Approach of complexity in nature: Entropic nonuniqueness
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] A ug axioms Review
Approach of Complexity in Nature:Entropic Nonuniqueness
Constantino Tsallis Centro Brasileiro de Pesquisas Fisicas, and National Institute for Science and Technology for ComplexSystems, Rua Xavier Sigaud 150, Rio de Janeiro-RJ 22290-180, Brazil; [email protected]; Tel.: +55-21-21417190 Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USAAcademic Editor: Hans J. HauboldReceived: 8 July 2016 ; Accepted: 8 August 2016; Published: date
Abstract:
Boltzmann introduced in the 1870s a logarithmic measure for the connection betweenthe thermodynamical entropy and the probabilities of the microscopic configurations of the system.His celebrated entropic functional for classical systems was then extended by Gibbs to the entirephase space of a many-body system and by von Neumann in order to cover quantum systems,as well. Finally, it was used by Shannon within the theory of information. The simplest expressionof this functional corresponds to a discrete set of W microscopic possibilities and is given by S BG = − k ∑ Wi = p i ln p i ( k is a positive universal constant; BG stands for Boltzmann–Gibbs). This relationenables the construction of BGstatistical mechanics, which, together with the Maxwell equationsand classical, quantum and relativistic mechanics, constitutes one of the pillars of contemporaryphysics. The BG theory has provided uncountable important applications in physics, chemistry,computational sciences, economics, biology, networks and others. As argued in the textbooks, itsapplication in physical systems is legitimate whenever the hypothesis of ergodicity is satisfied,i.e., when ensemble and time averages coincide. However, what can we do when ergodicity andsimilar simple hypotheses are violated, which indeed happens in very many natural, artificialand social complex systems. The possibility of generalizing BG statistical mechanics through afamily of non-additive entropies was advanced in 1988, namely S q = k − ∑ Wi = p qi q − , which recoversthe additive S BG entropy in the q → q is to be determined from mechanicalfirst principles, corresponding to complexity universality classes. Along three decades, this ideaintensively evolved world-wide (see the Bibliography in http://tsallis.cat.cbpf.br/biblio.htm) andled to a plethora of predictions, verifications and applications in physical systems and elsewhere.As expected, whenever a paradigm shift is explored, some controversy naturally emerged, as well,in the community. The present status of the general picture is here described, starting from itsdynamical and thermodynamical foundations and ending with its most recent physical applications. Keywords: complex systems; statistical mechanics; non-additive entropies; ergodicity breakdown
1. Introduction
In light of contemporary physics, the qualitative and quantitative study of nature may be doneat various levels, which here we refer to as microcosmos, mesocosmos and macrocosmos. At themacroscopic level, we have thermodynamics; at the microscopic level, we have mechanics (classical,quantum, relativistic mechanics, quantum chromodynamics) and the laws of electromagnetism,which enable in principle the full description of all of the degrees of freedom of the system; at themesoscopic level, we focus on the degrees of freedom of a typical particle, representing, in one wayor another, the behavior of most of the degrees of freedom of the system. The laws that govern themicrocosmos together with theory of probabilities are the basic constituents of statistical mechanics,
Axioms , xioms , , x 2 of 15 a theory, which then establishes the connections between these three levels of description of nature.At the microscopic level, we typically address classical or quantum equations of evolution with time,trajectories in phase space, Hamiltonians, Lagrangians, among other mathematical objects. At themesoscopic level, we address Langevin-like, master-like and Fokker–Planck-like equations. Finally,at the macroscopic level, we address the laws of thermodynamics with its concomitant Legendretransformations between the appropriate variables.In all of these theoretical approaches, the thermodynamical entropy S , introduced by Clausiusin 1865 [1] and its corresponding entropic functional S ( { p i } ) play a central role. In a stroke of genius,the first adequate entropic functional was introduced (for what we nowadays call classical systems)by Boltzmann in the 1870s [2,3] for a one-body phase space and was later on extended by Gibbs [4]to the entire many-body phase space. Half a century later, in 1932, von Neumann [5] extended theBoltzmann–Gibbs (BG) entropic functional to quantum systems. Finally, in 1942, Shannon showed[6] the crucial role that this functional plays in the theory of communication. The simplest expressionof this functional is that corresponding to a single discrete random variable admitting W possibilitieswith nonvanishing probabilities { p i } , namely: S BG = − k W ∑ i = p i ln p i (cid:16) W ∑ i = p i = (cid:17) (1)where k is a conventional positive constant (in physics, typically taken to be the Boltzmannconstant k B ). This expression enables, as is well known, the construction of what is usually referredto as (BG) statistical mechanics, a theory that is notoriously consistent with thermodynamics. To bemore precise, what is well established is that the BG thermostatistics is sufficient for satisfying theprinciples and structure of thermodynamics. Whether it is or not also necessary is a most importantquestion that we shall address later on in the present paper. This crucial issue and its interconnectionswith the Boltzmann and the Einstein viewpoints have been emphatically addressed by E.G.D. Cohenin his acceptance lecture of the 2004 Boltzmann Award [7].On various occasions, generalizations of the expression (1) have been advanced and studied inthe realm of information theory. In 1988, [8] (see also [9,10]) the generalization of the BG statisticalmechanics itself was proposed through the expression: S q = k − ∑ Wi = p qi q − = k W ∑ i = p i ln q p i (cid:16) W ∑ i = p i = q ∈ R ; S = S BG (cid:17) (2)where the q -logarithmic function is defined through ln q z ≡ z − q − − q (ln z = ln z ). Its inversefunction is defined as e zq ≡ [ + ( − q ) z ] − q ( e z = e z ) . Various predecessors of S q , q -exponentialsand q -Gaussians abound in the literature within specific historical contexts (see, for instance, [11] fora list with brief comments).
2. Additive Entropy versus Extensive Entropy
An entropic functional S ( { p i } ) is said to be additive (we are adopting Oliver Penrose’sdefinition [12]) if, for any two probabilistically independent systems A and B (i.e., p A + Bi , j = p Ai p Bj , ∀ ( i , j ) ), S ( A + B ) = S ( A ) + S ( B ) [ S ( A + B ) ≡ S ( { p A + Bi , j } ) ; S ( A ) ≡ S ( { p Ai } ) ; S ( B ) ≡ S ( { p Bj } )] (3) xioms , , x 3 of 15 It can be straightforwardly proven that S q satisfies: S q ( A + B ) k = S q ( A ) k + S q ( B ) k + ( − q ) S q ( A ) k S q ( B ) k (4)Consequently, S BG = S is additive, whereas S q is non-additive for q = S ( { p i } ) of a specific system (or a specific class of systems, with its N elementswith their corresponding correlations) is said to be extensive if:0 < lim N → ∞ S ( N ) N < ∞ (5)i.e., if S ( N ) grows like N for N >>
1, where N ∝ L d , d being the integer or fractal dimension of thesystem, and L its linear size.Let us emphasize that determining whether an entropic functional is additive is a very simplemathematical task (due to the hypothesis of independence), whereas determining if it is extensive fora specific system can be a very heavy one, sometimes even intractable. If all nonzero-probability events of a system constituted by N elements are equally probable, wehave p i = W ( N ) , ∀ i .In that case, S BG ( N ) = k ln W ( N ) and S q ( N ) = k ln q W ( N ) .Therefore, if the system satisfies W ( N ) ∝ µ N ( µ > N → ∞ ) (e.g., for independent coins, wehave W ( N ) = N ), referred to as the exponential class, we have that the additive entropy S BG is alsoextensive. Indeed, S BG ( N ) ∝ N . For all other values of q =
1, we have that the non-additive entropy S q is nonextensive.However, if we have instead a system such that W ( N ) ∝ N ρ ( ρ > N → ∞ ) , referred to as thepower-law class, we have that the non-additive entropy S q is extensive for: q = − ρ ( ρ > ) (6)Indeed, S − ρ ( N ) ∝ N . For all other values of q (including q = S q isnonextensive for this class; the extensive entropy corresponding to the limit ρ → ∞ precisely is theadditive S BG .Let us now mention another, more subtle, case where the nonzero probabilities are not equal [13].We consider a triangle of N ( N =
2, 3, 4, ... ) correlated binary random variables, say n heads and ( N − n ) tails ( n =
0, 1, 2, ..., N ) . The probabilities p N , n ( ∑ Nn = p N , n = ∀ N ) are different from zeroonly within a strip of width d (more precisely, for n =
0, 1, 2, ..., d ) ) and vanish everywhere else.This specific probabilistic model is asymptotically scale-invariant (i.e., it satisfies the so-called Leibniztriangle rule for N → ∞ ): see [13] for full details. For this strongly-correlated model, the non-additiveentropy S q is extensive for a unique value of q , namely: q = − d ( d =
1, 2, 3, ... ) (7)We see that the extensive entropy corresponding to the limit d → ∞ precisely is the additive S BG .These examples transparently show the important difference between entropic additivity andentropic extensivity. What has historically occurred is that, during 140 years, most physicistshave been focusing on systems that belong to the exponential class, typically either non-interactingsystems (ideal gas, ideal paramagnet) or short-range-interacting ones (e.g., d -dimensional Ising,XY and Heisenberg ferromagnets with first-neighbor interactions). Since for this class, but not so for xioms , , x 4 of 15 many others, the additive BG entropic functional is also extensive, a frequent confusion has emergedin the understanding of very many people and textbooks, which has led, and is unfortunately stillleading, to somehow considering additive and extensive as synonyms, which is definitively false( this error is so easy to make, such that, by inadvertence, the book [14] by Gell-Mann and myselfwas entitled Nonextensive Entropy, whereas it should have been entitled Non-additive Entropy;obviously, we definitively regret this misnomer).Further classes of systems do exist, for example the stretched exponential one, for which otherentropic functionals (e.g., S δ [15]) are necessary in order to achieve extensivity. Indeed, no value of q exists such that S q ( N ) ∝ N for this class. In fact, a plethora of entropic functionals are now availablein the information-theory literature (see, for instance, [16–29]). The entropic index q is to be determined from first principles, namely from the time evolution(in phase space, Hilbert space and analogous) of the state of the full system. This typically isan analytically hard task. Nevertheless, this task has been accomplished in some few cases. Letus briefly review some of them:1. The logistic map at its Feigenbaum point;2. The entropy of a subsystem of a ( + ) -dimensional system characterized by a central charge c at its quantum critical point;3. The entropy of a subsystem of a ( + ) -dimensional generalized isotropicLipkin–Meshkov–Glick model at its quantum critical point.For the logistic map x t + = − ax t ( < a < t =
0, 1, 2, ...; x t ∈ [ −
1, 1 ] , we have that a valueof q exists, such that S q asymptotically increases linearly with time, where the value of q is dictatedby the Lyapunov exponent being positive or zero, which in turn depends on the value of the externalparameter a . To be more precise, we assume the interval [ −
1, 1 ] of x divided into W tiny intervals(identified with i =
1, 2, ..., W ); we then place in one of those intervals many M initial conditions(with M >> W ); and finally, we iterate the map for each of these initial conditions. The numberof points M i ( t ) that are located at the i -th interval satisfy ∑ Wi = M i ( t ) = M , ∀ t . We define next theprobabilities p i ( t ) ≡ M i ( t ) / M , which enable the evaluation of the entropy S q ( t ) / k = − ∑ Wi = [ p i ( t )] q q − . Itcan be shown that a unique value of q exists such that K q ≡ lim t → ∞ lim W → ∞ lim M → ∞ S q ( t ) / kt is finite.For any value of q above this special one, the ratio K q vanishes, and for any value of q below thisspecial one, the ratio K q diverges.For all values of a such that the Lyapunov exponent λ is positive (i.e., in the presence of strongchaos, where the sensitivity to the initial conditions ξ ≡ lim ∆ x ( ) → ∆ x ( t ) ∆ x ( ) increases exponentiallywith time, ξ = e λ t ), we have that q =
1, and the ratio precisely equals the Lyapunov exponent (i.e., K = λ ; Pesin-like identity).In contrast, at the edge of chaos, i.e., for the value of a where successive bifurcations accumulate(sometimes referred to as the Feigenbaum point), i.e., a = q = a , we verify that ξ = e λ q tq , where a q -generalized version of the Pesin-like identityhas been rigorously established [31]. The edge of chaos of logistic-like maps provides a remarkableconnection of q -statistics with multifractals [30]. This is particularly welcome because the postulateof the entropy S q in order to have a basis for generalizing BG statistics was inspired precisely by thestructure of multifractals. The present status of our knowledge strongly suggests that a BG system xioms , , x 5 of 15 typically “lives” in a smoothly-occupied phase-space, whereas the systems obeying q -statistics “live”in hierarchically-occupied phase-spaces.Let us now address the entropy of an L -sized block of an N -sized quantum system at itsquantum critical point, belonging to the universality class, which is characterized by a central charge c (e.g., the universality classes of the short-range Ising and the short-range isotropic XY ferromagnetscorrespond respectively to c = c = S q is extensive for: q = √ + c − c (9)We verify that c → ∞ yields q = ( m , k ) , where m is the number of states of the model (e.g., if the system is constituted by s -sizedspins, we have m = s , s = k ( k =
0, 1, 2, ...) is the number of vanishing magnondensities. The entropy S q is extensive for: q = − m − k = − s − k ( m − k = s − k ≥ q ≥ ) (10)Notice that, in the limit s → ∞ , q = s -sized spins, we have [35]: q ≃ − ( s + ) (11)Before we proceed with analyzing thermodynamical aspects, let us stress that we have addressedhere two different types of linearities, the thermodynamical one (i.e., S q ( N ) ∝ N ) and the dynamicalone (i.e., S q ( t ) ∝ t ). Although the nature of these linearities is different and even the values of q , whichguarantee them, may be different (although possibly related), there are reasons to expect both to besatisfied on similar grounds: this question was in fact (preliminarily) addressed in [36] and elsewhere. Let us address here a question that frequently appears in the literature, generating some degreeof confusion. We refer to the discussion of Renyi entropy versus q -entropy on thermodynamical anddynamical grounds. The Renyi entropy [16] is defined as: S Rq ≡ ln ∑ Wi = p qi − q (cid:16) W ∑ i = p i = q ∈ R ; S R = S BG (cid:17) (12)hence: S Rq = ln [ + ( − q ) S q / k ] − q (13)It is straightforward to verify that S Rq ( S q ) is a monotonic function of S q , ∀ q . Consequently,under the same constraints, the extremization of S Rq yields precisely the same distribution as theextremization of S q (in total analogy with the trivial fact that maximizing, under the same constraints, S BG or say [ S BG ] yields one and the same BG exponential weight). This mathematical triviality is atthe basis of sensible confusion in the minds of some members of the community. Thermodynamicsand statistical mechanics is much more than a mere probability distribution, and the reader hassurely never seen, and this for more than one good reason, constructing a successful theory suchas thermodynamics by using say [ S BG ] instead of S BG . xioms , , x 6 of 15 To make things more precise, let us list now several important differences between S q and S Rq (see, for instance, [11] and the references therein).(i) Additivity: If A and B are two arbitrary probabilistically-independent systems, S Rq is additive, ∀ q , whereas S q satisfies the non-additive property in Equation (4).(ii) Concavity: S q ( { p i } ) is concave for all q >
0, whereas S Rq ( { p i } ) is concave only for 0 < q ≤ S q and S Rq are convex for q <
0. These properties have consequences for characterizing thethermodynamic stability of the system.(iii) Lesche stability: S q is Lesche-stable ∀ q >
0, whereas S Rq is Lesche-stable only for q =
1. Leschestability characterizes the experimental reproducibility of the entropy of a system.(iv) Pesin-like identity: For many physically important low-dimensional conservative or dissipativenonlinear dynamical systems with zero Lyapunov exponent, it is verified that, in the t → ∞ limit, S q ( t ) ∝ t for a unique special value of q =
1. This linearity property for t >> S Rq ( t ) ; indeed, for those systems, it can be easily verified that S Rq ( t ) ∝ ln t ( ∀ q ) . No dynamicalsystems are yet known for which S Rq ( t ) is linear for q =
1. This linearity enables, ∀ q , a naturalconnection with the coefficient (Lyapunov exponent for the q = N -sized quantum systems, it can be shown that afixed value of q = N → ∞ limit, S q ( N ) ∝ N , thus satisfying thenecessary thermodynamic extensivity for the entropy. For those systems, S Rq ( N ) ∝ ln N ( ∀ q ) ,which violates thermodynamics. For this statement, we have of course assumed that a(physically meaningful) limit q = N → ∞ limit. Various papers exist in theliterature that focus on situations such that a phenomenological index q can be defined, whichdepends on N (see, for instance, [37,38] and the references therein), but they remain out of thepresent scope, since their N → ∞ limit yields q = q = W ∝ e S BG / k , the exponentialfunction being the inverse of S BG / k = ln W (for equal probabilities), and for appropriateconstraints, it maximizes the entropy S BG . For q =
1, we have [39] W ∝ e S q / kq , where the q -exponential function precisely is the inverse of S q / k = ln q W (for equal probabilities), andfor appropriate constraints, it extremizes the entropy S q . In contrast with this property, thefactorizable likelihood function for the Renyi entropy is e S Rq , where the exponential function isthe inverse of S Rq = ln W (for equal probabilities), but it differs from the q -exponential function,which is the one that extremizes S Rq . These properties plausibly have consequences for the largedeviation theory of these systems (see the discussion about this theory below).
3. Why Must the Entropic Extensivity Be Preserved in All Circumstances?
Since we are ready to permit the entropic functional to be non-additive, should we not alsoallow for possible entropic nonextensivity? This question surely is a most interesting one, but tothe best of our understanding, the answer is no. Indeed, there exist at least two important reasonsfor always demanding the physical (thermodynamical) entropy of a given system to be extensive.One of them is based on the Legendre transformations structure of thermodynamics; the other oneis so suggested by the large deviations in some anomalous probabilistic models where the limitingdistributions are q -Gaussians. This argument has been developed in [11] and more recently in [15] (which we follownow). We briefly review this argument here. Let us first write a general Legendre transformation xioms , , x 7 of 15 form of a thermodynamical energy G of a generic d -dimensional system ( d being an integer orfractal dimension): G ( V , T , p , µ , H , . . . ) = U ( V , T , p , µ , H , . . . ) − TS ( V , T , p , µ , H , . . . ) (14) + pV − µ N ( V , T , p , µ , H , . . . ) − H M ( V , T , p , µ , H , . . . ) − · · · (15)where T , p , µ , H are the temperature, pressure, chemical potential and external magnetic fieldand U , S , V , N , M are the internal energy, entropy, volume, number of particles and magnetization.We may identify three types of variables, namely: (i) those that are expected to always be extensive( S , V , N , M , . . .), i.e., scaling with V ∝ L d , where L is a characteristic linear dimension of the system(notice the presence of N itself within this class); (ii) those that characterize the external conditionsunder which the system is placed ( T , p , µ , H , . . .), scaling with L θ ; and (iii) those that representenergies ( G , U ), scaling with L ǫ . Ordinary thermodynamical systems are those with θ = ǫ = d ;therefore, both the energies and the generically extensive variables scale with L d , and there is nodifference between Type (i) and (iii) variables, all of them being extensive in this case. There are,however, physical systems where ǫ = θ + d with θ =
0. Let us divide Equation (15) by L θ + d , namely, GL θ + d = UL θ + d − TL θ SL d + pL θ VL d − µ L θ NL d − HL θ ML d − · · · (16)If we consider now the thermodynamical L → ∞ limit, we obtain: e g = e u − e Ts + e pv − e µ n − e Hm − · · · (17)where, using a compact notation, ( e g , e u ) ≡ lim L → ∞ ( G , U ) / L θ + d represent the energies, ( s , v , n , m ) ≡ lim L → ∞ ( S , V , N , M ) / L d represent the usual extensive variables and ( e T , e p , e µ , e H ) ≡ lim L → ∞ ( T , p , µ , H ) / L θ correspond to the usually intensive ones. For a standard thermodynamicalsystem (e.g., a real gas ruled by a Lennard–Jones short-ranged potential, a simple metal, etc.) we have θ = ( e T , e p , e µ , e H ) = ( T , p , µ , H ) , i.e., the usual intensive variables), and ǫ = d (hence, ( e g , e u ) = ( g , u ) , i.e., the usual extensive variables); this is of course the case found in the textbooksof thermodynamics.The thermodynamic relations (15) and (16) put on an equal footing the entropy S , the volume V and the number of elements N , and the extensivity of the latter two variables is guaranteed bydefinition. In fact, a similar analysis can be performed using N instead of V since V ∝ N .An example of a nonstandard system with θ = r like 1/ r α ( α ≥ ) .For this system, we have θ = d − α whenever 0 ≤ α < d (see, for example, Figure 1 of [40]). Thispeculiar scaling occurs because the potential is not integrable, i.e., the integral R ∞ constant dr r d − r − α diverges for 0 ≤ α ≤ d ; therefore, the Boltzmann–Gibbs canonical partition function itself diverges.Gibbs was aware of this kind of problem and has pointed out [4] that whenever the partitionfunction diverges, the BG theory cannot be used because, in his words, “the law of distributionbecomes illusory”. The divergence of the total potential energy occurs for α ≤ d , which is referredto as long-range interactions. If α > d , which is the case of the d = α =
6, the integral does not diverge, and we recover thestandard behavior of short-range-interacting systems with the θ = r , has the advantage of illustratingclearly the thermodynamic relations (15) and (16) for the different scaling regimes, as shown inFigure 1. xioms , , x 8 of 15
0 1 a / d (long−range interactions) (short−range interactions) Intensive, e.g. , T, p, m , H (cid:181) L Extensive, e.g.,
G, U, S, N, V, M (cid:181) L d ( q „
0) ( q = 0) P s e udo − i n t e n s i v e , e . g ., T , p , m , H (cid:181) L q Extensive, e.g.,
S, N, V, M (cid:181) L d P s e udo − e x t e n s i v e , e . g ., G , U (cid:181) L d + q Figure 1.
Representation of the different scaling regimes of Equation (16) for classical d -dimensional systems. For attractive long-range interactions (i.e., 0 ≤ α / d ≤ α characterizingthe interaction range in a potential with the form 1/ r α ), we may distinguish three classes ofthermodynamic variables, namely, those scaling with L θ , named pseudo-intensive ( L is a characteristiclinear length; θ is a system-dependent parameter), those scaling with L d + θ , the pseudo-extensive ones(the energies), and those scaling with L d (which are always extensive). For short-range interactions(i.e., α > d ), we have θ =
0, and the energies recover their standard L d extensive scaling, fallingin the same class of S , N , V , etc., whereas the previous pseudo-intensive variables become trulyintensive ones (independent of L ); this is the region with two classes of variables that is coveredby the traditional textbooks of thermodynamics. From [15]. To summarize this crucial subsection, we may insist that what is thermodynamically relevantis that the entropy of a given system must be extensive, not that the entropic functional oughtto be additive. This is consistent with the fact that Einstein’s principle for the factorizability ofthe likelihood function is satisfied not only for the additive BG entropic functional, but also fornonadditive ones [39,41].
The so-called large deviation theory (LDT) [42] constitutes the mathematical counterpartof the heart of BG statistical mechanics, namely the famous canonical-ensemble BG factor e − β H ( N ) = e − N [ β h ( N )] with h ( N ) ≡ H ( N ) / N . Since, for short-range interactions, β h ( N ) is a thermodynamically-intensive quantity in the limit N → ∞ , we see that theBG weight represents an exponential decay with N . This exponential dependence is tobe associated [42–46] with the LDT probability P ( N ; x ) ≃ e − N r ( x ) , where Subindex 1in the rate function r ( x ) will soon become clear. Since r ( x ) is directly related to arelative entropy per particle (see, for instance, [43]), the quantity Nr ( x ) plays the role ofan extensive entropy.If we focus now on, say, a d -dimensional classical system involving two- body interactions whosepotential asymptotically decays at long distance r like − A / r α ( A > α ≥ ) , the canonical BGpartition function converges whenever the potential is integrable, i.e., for α / d > ≤ α / d ≤ ( α , d ) = (
1, 3 ) ; hence, α / d = f ( H N ) different from the exponential one, in order todescribe some specific stationary (or quasi-stationary) states differing from thermal equilibrium. TheHamiltonian H N generically scales like N ˜ N with ˜ N ≡ N − α / d − − α / d ≡ ln α / d N (with the q -logarithmicfunction defined as ln q z ≡ z − q − − q ; z >
0; ln z = ln z ). Notice that ( N → ∞ ) ˜ N ∼ N − α / d / ( − α / d ) xioms , , x 9 of 15 for 1 ≤ α / d <
1, ˜ N ∼ ln N for α / d = N ∼ ( α / d − ) for α / d >
1. The particular case α = N ∼ N , thus recovering the usual prefactor of mean field theories. The quantity β H N can berewritten as [( β ˜ N ) H N / ( N ˜ N )] N = [ ˜ β H N / ( N ˜ N )] N , where ˜ β ≡ β ˜ N ≡ k B ˜ T = ˜ N / k B T plays the roleof an intensive variable. The correctness of all of these scalings has been profusely verified in variouskinds of thermal, diffusive and geometrical (percolation) systems (see [11,45]). We see that, not onlyfor the usual case of short-range interactions, but also for long-range ones, [ ˜ β H N / ( N ˜ N )] plays a roleanalogous to an intensive variable. The q -exponential function e zq ≡ [ + ( − q ) z ] − q ( e z = e z )(and its associated q -Gaussian) has already emerged, in a considerable amount of nonextensive andsimilar systems, as the appropriate generalization of the exponential one (and its associated Gaussian).Therefore, it appears as rather natural to conjecture that, in some sense that remains to be preciselydefined, the LDT expression e − r N becomes generalized into something close to e − r q Nq ( q ∈ R ), wherethe generalized rate function r q is expected to be some generalized entropic quantity per particle. Asshown in Figures 2 and 3 (see the details in [45]), it is precisely this e − r q Nq behavior that emerges in astrongly correlated nontrivial model [43,45]. Since, as for the q = r q N appears to play the roleof a total entropy, this specific illustration is consistent with an extensive entropy. -6 -5 -4 -3 -2 -1 B(x=0.10)=4.57660794B(x=0.35)=173.038573 P ( N ; n / N < x ) Q=3/2=1/2=1 N x=0.35 0.564647 e - 0.033 N5/3 - 0.020 N5/3 x=0.10 0.828455 e - 0.480 N5/3 - 0.117 N5/3 Figure 2.
Comparison of the numerical data (dots) of [45] with a ( x ) e − r q Nq , where ( a ( x ) , r q ( x )) arepositive quantities. From [45]. xioms , , x 10 of 15 -6x10 -4x10 -2x10 Q=3/2=1/2=1 l n 5 / P ( N ; n / N < x ) x=0.35 x=0.10 N Figure 3.
The same data of Figure 2 in ( q -log)-linear representation. Let us stress that the uniqueasymptotically-power-law function, which provides straight lines at all scales of a ( q -log)-linearrepresentation, is the q -exponential function. The inset shows the results corresponding to N up to 50.From [45].
4. Further Applications and Final Words
A regularly-updated bibliography on the present subject can be found at [47]. At the same site,a selected list of theoretical, experimental, observational and computational papers can be found, aswell. From these very many papers, let us briefly mention here a few recent ones.For those systems that may be well described by a specific class of nonlinear homogeneous d = µ = ( − q ) , where µ is the exponent that characterizes the scaling between space and time (specifically thefact that x scales like t µ ) and q is the index of the q -Gaussian, which describes the paradigmaticsolution of the equation. Notice that q = µ = d -dimensional generalization of thatscaling, namely µ = + d ( − q ) [50]; hence, once again µ = q = q -statistics, or BG statistics, or even a combination of both emerges as a function of theunique external parameter ( K ) of the map. This and various other emergencies of q -Gaussian and q -exponential distributions in many natural, artificial and social complex systems are most probablyconnected with q -generalizations of the central limit theorem (see, for instance, [52–63]).Another q -statistical connection that certainly is interesting is the one with the so-called(asymptotically) scale-free networks. Indeed, their degree distribution has been shown in many casesto be given by p ( k ) ∝ e − k / κ q ( k being the number of links joining a given node), which plays the roleof the Boltzmann–Gibbs factor for short-range-interacting Hamiltonian systems. This connection wasalready established in the literature since one decade ago (see, for instance, [64,65]). Moreover, it hasbeen recently shown [66] that neither q nor κ depend independently on the dimensionality d and fromthe exponent α characterizing the range of the interaction, but, interestingly enough, only dependon the ratio α / d . Very many papers focus on the degree distributions of (asymptotically) scale-freenetworks from a variety of standpoints. For example, an interesting exactly solvable master-equationapproach is available in [67]. The novelty that we remind about in this mini-review is that the q -exponential degree distribution is here obtained from a simple entropic variational principle (undera constraint where the average degree plays the role of the internal energy in statistical mechanics). xioms , , x 11 of 15 High-energy physics has also been a field of many applications of q -statistics and relatedapproaches, such as Beck–Cohen superstatistics [68] and Mathai’s pathways (see [69–73] and thereferences therein). For example, a focus on the solar neutrino problem started long ago by Quaratiand collaborators [74–77] and has been revisited in several occasions, even recently [78,79]. In thearea of particle high-energy collisions, an intensive activity is currently in progress. It usuallyconcerns experiments performed at LHC/CERN (ALICE, ATLAS, CMS and LHCb Collaborations)and RHIC/Brookhaven (STAR and PHENIX Collaborations). As typical illustrations of suchmeasurements and their possible theoretical interpretations, let us mention [80–98]. A rich discussionabout the thermodynamical admissibility of the possible constraints under which the entropicfunctional can be optimized is also present in the literature (see, for instance, [10,11,83,99–101]).Many other systems (e.g., related to those mentioned in [102–105]) are awaiting for approachesalong the above and similar lines. They would be very welcome. Even so, we may say that thepresent status of the theory described herein is at a reasonably satisfactory stage of physical andmathematical understanding. Acknowledgments:
I warmly dedicate this review to Arak M. Mathai on his 80th anniversary. I am deeplyindebted to Hans J. Haubold, whose insights and encouragements have made this overview possible, and alsoto the anonymous referees who, through highly constructive remarks, made possible an improved version of thepresent manuscript. Finally, I acknowledge partial financial support from CNPq and Faperj (Brazilian agencies)and from the John Templeton Foundation (USA).
Conflicts of Interest:
The author declares no conflict of interest.
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