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Entropy

CONSTANTINO TSALLIS1,2

1 Centro Brasileiro de Pesquisas Físicas,

Rio de Janeiro, Brazil2 Santa Fe Institute, Santa Fe, USA

Article Outline

Glossary

Definition of the Subject

Introduction

Some Basic Properties

Boltzmann–Gibbs Statistical Mechanics

On the Limitations of Boltzmann–Gibbs Entropy

and Statistical Mechanics

The Nonadditive Entropy SqA Connection Between Entropy and Diffusion

Standard and q-Generalized Central Limit Theorems

Future Directions

Acknowledgments

Bibliography

Glossary

Absolute temperature Denoted T.

Clausius entropy Also called thermodynamic entropy.

Denoted S.

Boltzmann–Gibbs entropy Basis of Boltzmann–Gibbs

statistical mechanics. This entropy, denoted SBG, is ad-

ditive. Indeed, for two probabilistically independent

subsystemsA andB, it satisfies SBG(ACB) D SBG(A)CSBG(B).

Nonadditive entropy It usually refers to the basis of

nonextensive statistical mechanics. This entropy, de-

noted Sq, is nonadditive for q ¤ 1. Indeed, for twoprobabilistically independent subsystems A and B, it

satisfies Sq(A C B) ¤ Sq(A) C Sq(B) (q ¤ 1). For his-torical reasons, it is frequently (but inadequately) re-

ferred to as nonextensive entropy.

q-logarithmic and q-exponential functions Denoted

lnq x (ln1 x D ln x), and exq (ex1 D ex ), respectively.Extensive system So called for historical reasons. A more

appropriate name would be additive system. It is a sys-

tem which, in one way or another, relies on or is con-

nected to the (additive) Boltzmann–Gibbs entropy. Its

basic dynamical and/or structural quantities are ex-

pected to be of the exponential form. In the sense of

complexity, it may be considered a simple system.

Nonextensive system So called for historical reasons.

A more appropriate name would be nonadditive sys-

tem. It is a system which, in one way or another, relies

on or is connected to a (nonadditive) entropy such as

Sq(q ¤ 1). Its basic dynamical and/or structural quan-tities are expected to asymptotically be of the power-

law form. In the sense of complexity, it may be consid-

ered a complex system.

Definition of the Subject

Thermodynamics and statistical mechanics are among

the most important formalisms in contemporary physics.

They have overwhelming and intertwined applications in

science and technology. They essentially rely on two ba-

sic concepts, namely energy and entropy. The mathemati-

cal expression that is used for the first one is well known

to be nonuniversal; indeed, it depends on whether we are

say in classical, quantum, or relativistic regimes. The sec-

ond concept, and very specifically its connection with the

microscopic world, has been considered during well over

one century as essentially unique and universal as a physi-

cal concept. Although some mathematical generalizations

of the entropy have been proposed during the last forty

years, they have frequently been considered as mere prac-

tical expressions for disciplines such as cybernetics and

control theory, with no particular physical interpretation.

What we have witnessed during the last two decades is the

growth, among physicists, of the belief that it is not neces-

sarily so. In other words, the physical entropy would ba-

sically rely on the microscopic dynamical and structural

properties of the system under study. For example, for sys-

tems microscopically evolving with strongly chaotic dy-

namics, the connection between the thermodynamical en-

tropy and the thermostatistical entropy would be the one

2860 E Entropyfound in standard textbooks. But, for more complex sys-

tems (e. g., for weakly chaotic dynamics), it becomes ei-

ther necessary, or convenient, or both, to extend the tradi-

tional connection. The present article presents the ubiqui-

tous concept of entropy, useful even for systems for which

no energy can be defined at all, within a standpoint re-

flecting a nonuniversal conception for the connection be-

tween the thermodynamic and the thermostatistical en-

tropies. Consequently, both the standard entropy and its

recent generalizations, as well as the corresponding statis-

tical mechanics, are here presented on equal footing.

Introduction

The concept of entropy (from the Greek �� ����!, en

trepo, at turn, at transformation) was first introduced in

1865 by the German physicist and mathematician Rudolf

Julius Emanuel Clausius, Rudolf Julius Emanuel in or-

der to mathematically complete the formalism of classi-

cal thermodynamics [55], one of the most important the-

oretical achievements of contemporary physics. The term

was so coined to make a parallel to energy (from the Greek

����o& , energos, at work), the other fundamental con-

cept of thermodynamics. Clausius connection was given

by

dS D ıQT; (1)

where ıQ denotes an infinitesimal transfer of heat. In

other words, 1/T acts as an integrating factor for ıQ.

In fact, it was only in 1909 that thermodynamics was

finally given, by the Greek mathematician Constantin

Caratheodory, a logically consistent axiomatic formula-

tion.

In 1872, some years after Clausius proposal, the Aus-

trian physicist Ludwig Eduard Boltzmann introduced

a quantity, that he noted H, which was defined in terms

of microscopic quantities:

H �•

f (v) ln[ f (v)] dv ; (2)

where f (v)dv is the number of molecules in the veloc-

ity space interval dv. Using Newtonian mechanics, Boltz-

mann showed that, under some intuitive assumptions

(Stoßzahlansatz or molecular chaos hypothesis) regarding

the nature of molecular collisions, H does not increase

with time. Five years later, in 1877, he identified this quan-

tity with Clausius entropy through �kH � S, where k isa constant. In other words, he established that

S D �k•

f (v) ln[ f (v)] dv ; (3)

later on generalized into

S D �k“

f (q;p) ln[ f (q;p)] dq dp ; (4)

where (q;p) is called the�-space and constitutes the phase

space (coordinate q and momentum p) corresponding to

one particle.

Boltzmann’s genius insight – the first ever mathemat-

ical connection of the macroscopic world with the micro-

scopic one – was, during well over three decades, highly

controversial since it was based on the hypothesis of the

existence of atoms. Only a few selected scientists, like

the English chemist and physicist John Dalton, the Scot-

tish physicist and mathematician James Clerk Maxwell,

and the American physicist, chemist and mathematician

Josiah Willard Gibbs, believed in the reality of atoms and

molecules. A large part of the scientific establishment was,

at the time, strongly against such an idea. The intricate

evolution of Boltzmann’s lifelong epistemological strug-

gle, which ended tragically with his suicide in 1906, may

be considered as a neat illustration of Thomas Kuhn’s

paradigm shift, and the corresponding reaction of the sci-

entific community, as described in The Structure of Sci-

entific Revolutions. There are in fact two important for-

malisms in contemporary physics where the mathematical

theory of probabilities enters as a central ingredient. These

are statistical mechanics (with the concept of entropy as

a functional of probability distributions) and quantum

mechanics (with the physical interpretation of wave func-

tions and measurements). In both cases, contrasting view-

points and passionate debates have taken place alongmore

than one century, and continue still today. This is no sur-

prise after all. If it is undeniable that energy is a very deep

and subtle concept, entropy is even more. Indeed, energy

concerns the world of (microscopic) possibilities, whereas

entropy concerns the world of the probabilities of those

possibilities, a step further in epistemological difficulty.

In his 1902 celebrated book Elementary Principles of

Statistical Mechanics, Gibbs introduced the modern form

of the entropy for classical systems, namely

S D �kZ

d� f (q;p) ln[C f (q;p)] ; (5)

where � represents the full phase space of the system, thus

containing all coordinates and all momenta of its elemen-

tary particles, and C is introduced to take into account the

finite size and the physical dimensions of the smallest ad-

missible cell in � -space. The constant k is known today to

be a universal one, called Boltzmann constant, and given

by k D 1:3806505(24) � 10�23 Joule/Kelvin. The studies

Entropy E 2861of the German physicist Max Planck along Boltzmann and

Gibbs lines after the appearance of quantum mechanical

concepts, eventually led to the expression

S D k lnW ; (6)

which he coined as Boltzmann entropy. This expression is

carved on the stone of Boltzmann’s grave at the Central

Cemetery of Vienna. The quantity W is the total number

of microstates of the system that are compatible with our

macroscopic knowledge of it. It is obtained from Eq. (5)

under the hypothesis of an uniform distribution or equal

probabilities.

The Hungarian–American mathematician and physi-

cist Johann von Neumann extended the concept of BG en-

tropy in two steps – in 1927 and 1932 respectively –, in or-

der to also cover quantum systems. The following expres-

sion, frequently referred to as the von Neumann entropy,

resulted:

S D �k Tr � ln � ; (7)

� being the density operator (with Tr � D 1).Another important step was given in 1948 by the

American electrical engineer and mathematician Claude

Elwood Shannon. Having in mind the theory of digital

communications he explored the properties of the discrete

form

S D �kWX

iD1pi ln pi ; (8)

frequently referred to as Shannon entropy (withPW

iD1pi D 1). This form can be recovered from Eq. (5) for theparticular case for which the phase space density f (q;p) DPW

iD1 pi ı(q � qi) ı(p � pi). It can also be recovered fromEq. (7) when � is diagonal. We may generically refer to

Eqs. (5), (6), (7) and (8) as the BG entropy, noted SBG. It

is a measure of the disorder of the system or, equivalently,

of our degree of ignorance or lack of information about

its state. To illustrate a variety of properties, the discrete

form (8) is particularly convenient.

SomeBasic Properties

Non-negativity It can be easily verified that, in all cases,

SBG � 0, the zero value corresponding to certainty,i. e., pi D 1 for one of the W possibilities, and zerofor all the others. To be more precise, it is exactly so

whenever SBG is expressed either in the form (7) or in

the form (8). However, this property of non-negativity

may be no longer true if it is expressed in the form (5).

This violation is one of the mathematical manifesta-

tions that, at the microscopic level, the state of any

physical system exhibits its quantum nature.

Expansibility Also SBG(p1; p2; : : : ; pW ; 0) D SBG(p1; p2;: : : ; pW ), i. e., zero-probability events do not modify

our information about the system.

Maximal value SBG is maximized at equal probabilities,

i. e., for pi D 1/W ;8i. Its value is that of Eq. (6). Thiscorresponds to the Laplace principle of indifference or

principle of insufficient reason.

Concavity If we have two arbitrary probability distribu-

tions fpig and fp0ig for the same set ofW possibilities,we can define the intermediate probability distribution

p00i D � pi C (1 � �) p0i (0 < � < 1). It straightfor-wardly follows that SBG(fp00i g) � � SBG(fpig) C (1 ��) SBG(fp0ig). This property is essential for thermody-namics since it eventually leads to thermodynamic sta-

bility, i. e., to robustness with regard to energy fluctu-

ations. It also leads to the tendency of the entropy to

attain, as time evolves, its maximal value compatible

with our macroscopic knowledge of the system, i. e.,

with the possibly known values for the macroscopic

constraints.

Lesche stability or experimental robustness B. Lesche

introduced in 1982 [107] the definition of an interest-

ing property, which he called stability. It reflects the

experimental robustness that a physical quantity is ex-

pected to exhibit. In other words, similar experiments

should yield similar numerical results for the physi-

cal quantities. Let us consider two probability distribu-

tions fpig and fp0ig, assumed to be close, in the sensethat

PWiD1 jpi � p0i j < ı ; ı > 0 being a small number.

An entropic functional S(fpig) is said stable or exper-imentally robust if, for any given � > 0, a ı > 0 exists

such that jS(fpig) � S(fp0i g)j/Smax < � ;where Smax isthe maximal value that the functional can attain

(lnW in the case of SBG). This implies that limı!0limW!1(S(fpig) � S(fp0i g))/Smax D 0. As we shallsee soon, this property is much stronger than it seems

at first sight. Indeed, it provides a (necessary but not

sufficient) criterion for classifying entropic functionals

into physically admissible or not. It can be shown that

SBG is Lesche-stable (or experimentally robust).

Entropy production If we start the (deterministic) time

evolution of a generic classical system from an arbi-

trarily chosen point in its � phase space, it typically

follows a quite erratic trajectory which, in many cases,

gradually visits the entire (or almost) phase space. By

making partitions of this � -space, and counting the

frequency of visits to the various cells (and related

symbolic quantities), it is possible to define probabil-

2862 E Entropyity sets. Through them, we can calculate a sort of time

evolution of SBG(t). If the system is chaotic (sometimes

called strongly chaotic), i. e., if its sensitivity to the

initial conditions increases exponentially with time,

then SBG(t) increases linearly with t in the appropri-

ate asymptotic limits. This rate of increase of the en-

tropy is called Kolmogorov–Sinai entropy rate, and, for

a large class of systems, it coincides (Pesin identity

or Pesin theorem) with the sum of the positive Lya-

punov exponents. These exponents characterize the

exponential divergences, along various directions in

the � -space, of a small discrepancy in the initial con-

dition of a trajectory.

It turns out, however, that the Kolmogorov–Sinai en-

tropy rate is, in general, quite inconvenient for com-

putational calculations for arbitrary nonlinear dynam-

ical systems. In practice, another quantity is used in-

stead [102], usually referred to as entropy production

per unit time, which we note KBG. Its definition is as

follows. We first make a partition of the � -space into

many W cells (i D 1; 2; : : : ;W). In one of them, ar-bitrarily chosen, we randomly place M initial condi-

tions (i. e., an ensemble). As time evolves, the occu-

pancy of the W cells determines the set fM i(t)g, withPW

iD1 M i(t) D M. This set enables the definition ofa probability set with pi (t) � M i(t)/M, which in turndetermines SBG(t). We then define the entropy produc-

tion per unit time as follows:

KBG � limt!1

limW!1

limM!1

SBG(t)

t: (9)

Up to date, no theorem guarantees that this quan-

tity coincides with the Kolmogorov–Sinai entropy

rate. However, many numerical approaches of various

chaotic systems strongly suggest so. The same turns

out to occur with what is frequently referred in the lit-

erature as a Pesin-like identity. For instance, if we have

a one-dimensional dynamical system, its sensitivity to

the initial conditions � � limx(0)!0x(t)/x(0) istypically given by

�(t) D e�t ; (10)

wherex(t) is the discrepancy in the one-dimensional

phase space of two trajectories initially differing by

x(0), and � is the Lyapunov exponent (� > 0 corre-

sponds to strongly sensitive to the initial conditions, or

strongly chaotic, and � < 0 corresponds to strongly in-

sensitive to the initial conditions). The so-called Pesin-

like identity amounts, if � � 0, to

KBG D � : (11)

Additivity and extensivity If we consider a system A C Bconstituted by two probabilistically independent sub-

systems A and B, i. e., if we consider pACBi j D pAi pBj ,we immediately obtain from Eq. (8) that

SBG(A C B) D SBG(A) C SBG(B) : (12)

In other words, the BG entropy is additive [130]. If

our system is constituted by N probabilistically inde-

pendent identical subsystems (or elements), we clearly

have SBG(N) / N . It frequently happens, however,that the N elements are not exactly independent but

only asymptotically so in the N ! 1 limit. This is theusual case of many-body Hamiltonian systems involv-

ing only short-range interactions, where the concept of

short-range will be focused in detail later on. For such

systems, SBG is only asymptotically additive, i. e.,

0 < limN!1

SBG(N)

N< 1 : (13)

An entropy S(fpig) of a specific systems is said exten-sive if it satisfies

0 < limN!1

S(N)

N< 1 ; (14)

where no hypothesis at all is made about the possible

independence or weak or strong correlations between

the elements of the systemwhose entropy Swe are con-

sidering. Equation (13) amounts to say that the addi-

tive entropy SBG is extensive for weakly correlated sys-

tems such as the already mentioned many-body short-

range-interacting Hamiltonian ones. It is important to

clearly realize that additivity and extensivity are inde-

pendent properties. An additive entropy such as SBG is

extensive for simple systems such as the ones just men-

tioned, but it turns out to be nonextensive for other,

more complex, systems that will be focused on later

on. For many of these more complex systems, it is the

nonadditive entropy Sq (to be analyzed later on) which

turns out to be extensive for a non standard value of q

(i. e., q ¤ 1).

Boltzmann–Gibbs StatisticalMechanics

Physical systems (classical, quantum, relativistic) can be

theoretically described in very many ways, through mi-

croscopic, mesoscopic, macroscopic equations, reflecting

either stochastic or deterministic time evolutions, or even

both types simultaneously. Those systems whose time evo-

lution is completely determined by a well defined Hamil-

tonian with appropriate boundary conditions and ad-

Entropy E 2863missible initial conditions are the main purpose of an

important branch of contemporary physics, named sta-

tistical mechanics. This remarkable theory (or formalism,

as sometimes called), which for large systems satisfacto-

rily matches classical thermodynamics, was primarily in-

troduced by Boltzmann and Gibbs. The physical system

can be in all types of situations. Two paradigmatic such

situations correspond to isolation, and thermal contact

with a large reservoir called thermostat. Their stationary

state (t ! 1) is usually referred to as thermal equilib-rium. Both situations have been formally considered by

Gibbs within his mathematical formulation of statistical

mechanics, and they respectively correspond to the so-

calledmicro-canonical and canonical ensembles (other en-

sembles do exist, such as the grand-canonical ensemble,

appropriate for those situations in which the total num-

ber of elements of the system is not fixed; this is however

out of the scope of the present article).

The stationary state of themicro-canonical ensemble is

determined by pi D 1/W (8i, where i runs over all possi-ble microscopic states), which corresponds to the extrem-

ization of SBG with a single (and trivial) constraint, namely

WX

iD1pi D 1 : (15)

To obtain the stationary state for the canonical ensem-

ble, the thermostat being at temperature T, we must (typi-

cally) add one more constraint, namely

WX

iD1piEi D U ; (16)

where fEig are the energies of all the possible states of thesystem (i. e., eigenvalues of the Hamiltonian with the ap-

propriate boundary conditions). The extremization of SBGwith the two constraints above straightforwardly yields

pi De�ˇE i

Z(17)

Z �WX

jD1e�ˇE j (18)

with the partition function Z, and the Lagrange param-

eter ˇ D 1/kT . This is the celebrated BG distributionfor thermal equilibrium (or Boltzmann weight, or Gibbs

state, as also called), which has been at the basis of an

enormous amount of successes (in fluids, magnets, su-

perconductors, superfluids, Bose–Einstein condensation,

conductors, chemical reactions, percolation, among many

other important situations). The connection with classi-

cal thermodynamics, and its Legendre-transform struc-

ture, occurs through relations such as

1

TD @S@U

(19)

F � U � TS D � 1ˇln Z (20)

U D � @@̌

ln Z (21)

C � T @S@T

D @U@T

D �T @2F

@T2; (22)

where F, U and C are the Helmholtz free energy, the in-

ternal energy, and the specific heat respectively. The BG

statistical mechanics historically appeared as the first con-

nection between the microscopic and the macroscopic de-

scriptions of the world, and it constitutes one of the cor-

nerstones of contemporary physics. The Establishment re-

sisted heavily before accepting the validity and power of

Boltzmann’ s revolutionary ideas. In 1906 Boltzmann dra-

matically committed suicide, after 34 years that he had first

proposed the deep ideas that we are summarizing here. At

that early 20th century, few people believed in Boltzmann’s

proposal (among those few, wemust certainly mentionAl-

bert Einstein), andmost physicists were simply unaware of

the existence of Gibbs and of his profound contributions.

It was only half a dozen years later that the emerging new

generation of physicists recognized their respective genius

(thanks in part to various clarifications produced by Paul

Ehrenfest, and also to the experimental successes related

with Brownian motion, photoelectric effect, specific heat

of solids, and black-body radiation).

On the Limitations of Boltzmann–Gibbs Entropy

and StatisticalMechanics

Historical Background

As any other human intellectual construct, the applicabil-

ity of the BG entropy, and of the statistical mechanics to

which it is associated, naturally has restrictions. The un-

derstanding of present developments of both the concept

of entropy, and its corresponding statistical mechanics,

demand some knowledge of the historical background.

Boltzmann was aware of the relevance of the range

of the microscopic interactions between atoms and

molecules. He wrote, in his 1896 Lectures on Gas The-

ory [41], the following words:

When the distance at which two gas molecules inter-

act with each other noticeably is vanishingly small

relative to the average distance between a molecule

2864 E Entropyand its nearest neighbor—or, as one can also say,

when the space occupied by the molecules (or their

spheres of action) is negligible compared to the space

filled by the gas—then the fraction of the path of each

molecule during which it is affected by its interac-

tion with other molecules is vanishingly small com-

pared to the fraction that is rectilinear, or simply de-

termined by external forces. [ . . . ] The gas is “ideal”

in all these cases.

Also Gibbs was aware. In his 1902 book [88], he wrote:

In treating of the canonical distribution, we shall al-

ways suppose the multiple integral in equation (92)

[the partition function, as we call it nowadays] to

have a finite value, as otherwise the coefficient of

probability vanishes, and the law of distribution be-

comes illusory. This will exclude certain cases, but not

such apparently, as will affect the value of our results

with respect to their bearing on thermodynamics. It

will exclude, for instance, cases in which the system

or parts of it can be distributed in unlimited space

[ . . . ]. It also excludes many cases in which the en-

ergy can decrease without limit, as when the system

contains material points which attract one another

inversely as the squares of their distances. [ . . . ]. For

the purposes of a general discussion, it is sufficient to

call attention to the assumption implicitly involved in

the formula (92).

The extensivity/additivity of SBG has been challenged,

along the last century, by many physicists. Let us mention

just a few. In his 1936 Thermodynamics [82], Enrico Fermi

wrote:

The entropy of a system composed of several parts is

very often equal to the sum of the entropies of all the

parts. This is true if the energy of the system is the

sum of the energies of all the parts and if the work

performed by the system during a transformation is

equal to the sum of the amounts of work performed

by all the parts. Notice that these conditions are not

quite obvious and that in some cases they may not

be fulfilled. Thus, for example, in the case of a system

composed of two homogeneous substances, it will be

possible to express the energy as the sum of the ener-

gies of the two substances only if we can neglect the

surface energy of the two substances where they are

in contact. The surface energy can generally be ne-

glected only if the two substances are not very finely

subdivided; otherwise, it can play a considerable role.

Laszlo Tisza wrote, in his Generalized Thermodynam-

ics [178]:

The situation is different for the additivity postu-

late P a2, the validity of which cannot be inferred

from general principles. We have to require that the

interaction energy between thermodynamic systems

be negligible. This assumption is closely related to

the homogeneity postulate P d1. From the molecular

point of view, additivity and homogeneity can be ex-

pected to be reasonable approximations for systems

containing many particles, provided that the inter-

molecular forces have a short range character.

Corroborating the above, virtually all textbooks of quan-

tum mechanics contain the mechanical calculations cor-

responding to a particle in a square well, the harmonic

oscillator, the rigid rotator, a spin 1/2 in the presence of

a magnetic field, and the Hydrogen atom. In the textbooks

of statistical mechanics we can find the thermostatistical

calculations of all these systems . . . excepting the Hydro-

gen atom! Why? Because the long-range electron-proton

interaction produces an energy spectrum which leads to

a divergent partition function. This is but a neat illustra-

tion of the above Gibbs’ alert.

A Remark on the Thermodynamics

of Short- and Long-Range Interacting Systems

We consider here a simple d-dimensional classical fluid,

constituted by many N point particles, governed by the

Hamiltonian

H D K C V DNX

iD1

p2i2m

CX

i¤ jV (ri j) ; (23)

where the potential V(r) has, if it is attractive at short

distances, no singularity at the origin, or an integrable

singularity, and whose asymptotic behavior at infinity

is given by V(r) � �B/r˛ with B > 0 and ˛ � 0. Onesuch example is the d D 3 Lennard–Jones fluid, forwhich V(r) D A/r12 � B/r6(A > 0), i. e., repulsive at shortdistances and attractive at long distances. In this case

˛ D 6. Another example could be Newtonian gravita-tion with a phenomenological short-distance cutoff (i. e.,

V (r) ! 1 for r � r0 with r0 > 0. In this case, ˛ D 1.The full � -space of such a system has 2dN dimensions.

The total potential energy is expected to scale (assuming

a roughly homogeneous distribution of the particles) as

Upot(N)

N/ �B

Z 1

1

dr rd�1 r�˛ ; (24)

where the integral starts appreciably contributing above

a typical cutoff, here taken to be unity. This integral is finite

Entropy E 2865[ D �B/(˛ � d) ] for ˛/d > 1 (short-range interactions),and diverges for 0 � ˛/d � 1 (long-range interactions). Inother words, the energy cannot be generically character-

ized by Eq. (24), and we must turn onto a different and

more powerful estimation. Given the finiteness of the size

of the system, an appropriate one is, in all cases, given by

Upot(N)

N/ �B

Z N1/d

1

dr rd�1 r�˛ D �BdN? ; (25)

where

N? � N1�˛/d � 11 � ˛/d �

8

ˆ

ˆ

ˆ

ˆ

ˆ

<

ˆ

ˆ

ˆ

ˆ

ˆ

:

1

˛/d � 1 if ˛/d > 1 ;

lnN if ˛/d D 1 ;N1�˛/d

1 � ˛/d if 0 < ˛/d < 1 :(26)

Notice that N? D ln˛/d N where the q-log func-tion lnq x � (x1�q � 1)/(1 � q)(x > 0; ln1 x D ln x) willbe shown to play an important role later on. Satisfacto-

rily enough, Eqs. (26) recover the characterization with

Eq. (24) in the limit N ! 1, but they have the great ad-vantage of providing, for finite N, a finite value. This fact

will be now shown to enable to properly scale the macro-

scopic quantities in the thermodynamic limit (N ! 1),for all values of ˛/d � 0.

Let us address the thermodynamical consequences of

the microscopic interactions being short- or long-ranged.

To present a slightly more general illustration, we shall as-

sume from now on that our homogeneous and isotropic

classical fluid is made by magnetic particles. Its Gibbs free

energy is then given by

G(N; T; p;H) D U(N; T; p;H) � TS(N; T; p;H)C pV(N; T; p;H) � HM(N; T; p;H) ; (27)

where (T; p;H) correspond respectively to the tempera-

ture, pressure and external magnetic field, V is the volume

and M the magnetization. If the interactions are short-

ranged (i. e., if ˛/d > 1), we can divide this equation by N

and then take the N ! 1 limit. We obtain

g(T; p;H) D u(T; p;H) � Ts(T; p;H)C pv(T; p;H) � Hm(T; p;H) ; (28)

where g(T; p;H) � limN!1 G(N; T; p;H)/N , and anal-ogously for the other variables of the equation. If the in-

teractions were instead long-ranged (i. e., if 0 � ˛/d � 1),all these quantities would be divergent, hence thermody-

namically nonsense. Consequently, the generically correct

procedure, i. e. 8˛/d � 0, must conform to the followinglines:

limN!1

G(N; T; p;H)

NN?D lim

N!1U(N; T; p;H)

NN?

� limN!1

T

N?S(N; T; p;H)

N

C limN!1

p

N?V (N; T; p;H)

N

� limN!1

H

N?M(N; T; p;H)

N

(29)

hence

g(T?; p?;H?) D u(T?; p?;H?) � T?s(T?; p?;H?)C p?v(T?; p?;H?) � H?m(T?; p?;H?) ; (30)

where the definitions of T? and all the other variables are

self-explanatory (e. g., T? � T/N?). In other words, in or-der to have finite thermodynamic equations of states, we

must in general express them in the (T?; p?;H?) vari-

ables. If ˛/d > 1, this procedure recovers the usual equa-

tions of states, and the usual extensive (G;U; S;V ;M)

and intensive (T; p;H) thermodynamic variables. But, if

0 � ˛/d � 1, the situation is more complex, and we real-ize that three, instead of the traditional two, classes of ther-

modynamic variables emerge. We may call them exten-

sive (S;V ;M;N), pseudo-extensive (G;U) and pseudo-in-

tensive (T; p;H) variables. All the energy-type thermody-

namical variables (G; F;U) give rise to pseudo-extensive

ones, whereas those which appear in the usual Legendre

thermodynamical pairs give rise to pseudo-intensive ones

(T; p;H; �) and extensive ones (S;V ;M;N). See Figs. 1

and 2.

The possibly long-range interactions within Hamil-

tonian (23) refer to the dynamical variables themselves.

There is another important class of Hamiltonians, where

the possibly long-range interactions refer to the coupling

constants between localized dynamical variables. Such is,

for instance, the case of the following classical Hamilto-

nian:

H D K C V DNX

iD1

L2i2I

�X

i¤ j

Jx sxi s

xj C Jys

yi s

yj C Jzszi szj

r˛i j(˛ � 0) ; (31)

where fLig are the angular momenta, I the moment ofinertia, f(sxi ; s

yi ; s

zi )g are the components of classical ro-

tators, (Jx ; Jy ; Jz) are coupling constants, and rij runs

2866 E Entropy

Entropy, Figure 1

For long-range interactions (0 � ˛/d � 1) we have three classes

of thermodynamic variables, namely the pseudo-intensive (scal-

ing with N?), pseudo-extensive (scaling with NN?) and exten-

sive (scaling with N) ones. For short range interactions (˛/d > 1)

the pseudo-intensive variables become intensive (independent

from N), and the pseudo-extensive merge with the extensive

ones, all being now extensive (scaling with N), thus recovering

the traditional two textbook classes of thermodynamical vari-

ables

Entropy, Figure 2

The so-called extensive systems (˛/d > 1 for the classical ones)

typically involve absolutely convergent series, whereas the so-

called nonextensive systems (0 � ˛/d < 1 for the classical ones)

typically involve divergent series. Themarginal systems (˛/d D 1

here) typically involve conditionally convergent series, which

therefore depend on the boundary conditions, i. e., typically on

the external shape of the system. Capacitors constitute a notori-

ous example of the ˛/d D 1 case. The model usually referred to

in the literature as the Hamiltonian–Mean–Field (HMF) one lies

on the ˛ D 0 axis (8d > 0). The model usually referred to as

the d-dimensional ˛-XY model [19] lies on the vertical axis at ab-

scissa d (8˛ � 0)

over all distances between sites i and j of a d-dimen-

sional lattice. For example, for a simple hypercubic lattice

with unit crystalline parameter we have ri j D 1; 2; 3; : : : ifd D 1, ri j D 1;

p2; 2; : : : if d D 2, ri j D 1;

p2;

p3; 2; : : :

if d D 3, and so on. For such a case, we have that

N? �NX

iD2r�˛1i ; (32)

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 � 1corresponds to long-range ones.

The correctness of the present generalized thermo-

dynamical scalings has already been specifically checked

in many physical systems, such as a ferrofluid-like

model [97], Lennard–Jones-like fluids [90], magnetic sys-

tems [16,19,59,158], anomalous diffusion [66], percola-

tion [85,144].

Let us mention that, for the ˛ D 0 models (i. e., meanfield models), it is largely spread in the literature to divide

by N the potential term of the Hamiltonian in order to

make it extensive by force. Although mathematically ad-

missible (see [19]), this is obviously very unsatisfactory in

principle since it implies a microscopic coupling constant

which depends on N. What we have described here is the

thermodynamically proper way of eliminating the mathe-

matical 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 interac-

tions.

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-

chanics on the basis of an entropy Sq which general-

izes SBG. This entropy is defined as follows:

Sq � k1 �PWiD1 p

qi

q � 1 (q 2 R; S1 D SBG): (33)

For equal probabilities, this entropy takes the form

Sq D k lnq W (S1 D k lnW) ; (34)

where the q-logarithmic function has already been de-

fined.

Remark With the same or different prefactor, this en-

tropic form has been successively and independently in-

troduced in many occasions during the last decades.

J. Havrda and F. Charvat [92] were apparently the first to

ever introduce this form, though with a different prefactor

(adapted to binary variables) in the context of cybernet-

ics and information theory. I. Vajda [207], further studied

this form, quoting Havrda and Charvat. Z. Daroczy [74]

rediscovered this form (he quotes neitherHavrda–Charvat

Entropy E 2867nor Vajda). J. Lindhard and V. Nielsen [108] rediscovered

this form (they quote none of the predecessors) through

the property of entropic composability. B.D. Sharma and

D.P. Mittal [163] introduced a two-parameter form which

reproduces both Sq and Renyi entropy [145] as partic-

ular cases. A. Wehrl [209] mentions the form of Sq in

p. 247, quotes Daroczy, but ignores Havrda–Charvat, Va-

jda, Lindhard–Nielsen, and Sharma–Mittal. Myself I re-

discovered this form in 1985 with the aim of generalizing

Boltzmann–Gibbs statistical mechanics, but quote none of

the predecessors in the 1988 paper [183]. In fact, I started

knowing the whole story quite a few years later thanks to

S.R.A. Salinas and R.N. Silver, whowere the first to provide

me with the corresponding informations. Such rediscov-

eries can by no means be considered as particularly sur-

prising. Indeed, this happens in science more frequently

than usually realized. This point is lengthily and colorfully

developed by S.M. Stigler [167]. In p. 284, a most inter-

esting example is described, namely that of the celebrated

normal distribution. It was first introduced by Abraham

De Moivre in 1733, then by Pierre Simon de Laplace in

1774, then by Robert Adrain in 1808, and finally by Carl

Friedrich Gauss in 1809, nothing less than 76 years after

its first publication! This distribution is universally called

Gaussian because of the remarkable insights of Gauss con-

cerning the theory of errors, applicable in all experimen-

tal sciences. A less glamorous illustration of the same

phenomenon, but nevertheless interesting in the present

context, is that of Renyi entropy [145]. According to I.

Csiszar [64], p. 73, the Renyi entropy had already been es-

sentially introduced by Paul-Marcel Schutzenberger [161].

The entropy defined in Eq. (33) has the following main

properties:

(i) Sq is nonnegative (8q);(ii) Sq is expansible (8q > 0);(iii) Sq attains its maximal (minimal) value k lnq W for

q > 0 (for q < 0);

(iv) Sq is concave (convex) for q > 0 (for q < 0);

(v) Sq is Lesche-stable (8q > 0) [2];(vi) Sq yields a finite upper bound of the entropy pro-

duction per unit time for a special value of q, when-

ever the sensitivity to the initial conditions exhibits

an upper bound which asymptotically increases as

a power of time. For example, many D D 1 non-linear dynamical systems have a vanishing maximal

Lyapunov exponent �1 and exhibit a sensitivity to

the initial conditions which is (upper) bounded by

� D e�q tq ; (35)

with �q > 0, q < 1, the q-exponential function exq

being the inverse of lnq x. More explicitly (see Fig. 3)

exq �(

[1 C (1 � q) x]1

1�q if 1 C (1 � q)x > 0 ;0 otherwise :

(36)

Such systems have a finite entropy production per

unit time, which satisfies a q-generalized Pesin-like

identity, namely, for the construction described in

Sect. “Introduction”,

Kq � limt!1

limW!1

limM!1

Sq(t)

tD �q : (37)

The situation is in fact sensibly much richer than

briefly described here. For further details, see [27,28,

29,30,93,116,117,146,147,148,149,150,151,152].

(vii) Sq is nonadditive for q ¤ 1. Indeed, for indepen-dent subsystemsA and B, it can be straightforwardly

proved

Sq(A C B)k

D Sq(A)k

C Sq(B)k

C (1�q) Sq(A)k

Sq(B)

k;

(38)

or, equivalently,

Sq(ACB) D Sq(A)C Sq (B)C(1 � q)

kSq(A) Sq(B) ;

(39)

which makes explicit that (1 � q) ! 0 plays thesame role as k ! 1. Property (38), occasion-ally referred to in the literature as pseudo-additiv-

ity, can be called subadditivity (superadditivity) for

q > 1 (q < 1).

(viii) Sq D �k DqPW

iD1 pxi jxD1, where the 1909 Jackson

differential operator is defined as follows:

Dq f (x) �f (qx) � f (x)

qx � x (D1 f (x) D d f (x)/dx) :

(40)

(ix) An uniqueness theorem has been proved by San-

tos [159], which generalizes, for arbitrary q, that of

Shannon [162].

Let us assume that an entropic form S(fpi g) satisfiesthe following properties:

(a)

S(fpig) is a continuous function of fpig;(41)

2868 E Entropy(b)

S(pi D 1/W;8i)monotonically increaseswith the total number of possibilitiesW ;

(42)

(c)

S(A C B)k

D S(A)k

C S(B)k

C (1 � q) S(A)k

S(B)

k

if pACBi j D pAi pBj 8(i; j) ; with k > 0;(43)

(d)

S(fpig) D S(pL ; pM) C pqLS(fpi /pLg) C pqMS(fpi /pMg)

with pL �X

L terms

pi ; pM �X

M terms

pi (L C M D W) ;

and pL C pM D 1 :(44)

Then and only then [159] S(fpig) D Sq(fpig).(x) Another (equivalent) uniqueness theorem was

proved by Abe [1], which generalizes, for arbitrary q,

that of Khinchin [100].

Let us assume that an entropic form S(fpi g) satisfiesthe following properties:

(a)

S(fpi g) is a continuous function of fpig; (45)(b)

S(pi D 1/W;8i) monotonically increaseswith the total number of possibilitiesW ;

(46)

(c)

S(p1; p2 ; : : : ; pW ; 0) D S(p1; p2 ; : : : ; pW ) ;(47)

(d)

S(A C B)k

D S(A)k

C S(BjA)k

C (1 � q) S(A)k

S(BjA)k

where S(A C B) � S�n

pACBi j

o�

;

S(A) � S

0

@

8

<

:

WBX

jD1pACBi j

9

=

;

1

A ; and the conditional entropy

S(BjA) �PWA

iD1�

pAi�qS�n

pACBi j /pAi

o�

PWAiD1

�

pAi�q (k > 0)

(48)

Then and only then [1] S(fpig) D Sq(fpig).

Additivity Versus Extensivity of the Entropy

It is of great importance to distinguish additivity from

extensivity. An entropy S is additive [130] if its value

for a system composed by two independent subsys-

tems A and B satisfies S(A C B) D S(A) C S(B) (hence,for N independent equal subsystems or elements, we

have S(N) D NS(1)). Therefore, SBG is additive, andSq(q ¤ 1) is nonadditive. A substantially different mat-ter is whether a given entropy S is extensive for

a given system. An entropy is extensive if and only if

0 < limN!1 S(N)/N < 1. What matters for satisfacto-rily matching thermodynamics is extensivity not addi-

tivity. For systems whose elements are nearly indepen-

dent (i. e., essentially weakly correlated), SBG is extensive

and Sq is nonextensive. For systems whose elements are

strongly correlated in a special manner, SBG is nonexten-

sive, whereas Sq is extensive for a special value of q ¤ 1(and nonextensive for all the others).

Let us illustrate these facts for some simple ex-

amples of equal probabilities. If W(N) � A�N (A > 0,� > 1, and N ! 1), the entropy which is extensiveis SBG. Indeed, SBG(N) D k lnW(N) � (ln�)N / N (itis equally trivial to verify that Sq(N) is nonexten-

sive for any q ¤ 1). If W(N) � BN�(B > 0, � > 0, andN ! 1), the entropy which is extensive is S1�(1/�). In-deed, S1�(1/�)(N) � k�B1/�N / N (it is equally trivialto verify that SBG(N) / lnN, hence nonextensive). IfW(N) � C�N (C > 0,� > 1, ¤ 1, and N ! 1), thenSq(N) is nonextensive for any value of q. Therefore, in

such a complex case, one must in principle refer to some

other kind of entropic functional in order to match the ex-

tensivity required by classical thermodynamics.

Various nontrivial abstract mathematical models can

be found in [113,160,186,198,199] for which Sq(q ¤ 1) isextensive. Moreover, a physical realization is also avail-

able now [60,61] for a many-body quantum Hamiltonian,

namely the ground state of the following one:

H D �N�1X

iD1

�

(1 C )Sxi SxiC1 C (1 � )Syi S

yiC1�

� 2�NX

iD1Szi ; (49)

where � is a transverse magnetic field, and (Sxi ; Syi ; S

zi )

are Pauli matrices; for j j D 1 we have the Ising model,for 0 < j j < 1, we have the anisotropic XY model, and,for D 0, we have the isotropic XY model. The twoformer share the same symmetry and consequently be-

long to the same critical universality class (the Ising uni-

Entropy E 2869

Entropy, Figure 3

The q-exponential and q-logarithm functions in typical representations: a Linear-linear representation of exq; b Linear-linear repre-

sentation of e�xq ; c Log-log representation of y(x) D e�aq xq , solution of dy/dx D �aq y

q 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

charge c D 1/2), whereas the latter one belongs to a dif-ferent universality class (the XX one, which corresponds

to a central charge c D 1). At temperature T D 0 andN ! 1, this model exhibits a second-order phase tran-sition as a function of �. For the Ising model, the criti-

cal value is � D 1, whereas, for the XX model, the entireline 0 � � � 1 is critical. Since the system is at its groundstate (assuming a vanishingly small magnetic field compo-

nent in the x � y plane), it is a pure state (i. e., its densitymatrix �N is such that Tr �

2N D 1, 8N), hence the entropy

Sq(N)(8q > 0) is strictly zero. However, the situation isdrastically different for any L-sized block of the infinite

chain. Indeed, �L � TrN�L�N is such that Tr �2L < 1, i. e.,it is a mixed state, hence it has a nonzero entropy. The

block entropy Sq(L) � limN!1 Sq(N; L) monotonically

increases with L for all values of q. And it does so linearly

for

q Dp9 C c2 � 3

c; (50)

where c is the central charge which emerges in quan-

tum field theory [54]. In other words, 0 < limL!1S(p9Cc2�3)/c (L)/L < 1. Notice that q increases from zero

to unity when c increases from zero to infinity; q Dp37�

6 ' 0:083 for c D 1/2 (Ising model), q Dp10�3 ' 0:16

for c D 1 (isotropic XY model), q D 1/2 for c D 4 (dimen-sion of space-time), and q D (

p685 � 3)/26 ' 0:89 for

c D 26, related to string theory [89]. The possible phys-ical interpretation of the limit c ! 1 is still unknown,although it could correspond to some sort of mean field

approach.

2870 E EntropyNonextensive Statistical Mechanics

To generalize BG statistical mechanics for the canonical

ensemble, we optimize Sq with constraint (15) and also

WX

iD1PiEi D Uq ; (51)

where

Pi �pqi

PWjD1 p

qi

WX

iD1Pi D 1

!

(52)

is the so-called escort distribution [33]. It follows that

pi D P1/qi /PW

jD1 P1/qj . There are various converging rea-

sons for being appropriate to impose the energy constraint

with the fPig instead of with the original fpig. The full dis-cussion of this delicate point is beyond the present scope.

However, some of these intertwined reasons are explored

in [184]. By imposing Eq. (51), we follow [193], which in

turn reformulates the results presented in [71,183]. The

passage from one to the other of the various existing

formulations of the above optimization problem are dis-

cussed in detail in [83,193].

The entropy optimization yields, for the stationary

state,

pi De

�ˇq(E i�Uq)q

Z̄q; (53)

with

ˇq �ˇ

PWjD1 p

qj

; (54)

and

Z̄q �WX

i

e�ˇq (E i�Uq)q ; (55)

ˇ being the Lagrange parameter associated with the con-

straint (51). Equation (53) makes explicit that the proba-

bility distribution is, for fixed ˇq, invariant with regard to

the arbitrary choice of the zero of energies. The station-

ary state (or (meta)equilibrium) distribution (53) can be

rewritten as follows:

pi De

�ˇ 0qE iq

Z0q; (56)

with

Z0q �WX

jD1e

�ˇ 0qE jq ; (57)

and

ˇ0q �ˇq

1 C (1 � q)ˇqUq: (58)

The form (56) is particularly convenient for many appli-

cations where comparison with experimental or computa-

tional data is involved. Also, it makes clear that pi asymp-

totically decays like 1/E1/(q�1)i for q > 1, and has a cut-

off for q < 1, instead of the exponential decay with Ei for

q D 1.The connection to thermodynamics is established in

what follows. It can be proved that

1

TD @Sq@Uq

; (59)

with T � 1/(kˇ). Also we prove, for the free energy,

Fq � Uq � TSq D �1

ˇlnq Zq ; (60)

where

lnq Zq D lnq Z̄q � ˇUq : (61)

This relation takes into account the trivial fact that, in con-

trast with what is usually done in BG statistics, the en-

ergies fEig are here referred to Uq in (53). It can also beproved

Uq D �@

@̌lnq Zq ; (62)

as well as relations such as

Cq � T@Sq

@TD @Uq

@TD �T @

2Fq

@T2: (63)

In fact the entire Legendre transformation structure of

thermodynamics is q-invariant, which is both remarkable

and welcome.

A Connection Between Entropy and Diffusion

We review here one of the main common aspects of en-

tropy and diffusion. We shall present on equal footing

both the BG and the nonextensive cases [13,138,192,216].

Let us extremize the entropy

Sq D k1 �

R1�1 d(x/�) [� p(x)]

q

q � 1 (64)

with the constraintsZ 1

�1dx p(x) D 1 (65)

Entropy E 2871and

hx2iq �R1

�1 dx x2[p(x)]q

R1�1 dx [p(x)]

qD �2 ; (66)

� > 0 being some fixed value having the same physical di-

mensions of the variable x. We straightforwardly obtain

the following distribution:

pq(x) D8

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

<

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

:

1

�

�

q � 1�(3 � q)

�1/2 �

�

1

q � 1

�

�

�

3 � q2(q � 1)

�

1�

1 C q � 13 � q

x2

�2

�1/(q�1)

if 1 < q < 3 ;

1

�

1p2�

e�x2/2�2

if q D 1 ;

1

�

�

1 � q�(3 � q)

�1/2 �

�

5 � 3q2(1 � q)

�

�

�

2 � q1 � q

�

�

1 � 1 � q3 � q

x2

�2

�1/(1�q)

for jxj < � [(3 � q)/(1 � q)]1/2 ; and zero otherwise ,if q < 1 :

(67)

These distributions are frequently referred to as q-Gaus-

sians. For q > 1, they asymptotically have a power-law

tail (q � 3 is not admissible because the norm (65) can-not be satisfied); for q < 1, they have a compact sup-

port. For q D 1, the celebrated Gaussian is recovered; forq D 2, the Cauchy–Lorentz distribution is recovered; fi-nally, for q ! �1, the uniform distribution within theinterval [�1; 1] is recovered. For q D 3Cm1Cm , m being aninteger (m D 1; 2; 3; : : :), we recover the Student’s t-dis-tributions with m degrees of freedom [79]. For q D n�4n�2 ,n being an integer (n D 3; 4; 5; : : :), we recover the so-called r-distributions with n degrees of freedom [79]. In

other words, q-Gaussians are analytical extensions of Stu-

dent’s t- and r-distributions. In some communities they

are also referred to as the Barenblatt form. For q < 5/3,

they have a finite variance which monotonically increases

for q varying from �1 to 5/3; for 5/3 � q < 3, the vari-ance diverges.

Let us now make a connection of the above optimiza-

tion problem with diffusion. We focus on the following

quite general diffusion equation:

@ı p(x; t)

@tıD @@x

�

@U(x)

@xp(x; t)

�

C D @˛[p(x; t)]2�q

@jxj˛(0 < ı � 1; 0 < ˛ � 2; q < 3; t � 0) ; (68)

with a generic nonsingular potential U(x), and a gen-

eralized diffusion coefficient D which is positive (nega-

tive) for q < 2 (2 < q < 3). Several particular instances

of this equation have been discussed in the literature

(see [40,86,106,131,188] and references therein).

For example, the stationary state for ˛ D 2, 8ı, andany confining potential (i. e., limjxj!1 U(x) D 1) isgiven by [106]

p(x;1)q De

�ˇ [U(x)�U(0)]q

Z; (69)

Z �Z 1

�1dx e�ˇ [U(x)�U(0)]q ; (70)

1/ˇ � kT / jDj ; (71)

which precisely is the distribution obtained within nonex-

tensive statistical mechanics through extremization of Sq.

Also, the solution for ˛ D 2, ı D 1, U(x) D �k1x Ck22 x

2(8 k1, and k2 � 0), and p(x; 0) D ı(x) is givenby [188]

pq(x; t) De

�ˇ (t)[x�xM (t)]2q

Zq(t); (72)

ˇ(t)

ˇ(0)D�

Zq(0)

Zq(t)

�2

D��

1 � 1K2

e�t/��

C 1K2

��2/(3�q);

(73)

K2 �k2

2(2 � q)Dˇ(0)[Zq(0)]q�1; (74)

� � 1k2(3 � q)

; (75)

xM(t) �k1

k2C�

xM(0) �k1

k2

�

e�k2 t : (76)

In the limit k2 ! 0, Eq. (73) becomes

Zq(t) D˚

[Zq(0)]3�q C 2(2 � q)(3 � q)Dˇ(0)

[Zq(0)]2 t�1/(3�q)

; (77)

which, in the t ! 1 limit, yields1

ˇ(t)/ [Zq(t)]2 / t2/(3�q) : (78)

In other words, x2 scales like t , with

D 23 � q ; (79)

hence, for q > 1 we have > 1 (i. e., superdiffusion; in

particular, q D 2 yields D 2, i. e., ballistic diffusion),

2872 E Entropyfor q < 1 we have < 1 (i. e., subdiffusion; in particular,

q ! �1 yields D 0, i. e., localization), and naturally,for q D 1, we obtain normal diffusion. Four systems areknown for which results have been found that are con-

sistent with prediction (79). These are the motion of Hy-

dra viridissima [206], defect turbulence [73], simulation of

a silo drainage [22], and molecular dynamics of a many-

body long-range-interacting classical system of rotators

(˛ � XY model) [143]. For the first three, it has beenfound (q; ) ' (3/2; 4/3). For the latter one, relation (79)has been verified for various situations corresponding to

> 1.

Finally, for the particular case ı D 1 and U(x) D 0,Eq. (68) becomes

@p(x; t)

@tD D @

˛[p(x; t)]2�q

@jxj˛ (0 < ˛ � 2; q < 3) : (80)

The diffusion constant D just rescales time t. Only two pa-

rameters are therefore left, namely ˛ and q.

The linear case (i. e., q D 1) has two types of solutions:Gaussians for ˛ D 2, and Lévy- (or ˛-stable) distributionsfor 0 < ˛ < 2. The case ˛ D 2 corresponds to the CentralLimit Theorem, where the N ! 1 attractor of the sumsof N independent random variables with finite variance

precisely is a Gaussian. The case 0 < ˛ < 2 corresponds to

the sometimes called Levy–Gnedenko Central Limit Theo-

rem, where the N ! 1 attractor of the sums of N inde-pendent random variables with infinite variance (and ap-

propriate asymptotics) precisely is a Lévy distribution with

index ˛.

The nonlinear case (i. e., q ¤ 1) has solutions thatare q-Gaussians for ˛ D 2, and one might conjecture that,similarly, interesting solutions exist for 0 < ˛ < 2. Fur-

thermore, in analogy with the q D 1 case, one expects cor-responding q-generalized Central Limit Theorems to ex-

ist [187]. This is precisely what we present in the next Sec-

tion.

Standard and q-GeneralizedCentral Limit Theorems

The q-Product

It has been recently introduced (independently and virtu-

ally simultaneously) [43,125] a generalization of the prod-

uct, which is called q-product. It is defined, for x � 0 andy � 0, as follows:

x ˝q y �(

[x1�q C y1�q � 1]1/(1�q) if x1�q C y1�q > 1 ;0 otherwise :

(81)

It has, among others, the following properties:

it recovers the standard product as a particular instance,

i. e.,

x ˝1 y D xy ; (82)

it is commutative, i. e.,

x ˝q y D y ˝q x ; (83)

it is additive under q-logarithm, i. e.,

lnq(x ˝q y) D lnq x C lnq y (84)

(whereas we remind that lnq(x y) D lnq x C lnq y C (1 �q)(lnq x)(lnq y);

it has a (2 � q)-duality/inverse property, i. e.,

1/(x ˝q y) D (1/x) ˝2�q (1/y) ; (85)

it is associative, i. e.,

x ˝q (y ˝q z) D (x ˝q y) ˝q z D x ˝q y ˝q zD (x1�q C y1�q C z1�q � 2)1/(1�q) ;

(86)

it admits unity, i. e.,

x ˝q 1 D x : (87)

and, for q � 1, also a zero, i. e.,

x ˝q 0 D 0 (q � 1) : (88)

The q-Fourier Transform

We shall introduce the q-Fourier transform of a quite

generic function f (x) (x 2 R) as follows [140,189,202,203,204,205]:

Fq[ f ](�) �Z 1

�1dx eix�q ˝q f (x)

DZ 1

�1dx e

ix�[ f (x)]q�1

q f (x)

; (89)

where we have primarily focused on the case q � 1. Incontrast with the q D 1 case (standard Fourier transform),this integral transformation is nonlinear for q ¤ 1. It hasa remarkable property, namely that the q-Fourier trans-

form of a q-Gaussian is another q-Gaussian:

Fq

h

Nqp

ˇ e�ˇ x2

q

i

(�) D e�ˇ1 �2q1 ; (90)

Entropy E 2873with

Nq �

8

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

<

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

:

�

q � 1�

�1/2 �

�

1

q � 1

�

�

�

3 � q2(q � 1)

� if 1 < q < 3 ;

1p�

if q D 1 ;

3 � q2

�

1 � q�

�1/2 �

�

3 � q2(1 � q)

�

�

�

1

1 � q

� if q < 1 ;

(91)

and

q1 D z(q) �1 C q3 � q ; (92)

ˇ1 D1

ˇ2�qN

2(1�q)q (3 � q)

8: (93)

Equation (93) can be re-written as ˇp2�qˇ

1/p2�q

1 D[(N

2(1�q)q (3 � q))/8]1/

p2�q � K(q), which, for q D 1, re-

covers the well known Heisenberg-uncertainty-principle-

like relation ˇˇ1 D 1/4.If we iterate n times the relation z(q) in Eq. (92), we

obtain the following algebra:

qn(q) D2q C n(1 � q)2 C n(1 � q) (n D 0;˙1;˙2; : : : ) ; (94)

which can be conveniently re-written as

2

1 � qn(q)D 2

1 � q C n (n D 0;˙1;˙2; : : : ) : (95)

(See Fig. 4). We easily verify that qn(1) D 1 (8n),q˙1(q) D 1 (8q), as well as

1

qnC1D 2 � qn�1 : (96)

This relation connects the so called additive duality q !(2 � q) and multiplicative duality q ! 1/q, frequentlyemerging in all types of calculations in the literature.

Moreover, we see from Eq. (95) that multiple values of q

are expected to emerge in connection with diverse proper-

ties of nonextensive systems, i. e., in systems whose basic

entropy is the nonadditive one Sq. Such is the case of the

so called q-triplet [185], observed for the first time in the

magnetic field fluctuations of the solar wind, as it has been

revealed by the analysis of the data sent to NASA by the

spacecraft Voyager 1 [48].

Entropy, Figure 4

The q-dependence of qn(q) � q2;n(q)

q-Independent RandomVariables

Two random variables X [with density fX(x)] and Y [with

density fY (y)] having zero q-mean values (e. g., if fX(x)

and fY (y) are even functions) are said q-independent, with

q1 given by Eq. (92), if

Fq[X C Y](�) D Fq[X](�) ˝q1 Fq[Y](�) ; (97)i. e., if

Z 1

�1dz eiz�q ˝q fXCY (z) D

�Z 1

�1dx eix�q ˝q fX(x)

�

˝(1Cq)/(3�q)�Z 1

�1dy e

i y�q ˝q fX(y)

�

; (98)

with

fXCY (z) DZ 1

�1dx

Z 1

�1dy h(x; y) ı(x C y � z)

DZ 1

�1dx h(x; z � x)

DZ 1

�1dy h(z � y; y)

(99)

where h(x; y) is the joint density.

Clearly, q-independence means independence for

q D 1 (i. e., h(x; y) D fX(x) fY (y)), and implies a specialcorrelation for q ¤ 1. Although the full understanding ofthis correlation is still under progress, q-independence ap-

pears to be consistent with scale-invariance.

2874 E EntropyEntropy, Table 1

The attractors corresponding to the four basic cases, where the N variables that are being summed are q-independent (i. e., globally

correlated) with q1 D (1 C q)/(3 � q); �Q � (R

1

�1dx x2 [f (x)]Q)/(

R1

�1dx [f (x)]Q) 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 1 and 0 < ˛ < 2, it is a Lévy distribution L˛ � L1;˛ (the so called

Lévy-Gnedenko limit theorem); for ˛ D 2 and q ¤ 1, it is a q-Gaussian Gq � Lq;2 (the q-Central Limit Theorem; [203]); finally, for

q ¤ 1 and 0 < ˛ < 2, it is a generic (q; ˛)-stable distribution Lq;˛ ([204,205]). See [140,189] for typical illustrations of the four types

of attractors. Thedistribution L˛(x) remains, for 1 < ˛ < 2, close to aGaussian for jxjup to about xc(1; ˛), where itmakes a crossover

to a power-law. The distribution Gq(x) remains, for q > 1, close to a Gaussian for jxj up to about xc(q;2), where it makes a crossover

to a power-law. The distribution Lq;˛(x) remains, for q > 1 and ˛ < 2, close to a Gaussian for jxj up to about x(1)c (q; ˛), where it

makes a crossover to a power-law (intermediate regime), which lasts further up to about x(2)c (q; ˛), where it makes a second crossover

to another power-law (distant regime)

q D 1 [independent] q ¤ 1 (i. e., Q ¤ 1) [globally correlated]�Q < 1 G(x) Gq(x) D G(3q1�1)/(1Cq1)(x)(˛ D 2) [with same �1 of f (x)] [with same �Q of f (x)]

Gq(x) � G(x) if jxj � xc(q; 2)Gq(x) � Cq;2/jxj2/(q�1) if jxj � xc(q; 2)for q > 1, with limq!1 xc(q; 2) D 1

�Q ! 1 L˛ (x) Lq;˛ (x)(˛ < 2) [with same jxj ! 1 behavior of f (x)] [with same jxj ! 1 behavior of f (x)]

L˛ (x) � G(x) if jxj � xc(1; ˛) Lq;˛ � C(intermediate)q;˛ /jxj2(1�q)�˛(3�q)

2(q�1)

L˛ (x) � C1;˛/jxj1C˛ if jxj � xc(1; ˛) if x(1)c (q; ˛) � jxj � x(2)c (q; ˛)with lim˛!2 qc(1; ˛) D 1 Lq;˛ � C(distant)q;˛ /jxj

1C˛1C˛(q�1)

if jxj � x(2)c (q; ˛)

q-Generalized Central Limit Theorems

It is out of the scope of the present survey to provide the

details of the complex proofs of the q-generalized cen-

tral limit theorems. We shall restrict to the presentation

of their structure. Let us start by introducing a notation

which is important for what follows. A distribution is said

(q; ˛)-stable distribution Lq;˛(x) if its q-Fourier transform

Lq;˛(�) is of the form

Lq;˛(�) D a e�b j�j˛

q1

[a > 0; b > 0; 0 < ˛ � 2; q1 D (q C 1)/(3 � q)] :(100)

Consistently, L1;2 are Gaussians, L1;˛ are Lévy distribu-

tions, and Lq;2 are q-Gaussians.

We are seeking for the N ! 1 attractor associatedwith the sum of N identical and distinguishable random

variables each of them associated with one and the same

arbitrary symmetric distribution f (x). The random vari-

ables are independent for q D 1, and correlated in a spe-cial manner for q ¤ 1. To obtain the N ! 1 invariantdistribution, i. e. the attractor, the sum must be rescaled,

i. e., divided by Nı , where

ı D 1˛(2 � q) : (101)

For (˛; q) D (2; 1), we recover the traditional 1/pN

rescaling of Brownian motion. At the present stage, the

theorems have been established for q � 1 and are summa-rized in Table 1. The case q < 1 is still open at the time

at which these lines are being written. Two q < 1 cases

have been preliminarily explored numerically in [124]

and in [171]. The numerics seemed to indicate that the

N ! 1 limits would be q-Gaussians for both models.However, it has been analytically shown [94] that it is

not exactly so. The limiting distributions numerically are

amazingly close to q-Gaussians, but they are in fact differ-

ent. Very recently, another simple scale-invariant model

has been introduced [153], whose attractor has been ana-

lytically shown to be a q-Gaussian.

These q ¤ 1 theorems play for the nonadditive en-tropy Sq and nonextensive statistical mechanics the same

grounding role that the well known q D 1 theorems playfor the additive entropy SBG and BG statistical mechanics.

In particular, interestingly enough, the ubiquity of Gaus-

sians and of q-Gaussians in natural, artificial and social

systems may be understood on equal footing.

Future Directions

The concept of entropy permeates into virtually all quan-

titative sciences. The future directions could therefore

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 distribu-

tion p(k) (probability of a node having k links) which

is of the form

p(k) / 1(k0 C k)

( > 0; k0 > 0) ; (102)

or, equivalently,

p(k) / e�k/�q (q � 1; � > 0) ; (103)

with D 1/(q � 1) and k0 D �/(q � 1) (see Figs. 5and 6). This is not surprising since, if we associate

to each link an “energy” (or cost) and to each node

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-

ergies optimizing Sq precisely coincides with the de-

gree distribution. If, for any reason, we consider k

as the modulus of a d-dimensional vector k, the op-

timization of the functional Sq[p(k)] may lead to

p(k) / k� e�k/�q , where k� plays the role of a den-sity of states, �(d) being either zero (which repro-

duces Eq. (103)) or positive or negative. Several exam-

ples [12,39,76,91,165,172,173,212,213] already exist in

the literature; in particular, the Barabasi–Albert uni-

versality class D 3 corresponds to q D 4/3. A deeperunderstanding of this connectionmight enable the sys-

Entropy, Figure 6

Nongrowing dynamic network: a Cumulative degree distribu-

tion for typical values for the number N of nodes; b Same data

of a in the convenient representation linear q-log versus linear

with Zq(k) � lnq[Pq(> k)] � ([Pq(> k)]1�q � 1)/(1 � q) (the opti-

mal fitting with a q-exponential is obtained for the value of q

which has the highest value of the linear correlation r as indi-

cated in the inset; here this is qc D 1:84, which corresponds to

the slope �1.19 in a). See details in [172,173]

tematic calculation of several meaningful properties of

networks.

Nonlinear dynamical systems, self-organized criticality,

and cellular automata Various interesting phenomena

emerge in both low- and high-dimensional weakly

chaotic deterministic dynamical systems, either dis-

sipative or conservative. Among these phenomena

we have the sensitivity to the initial conditions and

the entropy production, which have been briefly ad-

dressed in Eq. (37) and related papers. But there

is much more, such as relaxation, escape, glassy

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 pos-

sible 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 clas-

sical Hamiltonians exhibits collective states whose

Lyapunov spectrum has a maximal value that vanishes

in the N ! 1 limit. As such, they constitute naturalcandidates for studying whether the concepts derived

from the nonadditive entropy Sq are applicable. A vari-

ety of properties have been calculated, through molec-

ular 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 ! 1) are typically observedbefore the terminal entrance into thermal equilib-

rium. Properties such as distribution of velocities

and angles, correlation functions, Lyapunov spectrum,

metastability, glassy states, aging, time-dependence of

the temperature in isolated systems, energy whenever

thermal contact with a large thermostat at a given

temperature is allowed, diffusion, order parameter,

and others, are typically focused on. An ongoing de-

bate exists, also involving Vlasov-like equations, Lyn-

den–Bell statistics, among others. The breakdown of

ergodicity that emerges in various situations makes

the whole discussion rich and complex. The activity

of the research nowadays in this area is illustrated

in papers such as [21,26,45,53,56,57,63,104,119,121,

Entropy, Figure 8

QuantumMonte Carlo simulations in [81]: aVelocity distribution

(superimposedwith a q-Gaussian);b Index q (superimposedwith

Lutz prediction [110], by courtesy of the authors)

126,127,132,133,134,135,136,142,169,200]. A quite re-

markable molecular-dynamical result has been ob-

tained for a paradigmatic long-range Hamiltonian: the

distribution of time averaged velocities sensibly differs

from that found for the ensemble-averaged velocities,

and has been shown to be numerically consistent with

a q-Gaussian [137], as shown in Fig. 7. This result pro-

vides strong support to a conjecture made long ago:

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–

Planck equations are currently being studied. Aspects

such as fractional derivatives, nonlinearities, space-de-

pendent diffusion coefficients are being focused on,

as well as their connections to entropic forms, and

associated generalized Langevin equations [20,23,24,

70,128,168,214]. Quite recently, computational (see

Fig. 8) and experimental (see Fig. 9) verifications of

Lutz’ 2003 prediction [110] have been exhibited [81],

namely about the q-Gaussian form of the velocity

distribution of cold atoms in dissipative optical lat-

tices, with q D 1 C 44ER/U0(ER and U0 being en-

ergy parameters of the optical lattice). These exper-

imental verifications are in variance with some of

those exhibited previously [96], namely double-Gaus-

sians. Although it is naturally possible that the ex-

perimental conditions have not been exactly equiv-

alent, this interesting question remains open at the

present time. A hint might be hidden in the recent

results [62] obtained for a quite different problem,

namely the size distributions of avalanches; indeed,

at a critical state, a q-Gaussian shape was obtained,

whereas, at a noncritical state, a double-Gaussian was

observed.

2878 E EntropyQuantum entanglement and quantum chaos The non-

local nature of quantum physics implies phenomena

that are somewhat analogous to those originated by

classical long-range interactions. Consequently, a va-

riety of studies are being developed in connection

with the entropy Sq [3,36,58,59,60,61,155,156,195].

The same happens with some aspects of quantum

chaos [11,180,210,211].

Astrophysics, geophysics, economics, linguistics, cog-

nitive psychology, and other interdisciplinary appli-

cations Applications are available and presently searched

in many areas of physics (plasmas, turbulence, nuclear

collisions, elementary particles, manganites), but also

in interdisciplinary sciences such astrophysics [38,47,

48,49,78,84,87,101,109,129,196], geophysics [4,5,6,7,8,

9,10,62,208], economics [25,50,51,52,80,139,141,197,

215], linguistics [118], cognitive psychology [181], and

others.

Global optimization, image and signal processing

Optimizing algorithms and related techniques for

signal and image processing are currently being de-

veloped using the entropic concepts presented in this

article [17,35,72,75,95,105,114,120,164,166,191].

Superstatistics and other generalizations The methods

discussed here have been generalized along a vari-

ety of lines. These include Beck–Cohen superstatis-

tics [32,34,65,190], crossover statistics [194,196], spec-

tral statistics [201]. Also, a huge variety of entropies

have been introduced which generalize in different

manners the BG entropy, or even focus on other pos-

sibilities. Their number being nowadays over forty, we

mention here just a few of them: see [18,44,69,98,99,

115].

Acknowledgments

Among the very many colleagues towards which I am

deeply grateful for profound and long-lasting comments

along many years, it is a must to explicitly thank S. Abe,

E.P. Borges, E.G.D. Cohen, E.M.F. Curado, M. Gell-Mann,

R.S. Mendes, A. Plastino, A.R. Plastino, A.K. Rajagopal, A.

Rapisarda and A. Robledo.

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