Journal of Computational and Applied Mathematics 227 (2009) 51–58

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Computational applications of nonextensive statistical mechanics$ Constantino Tsallis ∗ Centro Brasileiro de Pesquisas Físicas, Rua Xavier Sigaud 150, 22290-180 Rio de Janeiro, Brazil Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA article info a b s t r a c t

Article history: Computational applications of the nonextensive entropy Sq and nonextensive statistical Received 17 March 2008 mechanics, a current generalization of the Boltzmann–Gibbs (BG) theory, are briefly reviewed. The corresponding bibliography is provided as well. Keywords: © 2008 Elsevier B.V. All rights reserved. Nonextensive statistical mechanics Entropy Nonlinear dynamical systems Numerical applications

1. Introduction

Entropy is an ubiquitous concept [1]. Indeed, it emerges in both conservative and dissipative systems, in physical as well as nonphysical contexts. This comes from the fact that it is a functional of probability distributions, and can therefore be used with profit in a considerable variety of manners, for natural, artificial and even social systems. The concept of entropy naturally emerges in statistical mechanics, one of the most important theoretical approaches in contemporary physics. Specifically, Boltzmann–Gibbs (BG) statistical mechanics is essentially based on a specific connection between the Clausius thermodynamic entropy S and the set of probabilities {pi} (i = 1, 2,..., W ) of the microscopic configurations of the system. This connection is provided by the Boltzmann–Gibbs (or Boltzmann–Gibbs–von Neumann–Shannon) entropy SBG defined (for the discrete case) through XW SBG = −k pi ln pi, (1) i=1 where k is a positive constant, the most usual choices being either the Boltzmann constant kB, or just unity (k = 1). This entropic functional and its associated statistical mechanics address a large amount of interesting systems, and have been successfully used along over 130 years. However, very many complex systems escape to its domain of applicability. As an attempt to improve the situation, a generalization of Eq. (1) was proposed in 1988 [2]. See Table 1, where the q-logarithm is defined as x1−q − 1 lnq x ≡ (x > 0; q ∈ R; ln1 x = ln x), (2) 1 − q the inverse function, the q-exponential, being

1 x ≡ [ + − ] 1−q ∈ ; x = x eq 1 (1 q)x + (q R e1 e ), (3) [ ] = ≥ x = lnq x = ∀ with z + z if z 0, and zero otherwise. We verify that lnq eq eq x, x.

$ Invited paper. ∗ Corresponding address: Centro Brasileiro de Pesquisas Físicas, Rua Xavier Sigaud 150, 22290-180 Rio de Janeiro, Brazil. Tel.: +55 21 2141 7190; fax: +55 21 2141 7515. E-mail address: [email protected].

Table 1

The nonadditive entropy Sq (q 6= 1), and its particular, additive, limiting case SBG = S1 = ∀ PW = Entropy Equal probabilities (pi 1/W , i) Arbitrary probabilities ( i=1 pi 1) PW PW 1 q = 1 SBG = k ln WSBG = −k pi ln pi = k pi ln i=1 i=1 pi q 1−PW p i=1 i PW 1 ∀q Sq = k lnq WSq = k = k pi lnq q−1 i=1 pi

A+B = A B ∀ If we consider a system composed by two probabilistically independent subsystems A and B, i.e., pij pi pj (ij), we straightforwardly verify that

Sq(A + B) Sq(A) Sq(B) Sq(A) Sq(B) = + + (1 − q) . (4) k k k k k

Therefore, SBG is additive [3], and Sq (q 6= 1) is nonadditive. The entropy Sq leads to a natural generalization of BG statistical mechanics, currently referred to as nonextensive statistical mechanics,[2,4,5]. The nonextensive theory focuses on nonequilibrium stationary (or quasi-stationary) states in the same way the BG theory focuses on thermal equilibrium states. Recent reviews of the theory and its various applications (see, for instance, [6] for a recent review of astrophysical applications, whose lines the present review partially follows) can be found in [7–9], and a regularly updated bibliography is available at [10].

2. Extensivity of the nonadditive entropy Sq

From its very definition, additivity depends only on the functional form of the entropy. Extensivity also depends on the system. Indeed, the entropy S(N) of a given system composed by N (independent or correlated) elements is said extensive if limN→∞ S(N)/N is finite, i.e., if S(N) scales like N for large N, hence matching the classic thermodynamic behavior. Consequently, for systems whose elements are independent or weakly correlated, SBG is extensive, whereas Sq is nonextensive for q 6= 1. There are, in constrast, systems whose elements are strongly correlated, having as a consequence that Sq is extensive only for a special value of q, noted qent , with qent 6= 1. In other words, Sq is nonextensive for any value of q 6= qent . In particular, for such anomalous systems, SBG is nonextensive, hence inadequate for thermodynamical purposes at the limit N → ∞ (thermodynamic limit). There are finally systems that are even more complex, and for which there is no value of q such that Sq(N) is extensive. Such strongly anomalous systems remain outside the realm of nonextensive statistical mechanical concepts, and are not addressed here.

Abstract probabilistic models with either qent = 1 or qent 6= 1 are exhibited in [11]. Physical realizations are exhibited in [12] for strongly entangled systems. For example, the block entropy of magnetic chains in the presence of a transverse magnetic field at its critical value and at T = 0, belonging to the universality class characterized by the central charge c [13], yield [12] √ 9 + c2 − 3 qent = (c > 0). (5) c √ Therefore, for the one-dimensional Ising model with short-range interactions (hence c = 1/2), we have qent = 37 − 6 ' √0.08, and for the one-dimensional isotropic XY model with short-range interactions (hence c = 1), we have qent = 10 − 3 ' 0.16. At the c → ∞ limit, we recover the BG value qent = 1 (qent = 1/2 for c = 4; see [14] for c = 26).

3. Entropy production per unit time

Let us illustrate on one-dimensional unimodal nonlinear dynamical systems xt+1 = f (xt ) some interesting concepts, namely that of sensitivity to the initial conditions and that of entropy production per unit time. A system is said strongly chaotic (or just chaotic) if its Lyapunov exponent λ1 is positive, and weakly chaotic if it vanishes. The sensitivity to the initial conditions ξ is defined as

∆x(t) ξ(t) ≡ lim . (6) ∆x(0)→0 ∆x(0)

If λ1 > 0, we typically have

ξ(t) = eλ1t . (7)

The entropy production per unit time K1 in such a system is defined as

SBG(t) K1 ≡ lim lim lim , (8) t→∞ W →∞ M→∞ t C. Tsallis / Journal of Computational and Applied Mathematics 227 (2009) 51–58 53 where M is the number of initial conditions that have been randomly chosen within one of the W intervals in which the allowed phase space of the system has been partitioned. K1 typically is positive and satisfies a Pesin-like equality, i.e.,

K1 = λ1. (9)

For a large class of nonlinear functions fa(x), where a is a control parameter, the system has one or more values of a for which it is at the edge of chaos, with λ1 = 0. As anticipated, such systems are called weakly chaotic. ξ(t) may present a complex behavior which typically admits an upper bound. This upper bound has been shown, in many cases, to be given by [15–17]

= λqsen t ξ(t) eqsen , (10) where sen stands for sensitivity, qsen < 1, and λqsen > 0. Such systems satisfy [15,18], again as an upper bound, S (t) ≡ qsen Kqsen lim lim lim . (11) t→∞ W →∞ M→∞ t Although no proof is available, it has been conjectured [19] that

qsen = qent . (12)

4. q-generalization of the standard and the Lévy–Gnedenko Central Limit Theorems

The Central Limit Theorem (CLT) constitutes a mathematical building block of BG statistical mechanics. For example, under simple and appropriate constraints, the optimization of SBG yields a Gaussian (e.g., the Maxwellian distribution of velocities), which precisely is the attractor within the standard CLT. A slight extension yields Lévy distributions as well, whose second moment diverges. But all these distributions are obtained within the basic hypothesis of independence or quasi-independence, in the sense that the correlations do not resist up to large values of N. The Fokker–Planck counterpart of these facts can be focused on as follows. Consider the equation ∂p(x, t) ∂αp(x, t) = D (D > 0; 0 < α ≤ 2). (13) ∂t ∂|x|α Assume also that the initial distribution is a Dirac delta, i.e., p(x, 0) = δ(x). The exact solution, ∀(x, t), is a Gaussian if α = 2 (x scaling like t1/2), and a Lévy distribution if 0 < α < 2 (x scaling like t1/α). The equation being linear, these two results precisely correspond to the standard and the Lévy–Gnedenko CLT’s (finite and diverging second moment, respectively, of the distribution that is being summed up to N random variables). Let us now generalize Eq. (13) into the following nonlinear Fokker–Planck equation: ∂p(x, t) ∂α[p(x, t)]2−q = Dq (Dq ∈ R; 0 < α ≤ 2; q < 3). (14) ∂t ∂|x|α Assume once again that the initial distribution is a Dirac delta. The exact solutions are now [20,21] q-Gaussians if α = 2 1/(3−q) (x scaling like t ), the q-Gaussians being the distributions which optimize Sq under appropriate constraints. A more complex type of distributions, named (q, α)-stable distributions, emerge if 0 < α < 2 (the scaling between x and t going through a crossover between two different regimes). The corresponding CLT’s are the natural q-generalizations of the usual theorems, the main new ingredient in the hypothesis being now the fact that the N random variables are strongly correlated, in a manner currently referred to as q-independence. See [22–28] for details. The existence of these theorems might explain the ubiquity of q-Gaussians in natural, artificial and social systems. See, for instance, [29–36].

5. Computational applications of the nonadditive entropy Sq and nonextensive statistical mechanical concepts

Finally, let us briefly mention here some among the many current computational applications of the present theory (for further applications see [10]). The successful global optimization algorithm known as Simulated Annealing is based on a Gaussian visiting distribution and a Boltzmann weight for acceptance. It is sometimes referred to as the Boltzmann machine. With great computational advantages, it has been generalized into what is frequently referred to as the Generalized Simulated Annealing (GSA) [37,38], where the visitation of phase space is made through a qV -Gaussian, and the acceptance weight is a qA-exponential (where V and A stand respectively for visiting and acceptance). The best results are currently obtained for qV > 1 and qA < 1 [37–39]. Its first application in natural sciences was in theoretical chemistry [40]. Since then, a sensible amount of works have been dedicated to its use in very many disciplines, and even for further analysis, variations and improvements [41–125]. One of the important numerical steps for applying many among these methods is the efficient generation of random numbers following q-Gaussian distributions. This has been the purpose of various works as well [126–129]. Algorithms associated with ‘‘neural’’ networks have been worked out [130–133] with increased efficiency, as well as various other numerical applications [134–136]. A variety of works exist focusing on improved methods for processing signals, time series (nonlinear dynamics, epilepsy, earthquakes, economics) and images. Some of these and many others have been shown to be useful for biological and medical applications as well [137–195]. Related patents also exist [196–211]. 54 C. Tsallis / Journal of Computational and Applied Mathematics 227 (2009) 51–58

Acknowledgments

I am very grateful for warm and delightful hospitality in Kalamata by the organizers of the Conference in Numerical Analysis (3–7 September 2007, Greece). Partial financial support by CNPq and Faperj (Brazilian agencies) is acknowledged as well.

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