d + Au 200GeVK0S rarr 120587

01205870

d2N(2120587PTdPTdy)(G

eVc)minus2

60ndash88 times 10minus2

(a)

2 4 6 8 10 12

10minus5

10minus3

10minus1

10minus13

10minus15

10minus17

10minus19

10minus11

10minus9

10minus7

101

PT (GeVc)

d2N(2120587PTdPTdy)(G

eVc)minus2

(b)

2 4 6 8 10 12

10minus5

10minus3

10minus1

10minus13

10minus15

10minus17

10minus19

10minus11

10minus9

10minus7

101

PT (GeVc)

K0S rarr 12058701205870

Cu + Cu 200GeV

MinBias times 102

times 10minus20ndash20 times 220ndash60 times 02

60ndash94 times 005

d2N(2120587PTdPTdy)(G

eVc)minus2

pp data

(c)

2 4 6 8 10 12

10minus5

10minus3

10minus1

10minus13

10minus15

10minus17

10minus19

10minus11

10minus9

10minus7

101

PT (GeVc)

d2N(2120587PTdPTdy)(G

eVc)minus2

(d)

Figure 3 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 thefirst component and the second component respectively

lines denote the contributions of the first component and thesecond component respectively It is seen clearly that the softand hard interactions behave in the low and high transversemomentum of the identified particles The parameters 119879

1

1198792 and 119908 used in the calculations are given in Table 3 with

1205942dof The values of 119879

1are two to four times the values of

119879 or 1198791015840 The values of 1198792are about twice the values of 119879

1

because of the hard interaction In (9) the first componentis the contribution of soft process and the second component

is the contribution of hard processThedistribution in the lowtransverse momentum region is mainly contributed by thesoft processes The hard processes contribute high transversemomentums in the 119901

119879spectra For the two-component

distribution of the Boltzmann distribution the parameter 119908is used to denote the contribution of the soft process and 1minus 119908

is used to denote the contribution of the hard processIn summary we have compared Tsallis statistics and

the Boltzmann distribution in the analysis of the transverse

6 Advances in High Energy Physics

10minus5

10minus3

10minus1

10minus13

10minus11

10minus9

10minus7

101

1 2 3 4 5 6 7 8

MinBias times 2 times 103

0ndash20 times 102

20ndash40 times 2 times 101

40ndash60 times 2 times 100

60ndash88 times 5 times 10minus1

times 2 times 10minus1

PT (GeVc)

d + Au 200GeVKlowast0 rarr K+120587minus

Klowast0 rarr Kminus120587+

(Klowast0 + Klowast0)2

d2N(2120587PTdPTdy)(G

eVc)minus2

pp data

(a)

10minus5

10minus3

10minus1

10minus13

10minus11

10minus9

10minus7

101

1 2 3 4 5 6 7 8PT (GeVc)

d2N(2120587PTdPTdy)(G

eVc)minus2

(b)

1 2 3 4 5 6 7 8

10minus5

10minus3

10minus1

10minus13

10minus11

10minus9

10minus7

101

PT (GeVc)

MinBias times 800ndash20 times 4

40ndash60 times 05

20ndash40 times 160ndash94 times 03

times 0008

Cu + Cu 200GeVKlowast0 rarr K+120587minus

Klowast0 rarr Kminus120587+

(Klowast0 + Klowast0)2

d2N(2120587PTdPTdy)(G

eVc)minus2

pp data

(c)

1 2 3 4 5 6 7 8

10minus5

10minus3

10minus1

10minus13

10minus11

10minus9

10minus7

101

PT (GeVc)

d2N(2120587PTdPTdy)(G

eVc)minus2

(d)

Figure 4The same as Figure 2 but the solid lines are the results of (9) The dashed lines and the dotted lines denote the contributions of thefirst component and the second component respectively

momentum spectra of 1198700119878and 119870

lowast0 at midrapidity in 119889 +Au Cu + Cu and 119901 + 119901 collisions at radic119904

119873119873= 200GeV

The two methods can both describe the distribution of thefinal-state particles They have their own advantage andproper scope The two forms of Tsallis distribution canconsistently agree with the experimental points in the lowand high 119901

119879region Tsallis statistics is nonextensive statistics

[4] The parameter 119879 is temperature and the parameter 119902

summarily describes all features causing a departure fromthe Boltzmann-Gibbs statistics In [6] Var(119879)⟨119879⟩2 = 119902 minus

1 directly reflects intrinsic fluctuations of the temperatureHowever the Tsallis distribution also emerges from a numberof other dynamical mechanisms [22] The two-componentBoltzmann distribution can directly show the contribution ofthe soft interaction and the hard interaction in the observedspectra by the weight parameter 119908

Advances in High Energy Physics 7

Table 3 Values of 1198791 1198792 and 119908 taken in Figures 3 and 4 119879

1and 119879

2units are GeV

Figure Centrality 1198791

1198792

119908 1205942dof Figure Centrality 119879

11198792

119908 1205942dof

Figure 3(a)

MinBias 0380 0900 0996 1057

Figure 4(a)

MinBias 0300 0600 0977 12480ndash20 0380 0900 0998 0932 0ndash20 0310 0580 0974 089020ndash40 0390 0900 0996 0811 20ndash40 0310 0580 0950 078040ndash60 0400 0910 0996 0670 40ndash60 0310 0580 0974 060560ndash88 0410 0920 0996 0715 60ndash88 0320 0580 0940 0643pp data 0420 0930 0995 0600 pp data 0300 0600 0979 0580

Figure 3(c)

MinBias 0450 0900 0936 0858

Figure 4(c)

MinBias 0280 0580 0979 11500ndash20 0470 0920 0989 0705 0ndash20 0280 0580 0984 070020ndash60 0470 0920 0990 0426 20ndash40 0280 0580 0982 057060ndash94 0470 0920 0987 0395 40ndash60 0280 0600 0979 0352

mdash mdash mdash mdash mdash 60ndash94 0280 0600 0980 0320

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China under Grant no 11247250 no 11005071and no 10975095 and the Shanxi Provincial Natural ScienceFoundation under Grant no 2013021006

References

[1] L Adamczyk G Agakishiev M M Aggarwal et al ldquoDirectedflow of identified particles in Au+Au collisions at radic119878

119873119873=

200GeV at RHICrdquo Physical Review Letters vol 108 no 20Article ID 202301 6 pages 2012

[2] K Adcox S S Adler N N Ajitanand et al ldquoCentralitydependence of 120587+minus 119870+minus p and 119901

minus production from radic119904NN =

130GeV Au+Au collisions at RHICrdquo Physical Review Lettersvol 88 Article ID 242301 2002

[3] P S B Dev A Pilaftsis and U K Yang ldquoNew productionmechanism for heavy neutrinos at the LHCrdquo Physical ReviewLetters vol 112 no 8 Article ID 081801 5 pages 2014

[4] C Tsallis ldquoPossible generalization of Boltzmann-Gibbs statis-ticsrdquo Journal of Statistical Physics vol 52 no 1-2 pp 479ndash4871988

[5] B C Li Y ZWang FH Liu X JWen andY EDong ldquoParticleproduction in relativistic 119875119875(119875) and119860119860 collisions at RHIC andLHC energies with Tsallis statistics using the two-cylindricalmultisource thermal modelrdquo Physical Review D vol 89 ArticleID 054014 2014

[6] C Y Wong and G Wilk ldquoTsallis fits to 119901119879spectra and multiple

hard scattering in 119901119901 collisions at the LHCrdquo Physical Review Dvol 87 Article ID 114007 2013

[7] B C Li Y Z Wang and F H Liu ldquoFormulation of transversemass distributions in AundashAu collisions at radic119904

119873119873= 200

GeVnucleonrdquo Physics Letters B vol 725 no 4-5 pp 352ndash3562013

[8] M Rybczynski and Z Włodarczyk ldquoTsallis statistics approachto the transverse momentum distributions in pndashp collisionsrdquoThe European Physical Journal C vol 74 no 2 p 2785 2014

[9] F-H Liu Y-H Chen H-R Wei and B-C Li ldquoTransversemomentum distributions of final-state particles produced insoft excitation process in high energy collisionsrdquo Advances inHigh Energy Physics vol 2013 Article ID 965735 15 pages 2013

[10] B-C Li Y-Y Fu E-Q Wang L-L Wang and F-H LiuldquoTransverse momentum dependence of charged and strangehadron elliptic flows in CundashCu collisionsrdquo Journal of Physics GNuclear and Particle Physics vol 39 no 8 Article ID 0851092012

[11] A Adare S Afanasiev C Aidala et al ldquoMeasurement of1198700119878and

119870lowast0 in p+p d+Au and Cu+Cu collisions at radic119904NN = 200GeVrdquo

Physical Review C vol 90 Article ID 054905 2014[12] A Adare S Afanasiev C Aidala et al ldquoMeasurement of

neutral mesons in p+p collisions at radic119904 = 200GeV and scalingproperties of hadron productionrdquo Physical Review D vol 83Article ID 052004 2011

[13] J Adams V Eckardt J Putschke et al ldquo119870(892)lowast resonance

production in Au+Au and 119901+119901 collisions atradic119904119873119873

= 200GeVrdquoPhysical Review C vol 71 Article ID 064902 2005

[14] B I Abelev M M Aggarwal Z Ahammed et al ldquoHadronicresonance production in 119889 + Au collisions at radic119904

119873119873= 200 GeV

measured at the BNL relativistic heavy ion colliderrdquo PhysicalReview C vol 78 no 4 Article ID 044906 20 pages 2008

[15] M M Aggarwal Z Ahammed and A V AlakhverdyantsldquoKlowast0 production in Cu+Cu and Au+Au collisions at radic119904

119873119873=

624GeV and 200GeVrdquo Physical Review C vol 84 no 3 ArticleID 034909 2011

[16] A Adare S Afanasiev C Aidala et al ldquoIdentified chargedhadron production in119901+119901 collisions atradic119904 = 200 and 624GeVrdquoPhysical Review C vol 83 Article ID 064903 2011

[17] B I Abelev J Adams M M Aggarwal et al ldquoStrange particleproduction in 119901+119901 collisions atradic119904 = 200GeVrdquo Physical ReviewC vol 75 no 6 Article ID 064901 21 pages 2007

[18] G Aad B Abbott J Abdallah et al ldquoCharged-particle multi-plicities in 119901119901 interactions measured with the ATLAS detectorat the LHCrdquo New Journal of Physics vol 13 Article ID 0530332011

[19] K Aamodt N Abel U Abeysekara et al ldquoTransverse momen-tum spectra of charged particles in proton-proton collisions atradic119904 = 900GeV with ALICE at the LHCrdquo Physics Letters B vol693 no 2 pp 53ndash68 2010

[20] V Khachatryan A M Sirunyan A Tumasyan et alldquoTransverse-momentum and pseudorapidity distributions

8 Advances in High Energy Physics

of charged hadrons in pp collisions at radic119904 = 7 TeVrdquo PhysicalReview Letters vol 105 Article ID 022002 2010

[21] G Wilk and Z Włodarczyk ldquoInterpretation of the nonexten-sivity parameter q in some applications of Tsallis statistics andLevy distributionsrdquo Physical Review Letters vol 84 no 13 pp2770ndash2773 2000

[22] G Wilk and Z Wlodarczyk ldquoConsequences of temperaturefluctuations in observablesmeasured in high-energy collisionsrdquoThe European Physical Journal A vol 48 article 161 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Advances in High Energy Physics 5

MinBias times 102

0ndash20 times 10120ndash40

40ndash60 times 10minus1

times 10minus3pp data

2 4 6 8 10 12

10minus5

10minus3

10minus1

10minus13

10minus15

10minus17

10minus19

10minus11

10minus9

10minus7

101

PT (GeVc)

d + Au 200GeVK0S rarr 120587

01205870

d2N(2120587PTdPTdy)(G

eVc)minus2

60ndash88 times 10minus2

(a)

2 4 6 8 10 12

10minus5

10minus3

10minus1

10minus13

10minus15

10minus17

10minus19

10minus11

10minus9

10minus7

101

PT (GeVc)

d2N(2120587PTdPTdy)(G

eVc)minus2

(b)

2 4 6 8 10 12

10minus5

10minus3

10minus1

10minus13

10minus15

10minus17

10minus19

10minus11

10minus9

10minus7

101

PT (GeVc)

K0S rarr 12058701205870

Cu + Cu 200GeV

MinBias times 102

times 10minus20ndash20 times 220ndash60 times 02

60ndash94 times 005

d2N(2120587PTdPTdy)(G

eVc)minus2

pp data

(c)

2 4 6 8 10 12

10minus5

10minus3

10minus1

10minus13

10minus15

10minus17

10minus19

10minus11

10minus9

10minus7

101

PT (GeVc)

d2N(2120587PTdPTdy)(G

eVc)minus2

(d)

Figure 3 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 thefirst component and the second component respectively

lines denote the contributions of the first component and thesecond component respectively It is seen clearly that the softand hard interactions behave in the low and high transversemomentum of the identified particles The parameters 119879

1

1198792 and 119908 used in the calculations are given in Table 3 with

1205942dof The values of 119879

1are two to four times the values of

119879 or 1198791015840 The values of 1198792are about twice the values of 119879

1

because of the hard interaction In (9) the first componentis the contribution of soft process and the second component

is the contribution of hard processThedistribution in the lowtransverse momentum region is mainly contributed by thesoft processes The hard processes contribute high transversemomentums in the 119901

119879spectra For the two-component

distribution of the Boltzmann distribution the parameter 119908is used to denote the contribution of the soft process and 1minus 119908

is used to denote the contribution of the hard processIn summary we have compared Tsallis statistics and

the Boltzmann distribution in the analysis of the transverse

6 Advances in High Energy Physics

10minus5

10minus3

10minus1

10minus13

10minus11

10minus9

10minus7

101

1 2 3 4 5 6 7 8

MinBias times 2 times 103

0ndash20 times 102

20ndash40 times 2 times 101

40ndash60 times 2 times 100

60ndash88 times 5 times 10minus1

times 2 times 10minus1

PT (GeVc)

d + Au 200GeVKlowast0 rarr K+120587minus

Klowast0 rarr Kminus120587+

(Klowast0 + Klowast0)2

d2N(2120587PTdPTdy)(G

eVc)minus2

pp data

(a)

10minus5

10minus3

10minus1

10minus13

10minus11

10minus9

10minus7

101

1 2 3 4 5 6 7 8PT (GeVc)

d2N(2120587PTdPTdy)(G

eVc)minus2

(b)

1 2 3 4 5 6 7 8

10minus5

10minus3

10minus1

10minus13

10minus11

10minus9

10minus7

101

PT (GeVc)

MinBias times 800ndash20 times 4

40ndash60 times 05

20ndash40 times 160ndash94 times 03

times 0008

Cu + Cu 200GeVKlowast0 rarr K+120587minus

Klowast0 rarr Kminus120587+

(Klowast0 + Klowast0)2

d2N(2120587PTdPTdy)(G

eVc)minus2

pp data

(c)

1 2 3 4 5 6 7 8

10minus5

10minus3

10minus1

10minus13

10minus11

10minus9

10minus7

101

PT (GeVc)

d2N(2120587PTdPTdy)(G

eVc)minus2

(d)

Figure 4The same as Figure 2 but the solid lines are the results of (9) The dashed lines and the dotted lines denote the contributions of thefirst component and the second component respectively

momentum spectra of 1198700119878and 119870

lowast0 at midrapidity in 119889 +Au Cu + Cu and 119901 + 119901 collisions at radic119904

119873119873= 200GeV

The two methods can both describe the distribution of thefinal-state particles They have their own advantage andproper scope The two forms of Tsallis distribution canconsistently agree with the experimental points in the lowand high 119901

119879region Tsallis statistics is nonextensive statistics

[4] The parameter 119879 is temperature and the parameter 119902

summarily describes all features causing a departure fromthe Boltzmann-Gibbs statistics In [6] Var(119879)⟨119879⟩2 = 119902 minus

1 directly reflects intrinsic fluctuations of the temperatureHowever the Tsallis distribution also emerges from a numberof other dynamical mechanisms [22] The two-componentBoltzmann distribution can directly show the contribution ofthe soft interaction and the hard interaction in the observedspectra by the weight parameter 119908

Advances in High Energy Physics 7

Table 3 Values of 1198791 1198792 and 119908 taken in Figures 3 and 4 119879

1and 119879

2units are GeV

Figure Centrality 1198791

1198792

119908 1205942dof Figure Centrality 119879

11198792

119908 1205942dof

Figure 3(a)

MinBias 0380 0900 0996 1057

Figure 4(a)

MinBias 0300 0600 0977 12480ndash20 0380 0900 0998 0932 0ndash20 0310 0580 0974 089020ndash40 0390 0900 0996 0811 20ndash40 0310 0580 0950 078040ndash60 0400 0910 0996 0670 40ndash60 0310 0580 0974 060560ndash88 0410 0920 0996 0715 60ndash88 0320 0580 0940 0643pp data 0420 0930 0995 0600 pp data 0300 0600 0979 0580

Figure 3(c)

MinBias 0450 0900 0936 0858

Figure 4(c)

MinBias 0280 0580 0979 11500ndash20 0470 0920 0989 0705 0ndash20 0280 0580 0984 070020ndash60 0470 0920 0990 0426 20ndash40 0280 0580 0982 057060ndash94 0470 0920 0987 0395 40ndash60 0280 0600 0979 0352

mdash mdash mdash mdash mdash 60ndash94 0280 0600 0980 0320

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China under Grant no 11247250 no 11005071and no 10975095 and the Shanxi Provincial Natural ScienceFoundation under Grant no 2013021006

References

[1] L Adamczyk G Agakishiev M M Aggarwal et al ldquoDirectedflow of identified particles in Au+Au collisions at radic119878

119873119873=

200GeV at RHICrdquo Physical Review Letters vol 108 no 20Article ID 202301 6 pages 2012

[2] K Adcox S S Adler N N Ajitanand et al ldquoCentralitydependence of 120587+minus 119870+minus p and 119901

minus production from radic119904NN =

130GeV Au+Au collisions at RHICrdquo Physical Review Lettersvol 88 Article ID 242301 2002

[3] P S B Dev A Pilaftsis and U K Yang ldquoNew productionmechanism for heavy neutrinos at the LHCrdquo Physical ReviewLetters vol 112 no 8 Article ID 081801 5 pages 2014

[4] C Tsallis ldquoPossible generalization of Boltzmann-Gibbs statis-ticsrdquo Journal of Statistical Physics vol 52 no 1-2 pp 479ndash4871988

[5] B C Li Y ZWang FH Liu X JWen andY EDong ldquoParticleproduction in relativistic 119875119875(119875) and119860119860 collisions at RHIC andLHC energies with Tsallis statistics using the two-cylindricalmultisource thermal modelrdquo Physical Review D vol 89 ArticleID 054014 2014

[6] C Y Wong and G Wilk ldquoTsallis fits to 119901119879spectra and multiple

hard scattering in 119901119901 collisions at the LHCrdquo Physical Review Dvol 87 Article ID 114007 2013

[7] B C Li Y Z Wang and F H Liu ldquoFormulation of transversemass distributions in AundashAu collisions at radic119904

119873119873= 200

GeVnucleonrdquo Physics Letters B vol 725 no 4-5 pp 352ndash3562013

[8] M Rybczynski and Z Włodarczyk ldquoTsallis statistics approachto the transverse momentum distributions in pndashp collisionsrdquoThe European Physical Journal C vol 74 no 2 p 2785 2014

[9] F-H Liu Y-H Chen H-R Wei and B-C Li ldquoTransversemomentum distributions of final-state particles produced insoft excitation process in high energy collisionsrdquo Advances inHigh Energy Physics vol 2013 Article ID 965735 15 pages 2013

[10] B-C Li Y-Y Fu E-Q Wang L-L Wang and F-H LiuldquoTransverse momentum dependence of charged and strangehadron elliptic flows in CundashCu collisionsrdquo Journal of Physics GNuclear and Particle Physics vol 39 no 8 Article ID 0851092012

[11] A Adare S Afanasiev C Aidala et al ldquoMeasurement of1198700119878and

119870lowast0 in p+p d+Au and Cu+Cu collisions at radic119904NN = 200GeVrdquo

Physical Review C vol 90 Article ID 054905 2014[12] A Adare S Afanasiev C Aidala et al ldquoMeasurement of

neutral mesons in p+p collisions at radic119904 = 200GeV and scalingproperties of hadron productionrdquo Physical Review D vol 83Article ID 052004 2011

[13] J Adams V Eckardt J Putschke et al ldquo119870(892)lowast resonance

production in Au+Au and 119901+119901 collisions atradic119904119873119873

= 200GeVrdquoPhysical Review C vol 71 Article ID 064902 2005

[14] B I Abelev M M Aggarwal Z Ahammed et al ldquoHadronicresonance production in 119889 + Au collisions at radic119904

119873119873= 200 GeV

measured at the BNL relativistic heavy ion colliderrdquo PhysicalReview C vol 78 no 4 Article ID 044906 20 pages 2008

[15] M M Aggarwal Z Ahammed and A V AlakhverdyantsldquoKlowast0 production in Cu+Cu and Au+Au collisions at radic119904

119873119873=

624GeV and 200GeVrdquo Physical Review C vol 84 no 3 ArticleID 034909 2011

[16] A Adare S Afanasiev C Aidala et al ldquoIdentified chargedhadron production in119901+119901 collisions atradic119904 = 200 and 624GeVrdquoPhysical Review C vol 83 Article ID 064903 2011

[17] B I Abelev J Adams M M Aggarwal et al ldquoStrange particleproduction in 119901+119901 collisions atradic119904 = 200GeVrdquo Physical ReviewC vol 75 no 6 Article ID 064901 21 pages 2007

[18] G Aad B Abbott J Abdallah et al ldquoCharged-particle multi-plicities in 119901119901 interactions measured with the ATLAS detectorat the LHCrdquo New Journal of Physics vol 13 Article ID 0530332011

[19] K Aamodt N Abel U Abeysekara et al ldquoTransverse momen-tum spectra of charged particles in proton-proton collisions atradic119904 = 900GeV with ALICE at the LHCrdquo Physics Letters B vol693 no 2 pp 53ndash68 2010

[20] V Khachatryan A M Sirunyan A Tumasyan et alldquoTransverse-momentum and pseudorapidity distributions

8 Advances in High Energy Physics

of charged hadrons in pp collisions at radic119904 = 7 TeVrdquo PhysicalReview Letters vol 105 Article ID 022002 2010

[21] G Wilk and Z Włodarczyk ldquoInterpretation of the nonexten-sivity parameter q in some applications of Tsallis statistics andLevy distributionsrdquo Physical Review Letters vol 84 no 13 pp2770ndash2773 2000

[22] G Wilk and Z Wlodarczyk ldquoConsequences of temperaturefluctuations in observablesmeasured in high-energy collisionsrdquoThe European Physical Journal A vol 48 article 161 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

6 Advances in High Energy Physics

10minus5

10minus3

10minus1

10minus13

10minus11

10minus9

10minus7

101

1 2 3 4 5 6 7 8

MinBias times 2 times 103

0ndash20 times 102

20ndash40 times 2 times 101

40ndash60 times 2 times 100

60ndash88 times 5 times 10minus1

times 2 times 10minus1

PT (GeVc)

d + Au 200GeVKlowast0 rarr K+120587minus

Klowast0 rarr Kminus120587+

(Klowast0 + Klowast0)2

d2N(2120587PTdPTdy)(G

eVc)minus2

pp data

(a)

10minus5

10minus3

10minus1

10minus13

10minus11

10minus9

10minus7

101

1 2 3 4 5 6 7 8PT (GeVc)

d2N(2120587PTdPTdy)(G

eVc)minus2

(b)

1 2 3 4 5 6 7 8

10minus5

10minus3

10minus1

10minus13

10minus11

10minus9

10minus7

101

PT (GeVc)

MinBias times 800ndash20 times 4

40ndash60 times 05

20ndash40 times 160ndash94 times 03

times 0008

Cu + Cu 200GeVKlowast0 rarr K+120587minus

Klowast0 rarr Kminus120587+

(Klowast0 + Klowast0)2

d2N(2120587PTdPTdy)(G

eVc)minus2

pp data

(c)

1 2 3 4 5 6 7 8

10minus5

10minus3

10minus1

10minus13

10minus11

10minus9

10minus7

101

PT (GeVc)

d2N(2120587PTdPTdy)(G

eVc)minus2

(d)

Figure 4The same as Figure 2 but the solid lines are the results of (9) The dashed lines and the dotted lines denote the contributions of thefirst component and the second component respectively

momentum spectra of 1198700119878and 119870

lowast0 at midrapidity in 119889 +Au Cu + Cu and 119901 + 119901 collisions at radic119904

119873119873= 200GeV

The two methods can both describe the distribution of thefinal-state particles They have their own advantage andproper scope The two forms of Tsallis distribution canconsistently agree with the experimental points in the lowand high 119901

119879region Tsallis statistics is nonextensive statistics

[4] The parameter 119879 is temperature and the parameter 119902

summarily describes all features causing a departure fromthe Boltzmann-Gibbs statistics In [6] Var(119879)⟨119879⟩2 = 119902 minus

1 directly reflects intrinsic fluctuations of the temperatureHowever the Tsallis distribution also emerges from a numberof other dynamical mechanisms [22] The two-componentBoltzmann distribution can directly show the contribution ofthe soft interaction and the hard interaction in the observedspectra by the weight parameter 119908

Advances in High Energy Physics 7

Table 3 Values of 1198791 1198792 and 119908 taken in Figures 3 and 4 119879

1and 119879

2units are GeV

Figure Centrality 1198791

1198792

119908 1205942dof Figure Centrality 119879

11198792

119908 1205942dof

Figure 3(a)

MinBias 0380 0900 0996 1057

Figure 4(a)

MinBias 0300 0600 0977 12480ndash20 0380 0900 0998 0932 0ndash20 0310 0580 0974 089020ndash40 0390 0900 0996 0811 20ndash40 0310 0580 0950 078040ndash60 0400 0910 0996 0670 40ndash60 0310 0580 0974 060560ndash88 0410 0920 0996 0715 60ndash88 0320 0580 0940 0643pp data 0420 0930 0995 0600 pp data 0300 0600 0979 0580

Figure 3(c)

MinBias 0450 0900 0936 0858

Figure 4(c)

MinBias 0280 0580 0979 11500ndash20 0470 0920 0989 0705 0ndash20 0280 0580 0984 070020ndash60 0470 0920 0990 0426 20ndash40 0280 0580 0982 057060ndash94 0470 0920 0987 0395 40ndash60 0280 0600 0979 0352

mdash mdash mdash mdash mdash 60ndash94 0280 0600 0980 0320

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China under Grant no 11247250 no 11005071and no 10975095 and the Shanxi Provincial Natural ScienceFoundation under Grant no 2013021006

References

[1] L Adamczyk G Agakishiev M M Aggarwal et al ldquoDirectedflow of identified particles in Au+Au collisions at radic119878

119873119873=

200GeV at RHICrdquo Physical Review Letters vol 108 no 20Article ID 202301 6 pages 2012

[2] K Adcox S S Adler N N Ajitanand et al ldquoCentralitydependence of 120587+minus 119870+minus p and 119901

minus production from radic119904NN =

130GeV Au+Au collisions at RHICrdquo Physical Review Lettersvol 88 Article ID 242301 2002

[3] P S B Dev A Pilaftsis and U K Yang ldquoNew productionmechanism for heavy neutrinos at the LHCrdquo Physical ReviewLetters vol 112 no 8 Article ID 081801 5 pages 2014

[4] C Tsallis ldquoPossible generalization of Boltzmann-Gibbs statis-ticsrdquo Journal of Statistical Physics vol 52 no 1-2 pp 479ndash4871988

[5] B C Li Y ZWang FH Liu X JWen andY EDong ldquoParticleproduction in relativistic 119875119875(119875) and119860119860 collisions at RHIC andLHC energies with Tsallis statistics using the two-cylindricalmultisource thermal modelrdquo Physical Review D vol 89 ArticleID 054014 2014

[6] C Y Wong and G Wilk ldquoTsallis fits to 119901119879spectra and multiple

hard scattering in 119901119901 collisions at the LHCrdquo Physical Review Dvol 87 Article ID 114007 2013

[7] B C Li Y Z Wang and F H Liu ldquoFormulation of transversemass distributions in AundashAu collisions at radic119904

119873119873= 200

GeVnucleonrdquo Physics Letters B vol 725 no 4-5 pp 352ndash3562013

[8] M Rybczynski and Z Włodarczyk ldquoTsallis statistics approachto the transverse momentum distributions in pndashp collisionsrdquoThe European Physical Journal C vol 74 no 2 p 2785 2014

[9] F-H Liu Y-H Chen H-R Wei and B-C Li ldquoTransversemomentum distributions of final-state particles produced insoft excitation process in high energy collisionsrdquo Advances inHigh Energy Physics vol 2013 Article ID 965735 15 pages 2013

[10] B-C Li Y-Y Fu E-Q Wang L-L Wang and F-H LiuldquoTransverse momentum dependence of charged and strangehadron elliptic flows in CundashCu collisionsrdquo Journal of Physics GNuclear and Particle Physics vol 39 no 8 Article ID 0851092012

[11] A Adare S Afanasiev C Aidala et al ldquoMeasurement of1198700119878and

119870lowast0 in p+p d+Au and Cu+Cu collisions at radic119904NN = 200GeVrdquo

Physical Review C vol 90 Article ID 054905 2014[12] A Adare S Afanasiev C Aidala et al ldquoMeasurement of

neutral mesons in p+p collisions at radic119904 = 200GeV and scalingproperties of hadron productionrdquo Physical Review D vol 83Article ID 052004 2011

[13] J Adams V Eckardt J Putschke et al ldquo119870(892)lowast resonance

production in Au+Au and 119901+119901 collisions atradic119904119873119873

= 200GeVrdquoPhysical Review C vol 71 Article ID 064902 2005

[14] B I Abelev M M Aggarwal Z Ahammed et al ldquoHadronicresonance production in 119889 + Au collisions at radic119904

119873119873= 200 GeV

measured at the BNL relativistic heavy ion colliderrdquo PhysicalReview C vol 78 no 4 Article ID 044906 20 pages 2008

[15] M M Aggarwal Z Ahammed and A V AlakhverdyantsldquoKlowast0 production in Cu+Cu and Au+Au collisions at radic119904

119873119873=

624GeV and 200GeVrdquo Physical Review C vol 84 no 3 ArticleID 034909 2011

[16] A Adare S Afanasiev C Aidala et al ldquoIdentified chargedhadron production in119901+119901 collisions atradic119904 = 200 and 624GeVrdquoPhysical Review C vol 83 Article ID 064903 2011

[17] B I Abelev J Adams M M Aggarwal et al ldquoStrange particleproduction in 119901+119901 collisions atradic119904 = 200GeVrdquo Physical ReviewC vol 75 no 6 Article ID 064901 21 pages 2007

[18] G Aad B Abbott J Abdallah et al ldquoCharged-particle multi-plicities in 119901119901 interactions measured with the ATLAS detectorat the LHCrdquo New Journal of Physics vol 13 Article ID 0530332011

[19] K Aamodt N Abel U Abeysekara et al ldquoTransverse momen-tum spectra of charged particles in proton-proton collisions atradic119904 = 900GeV with ALICE at the LHCrdquo Physics Letters B vol693 no 2 pp 53ndash68 2010

[20] V Khachatryan A M Sirunyan A Tumasyan et alldquoTransverse-momentum and pseudorapidity distributions

8 Advances in High Energy Physics

of charged hadrons in pp collisions at radic119904 = 7 TeVrdquo PhysicalReview Letters vol 105 Article ID 022002 2010

[21] G Wilk and Z Włodarczyk ldquoInterpretation of the nonexten-sivity parameter q in some applications of Tsallis statistics andLevy distributionsrdquo Physical Review Letters vol 84 no 13 pp2770ndash2773 2000

[22] G Wilk and Z Wlodarczyk ldquoConsequences of temperaturefluctuations in observablesmeasured in high-energy collisionsrdquoThe European Physical Journal A vol 48 article 161 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Advances in High Energy Physics 7

Table 3 Values of 1198791 1198792 and 119908 taken in Figures 3 and 4 119879

1and 119879

2units are GeV

Figure Centrality 1198791

1198792

119908 1205942dof Figure Centrality 119879

11198792

119908 1205942dof

Figure 3(a)

MinBias 0380 0900 0996 1057

Figure 4(a)

MinBias 0300 0600 0977 12480ndash20 0380 0900 0998 0932 0ndash20 0310 0580 0974 089020ndash40 0390 0900 0996 0811 20ndash40 0310 0580 0950 078040ndash60 0400 0910 0996 0670 40ndash60 0310 0580 0974 060560ndash88 0410 0920 0996 0715 60ndash88 0320 0580 0940 0643pp data 0420 0930 0995 0600 pp data 0300 0600 0979 0580

Figure 3(c)

MinBias 0450 0900 0936 0858

Figure 4(c)

MinBias 0280 0580 0979 11500ndash20 0470 0920 0989 0705 0ndash20 0280 0580 0984 070020ndash60 0470 0920 0990 0426 20ndash40 0280 0580 0982 057060ndash94 0470 0920 0987 0395 40ndash60 0280 0600 0979 0352

mdash mdash mdash mdash mdash 60ndash94 0280 0600 0980 0320

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China under Grant no 11247250 no 11005071and no 10975095 and the Shanxi Provincial Natural ScienceFoundation under Grant no 2013021006

References

[1] L Adamczyk G Agakishiev M M Aggarwal et al ldquoDirectedflow of identified particles in Au+Au collisions at radic119878

119873119873=

200GeV at RHICrdquo Physical Review Letters vol 108 no 20Article ID 202301 6 pages 2012

[2] K Adcox S S Adler N N Ajitanand et al ldquoCentralitydependence of 120587+minus 119870+minus p and 119901

minus production from radic119904NN =

130GeV Au+Au collisions at RHICrdquo Physical Review Lettersvol 88 Article ID 242301 2002

[3] P S B Dev A Pilaftsis and U K Yang ldquoNew productionmechanism for heavy neutrinos at the LHCrdquo Physical ReviewLetters vol 112 no 8 Article ID 081801 5 pages 2014

[4] C Tsallis ldquoPossible generalization of Boltzmann-Gibbs statis-ticsrdquo Journal of Statistical Physics vol 52 no 1-2 pp 479ndash4871988

[5] B C Li Y ZWang FH Liu X JWen andY EDong ldquoParticleproduction in relativistic 119875119875(119875) and119860119860 collisions at RHIC andLHC energies with Tsallis statistics using the two-cylindricalmultisource thermal modelrdquo Physical Review D vol 89 ArticleID 054014 2014

[6] C Y Wong and G Wilk ldquoTsallis fits to 119901119879spectra and multiple

hard scattering in 119901119901 collisions at the LHCrdquo Physical Review Dvol 87 Article ID 114007 2013

[7] B C Li Y Z Wang and F H Liu ldquoFormulation of transversemass distributions in AundashAu collisions at radic119904

119873119873= 200

GeVnucleonrdquo Physics Letters B vol 725 no 4-5 pp 352ndash3562013

[8] M Rybczynski and Z Włodarczyk ldquoTsallis statistics approachto the transverse momentum distributions in pndashp collisionsrdquoThe European Physical Journal C vol 74 no 2 p 2785 2014

[9] F-H Liu Y-H Chen H-R Wei and B-C Li ldquoTransversemomentum distributions of final-state particles produced insoft excitation process in high energy collisionsrdquo Advances inHigh Energy Physics vol 2013 Article ID 965735 15 pages 2013

[10] B-C Li Y-Y Fu E-Q Wang L-L Wang and F-H LiuldquoTransverse momentum dependence of charged and strangehadron elliptic flows in CundashCu collisionsrdquo Journal of Physics GNuclear and Particle Physics vol 39 no 8 Article ID 0851092012

[11] A Adare S Afanasiev C Aidala et al ldquoMeasurement of1198700119878and

119870lowast0 in p+p d+Au and Cu+Cu collisions at radic119904NN = 200GeVrdquo

Physical Review C vol 90 Article ID 054905 2014[12] A Adare S Afanasiev C Aidala et al ldquoMeasurement of

neutral mesons in p+p collisions at radic119904 = 200GeV and scalingproperties of hadron productionrdquo Physical Review D vol 83Article ID 052004 2011

[13] J Adams V Eckardt J Putschke et al ldquo119870(892)lowast resonance

production in Au+Au and 119901+119901 collisions atradic119904119873119873

= 200GeVrdquoPhysical Review C vol 71 Article ID 064902 2005

[14] B I Abelev M M Aggarwal Z Ahammed et al ldquoHadronicresonance production in 119889 + Au collisions at radic119904

119873119873= 200 GeV

measured at the BNL relativistic heavy ion colliderrdquo PhysicalReview C vol 78 no 4 Article ID 044906 20 pages 2008

[15] M M Aggarwal Z Ahammed and A V AlakhverdyantsldquoKlowast0 production in Cu+Cu and Au+Au collisions at radic119904

119873119873=

624GeV and 200GeVrdquo Physical Review C vol 84 no 3 ArticleID 034909 2011

[16] A Adare S Afanasiev C Aidala et al ldquoIdentified chargedhadron production in119901+119901 collisions atradic119904 = 200 and 624GeVrdquoPhysical Review C vol 83 Article ID 064903 2011

[17] B I Abelev J Adams M M Aggarwal et al ldquoStrange particleproduction in 119901+119901 collisions atradic119904 = 200GeVrdquo Physical ReviewC vol 75 no 6 Article ID 064901 21 pages 2007

[18] G Aad B Abbott J Abdallah et al ldquoCharged-particle multi-plicities in 119901119901 interactions measured with the ATLAS detectorat the LHCrdquo New Journal of Physics vol 13 Article ID 0530332011

[19] K Aamodt N Abel U Abeysekara et al ldquoTransverse momen-tum spectra of charged particles in proton-proton collisions atradic119904 = 900GeV with ALICE at the LHCrdquo Physics Letters B vol693 no 2 pp 53ndash68 2010

[20] V Khachatryan A M Sirunyan A Tumasyan et alldquoTransverse-momentum and pseudorapidity distributions

8 Advances in High Energy Physics

of charged hadrons in pp collisions at radic119904 = 7 TeVrdquo PhysicalReview Letters vol 105 Article ID 022002 2010

[21] G Wilk and Z Włodarczyk ldquoInterpretation of the nonexten-sivity parameter q in some applications of Tsallis statistics andLevy distributionsrdquo Physical Review Letters vol 84 no 13 pp2770ndash2773 2000

[22] G Wilk and Z Wlodarczyk ldquoConsequences of temperaturefluctuations in observablesmeasured in high-energy collisionsrdquoThe European Physical Journal A vol 48 article 161 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

8 Advances in High Energy Physics

of charged hadrons in pp collisions at radic119904 = 7 TeVrdquo PhysicalReview Letters vol 105 Article ID 022002 2010

[21] G Wilk and Z Włodarczyk ldquoInterpretation of the nonexten-sivity parameter q in some applications of Tsallis statistics andLevy distributionsrdquo Physical Review Letters vol 84 no 13 pp2770ndash2773 2000

[22] G Wilk and Z Wlodarczyk ldquoConsequences of temperaturefluctuations in observablesmeasured in high-energy collisionsrdquoThe European Physical Journal A vol 48 article 161 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of