Nonadditive entropy: the concept and its use

Abstract

The entropic form $S_q$ is, for any $q \neq 1$, {\it nonadditive}. Indeed, for two probabilistically independent subsystems, it satisfies $S_q(A+B)/k=[S_q(A)/k]+[S_q(B)/k]+(1-q)[S_q(A)/k][S_q(B)/k] \ne S_q(A)/k+S_q(B)/k$. This form will turn out to be {\it extensive} for an important class of nonlocal correlations, if $q$ is set equal to a special value different from unity, noted $q_{ent}$ (where $ent$ stands for $entropy$). In other words, for such systems, we verify that $S_{q_{ent}}(N) \propto N (N>>1)$, thus legitimating the use of the classical thermodynamical relations. Standard systems, for which $S_{BG}$ is extensive, obviously correspond to $q_{ent}=1$. Quite complex systems exist in the sense that, for them, no value of $q$ exists such that $S_q$ is extensive. Such systems are out of the present scope: they might need forms of entropy different from $S_q$, or perhaps -- more plainly -- they are just not susceptible at all for some sort of thermostatistical approach. Consistently with the results associated with $S_q$, the $q$-generalizations of the Central Limit Theorem and of its extended L\'evy-Gnedenko form have been achieved. These recent theorems could of course be the cause of the ubiquity of $q$-exponentials, $q$-Gaussians and related mathematical forms in natural, artificial and social systems. All of the above, as well as presently available experimental, observational and computational confirmations -- in high energy physics and elsewhere --, are briefly reviewed. Finally, we address a confusion which is quite common in the literature, namely referring to distinct physical mechanisms {\it versus} distinct regimes of a single physical mechanism. Comment: Brief review to appear in "Statistical Power-Law Tails in High Energy Phenomena", ed. T.S. Biro, Eur. Phys. J. A (2009);10 pages including 3 figures

Full text

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arXiv:0812.4370v1 [physics.data-an] 23 Dec 2008
EPJ manuscript No.
(will be inserted by the editor)
Nonadditive entropy: the concept and its use⋆
Constantino Tsallisa
Centro Brasileiro de Pesquisas Fisicas
and National Institute of Science and Technology for Complex Systems
Xavier Sigaud 150, 22290-180 Rio de Janeiro-RJ, Brazil
and
Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA
Received: date / Revised version: date
Abstract. The thermodynamical concept of entropy was introduced by Clausius in 1865 in order to con-
struct the exact differential dS = δQ/T, where δQ is the heat transfer and the absolute temperature T
its integrating factor. A few years later, in the period 1872-1877, it was shown by Boltzmann that this
quantity can be expressed in terms of the probabilities associated with the microscopic configurations of
the system. We refer to this fundamental connection as the Boltzmann-Gibbs (BG) entropy, namely (in
its discrete form) SBG = −kPW
i=1pilnpi, where k is the Boltzmann constant, and {pi} the probabilities
corresponding to the W microscopic configurations (hencePW
cussed by Gibbs, von Neumann and Shannon, and constituting the basis of the celebrated BG statistical
mechanics, is additive. Indeed, if we consider a system composed by any two probabilistically independent
subsystems A and B (i.e., pA+B
ij
= pA
system is constituted by N equal elements which are are either independent or quasi-independent (i.e., not
too strongly correlated, in some specific nonlocal sense), this additivity guarantees SBG to be extensive
in the thermodynamical sense, i.e., that SBG(N) ∝ N in the N >> 1 limit. If, on the contrary, the cor-
relations between the N elements are strong enough, then the extensivity of SBG is lost, being therefore
incompatible with classical thermodynamics. In such a case, the many and precious relations described in
textbooks of thermodynamics become invalid. Along a line which will be shown to overcome this difficulty,
and which consistently enables the generalization of BG statistical mechanics, it was proposed in 1988 the
entropy Sq = k[1 −PW
theory, this and similar forms have in fact been repeatedly introduced before 1988. The entropic form
Sq is, for any q ?= 1, nonadditive. Indeed, for two probabilistically independent subsystems, it satisfies
Sq(A + B)/k = [Sq(A)/k] + [Sq(B)/k] + (1 − q)[Sq(A)/k][Sq(B)/k] ?= Sq(A)/k + Sq(B)/k. This form will
turn out to be extensive for an important class of nonlocal correlations, if q is set equal to a special value
different from unity, noted qent (where ent stands for entropy). In other words, for such systems, we verify
that Sqent(N) ∝ N (N >> 1), thus legitimating the use of the classical thermodynamical relations. Stan-
dard systems, for which SBG is extensive, obviously correspond to qent = 1. Quite complex systems exist in
the sense that, for them, no value of q exists such that Sq is extensive. Such systems are out of the present
scope: they might need forms of entropy different from Sq, or perhaps — more plainly — they are just
not susceptible at all for some sort of thermostatistical approach. Consistently with the results associated
with Sq, the q-generalizations of the Central Limit Theorem and of its extended L´ evy-Gnedenko form
have been achieved. These recent theorems could of course be the cause of the ubiquity of q-exponentials,
q-Gaussians and related mathematical forms in natural, artificial and social systems. All of the above, as
well as presently available experimental, observational and computational confirmations — in high energy
physics and elsewhere —, are briefly reviewed. Finally, we address a confusion which is quite common in
the literature, namely referring to distinct physical mechanisms versus distinct regimes of a single physical
mechanism.
i=1pi = 1). This entropic form, further dis-
ipB
j, ∀(i,j)), we verify that SBG(A + B) = SBG(A) + SBG(B). If a
i=1pq
i]/(q − 1)(q ∈ R; S1 = SBG). In the context of cybernetics and information
PACS. 05.20.-y Classical statistical mechanics; 02.50.Cw Probability theory; 05.90.+m Other topics in
statistical physics, thermodynamics, and nonlinear dynamical systems; 05.70.-a Thermodynamics
⋆To appear in Statistical Power-Law Tails in High Energy
Phenomena, ed. T.S. Biro, Eur. Phys. J. A (2009).
atsallis@cbpf.br
1 Introduction
The concept of entropy S, as well as its name, were intro-
duced in thermodynamics by Clausius in 1865 [1]. It was

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2Constantino Tsallis: Nonadditive entropy: the concept and its use
done on a purely macroscopic basis (the possible existence
of a microscopic world was merely speculative at the time,
with just a few incipient scientific indications), with the
purpose to associate with the heat transfer δQ an exact
differential. This was indeed achieved through the cele-
brated relation dS = δQ/T, where dS is the differential
entropy, and the absolute temperature T the integrating
factor.
A decade later, in the period 1872-1877, it was shown
by Boltzmann [2,3] that this quantity can be expressed
in terms of the probabilities associated with the micro-
scopic configurations of the system. We refer to this con-
nection — one of the deepest ever done in physics — as the
Boltzmann-Gibbs (BG) entropy. In its present (discrete)
form, it is written as follows
SBG= −k
W
?
i=1
pilnpi,(1)
where k is the Boltzmann constant (or some other conve-
nient value, e.g. k = 1, in areas outside physics, such as
information theory, cybernetics and others), and the prob-
abilities {pi} corresponding to the W microscopic config-
urations satisfy
W
?
i=1
pi= 1.(2)
This entropic form, further discussed by Gibbs [4], von
Neumann [5] and Shannon [6], constitutes the basis of
the BG statistical mechanics, one of the monuments of
contemporary physics. Eq. (1) satisfies a variety of conve-
nient mathematical properties (non-negativity, concavity,
expansibility, Lesche-stability, composability, Topsoe fac-
torizability, finite entropy production per unit time satis-
fying the Pesin identity). For equal probabilities, i.e., for
pi= 1/W (∀i), it attains its maximal value, namely
SBG= k lnW .(3)
In the present context, let us focus on its additivity prop-
erty. An entropy S({pi}) is said additive [7] if, for a system
composed by any two probabilistically independent subsys-
tems A and B (i.e., satisfying pA+B
1,2,...,WA; j = 1,2,...,WB), we verify
ij
= pA
ipB
j, ∀(i,j); i =
S(A + B) = S(A) + S(B),(4)
where S(A + B) ≡ S({pA+B
S(B) ≡ S({pB
SBG({pi}) given by Eq. (1) is additive. Due to this prop-
erty, the BG entropy of any system made of N equal and
independent elements satisfies
ij
}), S(A) ≡ S({pA
i}) and
j}). It is straightforward to verify that
SBG(N) = NSBG(1).(5)
This fact obviously complies with the classical thermody-
namical requirement for the entropy S to be extensive,
i.e., such that
S(N)
N
lim
N→∞
< ∞.(6)
Indeed, in such a case,
lim
N→∞
SBG(N)
N
= SBG(1) ≤ kln[W(1)], (7)
where W(1) is the number, assumed finite, of possible con-
figurations of one element.
If the system is constituted by N equal elements which
are not strictly independent, but quasi-independent in-
stead (i.e., not too strongly correlated, in some nonlo-
cal sense to be further clarified later on; typically for a
Hamiltonian many-body system whose elements interact
through short-range interactions, or which are weakly quan-
tum -entangled), the additivity of SBGguarantees its ex-
tensivity in the thermodynamical sense, i.e., that Eq. (6)
is satisfied.
If, on the contrary, the correlations between the N ele-
ments are strong enough (a feature which might typically
occur for nonergodic states, e.g., in Hamiltonian many-
body systems with long-range interactions, or which are
strongly quantum-entangled), then the extensivity of SBG
might be lost (at least at the level of a large subsystem
of a much larger system), being therefore incompatible
with classical thermodynamics. In such a case, many of
the useful relations described in textbooks of thermody-
namics may become invalid. It is precisely this pathologi-
cal class of systems the one which is addressed within the
thermostatistical theory usually referred to as nonexten-
sive statistical mechanics [8,18,19], described in the next
Section.
2 Nonadditive entropy and nonextensive
statistical mechanics
2.1 Nonadditive entropy Sq
As an attempt to generalize BG statistical mechanics, and
possibly provide a frame for handling some of the above
mentioned pathological systems, it was postulated in 1988
[8] the following entropy:
Sq= k1 −?W
i=1pq
i
q − 1
(q ∈ R; S1= SBG),(8)
A simple manner to obtain S1= SBGis through the use of
pq−1
i
= e(q−1)lnpi∼ 1+(q−1)lnpi. If q < 0, the sum must
be done only over configurations which have nonzero prob-
ability to occur. Entropy (8) can be conveniently rewritten
in the following alternative forms:
Sq= k
W
?
i=1
pilnq(1/pi) (9)
= −k
W
?
i=1
pq
ilnqpi
(10)
= −k
W
?
i=1
piln2−qpi,(11)

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Constantino Tsallis: Nonadditive entropy: the concept and its use3
where the q-logarithmic function is defined as follows:
lnqz ≡z1−q− 1
1 − q
(q ∈ R; z ≥ 0; ln1z = lnz).(12)
Sq attains its extremum (maximum for q > 0, and min-
imum for q < 0) for equal probabilities, and its value is
given by
Sq= k lnqW .(13)
It can be shown to satisfy, for independent subsystems,
Sq(A + B)
k
=Sq(A)
k
+Sq(B)
k
+(1−q)Sq(A)
k
Sq(B)
k
. (14)
Therefore, this entropy is generically nonadditive. It satis-
fies, nevertheless, most other properties (mentioned previ-
ously) of the entropy SBG. In other words, it constitutes a
sort of minimalistic generalization of SBG. From Eq. (14)
we obtain
Sq(A + B) = Sq(A) + Sq(B) +1 − q
k
Sq(A)Sq(B), (15)
which exhibits the equivalence between (q − 1) → 0 and
k → ∞. Since for stationary states (e.g., thermal equi-
librium), k appears multiplicatively accompanied by the
temperature T (i.e., in the form kT), k → ∞ turns out to
be equivalent to T → ∞. We may consider that it is here
where the fact emerges that, at the T → ∞ limit, all the
microcanonical, canonical, grand-canonical ensembles of
classical or quantum (Fermi-Dirac, Bose-Einstein, Gentile
parastatistics) statistics, q-statistics (as will become evi-
dent later on)), coincide, and coincide with the hypothesis
of equal probabilities for an isolated system.
Eq. (14) can be generalized in the presence of arbitrary
correlations between two sysbsystems A and B of a given
system. It becomes [9]
Sq(A + B)
k
=Sq(A)
k
+Sq(B|A)
k
+ (1 − q)Sq(A)
+ (1 − q)Sq(A|B)
k
Sq(B|A)
k
Sq(B)
k
=Sq(A|B)
k
+Sq(B)
kk
, (16)
where Sq(A + B) is to be calculated with the joint prob-
abilities
?pA+B
?pA
Sq(A|B) with the conditional probabilities
(analogously for Sq(B|A)): see [9] for full details. Eq. (16)
straightforwardly recovers Eq. (14) if A and B are inde-
pendent, hence Sq(A|B) = Sq(A) and Sq(B|A) = Sq(B).
It is precisely this nonadditivity the property which
enables thermodynamical extensivity. More precisely, if
both A and B are very large (i.e., NA>> 1 and NB>>
1), then a value of q, noted qent, might exist for which
Sqent(A + B) ∼ Sqent(A) + Sqent(B). In other words, if
a system has N >> 1 equal elements, it becomes possi-
ble that a special value of q exists such that generically
0 < limN→∞
?Sqent(N)/N?< ∞.
This interesting feature can be easily illustrated in the
case of equal probabilities, for which Eq. (13) holds. If
ij
?, Sq(A) with the marginal probabilities
j=1pA+B
ij
?(analogously for Sq(B)), and
i
?≡??WB
?pA+B
ij
/pB
j
?
W(N) ∼ CµN(with C > 0 and µ > 1), then qent = 1,
i.e., SBG(N) ∝ N
tions forbid many (typically most) microscopic configu-
rations to occur, then it might happen that the num-
ber Weff(N) of effective (or admissible) configurations
satisfies Weff(N) << W(N). If we have, in particular,
Weff∼ DNρ(with D > 0 and ρ ∈ R), then
qent= 1 −1
(N >> 1). But if strong correla-
ρ.
(17)
This type of highly restricted phase space may occur in
various systems, as has been numerically or analytically
illustrated in various examples. Let us briefly mention here
two of them that are analytically tractable, namely an
abstract probabilistic one and a physical one.
The probabilistic model consists in N correlated dis-
tinguishable binary variables [10]. The probabilities of the
2Nstates vanish excepting for ∼ (d+1)N of them (which
can be seen, in the classical Pascal-like triangular repre-
sentation, as a N-long “strip” whose width is (d + 1),
d being non-negative). This model asymptotically satis-
fies probabilistic scale-invariance (Leibnitz triangle rule)
in the limit N → ∞. It can be verified that
qent= 1 −1
d.
(18)
The physical model corresponds to a long ring of N
1/2 spins with ferromagnetic first-neighbor interactions at
zero temperature. The interactions are of the anisotropic
XY ones in the presence of a transverse external magnetic
field (i.e., along the Z direction) at its critical value. Two
well known universality classes are contained within such a
system, namely the Ising universality class (corresponding
to a central charge c = 1/2), and the isotropic XY uni-
versality class (corresponding to a central charge c = 1).
We consider a block of L successive spins among those
N spins, and address the entropy Sq(L) of the N → ∞
quantum system. More precisely, we are interested in
Sq(L) = k1 − Tr(ρL)q
q − 1
, (19)
where ρL≡ limN→∞Tr{N−L}ρ(N), ρ(N) being the den-
sity matrix associated with the system of N spins, and
where we have traced over all but the L successive spins.
We define, in this case, qentas the value of q for which the
block entropy Sq(L) is extensive, i.e., such that Sqent(L) ∝
L. Such value does exist [11], and it is given by qent =
√37−6 ≃ 0.0828 for c = 1/2, and qent=√10−3 ≃ 0.1623
for c = 1. By using a recent result within conformal quan-
tum field theory [12], these two values for qentcan be gen-
eralized for the entire class of (1 + 1)-dimensional models
characterized by a generic central charge c. It is obtained
[11]
√9 + c2− 3
qent=
c
.(20)
As we see, qentmonotonically increases from zero to unity
(the BG value!) when c increases from zero to infinity. For

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4Constantino Tsallis: Nonadditive entropy: the concept and its use
Table 1. QSS stands for quasi-stationary state ([16,17] and
references therein).
SYSTEMENTROPY SBG
(additive)
ENTROPY Sq (q < 1)
(nonadditive)
Short-range
interactions,
weakly
entangled
blocks, etc
EXTENSIVE
NONEXTENSIVE
Long-range
interactions
(QSS),
strongly
entangled
blocks, etcNONEXTENSIVE
EXTENSIVE
c = 4, one obtains q = 1/2, which has already emerged in
general relativistic problems [13]; the possible connection,
if there is one, remains however without explanation at
the present time. Also, for c = 26, which corresponds to
string theory [14], we obtain qent ≃ 0.8913. Finally, it
also remains presently without explanation the reason for
which c → ∞ leads to the BG result, i.e., qent= 1.
The previous system is a fermionic d = 1 one, for
which it is known that SBG(L) ∝ lnL. Results, though
only numerical, also exist for a bosonic d = 2 system [11].
Once again extensivity only occurs for qent< 1, whereas
SBG(L) ∝ L. Finally it is known that, for the black hole,
the BG entropy is proportional to the area L2, instead
of being proportional to the volume L3(see, for instance,
[15]); in fact, more generally, it is known, for d-dimensional
bosonic systems, the so-called area law, i.e., the fact that
SBG(L) ∝ Ld−1, which obviously violates classical ther-
modynamics. All these anomalies are believed to be a con-
sequence of strongly nonlocal quantum entanglement. In
fact all of the above results can be unified through the
following conjectural expression, ∀d:
SBG(L) ∝ ln2−dL ≡Ld−1− 1
d − 1
?= Ld∝ N .(21)
In all these cases, as for the d = 1 and the d = 2 above
described examples, it might well exist a value of qent<
1 such that thermodynamic extensivity is ensured, i.e.,
Sqent(L) ∝ Ld∝ N. This would of course mean that
Clausius entropy should be, for this class of anomalous
(sub)systems, identified with Sqentand not with SBG. The
plausible conjectural scenario is summarized in Table 1.
2.2 Nonextensive statistical mechanics
Since Sqgeneralizes SBGand maintains most of its math-
ematically convenient properties (e.g., concavity, Lesche
stability, among others), it is quite natural to attempt the
q-generalization of BG statistical mechanics itself. This
extended theory is usually referred to in the literature as
nonextensive statistical mechanics. It was first proposed
in 1988 [8], and later on connected to thermodynamics
[18,19]. It has since then received a considerable amount
of applications and verifications in natural, artificial and
social systems [20,21,22,23,24]. Some of its predictions
have been experimentally and observationally checked in
systems such as the motion of Hydra viridissima [25] and
cells [26,27,28], defect turbulence [29], solar wind [30,31,
32,33], cold atoms in optical dissipative lattices [34], dusty
plasma [35], silo drainage [36,37], high-energy physics (see,
e.g., [38,39,40,41,42]). They have also been checked an-
alytically and computationally in various nonlinear dy-
namical problems such as the edge of chaos of simple
unimodal dissipative maps [43,44,45,46,47,48,49,50,51,
52,53], and long-range-interacting many-body Hamilto-
nian systems [54,55,16,17]; also, various applications to
the so-called scale-free networks are available [56,57,58,
59,60,61,62,63,64]. In one way or another, most if not all
of these systems appear to share slow (power-law rather
than exponential) sensitivity to the initial conditions. In
other words, for classical systems, at the level of first prin-
ciples, BG statistical mechanical concepts are legitimate
and fruitful when the system exhibits a positive maximal
Lyapunov exponent (corresponding essentially to Boltz-
mann’s molecular chaos hypothesis), whereas vanishing
maximal Lyapunov exponent appears to be necessary (al-
though probably not sufficient) for the applicability of the
nonextensive statistical mechanical concepts. The mecha-
nisms that typically yield q-statistics involve, at the meso-
scopic level, non-Markovian processes [65], multiplicative
noise [66], nonlinear Fokker-Planck equations [67,68,69,
70], and similar ones.
Let us now briefly review, within the present theory,
two important stationary-state distributions, namely the
q-generalizationof the celebrated BG weight (discrete case),
and the q-generalization of the Gaussian distribution (con-
tinuous case).
To generalize the BG factor for the canonical ensemble
(i.e., a system in a stationary state due to its contact with
a “thermostat”) we follow [19]. We must extremize Sqas
given by Eq. (8) with the constraints
W
?
i=1
pi= 1, (22)
and
W
?
i=1
EiP(q)
i
= Uq, (23)
where the escort distribution {P(q)
i
} is defined through
[P(q)
i
]1/q
?W
P(q)
i
≡
pq
j=1pq
i
?W
j
, pi=
j=1[P(q)
j
]1/q,(24)
and Uq is a finite quantity characterizing the width of
the energy distribution {pi}; {Ei} are the eigenvalues of
the system Hamiltonian (with the chosen boundary condi-
tions). Notice, by the way, that constraint (22) can equiv-
alently be written as?W
i=1P(q)
i
= 1.

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Constantino Tsallis: Nonadditive entropy: the concept and its use5
Through the entropic optimization procedure, we ob-
tain straighforwardly
pi=e−βq(Ei−Uq)
q
¯Zq
,(25)
with
βq≡
β
?W
j=1pq
j
, (26)
and
¯Zq≡
W
?
i
e−βq(Ei−Uq)
q
,(27)
with the q-exponential function (inverse of the previously
defined q-logarithm) ez
(1 − q)z > 0, and zero otherwise (ez
Lagrange parameter associated with the constraint (23).
Eq. (25) makes explicit that the probability distribution
is, for fixed βq, invariant with regard to the arbitrary
choice of the zero of energies. The stationary state (or
(meta)equilibrium) distribution (25) can be rewritten as
follows:
pi=e
q≡ [1 + (1 − q)z]1/(1−q)if 1 +
1= ez), β being the
−β′
q
Z′q
qEi
,(28)
with
Z′
q≡
W
?
j=1
e
−β′
q
qEj
,(29)
and
β′
q≡
βq
1 + (1 − q)βqUq
. (30)
The form (28) is particularly convenient for many appli-
cations where comparison with experimental or computa-
tional data is involved. Also, it makes clear that piasymp-
totically decays like 1/E1/(q−1)
i
for q < 1, instead of the familiar exponential decay with
Eifor q = 1.
The connection to thermodynamics is established in
what follows. It can be proved that
for q > 1, and has a cutoff
1
T=∂Sq
∂Uq
, (31)
with T ≡ 1/(kβ). Also we can prove, for the free energy,
Fq≡ Uq− TSq= −1
βlnqZq,(32)
where
lnqZq= lnq¯Zq− βUq.(33)
This relation takes into account the trivial fact that, in
contrast with what is usually done in BG statistics, the
energies {Ei} are here referred to Uq in Eq. (25). It can
also be proved
Uq= −∂
∂βlnqZq,(34)
as well as relations such as
Cq≡ T∂Sq
∂T
=∂Uq
∂T
= −T∂2Fq
∂T2. (35)
In fact the entire Legendre transformation structure of
thermodynamics is q-invariant, which is both remarkable
and welcome.
As a final remark, let us stress an interesting feature
concerning q > 1 (power-law decay for pi). Let us assume
that the energy spectrum has a (quasi-continuous) den-
sity state g(E). The normalization condition (22) implies
that?dE g(E)p(E) is finite. Since, for q > 1, p(E) decays
as 1/E1/(q−1), it follows that g(E)/E1/(q−1)must be inte-
grable at infinity. This determines the maximal value of q
which is mathematically admissible in the present theory
(for instance, if g(E) is constant, then q < 2 must be satis-
fied). Let us now focus on the other constraint, namely Eq.
(23). We immediately see that Eg(E)[p(E)]qmust also be
integrable at infinity, i.e., Eg(E)/Eq/(q−1)must be inte-
grable, which implies the same limit for q as before! So,
for instance, if g(E) is a constant, both the normalization
and the energy constraints are finite for q < 2. For q ≥ 2
the entire theory becomes mathematically inadmissible.
Let us address now the continuous case which gener-
alizes the Gaussian distribution. We want to extremize
Sq= k1 −?dx[p(x)]q
q − 1
(36)
with the constraints
?
dxp(x = 1, (37)
and
?
dxx2P(q)(x) = σ2,(38)
where
P(q)(x) ≡
[p(x)]q
?dx[p(x)]q,(39)
σ2being a fixed (positive) quantity. We obtain
p(x) =
e−βx2
q
?dxe−βx2
q
,(40)
where β > 0 can be determined by using constraint (38).
The entire theory is valid for q < 3, above which both the
normalization and the q-variance (38) diverge. For q ≥ 1
the distribution is defined for all values of x; for q < 1 it
has a finite support. For q < 5/3 the standard variance
?dxx2p(x) is finite; for q ≥ 5/3 it diverges. See [71] for
a numerical comparison between variance and q-variance,
which exhibits the considerable convenience of the latter.
2.3 q-generalized central limit theorems
We focus here on the q-generalization of the Central Limit
Theorem (CLT). Let us remind what the standard CLT