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

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Why Q-Expectation Values Must Be Used in Nonextensive Statistical Mechanics Astrophys Space Sci (2006) 305:241–245 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) 305:241–245 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 ex- pectation 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 nor- malized 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) 305:241–245 243 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 en- tropies: one is of the Bregman type (Bregman, 1967) and where the other the Csisz´ar type (Csisz´ar, 1972).
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