Thermodynamic Derivation of the Tsallis and R\'enyi Entropy Formulas and the Temperature of Quark-Gluon Plasma

We derive Tsallis entropy, Sq, from universal thermostat independence and obtain the functional form of the corresponding generalized entropy-probability relation. Our result for finite thermostats interprets thermodynamically the subsystem temperature, T1, and the index q in terms of the temperature, T, entropy, S, and heat capacity, C of the reservoir as T1 = T exp(-S/C) and q = 1 - 1/C. In the infinite C limit, irrespective to the value of S, the Boltzmann-Gibbs approach is fully recovered. We apply this framework for the experimental determination of the original temperature of a finite thermostat, T, from the analysis of hadron spectra produced in high energy collisions, by analyzing frequently considered simple models of the quark-gluon plasma.

We derive Tsallis entropy, Sq, from universal thermostat independence and obtain the functional form of the corresponding generalized entropy-probability relation. Our result for finite thermostats interprets thermodynamically the subsystem temperature, T1, and the index q in terms of the temperature, T , entropy, S, and heat capacity, C of the reservoir as T1 = T exp(−S/C) and q = 1 − 1/C. In the infinite C limit, irrespective to the value of S, the Boltzmann -Gibbs approach is fully recovered. We apply this framework for the experimental determination of the original temperature of a finite thermostat, T , from the analysis of hadron spectra produced in high energy collisions, by analyzing frequently considered simple models of the quark-gluon plasma. A nonlinear entropy formula has been suggested by Rényi long ago and been applied to several areas in physics [1][2][3][4][5][6][7]. Another formula, the Tsallis entropy, has more recently been promoted as the keystone for a generalized thermodynamics, treating correlated physical systems [8][9][10][11]. A respectable amount of papers applying this idea to one or the other area in physics appeared [12][13][14][15][16][17]. Since from this entropy the canonical energy distribution is power-law tailed in place of the Boltzmann -Gibbs exponential, numerous high-energy distributions have been fitted using the Tsallis formula [18][19][20][21][22][23][24][25][26][27]. Its independence from the thermostat and the thermodynamical foundation behind the use of such a formula are interesting questions.
In our earlier works we investigated some general mathematical properties of alternative entropy formulas via their pairwise composition rules, and established that a scaled repetition of an arbitrary composition rule leads to an associative asymptotic composition rule of large subsystems [28]. All such rules are uniquely defined by a strict monotonic function, their formal logarithm. Recently we have also observed that -in connection to the zeroth law of the thermodynamics -the factorizability condition on the common entropy maximum [29] allows only for such rules [30]. We seek in this paper for the thermodynamical meaning of the q parameter generalizing the classical entropy formula, valid for q = 1. Some q = 1 parameter was calculated theoretically [31][32][33].
We found the thermodynamical interpretation of the entropy formula and its parameters on the analysis of the two-body thermodynamics of a single observed subsystem and a reservoir. For finite systems the microcanonical approach is the key to the physical interpretation. In the classical treatment subleading terms in a finite-energy expansion of the microcanonical entropy maximum are usually ignored, the reservoir is treated as constant in the canonical limit. A notable exception is the analysis of statistical fluctuations and their scaling in the thermodynamical limit [34][35][36][37][38][39][40]. We consider the correlation between subsystem and reservoir induced by the conservation of total energy while maximizing a monotonic function of the Boltzmann -Gibbs entropy, L(S). We seek for that very function, L, which counteracts finite size effects beyond the usual linear term, −βE i , in the Taylor-expansion of the L(S) = max principle.
We discuss now the thermal equilibrium of two systems, one with energy E 1 (subsystem) and the other with energy E − E 1 (reservoir), while their respective entropy contributions are combined by the general rule satisfying L(S 12 ) = L(S 1 ) + L(S 2 ). (1) Here we do not assume that the deviation from the simple additive rule would be small. For the sake of simplicity we consider here homogeneous rules, relevant for the cases when subsystem and reservoir are composed from the same matter (for details see Ref. [30]). The microcanonical condition for a maximal entropy state then defines the thermodynamical inverse temperature, requiring Varying the subsystem energy, E 1 , while keeping the total energy E fixed, we describe the thermal contact between subsystem and reservoir. This means that the derivative with respect to E 1 of the above expression (2) vanishes. Owing to the two E 1 -dependent contributions, it is equivalent to the statement that This equality, when taken in the E E 1 limit, usually defines the canonical approach. Now we would like to take into account effects to higher order in E 1 /E, and require that their leading term vanishes on the right hand side. The reservoir's entropy on the right hand side is Taylor-expanded: Collecting the coefficients of E 1 we arrive at

arXiv:1208.2533v2 [hep-ph] 21 Sep 2012
Here the first term on the right hand side is the familiar canonical (E 1 -independent) Lagrange multiplier, constituting the β 1 = β relation. Our key addition to the usual treatment is to require that the coefficient of the linear term in eq.(4) vanishes: This is a constraint for the L(S) function in general. Obviously, without considering L(S), the whole coefficient consisted only of S (E) as in the traditional approach, and nothing further could be done. We obtain the following condition: Since the left hand side of eq.(6) is a function of S, while the right hand side is a function of E, the left hand side must be treated as an S-independent constant by solving eq.(6) for L(S). This Universal Thermostat Independence (UTI) reads as The particular solution with L (0) = 1 and L(0) = 0 is given by The derivatives of the S(E) equation of state do have physical meaning: S (E) = 1/T and S (E) = −1/CT 2 are related to the traditional temperature and heat capacity of the reservoir. By using this we obtain a = 1/C.
The non-additivity parameter is simply the inverse heat capacity of the reservoir. For C → ∞ one has a → 0 and L(S) → S, so the Boltzmann -Gibbs formula is included by this limit. A connection between Tsallis entropy and constant heat capacity of the reservoir has been observed years ago [41,42]. Our result shows the background for this observation. The philosophy behind our approach is first to decide on the entropy formula by choosing L(S) generally and then to solve the maximization problem in terms of subsystem energies and corresponding probabilities. The knowledge gained from this analysis now will be generalized. Analog to a Gibbs ensemble, we extend a sum of two to a weighted sum of many. In this way the result of the two-body analysis in the form L(S 1 ) generalizes the classical entropy formula, S = − i P i ln P i , to This L-additive form of the generally non-additive entropy leads to the Tsallis entropy formula when applying eq. (8). In this way one obtains Here the coefficient of the second order correction, O E 2 1 /E 2 , vanishes for a = 1/C(E). By this we are led to the following entropy expression Our result applied to a Gibbs-ensemble with the relative occurrence frequency P i of states with energy E i , hence reads as Substituting eq.(12) we finally arrive at With the widespread notation q = 1 − a one obtains the Tsallis entropy formula It is suggestive to consider its inverse function, L −1 according to eq.(10). This delivers the Rényi entropy: Now the parameters β and a, defined in eqs. (5) and (9), are set by the physics of the finite-energy reservoir. The sign of the heat capacity, C, determines whether the q is smaller or larger than one. It may possibly carry an interesting message for the description of gravitating systems, with negative heat capacity. We proceed by noting that maximizing S Tsallis with respect to the P i weights of system instances with energy E i one obtains the canonical cut power-law distribution of energies: Using eqs.(5), (8), and (9) we rewrite this in the equivalent form, expressing the energy distribution in terms of the temperature, T , entropy, S and heat capacity, C of the ideal reservoir. The partition sum Z, obtained from normalization, is related to the Tsallis-entropy, L(S 1 ), and energy, E 1 , of the subsystem via its deformed logarithm: In the infinite heat capacity limit, irrespective to the value of S, formula (18) recovers the exponential distribution. The inverse logarithmic slope of the energy distribution, derived from it, is linear: with T 0 = T e −S/C Z 1/C (1 − 1/C). One concludes that the generalized entropy formula leads to a cut power-law energy distribution, based on a finite heat capacity reservoir. As such, it is a better approximation to the microcanonical distribution, than the canonical exponential. We demonstrate the usefulness of the above general results on the example of the thermal model to heavy-ion collisions. Experimental data from RHIC AuAu collision at 200 GeV deliver different T slope -s extrapolated to p T = 0 for different hadrons [19,43]. Considering that the energy at zero momentum is the rest mass in c = 1 units, a linear trend shows in the T slope (m i ) values, as seen in Fig.1. The open circles correspond to mesons, the filled ones to baryons in this figure. The steepness for mesons and baryons seem to be in the proportion 2 : 3 suggesting a quark coalescence hadronization picture, compatible with the factorization assumption P hadron (E) = P K i (E/K) with K = 2 and K = 3 for mesons and baryons, respectively. This scaling is acceptable and leads to T hadron slope (E) = T quark slope (E/K), with the common T 0 ≈ 48 MeV intersect in the formula (20). The valence quark matter heat capacity at RHIC AuAu collision tends to be C ≈ 4.5. Similar trends can be extracted from the analysis of fits to the ALICE data in 900 GeV pp collisions done in [44]: the values for the Tsallis-slope parameters, T 0 , are much lower than the canonical QCD phase transition temperature. Here we re-plotted the tabulated values given in [44] using the coalescence quark assumption, denoting mesons by open square boxes while baryons by filled boxes. We note, however, that these fits were performed in the very low p T range ( p T < 2.5 GeV/c) only, therefore the uncertainty of the fitted parameters is large.
In order to interpret this surprisingly low value for T 0 , we have to consider physical models of a finite thermostat, and calculate T 1 = 1/β 1 = T e −S/C , since lim C→∞ T 0 = T 1 for small subsystems in large reservoirs.
First we study the Stefan -Boltzmann formula supplemented with a bag constant, E/V = σT 4 +B in a volume V . Since the pressure is given by p = 1 3 σT 4 − B, the entropy is S = 4 3 σV T 3 . The heat capacity is the derivative of the energy with respect to temperature, At constant volume, V , this gives C V = 4σV T 3 = 3S and T 1V = T e −1/3 . At constant pressure the temperature cannot change in this model, so C p = ∞ and T 1P = T . Furthermore, considering an adiabatically expanding reservoir, a more realistic scenario in high-energy experiments, one deals with the heat capacity at constant entropy, C S = 3S(1 − T 4 * /T 4 )/4, with T * being the temperature where the pressure vanishes. In this case C S ≤ 3S/4 and T 1S ≤ T e −4/3 is the theoretical prediction. Figure 1 presents the inverse logarithmic slope, T slope , as a function of hadron masses and T 1 -lines for different physical models of the thermostat. Besides the three above described bag-model approaches we also indicate the classical Schwarzschild black hole, having C = −2S and T 1 = T e 1/2 . One inspects that this possibility is far from all experimental observations.
We note that theoretically a really constant heat capacity, C 0 , stems from the equation of state S(E) = C 0 ln(1 + E C0T0 ). The latter is a good ansatz for an effective equation of state of classical non-abelian gauge field systems on the lattice [45] and represents the high-E limit of Planck's S (E)-formula for thermal radiation.
Considering heat capacity in the above scenarios and a standard numerical value of T ≈ 167 MeV for the reservoir temperature, conjectured for the QGP at hadronization phenomenology and determined by lattice QCD calculations, one obtains T 1P = T = 167 MeV, T 1V = T e −1/3 ≈ 120 MeV and T 1S ≤ T e −4/3 ≈ 45 MeV characterizing the Tsallis-distribution of valence quarks.
The conjecture that in heavy ion collisions a statistical power-law energy distribution due to finite phase space availability corrections to the traditional canonical distribution may appear is further supported by the observation that the measure of non-additivity, a = 1/C, expressed by the inverse power in the fitted power-law tail, is reduced for increasing participant number [46]. The fitted power C is also tendentiously smaller in e + e − or pp than in heavy ion collisions [21]. Finally we realize that only the adiabatic scenario for the quark matter thermostat leads to T 0 values near to the ones extracted from experimental analysis by the coalescence assumption, T 1S ≈ T 0 ≈ 45 − 55 MeV.
In conclusion the Tsallis entropy formula is derived as the consequence of the following requirement: we seek for that non-additive entropy composition rule which cancels linearly energy-dependent corrections due to the finite E − E 1 energy in the reservoir to a subsystem's thermodynamical inverse temperature. This determines the composition rule and the entropy formula uniquely turns out to be the Tsallis entropy. This derivation explains the particular functional form of the Tsallis and Rényi for- mulas as generalized entropy expressions satisfying the UTI principle. With regard to the physical interpretation we have obtained q = 1 − 1/C, with C being the heat capacity of the total system with the conserved energy E. The canonical temperature of the subsystem becomes T 1 = e −S/C T with T (E) and S(E) being the traditional temperature and entropy of the finite reservoir, respectively. A preliminary analysis of experimental data on particle production seems to be sensitive to different physical assumptions about a QGP thermostat. Here the isentropic scenario performs best.