Why Q-Expectation Values Must Be Used in Nonextensive Statistical Mechanics

Astrophys Space Sci (2006) 305:241Ð245 DOI 10.1007/s10509-006-9198-5

ORIGINAL ARTICLE

Why q-Expectation Values Must be Used in Nonextensive Statistical Mechanics

Sumiyoshi Abe

Received: 2 May 2006 / Accepted: 2 June 2006 C Springer Science + Business Media B.V. 2006

Abstract There is a controversy in the area of nonextensive de Moura et al., 2000; Baldovin and Robledo, 2002; Johal statistical mechanics regarding the form of the expectation and Tirnakli, 2004), lattice Boltzmann models (Boghosian value of a physical quantity. Two definitions have been dis- et al., 2003), magnetism of colossal magnetoresistance man- cussed in the literature: one is the ordinary definition and the ganites (Reis et al., 2002), high-energy processes (Walton other is the normalized q-expectation value associated with and Rafelski, 2000; Bediaga et al., 2000; Beck, 2000; the escort distribution. Here, it is proved that the normalized Alberico et al., 2000; Navarra et al., 2003), quantum groups q-expectation value is the correct one to be employed. The (Abe, 1997, 1998, 2003b), cellular aggregates (Upadhyaya Shore-Johnson theorem is used to show that the formalism et al., 2001), L«evy flights (Prato and Tsallis, 1999; Abe and with the normalized q-expectation value is theoretically con- Rajagopal, 2000a), semiclassical dynamics in optical lat- sistent with minimum cross entropy principle, whereas the tices (Lutz, 2004), kinetics of charged particles (Rossani and ordinary expectation value has to be excluded. Scarfone, 2000), Internet traffic (Abe and Sukuki, 2003a), earthquakes (Abe and Suzuki, 2003a, 2005; Sotolongo- Keywords q-expectation values . nonextensive statistical Costa and Posadas, 2004), econophysical problems mechanics (Borland, 2002a,b; Kozuki and Fuchikami, 2003), and com- plex networks (Tadic and Thurner, 2004; Abe and Sukuki, 2004a; Wilk and Wlodarczyk, 2004). 1. Introduction In spite of these successes, the theoretical foundations of nonextensive statistical mechanics are yet to be established Nonextensive statistical mechanics (Abe and Okamoto, in some respects. Among others, the problem concerning the 2001; Kaniadakis et al., 2002; Gell-Mann and Tsallis, 2004; definition of expectation value of a physical quantity seems Kaniadakis and Lissia, 2004) pioneered by Tsallis (1988) to be poorly understood. There are two different definitions has been attracting continuous interest over the years. It is in the literature. One is the frequently employed definition, expected to offer a unified framework for describing com- which is the normalized q-expectation value (Tsallis et al., plex systems in their nonequilibrium stationary states, sys- 1998): tems with (multi)fractal and self-similar structures, long- range interacting systems, anomalous diffusion phenom- (nor) U =Hq = Pi εi , (1) ena, and so on. The worked examples include cosmic rays i (Tsallis et al., 2003), astrophysics and self-gravitating sys- (p )q tems (Lavagno et al., 1998; Taruya and Sakagami, 2003a,b), P = i , (2) i (p )q dynamical systems at the edge of chaos (Latora et al., 2000; j j

where Pi is called the escort distribution (Beck and Schl¬ogl, S. Abe 1993; Abe, 2003c) associated with the basic distribution p Institute of Physics, University of Tsukuba, Ibaraki 305-8571, i Japan and H is a physical random variable under consideration e-mail: [email protected] (e.g., the system energy) with its ith value εi .q is taken to be

Springer 242 Astrophys Space Sci (2006) 305:241Ð245 positive. The other definition is the ordinary one: (ord) −β pi εi − U , (5) i (ord) U =H= pi εi , (3) i where α and β are the Lagrange multipliers. After eliminating α (ord) , the resulting maximum entropy distribution p˜i is found which is preferred by some researchers. This may be due to be to the fact that the nonextensive statistical mechanical for- / − (ord) = + − ˜(ord) 1 (q 1) malisms with these two definitions of expectation value lead p˜i 1 (1 q)Sq to the stationary distribution of a similar kind (see Sec. 2). − 1/(q−1) q 1 (ord) Therefore, it is crucial to identify which the correct definition × 1 − β εi − U˜ , (6) is. q + It has been shown (Abe and Rajagopal, 2000) that, for a class of power-law distributions (i.e., the q-exponential where distribution with q > 1), only the normalized q-expectation β β = , q (7) value is consistent with the method of steepest descents for (ord) deriving canonical ensemble from microcanonical ensemble. i p˜i The situation, however, remains unclear for the other class of ≡ { , } ˜(ord) ˜ (ord) distributions with compact supports (i.e., the q-exponential and [a]+ max 0 a . In Equation (6), Sq and U are distribution with 0 < q < 1). calculated in terms of the distribution in Eq. (6) in the self- In this paper, we show that the definition to be employed referential manner. in nonextensive statistical mechanics is the normalized q- On the other hand, if the normalized q-expectation value expectation value. For this purpose, we study the proper- is employed, the functional to be maximized reads ties of two generalized relative entropies regarding the two different definitions of expectation value. Then, we use the (nor) [p; α, β] = Sq [p] − α pi − 1 Shore-Johnson theorem for minimum cross entropy (rela- i tive entropy) principle to prove that the formalism with the (p )q ε normalized q-expectation value is theoretically consistent, −β i i i − (nor) . (8) q U whereas the ordinary expectation value cannot be employed. j (p j )

The corresponding maximum entropy distribution is given by 2. Ordinary Expectation Value and Normalized q-Expectation Value 1 ∗ 1/(1−q) p˜(nor) = 1 − (1 − q)β ε − U˜ (nor) , (9) i ˜ (nor) i + First let us recall how nonextensive statistical mechanics de- Zq / − pends on the deﬁnition of expectation value. In nonexten- ˜ (nor) = + − ˜(nor) 1 (1 q) Zq 1 (1 q)Sq sive statistical mechanics, the Tsallis entropy indexed by q / − = − − β∗ ε − ˜ (nor) 1 (1 q) , (Tsallis, 1988) 1 (1 q) i U + (10) i k s [p] = B (p )q − 1 (4) where q 1 − q i i β β∗ = . q (11) (nor) is maximized under appropriate constraints on the normal- i p˜i ization condition of the probability distribution and the expectation value of a physical quantity under consideration. ˜(nor) Similarly to the case of the ordinary expectation value, Sq Here and hereafter, the Boltzmann constant, k , is set equal (nor) (nor) B and U˜ are the values of Sq and U calculated in terms of to unity for simplicity. (nor) the maximum entropy distribution p˜i in the self-referential If the ordinary expectation value is employed, then the manner. functional to be maximized is Clearly, the Tsallis entropy in Equation (4) and the normalized q-expectation value tend to the Boltzmann-Gibbs- =− (ord)[p; α, β] = S [p] − α p − 1 Shannon entropy S[p] i pi ln pi and the ordinary ex- q i → i pectation value in the limit q 1. Accordingly, both of

Springer Astrophys Space Sci (2006) 305:241Ð245 243

the distributions in Equations (6) and (9) converge to the Like H[p r], Iq [p r] and Kq [p r] are nonnegative and Boltzmann-Gibbs distribution p˜i ∼ exp(−βεi ) in such a lim- vanish if and only if pi = ri (∀i). This can be seen as fol- iting case. lows. Noticing that Iq [p r] admits the integral representation It is mentioned that they resemble in their forms (with (Naudts, 2004) opposite signs of the exponents). This point may be a reason pi why some researchers think that there is no essential reason q − − I [p r] = ds[sq 1 − (r )q 1], (17) to prefer the normalized -expectation value and the ordinary q − 1 i q q i ri expectation value may be used.

A point to be noticed is that in both cases the following we immediately see its nonnegativity. For Kq [p r], it is con- thermodynamic relations hold: venient to rewrite it as ∂ ˜(ord) 1 1−q Sq K [p r] = p [1 − (r /p ) ]. (18) = β, (12) q 1 − q i i i ∂U˜ (ord) i ∂ ˜(nor) S 1−q q = β, (13) Then, using the inequality, (1 − x )/(1 − q) ≥ 1 − x for ∂U˜ (nor) x > 0 and q > 0 with the equality for x = 1, we also see nonnegativity of Kq [p r]. In what follows, we discuss the which guarantee the existence of the thermodynamic physical meanings of Iq [p r] and Kq [p r]. Legendre-transform structure in both cases. However, it is Let us take the maximum entropy distributions as the ref- still an open problem in nonextensive statistical mechan- erence distributions in Iq [p r] and Kq [p r]. Taking into ics what the physical temperature is (Abe, 2001; Abe et al., account the exponents of the maximum entropy distributions 2001). in Equations (6) and (9) together with the dependencies of Iq [p r] and Kq [p r]onri , we can expect that Iq [p r] and Kq [p r] may be associated with the formalisms with the or- 3. Generalized relative entropies associated with dinary expectation value and the normalized q-expectation ordinary expectation value and normalized value, respectively. This is indeed the case, as we shall see q-expectation value below. = (ord) Substituting ri p˜i into Equation (14), we have Relative entropy has an important physical meaning as free (ord) = β (ord) − ˜ (ord) , energy difference. It is known in mathematical information Iq [p p˜ ] Fq Fq (19) theory that there exist two different kinds of relative entropies: one is of the Bregman type (Bregman, 1967) and where the other the Csisz«ar type (Csisz«ar, 1972). The Bregman-type relative entropy (Naudts, 2004) and 1 1 F(ord) = U (ord) − S , F˜ (ord) = U˜ (ord) − S˜(ord). (20) the Csisz«ar-type relative entropy (Abe, 1998; Tsallis, 1998b; q β q q β q Borland et al., 1998, 1999) associated with the Tsallis entropy = (nor) are respectively given by On the other hand, putting ri p˜i into Equation (15), we have 1 − − I [p r] = p [(p )q 1 − (r )q 1] (14) βˆ q − i i i K [p p˜(nor)] = F (nor) − F˜ (nor) , (21) q 1 i q (nor) q q q i p˜i q−1 − (pi − ri )(ri ) , i where 1 ∗ = − q 1−q , βˆ = β (p )q , (22) Kq [p r] − 1 (pi ) (ri ) (15) i 1 q i i (nor) = (nor) − 1 , ˜ (nor) = ˜ (nor) − 1 ˜(nor). Fq U Sq Fa U Sq (23) where ri is a reference distribution (i.e., prior). In the limit βˆ βˆ q → 1, both Iq [p r] and Kq [p r] tend to the Kullback- Leibler relative entropy Equations (19) and (21) imply that Iq [p r] and Kq [p r], in fact, give the “free energy differences” and, therefore, pi are identiﬁed with the generalized relative entropies associ- H[p r] = pi ln . (16) i ri ated with the ordinary expectation value and the normalized

Springer 244 Astrophys Space Sci (2006) 305:241Ð245 q-expectation value, respectively. We also mention that the Axiom III (System independence): It should not matter quantum mechanical counterpart of Kq [p r] has recently whether one accounts for independent information about been employed to prove the second law of thermodynamics independent systems separately in terms of their marginal (Abe and Rajagopal, 2003). distributions or in terms of the joint distribution. Now, convexity is one of the most important properties to Axiom IV (Subset independence): It should not matter be satisfied by any types of relative entropies. As can be seen, whether one treats independent subsets of the states of the Iq [p r] is convex in pi ,butnot inri . In marked contrast to this systems in terms of their separate conditional distributions flaw, Kq [p r] is, like the Kullback-Leibler relative entropy or in terms of the joint distribution. in Equation (16), jointly convex (Abe, 2003a, 2004). Axiom V (Expansibility): In the absence of new information, the prior (i.e., the reference distribution) should not be changed. λ λ ≤ λ , Kq a p(a) ar(a) a Kq [p(a) r(a)] (24) a a a These axioms are natural in the sense that all of them are fulfilled by the ordinary Kullback-Leibler relative entropy in λ > λ = where a 0 and a a 1. Equation (16), which gives rise to the free energy difference Finally, we mention that, like the Kullback-Leibler rel- in Boltzmann-Gibbs statistical mechanics. ative entropy, Kq [p r] is “composable” (Tsallis, 2001), For the Tsallis entropy in Equation (4), the axioms and but Iq [p r] is not. For factorized joint distributions of a uniqueness theorem are known in the literature (dos Santos, bipartite system (A, B), pij(A, B) = p(1)i (A)p(2) j (B) and 1997; Abe, 2000). In contrast to this fact, the above set of rij(A, B) = r(1)i (A)r(2) j (B), Kq [p(1) P(2) r(1)r(2)] satisfies axioms is quite general and not very restrictive. Accordingly, the following relation: they do not uniquely specify the explicit functional form of the relative entropy.

Kq [p(1) p(2) r(1)r(2)] = Kq [p(1) r(1)] + Kq [p(2) r(2)] Now, the Shore-Johnson theorem (Shore and Johnson 1980, 1981, 1983) states that the relative entropy J[p||r] +(q − 1)Kq [p(1) r(1)]Kq [p(2) r(2)], (25) with the prior ri and the posterior pi satisfying the axioms IÐV must have the following form: whereas such a closed relation does not exist for Iq [p(1) p(2) r(1)r(2)]. J[p||r] = pi h(pi /ri ), (26) i

4. Shore-Johnson theorem selects normalized where h(x) is some function. q-expectation value It is crucial to recognize that the function h(x) surely exists for Kq [p||r]: In this section, we shall see by using the Shore-Johnson 1 − theorem that Kq [p r] associated with the normalized q- h(x) = (1 − xq 1) . (27) expectation value leads to a consistent framework for mini- 1 − q mum cross entropy (i.e., relative entropy) principle, whereas || I [p r] corresponding to the ordinary expectation value does On the other hand, Iq [p r] cannot be recast to the form q || not. in Equation (26), since Iq [p r] may violate Axiom III. About a quarter a century ago, Shore and Johnson (1980, Therefore, we conclude that the Shore-Johnson theorem 1981, 1983) have presented a set of axioms for minimum supports the normalized q-expectation value and excludes cross entropy (i.e., relative entropy) principle. They have the possibility of using the ordinary expectation value from made an attempt to answer to the question: why the correct nonextensive statistical mechanics. rule of inference is to minimize relative entropy, in confor- mity with a vindication of Jaynes’ claim that every other rule will lead to contradiction (Uffink, 1995). 5. Concluding remarks The five key axioms are listed as follows. We have studied the meanings of the two different defini- Axiom I (Uniqueness): If the same problem is solved twice, tions of expectation value in nonextensive statistical me- then the same answer is expected to result both times. chanics, i.e., the ordinary expectation value and the normal- Axiom II (Invariance): The same answer is expected when ized q-expectation value. To identify which the correct def- the same problem is solved in two different coordinate inition is, we have discussed the corresponding two kinds systems, in which the posteriors in the two systems should of generalized relative entropies, which have the physical be related by the coordinate transformation. meanings as the “free energy differences”. It was shown that

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