arxiv:cond-mat/0304146v1 [cond-mat.stat-mech] 7 Apr 2003 Power law distribution of Rényi entropy for equilibrium systems having nonadditive energy Qiuping A. Wang Institut Supérieur des Matériaux du Mans, 44, Avenue F.A. Bartholdi, 72000 Le Mans, France Abstract Using Rényi entropy, an alternative statistics to Tsallis one for nonextensive systems at equilibrium is discussed. We show that it is possible to have the q-exponential distribution function for equilibrium nonextensive systems having nonadditive energy but additive entropy. PACS : 02.50.-r, 05.20.-y, 05.30.-d,05.70.-a 1 Introduction In the last decade, one of the important debates in statistics is about whether or not Boltzmann-Gibbs statistics (BGS) can be applied to nonextensive or nonadditive systems 1 with finite size or long range interaction, and whether there are alternatives to BGS for these systems. Polemics even take place within the Tsallis nonextensive statistical mechanics (NSM) to decide whether or not one should use nonadditive energy with nonadditive entropy. The reader can find a short comment on this problem in reference [1]. Rényi entropy[2] S R = ln i p q i 1 q, q 0 (1) is often associated with the study of multi-fractal and chaotic systems[3], where p i is the probability that the system is at the state labelled by i 1 A detailed discussion of these two concepts is given in [4] 1
(Boltzmann constant k=1). It is sometimes compared to Tsallis entropy[5] S T i = pq i 1, associated with a q-exponential distribution exp 1 q q (x) = [1 + (1 q)x] 1/(1 q), in the discussions of possible nonextensive statistics and the relative fundamental problem such as thermodynamic stability[6, 7, 8, 9]. Recently, Tsallis entropy is shown to be equivalent to Rényi one for systems at equilibrium[10, 11] (which is of course an open question[1]). Note that there is a monotonic relationship between these two entropies : S R = ln[1 + (1 q)st ] 1 q or S T 1 = e(1 q)sr, (2) 1 q and that, for complete probability distribution ( w i=1 p i = 1) in microcanonical ensemble, S R is identical to the Boltzmann entropy S : S R = S = ln w (3) since w i=1 p q i = w 1 q where w is the total number of states of the system. For other properties of S R, see [2, 3, 6]. Figure 1 shows the concavity of S R 0.7 0.6 q=0.1 0.5 q=1 S R (x) 0.4 q=10 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x Figure 1: The concavity of Rényi entropy S R for q > 0. 2
1 0.9 q=0.1 0.8 0.7 q=1 0.6 S T 0.5 0.4 0.3 q=5 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x Figure 2: The concavity of Tsallis entropy S T for q > 0. Due to the fact that S R is a monotonically increasing function of S T 2, they will reach the extremum together. One can hope that the maximum entropy (for q > 0[6]) will give same results with same constraints. Indeed, Rényi entropy has been used to derive, by maximum entropy method, the Tsallis q- exponential distribution within an additive energy formalism[12, 13]. Except for its extensive nature, this Rényi statistics is in some sense equivalent to the third version of Tsallis nonextensive statistics[14] with escort probability and presents the same problematics of the latter due to additive energy[1, 15]. Although 20 years ago Lesche[16] (and other authors recently[7, 17]) shown that Rényi entropy was not stable using special non-equilibrium probability distributions[18] and additive energy[7], in view of its importance in the study of chaos and fractal systems which are unusual in the sense that there is inhomogeneous space-time geometry, it would be interesting to see its behavior with nonadditive energy which, it should be noted, is consistent with the q-exponential distribution. 2 This can be illustrated by the following relationship : ds R ds = T 1+(1 q)s = dst T i pq i where i pq i is always positive. This fact should be taken into account in the study of thermodynamic stability. 3
In this paper, I will present a formal Rényi thermo-statistics for equilibrium systems with nonadditive energy required by the existence of thermodynamic equilibrium[19]. This formalism may be helpful for understanding the possible alternatives to BGS and to NSM for canonical systems with additive entropy but nonadditive energy[20] (as mentioned above, for microcanonical ensemble, S R = S, so that one should use Boltzmann statistics if entropy is additive). 2 Canonical distribution We suppose complete distribution w i=1 p i = 1 and U = w i=1 p i E i where U is the internal energy and E i the energy of the system at the state i. We will maximize as usual the following functional : We get F = ln i p q i w 1 q + α p i γ i=1 w p i E i. (4) i=1 p i [α γe i ] 1/(q 1). (5) Since S R recovers Boltzmann-Gibbs entropy S = w i=1 p i ln p i when q = 1, it is logical for us to require that Eq.(5) recovers the conventional exponential distribution for q = 1. This leads to p i = 1 Z [1 (q 1)βE i] 1/(q 1). (6) where (q 1)β = γ/α and Z = w i=1 [1 (q 1)βE i ] 1/(q 1). We will show the physical meaning of β later. Note that the second derivative d2 F = qp q 2 dp 2 i 0 for q 0. So the i stable maximum entropy is ensured for all equilibrium distributions. 3 Mixte character : nonadditive energy and additive entropy It has been shown[19] that, for thermal equilibrium to take place in nonextensive systems, the internal energy of the composite system must satisfy U(A + B) = U(A) + U(B) + λu(b)u(b) (7) 4
which means E ij (A + B) = E i (A) + E j (B) + λe i (B)E j (B) (8) where λ is a constant. Applying Eq.(8) to Eq.(6), we straightforwardly get the product joint probability for two subsystems A and B composing A + B : p ij (A + B) = p i (A)p j (B) (9) and the additivity of S R : S R (A + B) = S R (A) + S R (B). (10) If λ = (1 q)β. So S R is essentially different from S T, because in this case S T is nonadditive with S T (A + B) = S T (A) + S T (B) + (1 q)s T (A)S T (B). Note that we do not need independent or noninteracting or weakly interacting subsystems for establishing the additivity of S R or the nonadditivity of S T, as discussed in [19, 21, 22]). So in this formalism, we can deal with interacting systems with nonadditive energy but additive entropy. 4 Zeroth law and temperature It is easy to show that So we have or S R = ln Z + ln[1 + (1 q)βu]/(1 q). (11) β = [1 + (1 q)βu] SR U (12) 1 β = U (1 q)u. (13) SR Since [1 + (1 q)βu] is always positive (q-exponential probability cutoff), β has always the same sign as SR. β can be proved to be the effective inverse U temperature if we consider the zeroth law of thermodynamics. Let δs R be a small change of S R of the isolated composite system A + B. Equilibrium means δs R = 0. From Eq.(10), we have δs R (A) = δs R (B). However, 5
1 from Eq.(7), the energy conservation of A + B gives δu(a) = [1+(1 q)βu(a)] 1 δu(b). That leads to [1+(1 q)βu(b)] [1 + (1 q)βu(a)] SR (A) U(A) = [1 + (1 (B) q)βu(b)] SR U(B) (14) or β(a) = β(b) which characterizes the thermal equilibrium. 5 Some additive thermodynamic relations Due to the mixte character of this formalism with additive entropy and nonadditive energy, all the thermodynamic relations become nonlinear. In what follows, we will try to simplify this formal system and to give a linear form to this nonlinearity. Using the same machinery as in [22] which gives an extensive form to the nonextensive Tsallis statistics, we define an additive deformed energy E as follows : E = ln[1 + (1 q)βu]/(1 q)β (15) which is identical to U whenever q = 1. Note that E(A+B) = E(A)+E(B). In this way, Eq.(11) can be recast into So that β = SR. The first law can be written as E S R = ln Z + βe. (16) δe = TδS R + Y δx (17) where Y is the deformed pressure and X the coordinates (volume, surface...) and T = 1/β. The deformed free energy can be defined by F = E TS R = T ln Z, (18) so that Y = F X. The real pressure is Y R = [1 + (1 q)βu]y and the work is δw = Y R δx. The deformed heat is δq = TδS R and the real heat is δq R = [1 + (1 q)βu]δq 6
6 Grand-canonical distributions It is easy to get the grand-canonical ensemble distribution given by which gives p i = 1 Z [1 (q 1)β(E i µn i )] 1/(q 1), (19) S R = ln Z + ln[1 + (1 q)βu]/(1 q) + ln[1 (1 q)βωn]/(1 q). (20) Let M be the deformed particle number : M = ln[1 (1 q)βωn]/(1 q)βω, Eq.(20) becomes S R = ln Z + βe + βωm. (21) M must be additive, so that N is nonadditive satisfying N(A + B) = N(A) + N(B) + (1 q)βωn(a)n(b) (22) Due to the distribution function of Eq.(19), the quantum distributions will be identical to those in NSM[23]. 7 Conclusion In summary, the additive Rényi entropy is associated with a nonadditive energy to give an nonextensive statistics characterized by q-exponential distributions which are proved to be useful for many systems showing non Gaussian and power law distributions[24]. This formalism would be helpful as an alternative to Tsallis NSM for interacting nonextensive systems with additive information and entropy. I would like to emphasize in passing that the problem of the stability and observability of Rényi entropy must be considered seriously for systems at equilibrium and with nonadditive energy which is consistent with the distribution functions this entropy yields. A important point should be underlined following the result of the present work. Rényi entropy is identical to Boltzmann one for microcanonical ensemble. So, theoretically, its applicability to systems with nonadditive energy means that Boltzmann entropy may also be applied to nonextensive microcanonical systems as indicated by Gross[20]. Obviously, to the best of our knowledge, this cannot be true for canonical ensemble. 7
Acknowledgement The author thanks with great pleasure Professors S. Abe, D.H.E. Gross for valuable discussions on some points of this work and for bringing my attention to important references. References [1] Q.A. Wang, When is a quantity additive and when is it extensive, cond-mat/0303641 [2] A. Rényi, Calcul de probabilité,(paris, Dunod, 1966)P522. A. Wehrl, Rev. Mod. Phys., 50(1978)221 [3] C. Beck, Thermodynamics of chaotic systems(cambridge University Press, 1993) [4] H. Touchette, When is a quantity additive and when is it extensive, Physica A, 305(2002)84-88 [5] C. Tsallis, J. Stat. Phys., 52(1988)479 [6] E.M.F. Curado, and C. Tsallis, J. Phys. A: Math. Gen.24(1991)L69 [7] J.D. Ramshaw, Physics Letters A,198(1995)119 [8] G.R. Guerberoff, and G.A. Raggio, Phys. Lett. A, 214(1996)313 [9] T. Wada, Phys. Lett. A, 297(2002)334 [10] S. Abe, Physica A, 305(2002)62-68 [11] C. Tsallis and A.M.C. Souza, Constructing a statistical mechanics for Beck-Cohen superstatistics, cond-mat/0206044 [12] E.L. Lenzi, R.S. Mendes and L.R. da Silva, Physica A, 280(2000)337 [13] A.G. Bashkirov and A.V. Vityazev, Physica A, 277(2000)136 8
[14] F. Pennini, A.R. Plastino and A. Plastino, Physica A, 258(1998)446 C. Tsallis, R.S. Mendes and A.R. Plastino, Physica A, 261(1999)534; Silvio R.A. Salinas and C. Tsallis, Brazilian Journal of Physics(special issue: Nonadditive Statistical Mechanics and Thermodynamics), 29(1999). [15] Q.A. WANG, Euro. Phys. J. B, 26(2002)357 [16] B. Lesche, J. Stat. Phys., 27(1982)419 [17] S. Abe, Phys. Rev. E, 66(2002)046134 [18] S. Abe, private communication [19] Q.A. WANG, L. Nivanen, A. Le Méhauté and M. Pezeril, J. Phys. A, 35(2002)7003 [20] D.H.E Gross, Nonextensive hamiltonian systems follow Boltzmann s principle not Tsallis statistics-phase transition, second law of thermodynamics, Physica A, 305(2002)99-105 D.H.E Gross, Micro-canonical statistical mechanics of some nonextensive systems, Chaos, Solitons & Fractals, 13(2002)417-430 [21] Q.A. WANG, Phys. Lett. A, 300(2002)169 [22] Q.A. WANG and A. Le Méhauté, J. Math. Phys, 43(2002)5079 [23] Q.A. WANG, Quantum gas distributions prescribed by factorisation hypothesis of probability, Chaos, Solitons & Fractals, 14(2002)765 [24] Updated references concerning NSM can be found at http://tsallis.cat.cbpf.br/biblio.htm 9