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45. Tani J, Fukumura N (1997) Self-organizing internal represen- Nonadditive entropy It usually refers to the basis of tation in learning of navigation: A physical experiment by the nonextensive statistical mechanics. This entropy, de- mobile robot Yamabico. Neural Netw 10(1):153–159 noted Sq,isnonadditive for q 1. Indeed, for two 46. Tani J, Nolfi S (1999) Learning to perceive the world as articu- ¤ lated: An approach for hierarchical learning in sensory-motor probabilistically independent subsystems A and B,it satisﬁes S (A B) S (A) S (B)(q 1). For his- systems. Neural Netw 12:1131–1141 q C ¤ q C q ¤ 47. Tani J, Nishimoto R, Namikawa J, Ito M (2008) Co-developmen- torical reasons, it is frequently (but inadequately) re- tal learning between human and humanoid robot using a dy- ferred to as nonextensive entropy. namic neural network model. IEEE Trans Syst Man Cybern B. q-logarithmic and q-exponential functions Denoted Cybern 38:1 ln x (ln x ln x), and ex (ex ex ), respectively. 48. Varela FJ, Thompson E, Rosch E (1991) The Embodied mind: q 1 D q 1 D Cognitive science and human experience. MIT Press, Cam- Extensive system So called for historical reasons. A more bridge appropriate name would be additive system. It is a sys- 49. van Gelder TJ (1998) The dynamical hypothesis in cognitive sci- tem which, in one way or another, relies on or is con- ence. Behav Brain Sci 21:615–628 nected to the (additive) Boltzmann–Gibbs entropy. Its 50. Vaughan E, Di Paolo EA, Harvey I (2004) The evolution of control and adaptation in a 3D powered passive dynamic walker. basic dynamical and/or structural quantities are ex- In:PollackJ,BedauM,HusbandP,IkegamiT,WatsonR(eds) pected to be of the exponential form. In the sense of Proceedings of the Ninth International Conference on the Sim- complexity, it may be considered a simple system. ulation and Synthesis of Living Systems. MIT Press, Cambridge Nonextensive system So called for historical reasons. A more appropriate name would be nonadditive system. It is a system which, in one way or another, relies Entropy on or is connected to a (nonadditive) entropy such as S (q 1). Its basic dynamical and/or structural quan- q ¤ CONSTANTINO TSALLIS1,2 tities are expected to asymptotically be of the power- 1 Centro Brasileiro de Pesquisas Físicas, law form. In the sense of complexity, it may be consid- Rio de Janeiro, Brazil ered a complex system. 2 Santa Fe Institute, Santa Fe, USA

Definition of the Subject Article Outline Thermodynamics and statistical mechanics are among Glossary the most important formalisms in contemporary physics. Definition of the Subject They have overwhelming and intertwined applications in Introduction science and technology. They essentially rely on two ba- Some Basic Properties sic concepts, namely energy and entropy.Themathemati- Boltzmann–Gibbs Statistical Mechanics cal expression that is used for the first one is well known On the Limitations of Boltzmann–Gibbs Entropy to be nonuniversal; indeed, it depends on whether we are and Statistical Mechanics say in classical, quantum, or relativistic regimes. The sec- The Nonadditive Entropy S q ond concept, and very specifically its connection with the A Connection Between Entropy and Diffusion microscopic world, has been considered during well over Standard and q-Generalized Central Limit Theorems one century as essentially unique and universal as a physi- Future Directions cal concept. Although some mathematical generalizations Acknowledgments of the entropy have been proposed during the last forty Bibliography years, they have frequently been considered as mere prac- tical expressions for disciplines such as cybernetics and Glossary control theory, with no particular physical interpretation. Absolute temperature Denoted T. What we have witnessed during the last two decades is the Clausius entropy Also called thermodynamic entropy. growth, among physicists, of the belief that it is not neces- Denoted S. sarily so. In other words, the physical entropy would ba- Boltzmann–Gibbs entropy Basis of Boltzmann–Gibbs sically rely on the microscopic dynamical and structural statistical mechanics. This entropy, denoted SBG,isad- properties of the system under study. For example, for sys- ditive. Indeed, for two probabilistically independent tems microscopically evolving with strongly chaotic dy- subsystems A and B,itsatisfiesS (A B) S (A) namics, the connection between the thermodynamical en- BG C D BG C SBG(B). tropy and the thermostatistical entropy would be the one 2860 E Entropy

found in standard textbooks. But, for more complex sys- later on generalized into tems (e. g., for weakly chaotic dynamics), it becomes either necessary, or convenient, or both, to extend the tradi- S k f (q; p)ln[f (q; p)] dq dp ; (4) D tional connection. The present article presents the ubiqui- “ tous concept of entropy, useful even for systems for which where (q; p) is called the -space and constitutes the phase no energy can be defined at all, within a standpoint re- space (coordinate q and momentum p) corresponding to flecting a nonuniversal conception for the connection be- one particle. tween the thermodynamic and the thermostatistical en- Boltzmann’s genius insight – the first ever mathemat- tropies. Consequently, both the standard entropy and its ical connection of the macroscopic world with the micro- recent generalizations, as well as the corresponding statis- scopic one – was, during well over three decades, highly tical mechanics, are here presented on equal footing. controversial since it was based on the hypothesis of the existence of atoms. Only a few selected scientists, like Introduction the English chemist and physicist John Dalton, the Scot- The concept of entropy (from the Greek !, en tish physicist and mathematician James Clerk Maxwell, trepo, at turn, at transformation) was first introduced in and the American physicist, chemist and mathematician 1865 by the German physicist and mathematician Rudolf Josiah Willard Gibbs, believed in the reality of atoms and Julius Emanuel Clausius, Rudolf Julius Emanuel in or- molecules. A large part of the scientific establishment was, der to mathematically complete the formalism of classi- at the time, strongly against such an idea. The intricate cal thermodynamics [55], one of the most important the- evolution of Boltzmann’s lifelong epistemological strug- oretical achievements of contemporary physics. The term gle, which ended tragically with his suicide in 1906, may was so coined to make a parallel to energy (from the Greek be considered as a neat illustration of Thomas Kuhn’s o&, energos, at work), the other fundamental con- paradigm shift, and the corresponding reaction of the sci- cept of thermodynamics. Clausius connection was given entific community, as described in The Structure of Sci- by entific Revolutions. There are in fact two important formalisms in contemporary physics where the mathematical ıQ dS ; (1) theory of probabilities enters as a central ingredient. These D T are statistical mechanics (with the concept of entropy as where ıQ denotes an infinitesimal transfer of heat. In a functional of probability distributions) and quantum other words, 1/T acts as an integrating factor for ıQ. mechanics (with the physical interpretation of wave func- In fact, it was only in 1909 that thermodynamics was tions and measurements). In both cases, contrasting view- finally given, by the Greek mathematician Constantin points and passionate debates have taken place along more Caratheodory, a logically consistent axiomatic formula- than one century, and continue still today. This is no sur- tion. prise after all. If it is undeniable that energy is a very deep In 1872, some years after Clausius proposal, the Aus- and subtle concept, entropy is even more. Indeed, energy trian physicist Ludwig Eduard Boltzmann introduced concerns the world of (microscopic) possibilities,whereas a quantity, that he noted H, which was defined in terms entropy concerns the world of the probabilities of those of microscopic quantities: possibilities, a step further in epistemological difficulty. In his 1902 celebrated book Elementary Principles of H f (v)ln[f (v)] dv ; (2) Statistical Mechanics, Gibbs introduced the modern form • of the entropy for classical systems, namely where f (v)dv is the number of molecules in the velocity space interval dv. Using Newtonian mechanics, Boltz- S k d f (q; p)ln[Cf(q; p)] ; (5) D mann showed that, under some intuitive assumptions Z (Stoßzahlansatz or molecular chaos hypothesis)regarding where represents the full phase space of the system, thus thenatureofmolecularcollisions,H does not increase containing all coordinates and all momenta of its elemen- with time. Five years later, in 1877, he identified this quan- tary particles, and C is introduced to take into account the tity with Clausius entropy through kH S,wherek is finite size and the physical dimensions of the smallest ad- a constant. In other words, he established that missible cell in -space. The constant k is known today to be a universal one, called Boltzmann constant, and given S k f (v)ln[f (v)] dv ; (3) 23 D by k 1:3806505(24) 10 Joule/Kelvin. The studies • D Entropy E 2861 of the German physicist Max Planck along Boltzmann and This violation is one of the mathematical manifesta- Gibbs lines after the appearance of quantum mechanical tions that, at the microscopic level, the state of any concepts, eventually led to the expression physical system exhibits its quantum nature. Expansibility Also S (p ; p ;:::;p ; 0) S (p ; p ; BG 1 2 W D BG 1 2 S k ln W ; (6) :::; D pW ), i. e., zero-probability events do not modify our information about the system. which he coined as Boltzmann entropy. This expression is Maximal value SBG is maximized at equal probabilities, carved on the stone of Boltzmann’s grave at the Central i. e., for p 1/W ; i.ItsvalueisthatofEq.(6). This i D 8 Cemetery of Vienna. The quantity W is the total number corresponds to the Laplace principle of indifference or of microstates of the system that are compatible with our principle of insufficient reason. macroscopic knowledge of it. It is obtained from Eq. (5) Concavity If we have two arbitrary probability distribu- under the hypothesis of an uniform distribution or equal tions p and p for the same set of W possibilities, f i g f 0i g probabilities. we can define the intermediate probability distribution The Hungarian–American mathematician and physi- p p (1 ) p (0 <<1). It straightfor- 00i D i C 0i cist Johann von Neumann extended the concept of BG en- wardly follows that S ( p ) S ( p ) (1 BG f 00i g BG f i g C tropy in two steps – in 1927 and 1932 respectively –, in or- ) S ( p ). This property is essential for thermody- BG f 0i g der to also cover quantum systems. The following expres- namics since it eventually leads to thermodynamic sta- sion, frequently referred to as the von Neumann entropy, bility,i.e.,torobustness with regard to energy fluctu- resulted: ations. It also leads to the tendency of the entropy to attain, as time evolves, its maximal value compatible S k Tr ln ; (7) D with our macroscopic knowledge of the system, i. e., being the density operator (with Tr 1). with the possibly known values for the macroscopic D Another important step was given in 1948 by the constraints. American electrical engineer and mathematician Claude Lesche stability or experimental robustness B. Lesche Elwood Shannon. Having in mind the theory of digital introduced in 1982 [107] the definition of an interest- communications he explored the properties of the discrete ing property, which he called stability. It reflects the form experimental robustness that a physical quantity is expected to exhibit. In other words, similar experiments W should yield similar numerical results for the physi- S k p ln p ; (8) D i i cal quantities. Let us consider two probability distribu- i 1 XD tions pi and p0 , assumed to be close, in the sense f g f i g that W p p <ı;ı>0 being a small number. frequently referred to as Shannon entropy (with W i 1 j i 0i j i 1 An entropicD functional S( p )issaidstable or exper- p 1). This form can be recovered from Eq. (5)fortDhe P f i g i D P imentally robust if, for any given >0, a ı>0exists particular case for which the phase space density f (q; p) W D such that S( pi ) S( p0i ) /Smax <;where Smax is p ı(q qi) ı(p pi). It can also be recovered from j f g f g j i 1 i the maximal value that the functional can attain Eq.D(7)when is diagonal. We may generically refer to P (ln W inthecaseofSBG). This implies that limı 0 Eqs. (5), (6), (7)and(8)astheBG entropy,notedSBG.It ! limW (S( pi ) S( p0i ))/Smax 0. As we shall is a measure of the disorder of the system or, equivalently, !1 f g f g D see soon, this property is much stronger than it seems of our degree of ignorance or lack of information about at first sight. Indeed, it provides a (necessary but not its state. To illustrate a variety of properties, the discrete sufficient) criterion for classifying entropic functionals form (8) is particularly convenient. into physically admissible or not. It can be shown that SBG is Lesche-stable (or experimentally robust). Some Basic Properties Entropy production If we start the (deterministic) time Non-negativity It can be easily verified that, in all cases, evolution of a generic classical system from an arbi- S 0, the zero value corresponding to certainty, trarily chosen point in its phase space, it typically BG i. e., p 1 for one of the W possibilities, and zero follows a quite erratic trajectory which, in many cases, i D for all the others. To be more precise, it is exactly so gradually visits the entire (or almost) phase space. By whenever SBG is expressed either in the form (7)orin making partitions of this -space, and counting the the form (8). However, this property of non-negativity frequency of visits to the various cells (and related may be no longer true if it is expressed in the form (5). symbolic quantities), it is possible to define probabil- 2862 E Entropy

ity sets. Through them, we can calculate a sort of time Additivity and extensivity If we consider a system A B C evolution of SBG(t). If the system is chaotic (sometimes constituted by two probabilistically independent sub- called strongly chaotic), i. e., if its sensitivity to the systems A and B, i. e., if we consider pA B pA pB, ijC D i j initial conditions increases exponentially with time, we immediately obtain from Eq. (8)that then SBG(t)increaseslinearly with t in the appropriate asymptotic limits. This rate of increase of the en- SBG(A B) SBG(A) SBG(B) : (12) C D C tropy is called Kolmogorov–Sinai entropy rate,and,for a large class of systems, it coincides (Pesin identity In other words, the BG entropy is additive [130]. If or Pesin theorem) with the sum of the positive Lya- our system is constituted by N probabilistically inde- punov exponents. These exponents characterize the pendent identical subsystems (or elements), we clearly have S (N) N. It frequently happens, however, exponential divergences, along various directions in BG / the -space, of a small discrepancy in the initial con- that the N elements are not exactly independent but only asymptotically so in the N limit. This is the dition of a trajectory. !1 It turns out, however, that the Kolmogorov–Sinai en- usual case of many-body Hamiltonian systems involv- tropy rate is, in general, quite inconvenient for com- ing only short-range interactions, where the concept of putational calculations for arbitrary nonlinear dynam- short-range will be focused in detail later on. For such ical systems. In practice, another quantity is used in- systems, SBG is only asymptotically additive, i. e., stead [102], usually referred to as entropy production SBG(N) per unit time,whichwenoteKBG. Its definition is as 0 < lim < : (13) N N 1 follows. We first make a partition of the -space into !1 many W cells (i 1; 2;:::;W). In one of them, ar- An entropy S( p ) of a specific systems is said exten- D f i g bitrarily chosen, we randomly place M initial condi- sive if it satisfies tions (i. e., an ensemble). As time evolves, the occupancy of the W cells determines the set M (t) ,with S(N) f i g 0 < lim < ; (14) W M (t) M. This set enables the definition of N N 1 i 1 i D !1 a probaD bility set with p (t) M (t)/M,whichinturn P i i where no hypothesis at all is made about the possible determines S (t). We then define the entropy produc- BG independence or weak or strong correlations between tion per unit time as follows: the elements of the system whose entropy S we are con- SBG(t) sidering. Equation (13) amounts to say that the addi- KBG lim lim lim : (9) t W M t tive entropy SBG is extensive for weakly correlated sys- !1 !1 !1 tems such as the already mentioned many-body short- Up to date, no theorem guarantees that this quan- range-interacting Hamiltonian ones. It is important to tity coincides with the Kolmogorov–Sinai entropy clearly realize that additivity and extensivity are inde- rate. However, many numerical approaches of various pendent properties. An additive entropy such as SBG is chaotic systems strongly suggest so. The same turns extensive for simple systems such as the ones just men- out to occur with what is frequently referred in the lit- tioned, but it turns out to be nonextensive for other, erature as a Pesin-like identity. For instance, if we have more complex, systems that will be focused on later a one-dimensional dynamical system, its sensitivity to on. For many of these more complex systems, it is the the initial conditions limx(0) 0 x(t)/x(0) is ! nonadditive entropy Sq (to be analyzed later on) which typically given by turns out to be extensive for a non standard value of q (t) et ; (10) (i. e., q 1). D ¤ where x(t) is the discrepancy in the one-dimensional Boltzmann–Gibbs Statistical Mechanics phase space of two trajectories initially differing by x(0), and is the Lyapunov exponent (>0 corre- Physical systems (classical, quantum, relativistic) can be sponds to strongly sensitive to the initial conditions,or theoretically described in very many ways, through mi- strongly chaotic,and<0 corresponds to strongly in- croscopic, mesoscopic, macroscopic equations, reflecting sensitive to the initial conditions). The so-called Pesin- either stochastic or deterministic time evolutions, or even like identity amounts, if 0, to both types simultaneously. Those systems whose time evolution is completely determined by a well defined Hamil- KBG : (11) tonian with appropriate boundary conditions and ad- D Entropy E 2863 missible initial conditions are the main purpose of an other important situations). The connection with classi- important branch of contemporary physics, named sta- cal thermodynamics, and its Legendre-transform struc- tistical mechanics. This remarkable theory (or formalism, ture, occurs through relations such as as sometimes called), which for large systems satisfacto- 1 @S rily matches classical thermodynamics, was primarily in- (19) troduced by Boltzmann and Gibbs. The physical system T D @U 1 can be in all types of situations. Two paradigmatic such F U TS ln Z (20) situations correspond to isolation,andthermal contact Dˇ with a large reservoir called thermostat.Theirstationary @ U ln Z (21) state (t ) is usually referred to as thermal equilib- D@ˇ !1 rium. Both situations have been formally considered by @S @U @2F C T T ; (22) Gibbs within his mathematical formulation of statistical @T D @T D @T2 mechanics, and they respectively correspond to the so- called micro-canonical and canonical ensembles (other en- where F, U and C are the Helmholtz free energy,thein- sembles do exist, such as the grand-canonical ensemble, ternal energy,andthespecific heat respectively. The BG appropriate for those situations in which the total num- statistical mechanics historically appeared as the first con- ber of elements of the system is not fixed; this is however nection between the microscopic and the macroscopic de- out of the scope of the present article). scriptions of the world, and it constitutes one of the cor- The stationary state of the micro-canonical ensemble is nerstones of contemporary physics. The Establishment re- determined by p 1/W ( i,wherei runs over all possi- sisted heavily before accepting the validity and power of i D 8 ble microscopic states), which corresponds to the extrem- Boltzmann’ s revolutionary ideas. In 1906 Boltzmann dra- matically committed suicide, after 34 years that he had first ization of SBG with a single (and trivial) constraint, namely proposed the deep ideas that we are summarizing here. At W that early 20th century, few people believed in Boltzmann’s p 1 : (15) proposal (among those few, we must certainly mention Al- i D i 1 bert Einstein), and most physicists were simply unaware of XD To obtain the stationary state for the canonical ensem- the existence of Gibbs and of his profound contributions. ble, the thermostat being at temperature T,wemust(typi- It was only half a dozen years later that the emerging new cally) add one more constraint, namely generation of physicists recognized their respective genius (thanks in part to various clarifications produced by Paul W Ehrenfest, and also to the experimental successes related p E U ; (16) with Brownian motion, photoelectric effect, specific heat i i D i 1 of solids, and black-body radiation). XD where E are the energies of all the possible states of the f i g On the Limitations of Boltzmann–Gibbs Entropy system (i. e., eigenvalues of the Hamiltonian with the ap- and Statistical Mechanics propriate boundary conditions). The extremization of SBG with the two constraints above straightforwardly yields Historical Background As any other human intellectual construct, the applicabil- ˇ Ei e pi (17) ity of the BG entropy, and of the statistical mechanics to D Z which it is associated, naturally has restrictions. The un- W derstanding of present developments of both the concept Z e ˇ E j (18) of entropy, and its corresponding statistical mechanics, j 1 XD demand some knowledge of the historical background. Boltzmann was aware of the relevance of the range with the partition function Z, and the Lagrange param- of the microscopic interactions between atoms and eter ˇ 1/kT.ThisisthecelebratedBG distribution D molecules. He wrote, in his 1896 Lectures on Gas The- for thermal equilibrium (or Boltzmann weight,orGibbs ory [41], the following words: state, as also called), which has been at the basis of an enormous amount of successes (in fluids, magnets, su- When the distance at which two gas molecules inter- perconductors, superfluids, Bose–Einstein condensation, act with each other noticeably is vanishingly small conductors, chemical reactions, percolation, among many relative to the average distance between a molecule 2864 E Entropy

and its nearest neighbor—or, as one can also say, The situation is different for the additivity postu- when the space occupied by the molecules (or their late P a2, the validity of which cannot be inferred spheres of action) is negligible compared to the space from general principles. We have to require that the filled by the gas—then the fraction of the path of each interaction energy between thermodynamic systems molecule during which it is affected by its interac- be negligible. This assumption is closely related to tion with other molecules is vanishingly small com- the homogeneity postulate P d1. From the molecular pared to the fraction that is rectilinear, or simply de- point of view, additivity and homogeneity can be ex- termined by external forces. [ ... ] The gas is “ideal” pected to be reasonable approximations for systems in all these cases. containing many particles, provided that the inter- molecular forces have a short range character. Also Gibbs was aware. In his 1902 book [88], he wrote: In treating of the canonical distribution, we shall al- Corroborating the above, virtually all textbooks of quan- ways suppose the multiple integral in equation (92) tum mechanics contain the mechanical calculations cor- [the partition function, as we call it nowadays] to responding to a particle in a square well, the harmonic have a finite value, as otherwise the coefficient of oscillator, the rigid rotator, a spin 1/2 in the presence of probability vanishes, and the law of distribution be- a magnetic field, and the Hydrogen atom. In the textbooks comes illusory. This will exclude certain cases, but not of statistical mechanics we can find the thermostatistical such apparently, as will affect the value of our results calculations of all these systems ... excepting the Hydro- with respect to their bearing on thermodynamics. It gen atom! Why? Because the long-range electron-proton will exclude, for instance, cases in which the system interaction produces an energy spectrum which leads to or parts of it can be distributed in unlimited space a divergent partition function. This is but a neat illustra- [ ... ]. It also excludes many cases in which the en- tion of the above Gibbs’ alert. ergy can decrease without limit, as when the system contains material points which attract one another A Remark on the Thermodynamics inversely as the squares of their distances. [ ... ]. For of Short- and Long-Range Interacting Systems the purposes of a general discussion, it is sufficient to We consider here a simple d-dimensional classical fluid, call attention to the assumption implicitly involved in constituted by many N point particles, governed by the the formula (92). Hamiltonian The extensivity/additivity of SBG has been challenged, N along the last century, by many physicists. Let us mention p2 H K V i V(r ) ; (23) just a few. In his 1936 Thermodynamics [82], Enrico Fermi D C D 2m C ij i 1 i j wrote: XD X¤ The entropy of a system composed of several parts is where the potential V(r) has, if it is attractive at short very often equal to the sum of the entropies of all the distances, no singularity at the origin, or an integrable parts. This is true if the energy of the system is the singularity, and whose asymptotic behavior at infinity ˛ sum of the energies of all the parts and if the work is given by V(r) B/r with B > 0and˛ 0. One performed by the system during a transformation is such example is the d 3 Lennard–Jones fluid, for D equal to the sum of the amounts of work performed which V(r) A/r12 B/r6(A > 0), i. e., repulsive at short D by all the parts. Notice that these conditions are not distances and attractive at long distances. In this case quite obvious and that in some cases they may not ˛ 6. Another example could be Newtonian gravita- D be fulfilled. Thus, for example, in the case of a system tion with a phenomenological short-distance cutoff (i. e., composed of two homogeneous substances, it will be V(r) for r r0 with r0 > 0. In this case, ˛ 1. !1 D possible to express the energy as the sum of the ener- The full -space of such a system has 2dN dimensions. gies of the two substances only if we can neglect the The total potential energy is expected to scale (assuming surface energy of the two substances where they are a roughly homogeneous distribution of the particles) as in contact. The surface energy can generally be ne- Upot(N) glected only if the two substances are not very finely B 1 drrd 1 r ˛ ; (24) N / subdivided; otherwise, it can play a considerable role. Z1 Laszlo Tisza wrote, in his Generalized Thermodynam- where the integral starts appreciably contributing above ics [178]: a typical cutoff, here taken to be unity. This integral is finite Entropy E 2865

[ B/(˛ d)]for˛/d > 1(short-range interactions), procedure, i. e. ˛/d 0, must conform to the following D 8 and diverges for 0 ˛/d 1(long-range interactions). In lines: other words, the energy cannot be generically character- G(N; T; p; H) U(N; T; p; H) ized by Eq. (24), and we must turn onto a different and lim lim N NN? D N NN? more powerful estimation. Given the finiteness of the size !1 !1 T S(N; T; p; H) of the system, an appropriate one is, in all cases, given by lim N N? N !1 (29) N1/d ; ; ; Upot(N) d 1 ˛ B ? p V(N T p H) B drr r N ; (25) lim C N N? N N / 1 Dd !1 Z H M(N; T; p; H) where lim N N? N !1 1 if ˛/d > 1; hence ˛ 1 ˛/d /d 1 ? N 1 8 N ˆln N if ˛/d 1; ? ? ? ? ? ? ? ? ? ? 1 ˛/d ˆ D g(T ; p ; H ) u(T ; p ; H ) T s(T ; p ; H ) ˆ 1 ˛/d D < N p?v(T?; p?; H?) H?m(T?; p?; H?) ; (30) if 0 <˛/d < 1 : C ˆ 1 ˛/d ˆ ? ˆ (26) where the definitions of T and all the other variables are : ? ? ? self-explanatory (e. g., T T/N ). In other words, in or- Notice that N ln˛/d N where the q-log func- 1 q D der to have finite thermodynamic equations of states, we tion ln x (x 1)/(1 q)(x > 0; ln x ln x)will ? ? ? q 1 D must in general express them in the (T ; p ; H )vari- be shown to play an important role later on. Satisfacto- ables. If ˛/d > 1, this procedure recovers the usual equa- rily enough, Eqs. (26) recover the characterization with tions of states, and the usual extensive (G; U; S; V ; M) Eq. (24)inthelimitN , but they have the great ad- !1 and intensive (T; p; H) thermodynamic variables. But, if vantage of providing, for finite N,afinite value. This fact 0 ˛/d 1, the situation is more complex, and we real- will be now shown to enable to properly scale the macro- ize that three, instead of the traditional two, classes of ther- scopic quantities in the thermodynamic limit (N ), !1 modynamic variables emerge. We may call them exten- for all values of ˛/d 0. sive (S; V; M; N), pseudo-extensive (G; U)andpseudo-in- Let us address the thermodynamical consequences of tensive (T; p; H) variables. All the energy-type thermody- the microscopic interactions being short- or long-ranged. namical variables (G; F; U) give rise to pseudo-extensive To present a slightly more general illustration, we shall as- ones, whereas those which appear in the usual Legendre sume from now on that our homogeneous and isotropic thermodynamical pairs give rise to pseudo-intensive ones classical fluid is made by magnetic particles. Its Gibbs free (T; p; H;) and extensive ones (S; V; M; N). See Figs. 1 energy is then given by and 2. The possibly long-range interactions within Hamil- G(N; T; p; H) U(N; T; p; H) TS(N; T; p; H) D tonian (23) refer to the dynamical variables themselves. pV (N; T; p; H) HM(N; T; p; H) ; (27) C There is another important class of Hamiltonians, where the possibly long-range interactions refer to the coupling where (T; p; H) correspond respectively to the tempera- constants between localized dynamical variables.Suchis, ture, pressure and external magnetic field, V is the volume for instance, the case of the following classical Hamilto- and M the magnetization. If the interactions are short- nian: ranged (i. e., if ˛/d > 1), we can divide this equation by N and then take the N limit. We obtain !1 N L2 H K V i D C D 2I g(T; p; H) u(T; p; H) Ts(T; p; H) i 1 D XD pv(T; p; H) Hm(T; p; H) ; (28) x x y y z z Jx si s j Jysi s j Jz si s j C C C (˛ 0) ; (31) r˛ where g(T; p; H) limN G(N; T; p; H)/N, and anal- i j ij !1 ogously for the other variables of the equation. If the in- X¤ teractions were instead long-ranged (i. e., if 0 ˛/d 1), where Li are the angular momenta, I the moment of f g y all these quantities would be divergent, hence thermody- inertia, (sx ; s ; sz ) are the components of classical ro- f i i i g namically nonsense. Consequently, the generically correct tators, (Jx ; Jy; Jz ) are coupling constants, and rij runs 2866 E Entropy

which has in fact the same asymptotic behaviors as in- dicated in Eq. (26). In other words, here again ˛/d > 1 corresponds to short-range interactions, and 0 ˛/d 1 corresponds to long-range ones. The correctness of the present generalized thermodynamical scalings has already been specifically checked in many physical systems, such as a ferrofluid-like model [97], Lennard–Jones-like fluids [90], magnetic systems [16,19,59,158], anomalous diffusion [66], percola- Entropy, Figure 1 For long-range interactions (0 ˛/d 1) we have three classes tion [85,144]. Let us mention that, for the ˛ 0 models (i. e., mean of thermodynamic variables,namelythepseudo-intensive (scal- D ing with N?), pseudo-extensive (scaling with NN?)andexten- field models), it is largely spread in the literature to divide sive (scaling with N) ones. For short range interactions (˛/d > 1) by N the potential term of the Hamiltonian in order to the pseudo-intensive variables become intensive (independent make it extensive by force. Although mathematically ad- from N), and the pseudo-extensive merge with the extensive ones, all being now extensive (scaling with N), thus recovering missible (see [19]),thisisobviouslyveryunsatisfactoryin the traditional two textbook classes of thermodynamical vari- principle since it implies a microscopic coupling constant ables which depends on N.Whatwehavedescribedhereisthe thermodynamically proper way of eliminating the mathematical difficulties emerging in the models in the presence of long-range interactions. Last but not least, we verify a point which is crucial for the developments here below, namely that the entropy S is expected to be extensive no matter the range of the interactions.

The Nonadditive Entropy Sq Introduction and Basic Properties The possibility was introduced in 1988 [183](see also [42,112,157,182]) to generalize the BG statistical me- chanicsonthebasisofanentropySq which general- Entropy, Figure 2 izes SBG. This entropy is defined as follows: The so-called extensive systems (˛/d > 1 for the classical ones) typically involve absolutely convergent series,whereastheso- W q 1 i 1 pi called nonextensive systems (0 ˛/d < 1 for the classical ones) S k D (q R; S S ): (33) q q 1 2 1 D BG typically involve divergent series. The marginal systems (˛/d D 1 P here) typically involve conditionally convergent series,which therefore depend on the boundary conditions, i. e., typically on For equal probabilities, this entropy takes the form the external shape of the system. Capacitors constitute a notori- ous example of the ˛/d D 1 case. The model usually referred to Sq k lnq W (S1 k ln W) ; (34) in the literature as the Hamiltonian–Mean–Field (HMF) one lies D D on the ˛ D 0axis(8d > 0). The model usually referred to as where the q-logarithmic function has already been de- the d-dimensional ˛-XY model [19] lies on the vertical axis at ab- fined. scissa d (8˛ 0) Remark With the same or different prefactor, this en- over all distances between sites i and j of a d-dimen- tropic form has been successively and independently in- sional lattice. For example, for a simple hypercubic lattice troduced in many occasions during the last decades. with unit crystalline parameter we have r 1; 2; 3;:::if J. Havrda and F. Charvat [92] were apparently the first to ij D d 1, r 1; p2; 2;:::if d 2, r 1; p2; p3; 2;::: ever introduce this form, though with a different prefactor D ij D D ij D if d 3, and so on. For such a case, we have that (adapted to binary variables) in the context of cybernet- D ics and information theory. I. Vajda [207], further studied N ? ˛ this form, quoting Havrda and Charvat. Z. Daroczy [74] N r ; (32) 1i rediscovered this form (he quotes neither Havrda–Charvat i 2 XD Entropy E 2867

x nor Vajda). J. Lindhard and V. Nielsen [108] rediscovered with q > 0, q < 1, the q-exponential function eq this form (they quote none of the predecessors) through being the inverse of lnq x.Moreexplicitly(seeFig.3) the property of entropic composability. B.D. Sharma and 1 1q D.P. Mittal [163] introduced a two-parameter form which x [1 (1 q) x] if 1 (1 q)x > 0; reproduces both S and Renyi entropy [145]aspartic- eq C C q ( 0otherwise: ular cases. A. Wehrl [209] mentions the form of Sq in p. 247, quotes Daroczy, but ignores Havrda–Charvat, Va- (36) jda, Lindhard–Nielsen, and Sharma–Mittal. Myself I re- Such systems have a finite entropy production per discovered this form in 1985 with the aim of generalizing unit time, which satisfies a q-generalized Pesin-like Boltzmann–Gibbs statistical mechanics, but quote none of identity, namely, for the construction described in the predecessors in the 1988 paper [183]. In fact, I started Sect. “Introduction”, knowing the whole story quite a few years later thanks to Sq(t) S.R.A. Salinas and R.N. Silver, who were the first to provide Kq lim lim lim q : (37) t W M t D me with the corresponding informations. Such rediscov- !1 !1 !1 eries can by no means be considered as particularly sur- The situation is in fact sensibly much richer than prising. Indeed, this happens in science more frequently briefly described here. For further details, see [27,28, than usually realized. This point is lengthily and colorfully 29,30,93,116,117,146,147,148,149,150,151,152]. developed by S.M. Stigler [167]. In p. 284, a most inter- (vii) S is nonadditive for q 1. Indeed, for indepen- q ¤ esting example is described, namely that of the celebrated dent subsystems A and B, it can be straightforwardly normal distribution.ItwasfirstintroducedbyAbraham proved De Moivre in 1733, then by Pierre Simon de Laplace in Sq(A B) Sq(A) Sq(B) Sq (A) Sq(B) 1774, then by Robert Adrain in 1808, and finally by Carl C (1 q) ; Friedrich Gauss in 1809, nothing less than 76 years after k D k C k C k k its first publication! This distribution is universally called (38) Gaussian because of the remarkable insights of Gauss con- or, equivalently, cerning the theory of errors, applicable in all experimental sciences. A less glamorous illustration of the same (1 q) S (A B) S (A) S (B) S (A) S (B) ; phenomenon, but nevertheless interesting in the present q C D q C q C k q q context, is that of Renyi entropy [145]. According to I. (39) Csiszar [64], p. 73, the Renyi entropy had already been essentially introduced by Paul-Marcel Schutzenberger [161]. which makes explicit that (1 q) 0playsthe ! same role as k . Property (38), occasion- !1 The entropy defined in Eq. (33) has the following main ally referred to in the literature as pseudo-additiv- properties: ity, can be called subadditivity (superadditivity)for q > 1(q < 1). (i) S is nonnegative ( q); W x q (viii) Sq kDq i 1 pi x 1, where the 1909 Jackson 8 D D j D (ii) Sq is expansible ( q > 0); differential operator is defined as follows: 8 P (iii) Sq attains its maximal (minimal) value k lnq W for q > 0(forq < 0); f (qx) f (x) Dq f (x) (D1 f (x) d f (x)/dx) : (iv) S is concave (convex) for q > 0(forq < 0); qx x D q (v) S is Lesche-stable ( q > 0) [2]; (40) q 8 (vi) Sq yields a finite upper bound of the entropy production per unit time for a special value of q,when- (ix) An uniqueness theorem has been proved by San- ever the sensitivity to the initial conditions exhibits tos [159], which generalizes, for arbitrary q,thatof an upper bound which asymptotically increases as Shannon [162]. a power of time. For example, many D 1non- Let us assume that an entropic form S( pi )satisfies D f g linear dynamical systems have a vanishing maximal the following properties: (a) Lyapunov exponent 1 and exhibit a sensitivity to the initial conditions which is (upper) bounded by S( p ) is a continuous function of p ; f i g f i g t (41) e q ; (35) D q 2868 E Entropy

(b) Additivity Versus Extensivity of the Entropy

S(p 1/W; i) monotonically increases It is of great importance to distinguish additivity from i D 8 with the total number of possibilities W; extensivity.AnentropyS is additive [130]ifitsvalue (42) for a system composed by two independent subsystems A and B satisfies S(A B) S(A) S(B)(hence, (c) C D C for N independent equal subsystems or elements, we S(A B) S(A) S(B) S(A) S(B) have S(N) NS(1)). Therefore, S is additive, and C (1 q) D BG k D k C k C k k S (q 1) is nonadditive. A substantially different mat- q ¤ A B A B ter is whether a given entropy S is extensive for if pijC pi p j (i; j) ; with k > 0; D 8 a given system. An entropy is extensive if and only if (43) 0 < limN S(N)/N < . What matters for satisfacto- !1 1 (d) rily matching thermodynamics is extensivity not addi- q q tivity. For systems whose elements are nearly indepen- S( pi ) S(pL ; pM) pL S( pi /pL ) pM S( pi /pM ) f g D C f g C f g dent (i. e., essentially weakly correlated), SBG is extensive with p p ; p p (L M W) ; L i M i C D and Sq is nonextensive. For systems whose elements are L terms M terms X X strongly correlated in a special manner, SBG is nonexten- and pL pM 1 : sive, whereas S is extensive for a special value of q 1 C D q ¤ (44) (and nonextensive for all the others). Let us illustrate these facts for some simple ex- Then and only then [159] S( p ) S ( p ). f i g D q f i g amples of equal probabilities. If W(N) AN (A > 0, (x) Another (equivalent) uniqueness theorem was >1, and N ), the entropy which is extensive proved by Abe [1], which generalizes, for arbitrary q, !1 is S . Indeed, S (N) k ln W(N) (ln )N N (it that of Khinchin [100]. BG BG D / is equally trivial to verify that S (N)isnonexten- Let us assume that an entropic form S( p )satisfies q f i g sive for any q 1). If W(N) BN(B > 0, >0, and the following properties: ¤ N ), the entropy which is extensive is S1 (1/).In- (a) !1 1/ deed, S1 (1/)(N) kB N N (it is equally trivial / S( pi ) is a continuous function of pi ; (45) to verify that SBG(N) ln N, hence nonextensive). If f g f g / (b) W(N) CN (C > 0, >1, 1, and N ), then ¤ !1 Sq(N) is nonextensive for any value of q. Therefore, in S(p 1/W; i) monotonically increases i D 8 such a complex case, one must in principle refer to some with the total number of possibilities W; other kind of entropic functional in order to match the ex- (46) tensivity required by classical thermodynamics. (c) Various nontrivial abstract mathematical models can be found in [113,160,186,198,199]forwhichS (q 1) is q ¤ S(p ; p ;:::;p ; 0) S(p ; p ;:::;p ); extensive. Moreover, a physical realization is also avail- 1 2 W D 1 2 W (47) able now [60,61] for a many-body quantum Hamiltonian, namely the ground state of the following one: (d)

N 1 S(A B) S(A) S(B A) S(A) S(B A) C j (1 q) j H x x y y D C C (1 )Si Si 1 (1 )Si Si 1 k k k k k D C C C i 1 C where S(A B) S pA B ; XD C ijC N n o 2 Sz ; (49) WB i S(A) S pA B ; and the conditional entropy i 1 08 ijC 91 XD j 1 0) j jD j WA n A q o for 0 < < 1, we have the anisotropic XY model, and, P i 1 pi j j D for 0, we have the isotropic XY model. The two (48) D P former share the same symmetry and consequently be- Then and only then [1] S( p ) S ( p ). long to the same critical universality class (the Ising uni- f i g D q f i g Entropy E 2869

Entropy, Figure 3 The q-exponential and q-logarithm functions in typical representations: a Linear-linear representation of ex ; b Linear-linear repre- q x aq x q sentation of eq ; c Log-log representation of y(x) D eq ,solutionofdy/dx Daq y with y(0) D 1; d Linear-linear representation of Sq D lnq W (value of the entropy for equal probabilities)

versality class, which corresponds to a so-called central increases with L for all values of q.Anditdoessolinearly charge c 1/2), whereas the latter one belongs to a dif- for D ferent universality class (the XX one, which corresponds p9 c2 3 to a central charge c 1). At temperature T 0and q C ; (50) D D D c N , this model exhibits a second-order phase tran- !1 sition as a function of . For the Ising model, the criti- where c is the central charge which emerges in quan- cal value is 1, whereas, for the XX model, the entire tum field theory [54]. In other words, 0 < limL D !1 line 0 1 is critical. Since the system is at its ground S(p9 c2 3)/c (L)/L < .Noticethatq increases from zero C 1 state (assuming a vanishingly small magnetic field compo- to unity when c increases from zero to infinity; q p37 D nent in the x y plane), it is a pure state (i. e., its density 6 0:083 for c 1/2 (Ising model), q p10 3 0:16 2 ' D D ' matrix N is such that Tr 1, N), hence the entropy for c 1(isotropicXY model), q 1/2 for c 4(dimen- N D 8 D D D Sq(N)( q > 0) is strictly zero. However, the situation is sion of space-time), and q (p685 3)/26 0:89 for 8 D ' drastically different for any L-sized block of the infinite c 26, related to string theory [89]. The possible phys- 2 D chain. Indeed, L TrN LN is such that Tr L < 1, i. e., ical interpretation of the limit c is still unknown, !1 it is a mixed state, hence it has a nonzero entropy. The although it could correspond to some sort of mean field block entropy Sq(L) limN Sq(N; L) monotonically approach. !1 2870 E Entropy

Nonextensive Statistical Mechanics and

To generalize BG statistical mechanics for the canonical ˇq ˇ0 : (58) ensemble, we optimize Sq with constraint (15)andalso q 1 (1 q)ˇ U C q q W The form (56) is particularly convenient for many appli- P E U ; (51) i i D q cations where comparison with experimental or computa- i 1 XD tional data is involved. Also, it makes clear that pi asymp- where 1/(q 1) totically decays like 1/Ei for q > 1, and has a cut- q off for q < 1, instead of the exponential decay with E for p W i P i P 1 (52) q 1. i W q i D D j 1 pi i 1 ! The connection to thermodynamics is established in D XD what follows. It can be proved that is the so-cPalled escort distribution [33]. It follows that 1/q W 1/q pi P / P . There are various converging rea- 1 @S D i j 1 j q sons for being Dappropriate to impose the energy constraint ; (59) P T D @Uq with the P instead of with the original p .Thefulldis- f i g f i g cussion of this delicate point is beyond the present scope. with T 1/(kˇ). Also we prove, for the free energy, However, some of these intertwined reasons are explored 1 in [184]. By imposing Eq. (51), we follow [193], which in F U TS ln Z ; (60) q q q Dˇ q q turn reformulates the results presented in [71,183]. The passage from one to the other of the various existing where formulations of the above optimization problem are dis- cussedindetailin[83,193]. lnq Zq lnq Z¯q ˇUq : (61) D The entropy optimization yields, for the stationary state, This relation takes into account the trivial fact that, in contrast with what is usually done in BG statistics, the en- ˇq(Ei Uq) e ergies Ei are here referred to Uq in (53). It can also be q f g pi ; (53) proved D Z¯q @ with U ln Z ; (62) q D@ˇ q q ˇ ˇ ; (54) q W q as well as relations such as j 1 p j D 2 @Sq @Uq @ Fq and P C T T : (63) q @T D @T D @T2 W ˇq (Ei Uq) Z¯q eq ; (55) In fact the entire Legendre transformation structure of thermodynamics is q-invariant, which is both remarkable Xi and welcome. ˇ being the Lagrange parameter associated with the constraint (51). Equation (53) makes explicit that the proba- A Connection Between Entropy and Diffusion bility distribution is, for fixed ˇq, invariant with regard to the arbitrary choice of the zero of energies. The station- We review here one of the main common aspects of en- ary state (or (meta)equilibrium) distribution (53)canbe tropy and diffusion. We shall present on equal footing rewritten as follows: both the BG and the nonextensive cases [13,138,192,216]. 0 Let us extremize the entropy ˇq Ei eq p ; (56) q i D Z 1 1 d(x/)[ p(x)] q0 Sq k 1 (64) D R q 1 with with the constraints W 0 ˇq E j Z e ; (57) 1 q0 q dxp(x) 1 (65) j 1 D XD Z1 Entropy E 2871 and with a generic nonsingular potential U(x), and a gen-

2 q eralized diffusion coefficient D which is positive (nega- 1 dxx [p(x)] x2 2 ; (66) tive) for q < 2(2< q < 3). Several particular instances q 1 q h i R 1 dx [p(x)] D of this equation have been discussed in the literature 1 (see [40,86,106,131,188] and references therein). >0 being soRme fixed value having the same physical di- For example, the stationary state for ˛ 2, ı,and mensions of the variable x. We straightforwardly obtain D 8 any confining potential (i. e., lim U(x) )is the following distribution: x D1 given by [106] j j!1

pq(x) ˇ [U(x) U(0)] D eq 1 p(x; )q ; (69) 1 D Z 1 q 1 1/2 q 1 1 8 (3 q) 3 q q 1 x2 1/(q 1) 1 ˇ [U(x) U(0)] Z dx eq ; (70) ˆ 2(q 1) 1 2 ˆ C 3 q ˆ Z1 ˆ if 1 < q < 3; ˆ 1/ˇ kT D ; (71) ˆ /j j ˆ 1 1 x2/22 ˆ e which precisely is the distribution obtained within nonex- ˆ p2 ˆ ˆ if q 1; tensive statistical mechanics through extremization of Sq. <ˆ D 5 3q Also, the solution for ˛ 2, ı 1, U(x) k1x k2 2 D D D C 1/2 2 1/(1q) ; ı ˆ 1 1 q 2(1 q) 1 q x 2 x ( k1,andk2 0), and p(x 0) (x) is given ˆ 1 8 D ˆ 2 by [188] ˆ (3 q) 2 q 3 q ˆ ˆ 1 q 2 ˆ 1/2 ˇ(t)[x xM (t)] ˆfor x <[(3 q)/(1 q)] ; and zero otherwise , eq ˆ j j pq(x; t) ; (72) ˆ if q < 1 : D Z (t) ˆ q ˆ 2 :ˆ (67) ˇ(t) Zq(0) ˇ(0) D Zq(t) These distributions are frequently referred to as q-Gaus- (73) 2/(3 q) sians.Forq > 1, they asymptotically have a power-law 1 1 1 e t/ ; tail (q 3 is not admissible because the norm (65)can- D K C K 2 2 not be satisfied); for q < 1, they have a compact sup- k K 2 ; (74) port. For q 1, the celebrated Gaussian is recovered; for 2 2(2 q)Dˇ(0)[Z (0)]q 1 D q q 2, the Cauchy–Lorentz distribution is recovered; fi- 1 D nally, for q , the uniform distribution within the ; (75) k2(3 q) !1 3 m interval [ 1; 1] is recovered. For q 1Cm , m being an D k1 k1 k2 t integer (m 1; 2; 3;:::), we recover theCStudent’s t-dis- xM(t) xM(0) e : (76) D k2 C k2 tributions with m degrees of freedom [79]. For q n 4 , D n2 n being an integer (n 3; 4; 5;:::), we recover the so- In the limit k 0, Eq. (73) becomes D 2 ! called r-distributions with n degrees of freedom [79]. In other words, q-Gaussians are analytical extensions of Stu- Z (t) [Z (0)]3 q 2(2 q)(3 q)Dˇ(0) q D q C dent’s t-andr-distributions. In some communities they 2 1/(3 q) ˚ [Z (0)] t ; (77) are also referred to as the Barenblatt form. For q < 5/3, q they have a finite variance which monotonically increases which, in the t limit, yields for q varying from to 5/3; for 5/3 q < 3, the vari- !1 1 ance diverges. 1 2 2/(3 q) [Zq(t)] t : (78) Let us now make a connection of the above optimiza- ˇ(t) / / tion problem with diffusion. We focus on the following In other words, x2 scales like t ,with quite general diffusion equation: 2 ; (79) @ı p(x; t) @ @U(x) @˛[p(x; t)]2 q D 3 q p(x; t) D @tı D @x @x C @ x ˛ j j hence, for q > 1wehave >1(i.e.,superdiffusion;in (0 <ı 1; 0 <˛ 2; q < 3; t 0) ; (68) particular, q 2yields 2, i. e., ballistic diffusion), D D 2872 E Entropy

for q < 1wehave <1(i.e.,subdiffusion;inparticular, It has, among others, the following properties: q yields 0, i. e., localization), and naturally, it recovers the standard product as a particular instance, !1 D for q 1, we obtain normal diffusion.Foursystemsare i. e., D known for which results have been found that are consistent with prediction (79). These are the motion of Hy- x y xy; (82) ˝1 D dra viridissima [206], defect turbulence [73], simulation of a silo drainage [22], and molecular dynamics of a many- it is commutative,i.e., body long-range-interacting classical system of rotators (˛ XY model) [143]. For the first three, it has been x q y y q x ; (83) ˝ D ˝ found (q; ) (3/2; 4/3). For the latter one, relation (79) ' has been verified for various situations corresponding to it is additive under q-logarithm,i.e., >1. Finally, for the particular case ı 1andU(x) 0, lnq(x q y) lnq x lnq y (84) D D ˝ D C Eq. (68) becomes (whereas we remind that lnq(xy) lnq x lnq y (1 @p(x; t) @˛[p(x; t)]2 q D C C q)(lnq x)(lnq y); D ˛ (0 <˛ 2; q < 3) : (80) @t D @ x it has a (2 q)-duality/inverse property, i. e., j j The diffusion constant D just rescales time t.Onlytwopa- 1/(x q y) (1/x) 2 q (1/y) ; (85) rameters are therefore left, namely ˛ and q. ˝ D ˝ Thelinearcase(i.e.,q 1)hastwotypesofsolutions: D Gaussians for ˛ 2, and Lévy-(or˛-stable) distributions it is associative,i.e., D for 0 <˛<2. The case ˛ 2 corresponds to the Central D x (y z) (x y) z x y z Limit Theorem,wheretheN attractor of the sums ˝q ˝q D ˝q ˝q D ˝q ˝q !1 (86) of N independent random variables with finite variance (x1 q y1 q z1 q 2)1/(1 q) ; D C C precisely is a Gaussian. The case 0 <˛<2 corresponds to the sometimes called Levy–Gnedenko Central Limit Theo- it admits unity,i.e., rem,wheretheN attractor of the sums of Ninde- !1 pendent random variables with infinite variance (and ap- x q 1 x : (87) ˝ D propriate asymptotics) precisely is a Lévy distribution with index ˛. and, for q 1, also a zero,i.e., The nonlinear case (i. e., q 1) has solutions that ¤ are q-Gaussians for ˛ 2, and one might conjecture that, x q 0 0(q 1) : (88) D ˝ D similarly, interesting solutions exist for 0 <˛<2. Fur- thermore, in analogy with the q 1 case, one expects cor- D The q-Fourier Transform responding q-generalized Central Limit Theorems to exist [187]. This is precisely what we present in the next Sec- We shall introduce the q-Fourier transform of a quite generic function f (x)(x R) as follows [140,189,202, tion. 2 203,204,205]: Standard and q-Generalized Central Limit Theorems F [ f ]() 1 dx eix f (x) The q-Product q q ˝q Z1 ; (89) It has been recently introduced (independently and virtu- ix[ f (x)]q1 1 dx e f (x) ally simultaneously) [43,125] a generalization of the prod- D q Z uct, which is called q-product.Itisdefined,forx 0and 1 wherewehaveprimarilyfocusedonthecaseq 1. In y 0, as follows: contrast with the q 1 case (standard Fourier transform), D x y this integral transformation is nonlinear for q 1. It has q ¤ ˝ a remarkable property, namely that the q-Fourier trans- [x1 q y1 q 1]1/(1 q) if x1 q y1 q > 1; C C form of a q-Gaussian is another q-Gaussian: ( 0otherwise: 2 2 (81) F N ˇ e ˇ x () e ˇ1 ; (90) q q q D q1 h p i Entropy E 2873 with

1 q 1 1/2 q 1 if 1 < q < 3 ; 8 3 q ˆ ˆ 2(q 1) ˆ 1 ˆ ; Nq ˆ if q 1 ˆ p D <ˆ 3 q 3 q 1 q 1/2 2(1 q) ˆ if q < 1 ; ˆ 2 1 ˆ ˆ 1 q ˆ ˆ (91) :ˆ and 1 q q z(q) C ; (92) 1 D 3 q 2(1 q) 1 Nq (3 q) ˇ : 1 2 q (93) D ˇ 8 Entropy, Figure 4 The q-dependence of qn(q) q2;n(q) p2 q 1/p2 q Equation (93) can be re-written as ˇ ˇ1 2(1 q) 1/ 2 q D [(N (3 q))/8] p K(q), which, for q 1, re- q D q-Independent Random Variables covers the well known Heisenberg-uncertainty-principle- like relation ˇˇ 1/4. Two random variables X [with density fX(x)] and Y [with 1 D If we iterate n times the relation z(q)inEq.(92), we density fY (y)] having zero q-mean values (e. g., if fX(x) obtain the following algebra: and fY (y) are even functions) are said q-independent,with q1 given by Eq. (92), if 2q n(1 q) q (q) C (n 0; 1; 2;:::) ; (94) Fq[X Y]() Fq[X]() q1 Fq[Y]() ; (97) n D 2 n(1 q) D ˙ ˙ C D ˝ C i. e., if which can be conveniently re-written as 1 iz dz eq q fX Y (z) 2 2 ˝ C D n (n 0; 1; 2;:::) : (95) Z1 1 qn(q) D 1 q C D ˙ ˙ 1 ix dx eq q fX(x) (1 q)/(3 q) ˝ ˝ C Z1 (See Fig. 4). We easily verify that qn(1) 1( n), D 8 1 iy q (q) 1( q), as well as dy eq q fX(y) ; (98) ˙1 D 8 ˝ Z1 1 with 2 qn 1 : (96) qn 1 D 1 1 C fX Y (z) dx dyh(x; y) ı(x y z) C D C This relation connects the so called additive duality q Z1 Z1 ! 1 (2 q)andmultiplicative duality q 1/q,frequently dxh(x; z x) (99) ! D emerging in all types of calculations in the literature. Z1 Moreover, we see from Eq. (95) that multiple values of q 1 dyh(z y; y) D are expected to emerge in connection with diverse proper- Z1 ties of nonextensive systems, i. e., in systems whose basic where h(x; y)isthejointdensity. entropy is the nonadditive one Sq.Suchisthecaseofthe Clearly, q-independence means independence for so called q-triplet [185], observed for the first time in the q 1(i.e.,h(x; y) f (x) f (y)), and implies a special D D X Y magnetic field fluctuations of the solar wind, as it has been correlation for q 1. Although the full understanding of ¤ revealed by the analysis of the data sent to NASA by the this correlation is still under progress, q-independence ap- spacecraft Voyager 1 [48]. pears to be consistent with scale-invariance. 2874 E Entropy

Entropy, Table 1 The attractors corresponding to the four basic cases, where the N variables that are being summed are q-independent (i. e., globally 1 2 Q 1 Q correlated) with q1 D (1 C q)/(3 q); Q (R1 dxx [f(x)] )/(R1 dx [f(x)] ) with Q 2q 1. The attractor for (q;˛) D (1; 2) is a Gaussian G(x) L1;2 (standard Central Limit Theorem); for q D 1and0<˛<2, it is a Lévy distribution L˛ L1;˛ (the so called Lévy-Gnedenko limit theorem); for ˛ D 2andq ¤ 1, it is a q-Gaussian Gq Lq;2 (the q-Central Limit Theorem; [203]); finally, for q ¤ 1and0<˛<2, it is a generic (q;˛)-stable distribution Lq;˛ ([204,205]). See [140,189] for typical illustrations of the four types of attractors. The distribution L˛(x)remains,for1<˛<2, close to a Gaussian for jxj up to about xc(1;˛), where it makes a crossover to a power-law. The distribution Gq(x)remains,forq > 1, close to a Gaussian for jxj up to about xc(q; 2), where it makes a crossover (1) to a power-law. The distribution Lq;˛(x)remains,forq > 1and˛<2, close to a Gaussian for jxj up to about xc (q;˛), where it (2) makes a crossover to a power-law (intermediate regime), which lasts further up to about xc (q;˛), where it makes a second crossover to another power-law (distant regime)

q 1[independent] q 1(i. e.,Q 1) [globally correlated] D ¤ ¤ Q < G(x) Gq(x) G(3q 1)/(1Cq )(x) 1 D 1 1 (˛ 2) [with same 1 of f(x)] [with same Q of f(x)] D Gq(x) G(x)if x xc(q; 2) 2/(qj 1)j Gq(x) Cq;2/ x if x xc(q; 2) j j j j for q > 1, with limq!1 xc(q; 2) D1 Q L˛(x) Lq;˛ (x) !1 (˛<2) [with same x behavior of f(x)] [with same x behavior of f(x)] j j!1 j j!1 2(1q)˛(3q) (intermediate) 2(q1) L˛ (x) G(x)if x xc(1;˛) Lq;˛ Cq;˛ / x 1Cj˛j (1) j j (2) L˛ (x) C1;˛/ x if x xc(1;˛) if xc (q;˛) x xc (q;˛) j j j j j j 1C˛ (distant) 1C˛(q1) with lim˛!2 qc(1;˛) Lq;˛ Cq;˛ / x D1 (2)j j if x xc (q;˛) j j

q-Generalized Central Limit Theorems For (˛; q) (2; 1), we recover the traditional 1/pN D rescaling of Brownian motion. At the present stage, the It is out of the scope of the present survey to provide the theorems have been established for q 1 and are summa- detailsofthecomplexproofsoftheq-generalized cen- rized in Table 1.Thecaseq < 1 is still open at the time tral limit theorems. We shall restrict to the presentation at which these lines are being written. Two q < 1cases of their structure. Let us start by introducing a notation have been preliminarily explored numerically in [124] which is important for what follows. A distribution is said and in [171]. The numerics seemed to indicate that the (q;˛)-stable distribution Lq;˛(x)ifitsq-Fourier transform N limits would be q-Gaussians for both models. Lq;˛()isoftheform !1 However, it has been analytically shown [94]thatitis not exactly so. The limiting distributions numerically are b ˛ L ;˛() a e q D q1 j j amazingly close to q-Gaussians, but they are in fact diﬀer- [a > 0; b > 0; 0 <˛ 2; q (q 1)/(3 q)] : ent. Very recently, another simple scale-invariant model 1 D C (100) has been introduced [153], whose attractor has been analytically shown to be a q-Gaussian. Consistently, L are Gaussians, L are Lévy distribu- These q 1 theorems play for the nonadditive en- 1;2 1;˛ ¤ tions, and Lq;2 are q-Gaussians. tropy Sq and nonextensive statistical mechanics the same We are seeking for the N attractor associated grounding role that the well known q 1theoremsplay !1 D with the sum of N identical and distinguishable random for the additive entropy SBG and BG statistical mechanics. variables each of them associated with one and the same In particular, interestingly enough, the ubiquity of Gaus- arbitrary symmetric distribution f (x). The random vari- sians and of q-Gaussians in natural, artiﬁcial and social ables are independent for q 1, and correlated in a spe- systems may be understood on equal footing. D cial manner for q 1. To obtain the N invariant ¤ !1 distribution, i. e. the attractor, the sum must be rescaled, Future Directions i. e., divided by Nı ,where The concept of entropy permeates into virtually all quan- 1 titative sciences. The future directions could therefore ı : (101) D ˛(2 q) be very varied. If we restrict, however, to the evidence Entropy E 2875

Entropy, Figure 5 Snapshot of a nongrowing dynamic network with N D 256 nodes (see details in [172], by courtesy of the author) presently available, the main lines along which evolution occurs are: Networks Many of the so-called scale-free networks, among others, systematically exhibit a degree distribution p(k)(probability of a node having k links) which is of the form 1 p(k) ( >0; k0 > 0) ; (102) / (k k) Entropy, Figure 6 0 C Nongrowing dynamic network: a Cumulative degree distribu- or, equivalently, tion for typical values for the number N of nodes; b Same data of a in the convenient representation linear q-log versus linear k/ 1q p(k) eq (q 1; >0) ; (103) with Zq(k) lnq[Pq(> k)] ([Pq(> k)] 1)/(1 q)(theopti- / mal fitting with a q-exponential is obtained for the value of q with 1/(q 1) and k /(q 1) (see Figs. 5 which has the highest value of the linear correlation r as indi- D 0 D and 6). This is not surprising since, if we associate cated in the inset;herethisisqc D 1:84, which corresponds to to each link an “energy” (or cost) and to each node the slope 1.19 in a). See details in [172,173] half of the “energy” carried by its links (the other half being associated with the other nodes to which any speciﬁc node is linked), the distribution of en- tematic calculation of several meaningful properties of ergies optimizing Sq precisely coincides with the de- networks. gree distribution. If, for any reason, we consider k Nonlinear dynamical systems, self-organized criticality, as the modulus of a d-dimensional vector k,theop- and cellular automata Various interesting phenomena timization of the functional Sq[p(k)] may lead to emerge in both low- and high-dimensional weakly p(k) k e k/ ,wherek plays the role of a den- chaotic deterministic dynamical systems, either dis- / q sity of states, (d) being either zero (which repro- sipative or conservative. Among these phenomena duces Eq. (103)) or positive or negative. Several exam- we have the sensitivity to the initial conditions and ples [12,39,76,91,165,172,173,212,213] already exist in the entropy production, which have been brieﬂy ad- the literature; in particular, the Barabasi–Albert uni- dressed in Eq. (37) and related papers. But there versality class 3 corresponds to q 4/3. A deeper is much more, such as relaxation, escape, glassy D D understanding of this connection might enable the sys- states, and distributions associated with the station- 2876 E Entropy

Entropy, Figure 7 Distribution of velocities for the HMF model at the quasi-stationary state (whose duration appears to diverge when N !1). The blue curves indicate a Gaussian, for comparison. See details in [137]

ary state [14,15,31,46,62,67,68,77,103,111,122,123,154, 170,174,176,177,179]. Also, recent numerical indi- cations suggest the validity of a dynamical version of the q-generalized central limit theorem [175]. The possible connections between all these various properties is still in its infancy. Long-range-interacting many-body Hamiltonians A wide class of long-range-interacting N-body classical Hamiltonians exhibits collective states whose Lyapunov spectrum has a maximal value that vanishes in the N limit. As such, they constitute natural !1 candidates for studying whether the concepts derived from the nonadditive entropy Sq are applicable. A variety of properties have been calculated, through molecular dynamics, for various systems, such as Lennard– Jones-like fluids, XY and Heisenberg ferromagnets, gravitational-like models, and others. One or more long-standing quasi-stationary states (infinitely long- standing in the limit N ) are typically observed !1 before the terminal entrance into thermal equilib- Entropy, Figure 8 rium. Properties such as distribution of velocities Quantum Monte Carlo simulations in [81]: a Velocity distribution and angles, correlation functions, Lyapunov spectrum, (superimposed with a q-Gaussian); b Index q (superimposed with metastability, glassy states, aging, time-dependence of Lutz prediction [110], by courtesy of the authors) the temperature in isolated systems, energy whenever thermal contact with a large thermostat at a given 126,127,132,133,134,135,136,142,169,200]. A quite re- temperature is allowed, diffusion, order parameter, markable molecular-dynamical result has been ob- and others, are typically focused on. An ongoing de- tained for a paradigmatic long-range Hamiltonian: the bate exists, also involving Vlasov-like equations, Lyn- distribution of time averaged velocities sensibly differs den–Bell statistics, among others. The breakdown of from that found for the ensemble-averaged velocities, ergodicity that emerges in various situations makes and has been shown to be numerically consistent with the whole discussion rich and complex. The activity a q-Gaussian [137],asshowninFig.7.Thisresultpro- of the research nowadays in this area is illustrated vides strong support to a conjecture made long ago: in papers such as [21,26,45,53,56,57,63,104,119,121, see Fig. 4 at p. 8 of [157]. Entropy E 2877

Entropy, Figure 9 Experiments in [81]: a Velocity distribution (superimposed with a q-Gaussian); b Index q as a function of the frequency; c Velocity distribution (superimposed with a q-Gaussian; the red curve is a Gaussian); d Tail of the velocity distribution (superimposed with the asymptotic power-law of a q-Gaussian). [By courtesy of the authors]

Stochastic differential equations Quite generic Fokker– ergy parameters of the optical lattice). These exper- Planck equations are currently being studied. Aspects imental verifications are in variance with some of such as fractional derivatives, nonlinearities, space-de- those exhibited previously [96], namely double-Gaus- pendent diffusion coefficients are being focused on, sians. Although it is naturally possible that the ex- as well as their connections to entropic forms, and perimental conditions have not been exactly equiv- associated generalized Langevin equations [20,23,24, alent, this interesting question remains open at the 70,128,168,214]. Quite recently, computational (see present time. A hint might be hidden in the recent Fig. 8) and experimental (see Fig. 9) verifications of results [62] obtained for a quite different problem, Lutz’ 2003 prediction [110] have been exhibited [81], namely the size distributions of avalanches; indeed, namely about the q-Gaussian form of the velocity at a critical state, a q-Gaussian shape was obtained, distribution of cold atoms in dissipative optical lat- whereas, at a noncritical state, a double-Gaussian was tices, with q 1 44E /U (E and U being en- observed. D C R 0 R 0 2878 E Entropy

Quantum entanglement and quantum chaos The non- 4. Abe S, Suzuki N (2003) Law for the distance between succes- local nature of quantum physics implies phenomena sive earthquakes. J Geophys Res (Solid Earth) 108(B2):2113 that are somewhat analogous to those originated by 5. Abe S, Suzuki N (2004) Scale-free network of earthquakes. Eu- rophys Lett 65:581–586 classical long-range interactions. Consequently, a va- 6. Abe S, Suzuki N (2005) Scale-free statistics of time interval be- riety of studies are being developed in connection tween successive earthquakes. Physica A 350:588–596 with the entropy Sq [3,36,58,59,60,61,155,156,195]. 7. Abe S, Suzuki N (2006) Complex network of seismicity. Prog The same happens with some aspects of quantum Theor Phys Suppl 162:138–146 chaos [11,180,210,211]. 8. Abe S, Suzuki N (2006) Complex-network description of seismicity. Nonlinear Process Geophys 13:145–150 Astrophysics, geophysics, economics, linguistics, cog- 9. Abe S, Sarlis NV, Skordas ES, Tanaka H, Varotsos PA (2005) Op- nitive psychology, and other interdisciplinary appli- timality of natural time representation of complex time series. cations Applications are available and presently searched Phys Rev Lett 94:170601 in many areas of physics (plasmas, turbulence, nuclear 10. Abe S, Tirnakli U, Varotsos PA (2005) Complexity of seismicity collisions, elementary particles, manganites), but also and nonextensive statistics. Europhys News 36:206–208 11. Abul AY-M (2005) Nonextensive random matrix theory ap- in interdisciplinary sciences such astrophysics [38,47, proach to mixed regular-chaotic dynamics. Phys Rev E 48,49,78,84,87,101,109,129,196], geophysics [4,5,6,7,8, 71:066207 9,10,62,208], economics [25,50,51,52,80,139,141,197, 12. Albert R, Barabasi AL (2000) Phys Rev Lett 85:5234–5237 215], linguistics [118], cognitive psychology [181], and 13. 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