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

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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 .
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