dynamical scenario for nonextensive statistical mechanics

Physica A 340 (2004) 1–10www.elsevier.com/locate/physa

Dynamical scenario for nonextensivestatistical mechanicsConstantino Tsallisa;b;∗

aCentro Brasileiro de Pesquisas Fisicas, Rua Xavier Sigaud 150, Rio de Janeiro RJ, 22290-180, BrazilbSanta Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA

Abstract

Statistical mechanics can only be ultimately justi&ed in terms of microscopic dynamics (clas-sical, quantum, relativistic, or any other). It is known that Boltzmann–Gibbs statistics is basedon the hypothesis of exponential sensitivity to the initial conditions, mixing and ergodicityin Gibbs �-space. What are the corresponding hypothesis for nonextensive statistical mechan-ics? A scenario for answering such question is advanced, which naturally includes the a prioridetermination of the entropic index q, as well as its cause and manifestations, for say many-bodyHamiltonian systems, in (i) sensitivity to the initial conditions in Gibbs �-space, (ii) relaxationof macroscopic quantities towards their values in anomalous stationary states that di4er from theusual thermal equilibrium (e.g., in some classes of metastable or quasi-stationary states), and(iii) energy distribution in the �-space for the same anomalous stationary states.c© 2004 Elsevier B.V. All rights reserved.

PACS: 05.70.Ln; 05:45: − a; 05:70: − a; 05.90.+m

Keywords: Nonlinear dynamics; Nonextensive statistical mechanics; Metastable states; Mixing; Weak chaos

1. Introduction

There is no reasonable doubt that it was, at least intuitively, neat and clear forLudwig Boltzmann and for Albert Einstein [1,2] that statistical mechanics descends,in one way or another, from microscopic dynamics. It should be so for all essentialconcepts of the theory, including the celebrated expression for the entropy, namely

∗ Centro Brasileiro de Pesquisas Fisicas, Xavier Sigaud 150, Rio de Janeiro RJ 2290-180, Brazil.Tel.: +55-21-2141-7190; fax: +55-21-2141-7515.

E-mail address: [email protected] (C. Tsallis).

0378-4371/$ - see front matter c© 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.physa.2004.03.072

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2 C. Tsallis / Physica A 340 (2004) 1–10

(written here for a discrete case)

SBG =−kW∑i=1

pi lnpi (1)

from now on referred to as the Boltzmann–Gibbs (BG) entropy. However, and in-terestingly enough, it is until now unknown how to start from say Newton’s lawF =ma and end up with this remarkable functional form and its associated variationalprinciple. It is Boltzmann’s magni&cent intuition which made him to propose this spe-ci&c microscopic expression for Clausius’ thermodynamic entropy. Expression which istotally consistent with Newton’s law, although rigorously speaking we still do not knowwhy.If such is the situation for SBG, it is clear that the situation has to be even more

intriguing—to say the least!—for any generalization of SBG, such as [3]

Sq = k1− ∑W

i=1 pqi

q − 1(q∈R; S1 = SBG) : (2)

Even if we do not yet know how SBG descends from dynamics and what restrictivepremises the microscopic dynamics must satisfy, it has been profusely veri&ed that SBGis the natural entropic form everytime we have nonintegrable and suNciently chaoticdynamics (i.e., positive Lyapunov exponents), known to yield quick mixing and even-tually ergodicity in phase space. A natural question arises: what happens if the systemis nonintegrable but all Lyapunov exponents vanish? More precisely, what happenswhen the sensitivity to the initial conditions diverges less than exponentially? It seemsplausible to imagine that the answer depends on how the sensitivity diverges asymp-totically, a power-law?, a slow logarithmic divergence?, some other type? The casethat we focus on in the present paper, and generically in nonextensive statistical me-chanics, is the ubiquitous power-law. It is for such dynamics that one imagines that Sqwould be the appropriate physical entropy to be used for various purposes, includingthe bridging to thermodynamics. Furthermore, one imagines that the appropriate valueof the entropic index q should reOect the exponent of the power-law, thus constitutingthe basis for universality classes of nonextensivity.

2. A possible triangle for the entropic index q

Although most of the notions that we shall use here occur in all types of dynamics,including quantum ones, we shall focus on classical systems for simplicity.

2.1. Sensitivity to the initial conditions

Let us start with the notion of sensitivity to the initial conditions. At the presentpoint it is enough, as an illustration, to consider a one-dimensional nonlinear dynamicalsystem (e.g., an unimodal map like the logistic one) characterized by x(t). We de&nethe sensitivity (to the initial conditions) � ≡ limPx(0)→0 Px(t)=Px(0), where Px(t) ≡

C. Tsallis / Physica A 340 (2004) 1–10 3

x(t) − x′(t) denotes the di4erence between two trajectories x(t) and x′(t) that areinitially close to each other. The following di4erential equation is typically satis&ed:

d�dt= �1� ; (3)

where �1 is the Lyapunov exponent (or essentially the largest one in more generalsituations). The solution of course is

�(t) = e�1t ; (4)

hence ln � ˙ t. This solution generically holds everytime �1 �= 0. But if we have�1 = 0 (referred to as weak chaos if there is mixing, and integrability otherwise), weexpect the adequate di4erential equation to typically be the following generalization ofEq. (3):

d�dt= �qsen�

qsen ; (5)

where sen stands for sensitivity. Its solution is given by

�(t) = e�qsen tqsen ; (6)

hence lnqsen �˙ t , with [4]

exq ≡ [1 + (1− q)x]1=(1−q) (ex1 = ex) ; (7)

and

lnq x ≡ x1−q − 11− q

(ln1 x = ln x) : (8)

(We remind that exq is always real and nonnegative; for q¡ 1, it vanishes for all valuesof x such that 1 + (1− q)x6 0).If �1 = 0 and, in spite of that, there is mixing in the full dynamical space of the

system, we typically expect qsen ¡ 1 and �qsen ¿ 0; if there is no mixing, we typicallyexpect qsen ¿ 1 and �qsen ¡ 0. It is of course possible in even more pathological cases(and therefore more rare in natural or arti&cial systems) that both �1 and �qsen vanish,but such cases are out of the scope of the present discussion, centered on power-laws(and not on even weaker behaviors such as the logarithmic ones).It is by now well known that an illustration of the situation we are focusing on

here is provided by the edge of chaos of the logistic map: qsen=0:2445 : : : and �qsen =1=(1− qsen) = 1:3236 : : : [5,6]. Many other paradigmatic dynamical systems are knownor expected to also enter, in one way or another, into the same category. This isthe case of globally coupled standard maps close to integrability ([7–9] and referencestherein) or long-range-interacting many-body Hamiltonian systems ([10] and referencestherein).Let us &nally mention that, at least for simple one-dimensional maps, qsen also

enters prominently in basic properties such as the entropy production per unit time[11] (essentially the Kolmogorov–Sinai entropy rate), and the values at which themultifractal function (associated with the edge of chaos) may vanish [12].


2.2. Relaxation

We de&ne the quantity

�(t) ≡ O(t)−O(∞)O(0)−O(∞)

; (9)

where O is a macroscopic observable relaxing towards its value at a stationary state(thermal equilibrium, or some other). The associated di4erential equation very fre-quently is

d�dt

=− 1�1� (10)

�1¿ 0 being the relaxation time. The solution of this equation is

�(t) = e−t=�1 : (11)

The quantity �1 is in principle expected to depend on the observable O. However, inall generic cases one expects (according to Krylov’s classical remarks) 1=�1 ∼ �1 if�1¿ 0. This is quite natural since the relaxation of a macroscopic quantity reOectsnothing but the exploration that the system makes of its full dynamical space in itssearch of a possible stationary state (given some class of initial conditions), and thisoccurs at a rhythm dictated by the Lyapunov exponent �1 (or, in general, by the fullLyapunov spectrum). In particular, if �1 (or the maximal Lyapunov exponent) vanishes,we expect �1 to diverge, thus making Eq. (11) inappropriate. It is quite plausible that,in such situation, Eq. (10) becomes generalized into

d�dt

=− 1�qrel

�qrel ; (12)

where rel stands for relaxation. The solution is of course given by

�(t) = e−t=�qrelqrel (�qrel ¿ 0) (13)

being typically qrel ¿ 1. It could well happen that qrel ¿ 1 depends on the speci&cobservable O we might be interested in, but even if such is the case, we expectall possible qrel’s to be simply inter-related. Moreover, we expect them to be essen-tially characterized by the speed at which the nonlinear dynamics makes the system toapproach its stable (or metastable) stationary (or quasistationary) state, given the classof initial conditions within which the system has been started. More speci&cally, when�1 = 0, we expect simple hypothesis such as ergodicity in full phase space (basic forthe validity of BG statistical mechanics) to be not ful&lled. Even more, we expect(under conditions to be determined) the system to have a tendency of lengthily livingin some kind of attractor or pseudo-attractor, whose Lebesgue measure in full dynam-ical space would typically be zero, as it is the case of (multi)fractals or scale-freenetworks. If we assume an ensemble whose Lebesgue measure is di4erent from zeroat t = 0 inside a special region of full space, we might observe a shrinking of theLebesgue measure as time goes on. The loss of Lebesgue measure would possibly oc-cur through a q-exponential function whose value of q would be noted qrel. From nowon, we reserve the notation qrel for the shrinking of Lebesgue measure, if such is the


case. All other qrel’s would be simply related to this one. Let us emphasize that thepresent scenario does not exclude that, at even larger times, the system escapes fromthis (multi)fractal “prison” and ultimately satis&es ergodicity, thus making a crossoverfrom qrel to q=1. This would in fact be the case unless some limiting situations (seeRef. [8]) are taken which strictly guarantee that �1 vanishes. When such escape ispossible, aging phenomena might be present [13,14], aging being presumably relatedto a slow collective dynamics which would prepare the escape from the “prison”.A typical illustration of the picture we have described in the present subsection

is, as before, the edge of chaos of the logistic map, where it is qrel � 2:4 [15].Other paradigmatic illustrations should include the systems mentioned in the previoussubsection, but this remains to be veri&ed.Let us &nally mention that, at least for simple one-dimensional maps, qrel also

emerges [15] in basic properties such as the fractal dimension (maximum of the mul-tifractal function) of the attractor at the edge of chaos.

2.3. Stationary state

Let us now address what may in some sense be considered as the most useful thermo-statistical function, namely the distribution of total energies in Gibbs �-space of a ther-modynamical system in contact with a canonical thermostat &xing thetemperature T .Let us &rst consider a Hamiltonian system whose elements interact only locally,

or almost not at all (e.g., the ideal gas). A typical example of local interaction isa classical two-body (attractive) potential which has no singularity at the origin, andwhich asymptotically decays with distance r as −1=r� with �=d¿ 1, where d is thespace dimension of the system; for instance, for the usual Lennard–Jones Ouid, wehave �=6 and d=3. We know that the relevant stationary state is thermal equilibriumand that

pi =e−!1Ei

Z1(!1 ≡ 1=kT ;Z1 ≡

∑j

e−!1Ej) ; (14)

where {Ei} is the set of eigenvalues of the Hamiltonian with speci&c boundary con-ditions. This celebrated distribution can be obtained through the textbook procedurewhich consists in optimizing SBG under the constraints

∑i pi = 1 and

∑i piEi = U1,

where U1 is the total internal energy. It seems to be realized by only a few physiciststhat this distribution can also be obtained, albeit without formal justi&cation up to now,through a di4erential equation, namely

d(piZ1)dEi

=−!1(piZ1) : (15)

The black-body radiation law was &rst found, and published in October 1900 [16],by Max Planck through an heuristic di4erential equation, precisely one which, withappropriate variables, contains Eq. (15) as a particular case! This is known only bythose interested in the history of science. And it has not been incorporated in stan-dard textbooks presumably because of the lack of formal justi&cation. Nevertheless the


mathematical fact is there, and in my opinion it is perhaps not a mere coincidence.Indeed, since the sensitivity to the initial conditions follows an exponential law, andthe same typically does its prominent consequence namely the relaxation of macro-scopic thermostatistical quantities, it cannot be a very great surprise that the same doeshappen to the distribution associated with the stationary state towards which the systemrelaxes.For Hamiltonian systems whose elements do not interact locally but globally (e.g.,

the classical systems mentioned previously but with 06 �=d6 1, in which case thepotential is not integrable), it seems quite plausible that Eq. (15) becomes generalizedinto

d(piZqstat )dEi

=−!qstat (piZqstat )qstat ; (16)

where stat stands for stationary state, and !qstat is the adequate inverse temperature(in units of k). The solution is given by

pi =e−!qstat Eiqstat

Zqstat(!qstat ≡ 1=kTstat ;Zqstat ≡

∑j

e−!qstat Ejqstat ) ; (17)

and one typically expects qstat ¿ 1. It is in fact precisely this solution which emerges[3] by optimizing, through conveniently written norm and energy constraints, Sq insteadof SBG.There is molecular dynamical numerical evidence [17] for some classical long-range-

interacting many-body inertial rotators on d-dimensional lattices that they might sat-isfy, in some robust metastable states in the presence of a thermostat, the total en-ergy distribution indicated in Eq. (17). But, even at the numerical level, the problemis uneasy, though not impossible, to study. This is the subject of ongoing e4orts.The strategy which is followed is to consider an isolated system of N rotators at a&xed total energy. Then consider once for ever a subset of M rotators among theN available. Finally, one must numerically approach the t → ∞ limit of the M →∞ limit of the N → ∞ limit. The whole procedure must be repeated many times(Nrealizations → ∞, mathematically speaking) for realizations di4ering in the precisemulti-particle initial conditions, always within some speci&c and nontrivial class of ini-tial conditions, and then averaging the results. The distribution obtained in this manner(i.e., limt→∞ limM→∞ limN→∞ limNrealizations→∞) is to be compared with Eq. (17).Let us make one more remark, which concerns the distributions of momenta of these

rotators. If we denote with H({Li}; {+i}) the Hamiltonian associated with say planarrotators, the distribution in �-space is given by

p({Li}; {+i})˙ e−!qstatH({Li};{+i})qstat : (18)

Then the one-body angular momentum marginal distribution p(L) is given by

p(L) =∫(UN

i=2 dLi)(UNi=1 d+i)p({Li}; {+i}) ;

˙∫(UN

i=2 dLi)(UNi=1 d+i)e

−!qstatH({Li};{+i})qstat : (19)


-2 -1 0 1 20

0.5

1

1.5

2

2.5

3

t = 0

-2 -1 0 1 20

0.5

1

1.5

2

2.5

3

t = 101

-2 -1 0 1 20

0.5

1

1.5

2

2.5

3

t = 102

-2 -1 0 1 20

0.5

1

1.5

2

2.5

3

t = 103

-2 -1 0 1 20

0.2

0.4

0.6

0.8

t = 104

-2 -1 0 1 20

0.2

0.4

0.6

0.8

t = 105

-2 -1 0 1 20

0.2

0.4

0.6

0.8

t = 106

-2 -1 0 1 20

0.2

0.4

0.6

0.8

t = 107

PD

F

p

Fig. 1. Snapshots of the evolution of PDF for momenta in the (L; +) − space (i.e., in --space), startingwith a “hot water bag”. In solid black circles, the instantaneous PDF at time t = 10k (with k = 0; 1; : : : 7),for M = 500 spins inside an N = 5000 spin system. In empty circles, the average of 103 realizations forN = 105 spins (all averages made during the QSS plateau). In dashed line, the qL-exponential &tting curvewith T = TQSS = 0:38 and qL = 3:7. Finally, in solid line, the analytical Gaussian PDF. These last threecurves are the same in every frame, and are plotted for reference. From Ref. [8].


We do not yet know what is the result for this distribution but it could also be aqL-exponential in L2 with qL¿ 1 related to qstat is some still unknown manner. Oneexpects (essentially because the integral of a q-exponential is another q-exponential) theindex qL to either coincide with qstat or be a simple function of it (see Ref. [18]). In anycase, the available numerical evidence is not inconsistent with such possibilities ([8]and references therein). See, for instance, Fig. 1 (from Ref. [8]), where for the &rst timethe momentum distribution is calculated (through molecular dynamics) for a systemwhich is sensibly closer to a canonical ensemble (contact with a large thermostat) thanthe microcanonical ensemble considered up to now in the literature for such systems.

3. Final remarks

The q-indices discussed in Section 2 and their possible interconnections areschematically represented in Fig. 2. It is our scenario that they are di4erent “faces” ofthe same basic phenomenon, namely how the system “likes” to evolve and mix in partor all of its full phase space (Gibbs �-space for Hamiltonian systems) given its initialconditions. If the system is strongly chaotic (i.e., positive Lyapunov exponents), es-sentially satisfying Boltzmann’s “molecular chaos hypothesis”, then the system will bemixing and ergodic allover the entire phase space. BG statistical mechanics is expectedto appropriately describe the thermal statistics of the system, the associated microscopicexpression for the entropy undoubtedly being SBG. But if the system is only weakly

SENSITIVITY

RELAXATION STATIONARY STATE(qrel)

(qsen)

(qstat)

Fig. 2. The triangle of the basic values of q, namely those associated with sensitivity to the initial conditions,relaxation and stationary state. For the most relevant situations we expect qsen6 1, qrel¿ 1 and qstat¿ 1.These indices are presumably inter-related since they all descend from the particular dynamical explorationthat the system does of its full phase space. For example, for long-range Hamiltonian systems characterizedby the decay exponent � and the dimension d, it could be that qstat decreases from a value above unity(e.g., 2 or 3

2 ) to unity when �=d increases from zero to unity. For such systems one expects relations likethe (particularly simple) qstat = qrel = 2 − qsen or similar ones. In any case, it is clear that, for �=d¿ 1(i.e., when BG statistics is known to be the correct one), one has qstat = qrel = qsen = 1. All the weaklychaotic systems focused on here are expected to have well de&ned values for qsen and qrel, but only thoseassociated with a Hamiltonian are expected to also have a well de&ned value for qstat .


chaotic (i.e., zero Lyapunov exponents, but not integrable), then the occupancy ofphase space might be much more complex everytime the system is started within spe-ci&c nonzero-Lebesgue-measure regions of phase space. The region of phase spacewhere the system may lengthily evolve before eventually occupying the entire spacemight have a (multi)fractal or hierarchical structure of the type currently referred toas scale-free network (see [19] and references therein). As long as the system remainsin that region, its thermostatistics might be adequately be described within nonexten-sive statistical mechanics, the associated microscopic expression for the thermodynamicentropy presumably being Sq with a speci&c value of q. During the metastable orquasi-stationary state we would then have (qsen; qrel; qstat) �= (1; 1; 1), and at later times(in many cases, in&nitely later!) a crossover would occur to qstat = qrel= qsen=1. ThisconOuence onto a single value of q for BG statistics might be at the origin of oc-casional confusion within the interested community. Indeed, three essentially di4erent(though related) concepts just happen to coincide within the BG formalism.

Acknowledgements

It is a deep pleasure to thank the creators and organizers of this wonderful meet-ing in Cagliari (Villasimius) where we all received exceptionally warm hospitality.Although they were many and very valuable, it is for me but a matter of elementaryjustice to gratefully name those who contributed the most: P. Quarati, G. Kaniadakis,A. Rapisarda, M. Lissia and E.G.D. Cohen.Partial support by FAPERJ, CNPq and PRONEX (Brazilian agencies) is gratefully

acknowledged.

References

[1] A. Einstein, Annalen der Physik 33 (1910) 1275 [“Usually W is put equal to the number ofcomplexions: : : In order to calculate W , one needs a complete (molecular-mechanical) theory of thesystem under consideration. Therefore it is dubious whether the Boltzmann principle has any meaningwithout a complete molecular-mechanical theory or some other theory which describes the elementaryprocesses. S = (R=N )logW+ const. seems without content, from a phenomenological point of view,without giving in addition such an Elementartheorie.”] (Translation: Abraham Pais, Subtle is the Lord: : : Oxford University Press, 1982)].

[2] E.G.D. Cohen, Physica A 305 (2002) 19;M. Baranger, Physica A 305 (2002) 27.

[3] C. Tsallis, J. Stat. Phys. 52 (1988) 479;E.M.F. Curado, C. Tsallis, J. Phys. A 24 (1991) L69 [Corrigenda: 24 (1991) 3187 (1991) and 25(1992) 1019];C. Tsallis, R.S. Mendes, A.R. Plastino, Physica A 261 (1998) 534 ; For a recent review seeM. Gell-Mann, C. Tsallis (Eds.), Nonextensive Entropy—Interdisciplinary Applications; OxfordUniversity Press, New York, 2004; For full bibliography see http://tsallis.cat.cbpf.br/biblio.htm..

[4] C. Tsallis, Quimica Nova 17 (1994) 468;see also C. Tsallis, in: S. Abe, Y. Okamoto (Eds.), Nonextensive Statistical Mechanics and itsApplications, Series Lecture Notes in Physics, Springer, Heidelberg, 2001, 3p. (Appendix).

[5] C. Tsallis, A.R. Plastino, W.-M. Zheng, Chaos, Solitons and Fractals 8 (1997) 885.

http://tsallis.cat.cbpf.br/biblio.htm


[6] F. Baldovin, A. Robledo, Phys. Rev. E 66 (2002) 045104(R);F. Baldovin, A. Robledo, cond-mat/0304410 Phys. Rev. E (2004), in press.

[7] F. Baldovin, E. Brigatti, C. Tsallis, Phys. Lett. A 320 (2004) 254.[8] F. Baldovin, L.G. Moyano, A.P. Majtey, A. Robledo, C. Tsallis, Physica A (2004), these proceedings,

doi:10.1016/j.physa.2004.04.009.[9] G.F.J. Ananos, F. Baldovin, C. Tsallis, Anomalous sensitivity to initial conditions and entropy production

in standard maps: Nonextensive approach, cond-mat/0403656.[10] C. Anteneodo, C. Tsallis, Phys. Rev. Lett. 80 (1998) 5313;

A. Campa, A. Giansanti, D. Moroni, C. Tsallis, Phys. Lett. A 286 (2001) 251;B.J.C. Cabral, C. Tsallis, Phys. Rev. E 66 (2002) 065101(R);F.D. Nobre, C. Tsallis, Phys. Rev. E 68 (2003) 036115;E.P. Borges, C. Tsallis, A. Giansanti, D. Moroni, Dinamica de um sistema nao extensivo de rotoresclassicos anisotropicos acoplados, in: T. Tome (Ed.), Tendencias da Fisica Estatistica no Brasil, volumehonoring S.R.A. Salinas, Editora Livraria da Fisica, Sao Paulo, 2003, p. 84.

[11] V. Latora, M. Baranger, A. Rapisarda, C. Tsallis, Phys. Lett. A 273 (2000) 97.[12] M.L. Lyra, C. Tsallis, Phys. Rev. Lett. 80 (1998) 53.[13] M.A. Montemurro, F. Tamarit, C. Anteneodo, Phys. Rev. E 67 (2003) 031106.[14] A. Robledo, cond-mat/0307285.[15] F.A.B.F. de Moura, U. Tirnakli, M.L. Lyra, Phys. Rev. E 62 (2000) 6361;

E.P. Borges, C. Tsallis, G.F.J. Ananos, P.M.C. Oliveira, Phys. Rev. Lett. 89 (2002) 254103.[16] M. Planck, Verhandlungen der Deutschen Physikalischen Gessellschaft 2 (1900) 202 and 237, [English

translation: D. ter Haar, S.G. Brush, Planck’s Original Papers in Quantum Physics, Taylor and Francis,London, 1972].

[17] V. Latora, A. Rapisarda, C. Tsallis, Phys. Rev. E 64 (2001) 056134;C. Tsallis, A. Rapisarda, V. Latora, F. Baldovin, in: T. Dauxois, S. Ru4o, E. Arimondo, M. Wilkens(Eds.), Dynamics and Thermodynamics of Systems with Long-Range Interactions, Lecture Notes inPhysics, Vol. 602, Springer, Berlin, 2002;A. Pluchino, V. Latora, A. Rapisarda, Spin-glass phase in the Hamiltonian Mean Field model, PhysicaA (2004), in press, cond-mat/0306374;C. Tsallis, What should a statistical mechanics satisfy to reOect nature?, in: H.L. Swinney, C. Tsallis(Eds.), Anomalous Distributions, Nonlinear Dynamics and Nonextensivity, Physica D (2004), in press[cond-mat/0403012];L.G. Moyano, F. Baldovin, C. Tsallis, cond-mat/0305091.

[18] R.S. Mendes, C. Tsallis, Phys. Lett. A 285 (2001) 273.[19] R. Albert, A.L. Barabasi, Phys. Rev. Lett. 85 (2000) 5234.

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dynamical scenario for nonextensive statistical mechanics

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