Equivalence of the Phenomenological Tsallis Distribution to the Transverse Momentum Distribution of Q-Dual Statistics

Eur. Phys. J. A (2020) 56:106 https://doi.org/10.1140/epja/s10050-020-00117-9

Letter

Equivalence of the phenomenological Tsallis distribution to the transverse momentum distribution of q-dual statistics

A. S. Parvan1,2,a 1 Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Russia 2 Department of Theoretical Physics, Horia Hulubei National Institute of Physics and Nuclear Engineering, Bucharest-Magurele, Romania

Received: 29 October 2019 / Accepted: 9 March 2020 / Published online: 6 April 2020 © The Author(s) 2020 Communicated by Tamas Biro

Abstract In the present work, we have found that the was proposed in another format (see Refs. [15,16]) phenomenological Tsallis distribution (which nowadays is ⎡ ⎤ q ∞ 1−q largely used to describe the transverse momentum distribu- p2 + m2 1 dσ ⎣ L T ⎦ tions of hadrons measured in pp collisions at high ener- = cpT dpL 1 − (1 − q) , (2) σ dpT T gies) is consistent with the basis of the statistical mechan- 0 ics if it belongs to the q-dual nonextensive statistics instead where pL is the longitudinal momentum. However, further of the Tsallis one. We have defined the q-dual statistics we will discuss only the function (1)asitiswidelyusedin based on the q-dual entropy which was obtained from the high-energy physics. Tsallis entropy under the multiplicative transformation of In high-energy physics, the phenomenological transverse → / the entropic parameter q 1 q. We have found that the momentum distribution (1) is associated with the Tsallis phenomenological Tsallis distribution is equivalent to the statistics based on the Tsallis entropy [17] transverse momentum distribution of the q-dual statistics in q − the zeroth term approximation. Since the q-dual statistics pi pi S = , pi = 1, (3) is properly defined, it provides a correct link between the 1 − q i i phenomenological Tsallis distribution and the second law of thermodynamics. where pi is the probability of the i-th microscopic state of the system and q ∈ R is a real parameter taking values 0 < q < ∞ Now, the phenomenological single-particle Tsallis distribu- . Note that there is also another supplementary definition tion [1,2] is used for the description of the experimental data of the statistical entropy named as the escort entropy, which on the transverse momentum distributions of hadrons created is written in the form as [18,19] in the proton-proton and heavy-ion collisions at RHIC and / −q 1 − p1 q LHC energies [3–14]. The phenomenological Tsallis trans- i i SE = . (4) verse momentum distribution for the Maxwell–Boltzmann q − 1 statistics of particles introduced in Refs. [1,2] has the form However, in the present study, we do not use it, because the escort entropy (4) is not suitable for the search of the origin d2 N gV of the transverse momentum distribution (1). = p m y 2 T T cosh dpT dy (2π) The Tsallis generalized statistical mechanics is con- q − structed on the base of the statistical entropy (3), different mT cosh y − μ 1 q × 1 − (1 − q) , (1) definitions of the expectation values of thermodynamic quan- T tities and the second law of thermodynamics [17]. There is at least three classification schemes of the Tsallis statistical where m is the transverse mass of particles. Initially, the T mechanics depending on the choice of the definition of the phenomenological Tsallis transverse momentum distribution expectation values of the thermodynamic quantities [18]. The Tsallis-1 statistics deals with the standard linear definition of = the expectation values, A i pi Ai . The Tsallis-2 statis- a e-mail: [email protected] (corresponding author) tics is defined on the base of the generalized expectation val- 123 106 Page 2 of 5 Eur. Phys. J. A (2020) 56 :106

= q < < ∞ → / ues of the thermodynamic quantities, A i pi Ai , and to 1 q under the rule q 1 q. The comprehensive the Tsallis-3 statistics is associated with the expectation val- description of the properties of the Tsallis entropy (3) can be = q / q ues with the escort probabilities as A i pi Ai i pi . found, for example, in Ref. [19]. In spite of the success of the function (1) in the applica- The statistical mechanics based on this entropy is defined tions its connection with the basis of the statistical mechanics by Eq. (5) with the probabilities pi of the microstates of the is still under the question. The phenomenological single- system normalized to unity particle Tsallis distribution (1) was obtained using the ϕ = − = method of the maximization of generalized entropy of the pi 1 0(6) ideal gas instead of the general Tsallis entropy (3). This i method is correct only for the usual Boltzmann-Gibbs statis- and by the standard expectation values tics, but for the Tsallis statistics it is under the question A= p A . (7) [20]. Nevertheless, the correct connection of the distribu- i i i tion function (1) with the Tsallis entropy (3) was established in Ref. [20]. In that paper, it was rigorously demonstrated Here, the definition of the expectation values (7) is consistent that the phenomenological transverse momentum distribu- with probability normalization constraint (6). Let us consider tion (1) corresponds to the zeroth term approximation of the the grand canonical ensemble. The thermodynamic potential Ω Tsallis-2 statistics. In the case of massive particles, this result of the grand canonical ensemble is was confirmed in Ref. [21]. However, the Tsallis-2 statistics Ω =H−TS− μN ⎡ ⎤ is improperly defined as the generalized expectation values 1 −1 = q p q − 1 of the thermodynamic quantities, A i p Ai ,inthis ⎣ i ⎦ i = pi Ei − μNi − Tq , (8) formalism are not consistent with probability normalization q − 1 = i condition, i pi 1 (see Ref. [18]). This means that the = Tsallis-2 statistics disagrees with the probability theory and where H i pi Ei is the mean energy of the system, = the second law of thermodynamics. Thus considering the N i pi Ni is the mean number of particles of the sys- phenomenological transverse momentum distribution (1)as tem, and Ei and Ni are the energy and the number of particles, a single-particle distribution function of the Tsallis statistics respectively, in the ith microscopic state of the system. based on the Tsallis entropy (3) is a serious lack, because in The unknown probabilities {pi } are obtained from the this case the distribution (1) is obtained on the base of the second law of thermodynamics (the principle of maximum erroneous general premises. entropy). In the grand canonical ensemble they are found In the present study we demonstrate that this problem of from the constrained local extrema of the thermodynamic connection of the phenomenological transverse momentum potential (8) by the method of the Lagrange multipliers (see, distribution (1) with the basis of the statistical mechanics for example, Refs. [22–24]): can be uniquely solved by introducing the new nonextensive Φ = Ω − λϕ, (9) statistics on the base of the q-dual entropy instead of the ∂Φ Tsallis entropy (3). = 0, (10) ∂p q-dual statistics The q-dual entropy is obtained from the i Tsallis entropy (3) by the multiplicative transformation q → where Φ is the Lagrange function and λ is an arbitrary real 1/q. Then, we have constant. Substituting Eqs. (6) and (8) into Eqs. (9), (10) and using Eq. (6) again, we obtain the normalized equilibrium 1/q p − pi probabilities of the grand canonical ensemble of the q-dual S = q i , (5) q − 1 statistics as i q − − Ei + μNi 1 q where q ∈ R is a real parameter taking values 0 < q < pi = 1 + (1 − q) (11) T ∞. In the Gibbs limit q → 1, the entropy (5) recovers the =− and Boltzmann-Gibbs-Shannon entropy, S i pi ln pi .The q properties of the q-dual entropy are the same as those of the − E + μN 1−q 1 + (1 − q) i i = 1, (12) Tsallis entropy (3) but at different values of the parameter q. T The values of the Tsallis entropy for q < 1 correspond to i the values of entropy (5)forq > 1 and, inversely, the values where ≡ λ−T and ∂ Ei /∂pi = ∂ Ni /∂pi = 0. In the Gibbs of entropy (3)forq > 1 correspond to the values of the limit q → 1, the probability pi = exp[( − Ei + μNi )/T ], < =− q-dual entropy for q 1. Therefore, for the entropy (5)the where T ln Z is the thermodynamic potential of grand = [−( − μ )/ ] properties of the Tsallis entropy (3) are completely preserved canonical ensemble and Z i exp Ei Ni T is with respect to the replacement of the ranges of 0 < q < 1 the partition function. 123 Eur. Phys. J. A (2020) 56 :106 Page 3 of 5 106

Substituting Eq. (11)into(7), we obtain the statistical aver- tics of particles in the grand canonical ensemble. The trans- ages of the q-dual statistics in the grand canonical ensemble verse momentum distribution of particles is related to the as mean occupation numbers of the ideal gas as q − + μ 1−q 2π Ei Ni 2 A= Ai 1 + (1 − q) , (13) d N V = dϕp ε n σ , (20) T ( π)3 T p p i dpT dy 2 σ 0 where the norm function is the solution of Eq. (12). Note ε = that the quantities (11)–(13)oftheq-dual statistics exactly where p mT cosh y is the single-particle energy, pT and y recover the corresponding quantities of the Tsallis-1 statis- are the transverse momentum and rapidity variables, respec- = 2 + 2 tics under the multiplicative transformation of the entropic tively, and mT pT m . The mean occupation numbers parameter q → 1/q (see Ref. [23]). npσ are calculated from Eq. (18) using the mean occupation Let us rewrite the norm equation (12) and the statistical numbers of the ideal gas of the Boltzmann-Gibbs statistics: averages (13) in the integral representation in the case when β = 1 . the parameter q > 1. Using the integral representation of npσ G β(ε −μ) (21) e p + η the Gamma-function [25], we can rewrite the norm equation (12)forq > 1inthefollowingform Considering Eqs. (18), (20) and (21), we obtain ∞ −Ω (β) ∞ − +( − ) G 2 1 1 t 1 1 q T d N gV 1 q−1 = p m cosh y t e dt = 1(14) ( π)2 T T q dpT dy 2 !Γ q Γ n=0 n q−1 q−1 0 ∞ n 1 −t 1+(1−q) −β ΩG β or × q−1 T , t e β( −μ) dt (22) e mT cosh y + η ∞ ∞ 0 1 1 −t 1+(1−q) t q−1 e T q where = n!Γ n 0 q−1 0 1 η n −β (εp−μ) −β ΩG β dt = 1, (15) − β ΩG β = ln 1 + ηe (23) p,σ where β = t(q − 1)/T and is the thermodynamic potential for the relativistic ideal gas 1 Ω β =− Z β , of the Boltzmann–Gibbs statistics of microstates and εp = G β ln G (16) 2 + 2 η =− , , p m . The norm function for 1 0 1 is calcu- −β (Ei −μNi ) ZG β = e . (17) lated from the norm equation (15)usingEq.(23). i Maxwell–Boltzmann statistics of particles. In the limit The statistical averages (13) in the integral representation for η → 0, the thermodynamic potential of the ideal gas of the q > 1 can be rewritten as Boltzmann-Gibbs statistics (23) can be written as ∞ −Ω (β) 2 − +( − ) G gV m β μ 1 1 t 1 1 q T Ω β =− e K β m , (24) q−1 G 2 2 2 A= t e AG β dt 2π β q Γ − q 1 0 where Kν(z) is the modiﬁed Bessel function of the second ∞ ∞ kind. Substituting Eq. (24)into(15), we obtain the norm 1 1 −t 1+(1−q) = t q−1 e T equation for the Maxwell-Boltzmann statistics of particles !Γ q n=0 n q−1 as 0 n ∞ ∞ × −β ΩG β AG β dt, (18) n +μn ω 1 1 −n −t 1+(1−q) t q−1 e T where n! q n=0 Γ − q 1 0 1 −β( −μ ) β = Ei Ni . ( − ) n A G Ai e (19) t q 1 m Z (β ) K2 dt = 1, (25) G i T Equation (18) links the statistical averages of the q-dual where statistics with the corresponding statistical averages (19)of gVTm2 1 the Boltzmann-Gibbs statistics. ω = . (26) π 2 − Transverse momentum distribution Let us consider the 2 q 1 relativistic ideal gas with the Fermi–Dirac (η = 1), Bose– Substituting Eq. (24)into(22) and taking the limit η = Einstein (η =−1) and Maxwell–Boltzmann (η = 0) statis- 0, we obtain the transverse momentum distribution for the 123 106 Page 4 of 5 Eur. Phys. J. A (2020) 56 :106

Maxwell–Boltzmann statistics of particles as term approximation of both the q-dual nonextensive statis- ∞ tics (cf. Eqs. (31) and (33) of Ref. [1] along with Eq. (29)of d2 N gV ωn 1 = p m y the present paper) and the Tsallis statistics [21]. 2 T T cosh dpT dy (2π) n! Γ q To summarize, we have introduced the q-dual entropy n=0 q−1 ∞ which is defined from the Tsallis entropy by the multiplica- −m cosh y+μ(n+1) → / 1 −n −t 1+(1−q) T tive transformation of the entropic parameter q 1 q. × t q−1 e T The q-dual nonextensive statistics based on this statistical 0 entropy and consistently defined expectation values of ther- n t(q − 1)m modynamic quantities was formulated in the framework of × K dt. (27) 2 T the grand canonical ensemble. The thermodynamic potential was obtained from the fundamental thermodynamic potential Zeroth term approximation Let us find the transverse by the Legendre transform. The probabilities of microstates momentum distribution of particles in the zeroth term of the system were derived from the second law of ther- approximation, which was introduced in Ref. [20]. In this modynamics (the Jaynes maximum entropy principle) by approximation we retain only the zeroth term (n = 0) in the the constrained local extrema of the statistical thermody- series expansion. Taking n = 0inEq.(15), we obtain that namic potential. The norm equation and statistical averages the norm function = 0. Substituting = 0 into Eq. (22) were expressed analytically in terms of the corresponding and considering only the zeroth term and the equation Boltzmann–Gibbs quantities using the integral representa- ∞ 1 − ( + ) tion and the exponential function containing the thermo- = (−η)ke x k 1 , (28) ex + η dynamic potential was expanded into the series. We have k=0 obtained the exact analytical formulae for the transverse where |e−x | < 1, we obtain the transverse momentum dis- momentum distributions of the relativistic massive particles tribution in the zeroth term approximation as following the Bose–Einstein, Fermi–Dirac and Maxwell– Boltzmann statistics in the framework of q-dual statistics d2 N gV in the grand canonical ensemble. = p m y 2 T T cosh dpT dy (2π) In the present study, we have found that the phenomeno- ∞ q − μ 1−q logical Tsallis distribution for the Maxwell–Boltzmann × (−η)k − ( + )( − ) mT cosh y 1 k 1 1 q statistics of particles introduced in Refs. [1,2] is equiva- = T k 0 lent to the transverse momentum distribution of the q-dual for η =−1, 0, 1 (29) nonextensive statistics in the zeroth term approximation. This or demonstrates that the correct link between the phenomenological Tsallis distribution and the fundamental theory of d2 N gV = p m y the statistical mechanics is provided by the q-dual entropy 2 T T cosh dpT dy (2π) instead of the Tsallis one. The q-dual statistics is properly q m cosh y − μ 1−q defined in comparison with the Tsallis-2 statistics as the × 1 − (1 − q) T T expectation values of this formalism are consistent with prob- for η = 0. (30) ability normalization condition. Thus we have found that the well-known phenomenological single-particle Tsallis distri- The transverse momentum distribution (30) exactly coin- bution is thermodynamically founded only if it belongs to cides with the phenomenological Tsallis distribution (1) the q-dual nonextensive statistics instead of the Tsallis one. introduced in [1,2] (see Eq. (56) in Ref. [1]). Thus, the phenomenological Tsallis distribution (1) for the Maxwell– Acknowledgements This work was supported in part by the joint Boltzmann statistics of particles exactly corresponds to research projects of JINR and IFIN-HH. the zeroth term approximation of the q-dual nonextensive Data Availability Statement This manuscript has no associated data statistics based on the entropy (5). Moreover, the trans- or the data will not be deposited. [Authors’ comment: There is no data verse momentum distribution (30)oftheq-dual nonexten- to analyze.]. sive statistics recovers the transverse momentum distribu- Open Access This article is licensed under a Creative Commons Attri- tion of the Tsallis-1 statistics in the zeroth term approxi- bution 4.0 International License, which permits use, sharing, adaptation, mation under the transformation of the entropic parameter distribution and reproduction in any medium or format, as long as you q → 1/q [20,21]. However, the phenomenological Tsal- give appropriate credit to the original author(s) and the source, pro- vide a link to the Creative Commons licence, and indicate if changes lis distributions for the quantum (Fermi–Dirac and Bose– were made. The images or other third party material in this article Einstein) statistics of particles introduced in Ref. [1] do not are included in the article’s Creative Commons licence, unless indi- recover the transverse momentum distributions in the zeroth cated otherwise in a credit line to the material. If material is not 123 Eur. Phys. J. A (2020) 56 :106 Page 5 of 5 106 included in the article’s Creative Commons licence and your intended 10. L. Marques, J. Cleymans, A. Deppman, Phys. Rev. D 91, 054025 use is not permitted by statutory regulation or exceeds the permit- (2015) ted use, you will need to obtain permission directly from the copy- 11. T. Bhattacharyya, J. Cleymans, A. Khuntia, P. Pareek, R. Sahoo, right holder. 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