Physica A 344 (2004) 631–636www.elsevier.com/locate/physa

Critical �uctuations, intermittent dynamics andTsallis statisticsAlberto Robledo

Instituto de F�isica, Universidad Nacional Aut�onoma de M�exico, Apartado Postal 20-364,M�exico 01000 D.F., Mexico

Available online 4 July 2004

Abstract

It is pointed out that the dynamics of the order parameter at a thermal critical point obeys theprecepts of the nonextensive Tsallis statistics. We arrive at this conclusion by putting togethertwo well-de-ned statistical–mechanical developments. The -rst is that critical �uctuations arecorrectly described by the dynamics of an intermittent nonlinear map. The second is that inter-mittency in the neighborhood of a tangent bifurcation in such map rigorously obeys nonextensivestatistics. We comment on the implications of this result.c© 2004 Elsevier B.V. All rights reserved.

PACS: 64.60.Ht; 75.10.Hk; 05.45.−a; 05.10.Cc

Keywords: Critical �uctuations; Intermittency; Nonextensive statistics; Anomalous stationary states

1. Introduction

The modern theory of critical phenomena [1–4] has a distinguished history [4] ofachievements in understanding the source of the scaling and universality associatedto continuous phase transitions. The introduction of renormalization group (RG) con-cepts provided the means for calculating exponents, scaling functions and marginaldimensionalities of critical points in thermal and other statistical–mechanical models.Moreover, the RG methods have been fruitful in other areas of physics, in condensedmatter problems, in non-linear dynamics and in other -elds [4]. The success of theRG strategy in handling problems involving many length scales is illustrated by theuse of a coarse-grained free energy or e<ective action. For instance, the equilibrium

E-mail address: robledo@-sica.unam.mx (A. Robledo).

0378-4371/$ - see front matter c© 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.physa.2004.06.043

632 A. Robledo / Physica A 344 (2004) 631–636

con-gurations of Ising spins at the critical temperature shows magnetic domains on allsize scales and these are suitably studied by means of the Landau–Ginzburg–Wilson(LGW) continuous spin model [1–4].Here we may add yet other universal aspect to critical point properties, the non-

extensivity of order-parameter �uctuations (as explained in more detail below and inRef. [5]). This previously unidenti-ed property develops as the size of subsystemsor domains of a thermal system at its critical point is allowed to become in-nitelylarge. These domains have been shown [6–9], via the use of the LGW free energy, topossess intermittency properties, and these, in turn, are seen to comply [10,11] withthe presumptions of the nonextensive generalization [12,13] of the Boltzmann–Gibbs(BG) statistical mechanics.In a series of papers [6–9] a bridge has been built connecting the equilibrium dynam-

ics of �uctuations at an ordinary thermal critical point with the intermittent dynamicsof critical nonlinear maps at a tangent bifurcation. With the initial aim of investigatingthe origin of the relationship between the fractal geometry of clusters of the orderparameter and the critical exponents of the phase transition, a theoretical approach wasdevised that lead to the evaluation of the partition function of a cluster or domainwhere the dominant contributions arise from a singularity (similar to an instanton) lo-cated in the space outside it [6,7]. Subsequently [8,9], a nonlinear map for the averageorder parameter was constructed whose dynamics reproduce the averages of the thermalcritical properties. This map has as a main feature the tangent bifurcation and so timeevolution is intermittent.Recently [10,11], we have shown that the known exact static solution of the RG

equations for the tangent bifurcation in nonlinear maps also describes the dynamicsof iterates. The -xed-point expressions have the speci-c form that corresponds to thetemporal evolution of ensembles of iterates prescribed by the nonextensive formalism.The proof rests on the derivation of the sensitivity to initial conditions �t exclusivelyfrom RG procedures without approximations followed by comparison with the nonex-tensive �t . The study of the intermittency transition has been expanded via detailedderivation of their q-generalized Lyapunov coeGcients �q and interpretation of thedi<erent types of sensitivity �t [11]. Likewise, the properties of the intricate trajecto-ries at the edge of chaos in unimodal maps have been analytically obtained leadingtoo to the determination of �q and interpretation of the dynamics at the strangeattractor [14,15].We combine the results mentioned in the previous two paragraphs to point out

the nonextensive nature of critical �uctuations. This conclusion relates to the questof the physical circumstances for which BG statistics fails to be applicable and itsnonextensive generalization might o<er correct descriptions. These anomalous situationsare signalled by the vanishing of the Lyapunov coeGcients (a single coeGcient �1 = 0for one-dimensional maps) and exhibit nonergodicity or unusual phase space mixing[10–14,16]. At an intermittency transition hindered or incomplete mixing in phase spacearises from the special ‘tangency’ shape of the map at its origin. This has the e<ect ofcon-ning or expelling trajectories causing irregular phase-space sampling, in contrastto the thorough coverage in generic states with �1 ¿ 0. The occurrence of anomalousnonextensive states appears to be related with a nonuniform convergence of limits, such

A. Robledo / Physica A 344 (2004) 631–636 633

as the thermodynamic and in-nite time limits. Here we comment on the nonuniformconvergence associated to the description of critical domains as obtained from the LGWfree energy.Below we expand our arguments.

2. Criticality and intermittency

We brie�y recall basic elements and results of the approach in Refs. [6–9]. Thestarting point is the partition function of the d-dimensional system at criticality, Z =∫

D[] exp(−c[]); where c[]=g1∫

� dV[12 (∇)2 + g2||�+1

]is the critical LGW

free energy of a system of d-dimensional volume �, is the order parameter (e.g.magnetization per unit volume) and � is the critical isotherm exponent. The partitionfunction Z was evaluated for a subsystem of size V ∼ Rd and by approximatingits path integral as a summation over the saddle-point con-gurations of c[]—anapproximation valid for g1�1. Integration of the Euler–Lagrange equation associatedto the saddle points of c[], and identi-cation of dominant saddle points, leads topower-law magnetization pro-les for (spherically symmetric) critical clusters (r), and-nally, to the evaluation of the free energy c[] and the partition function Z inclosed form [6,7]. In doing this it was important to notice that only con-gurationswith r0�R, where r0 is a system-dependent reference position r0=r0(g2; �; (0)), havea nonvanishing contribution to the path integration [6,7]. These con-gurations vanishfor the in-nite size system, so there is nonuniform convergence in relation to the limitsR → ∞ and r0 → ∞, a feature that is signi-cant for our connection with q-statistics.Based on the above results the fractal geometry of the critical clusters was determinedand its relationship with the exponent � was derived. The power-law dependence ofthe magnetization on the cluster radius was identi-ed with the fractal dimension of thegeometry of the cluster [6,7]. Multifractal properties appear when global, rather than asingle cluster properties are considered.Subsequent to this development, a link was revealed [8,9] between the �uctuation

properties of a critical system described as above and the dynamics of marginallychaotic intermittent maps. By considering the space-averaged magnetization �=

∫V (x)

dV , the statistical weight �(�) = exp(−c[�])=Z , where c[�] ∼ g1g2��+1 and Z =∫d� exp(−c[�]), was seen to be the invariant density of a statistically equivalent

one-dimensional map. The functional form of this map was obtained as the solutionof an inverse Frobenius–Perron problem [8,17]. For small values of � the map hasthe form �n+1 = �n + u��+1

n + �, where the amplitude u depends on g1, g2 and �,and the shift parameter � ∼ R−d. This map can be recognized as that describing theintermittency route to chaos in the vicinity of a tangent bifurcation [18]. The completeform of the map displays a superexponentially decreasing region that takes back theiterate close to the origin in one step. Thus the parameters of the thermal systemdetermine the dynamics of the map. Averages of order-parameter critical con-gurationsare equivalent to iteration time averages along the trajectories of the map close to thetangent bifurcation. The mean number of iterations in the laminar region was seen tobe related to the mean magnetization within a critical cluster of radius R. There is a

634 A. Robledo / Physica A 344 (2004) 631–636

corresponding power law dependence of the duration of the laminar region on the shiftparameter � of the map [8]. For � ¿ 0 the (small) Lyapunov exponent is simply relatedto the critical exponent � [7].

3. Intermittency and nonextensivity

Next, in a few words, we recall the nonextensive nature of the nonlinear dynamicsat the tangent bifurcation [10,11]. At this transition, the intermittency route to chaos,the ordinary Lyapunov exponent �1 vanishes and the sensitivity to initial conditions�t ≡ |dxt=dx0| (where xt is the orbit position at time t given the initial position x0 attime t =0) is no longer given by the BG law �t =exp(�1t) but acquires either a poweror a super-exponential law [11]. The nonextensive formalism indicates that �t is givenby the q-exponential expression �t = expq(�qt) ≡ [1 − (q − 1)�qt]−1=(q−1), containingthe entropic index q and the q-generalized Lyapunov coeGcient �q. Also, according tothe generalized theory, the q-Pesin identity Kq =�q replaces the ordinary Pesin identityK1 = �1, �1 ¿ 0, where Kq ≡ t−1[Sq(0) − Sq(t)] and K1 ≡ t−1[S1(t) − S1(0)] are therates of entropy increment based on the Tsallis entropy Sq ≡ ∑

i pi lnq(1=pi) = (q −1)−1[1−∑W

i pqi ], (where lnq y ≡ (y1−q −1)=(1−q) is the inverse of expq(y)) and the

BG entropy S1(t)=−∑Wi=1 pi(t) lnpi(t), respectively. (Above, pi(t) is the distribution

obtained from the relative frequencies with which the positions of an ensemble oftrajectories occur within cells i = 1; : : : ; W at iteration time t.) In the limit q → 1 theexpressions for the nonextensive theory reduce to the ordinary BG expressions. SeeRefs. [10,14] for a more rigorous description of the Pesin identity and related issues.Assisted by the known RG treatment for the tangent bifurcation [18], the formula

for �t has been rigorously derived [10,11] and found to comply with the q-exponentialform as given above. Also the validity of the q-Pesin identity has been substantiatedfor this problem [10]. The tangent bifurcation is usually studied by means of the mapf(x)= �+ x+u|x|z +O(|x|z), u ¿ 0, in the limit � → 0. The associated RG -xed-pointmap x′=f∗(x) was found to be x′=x expz(uxz−1)=x[1−(z−1)uxz−1]−1=(z−1), �=0; asit satis-es f∗(f∗(x))=$−1f∗($x) with $=21=(z−1) and has a power-series expansion inx that coincides with f(x) in the two lowest-order terms. (Above xz−1 ≡ |x|z−1 sgn(x)).The long time dynamics is readily derived from f∗(x), one obtains �t(x0) = [1− (z −1)axz−1

0 t]−z=(z−1); and so, q = 2 − z−1 and �q(x0) = zaxz−10 [10,11]. When q ¿ 1 the

left-hand side (x ¡ 0) of the tangent bifurcation map exhibits a weak insensitivity toinitial conditions, i.e., power-law convergence of orbits. However at the right-hand side(x ¿ 0) of the bifurcation the argument of the q-exponential becomes positive and thisresults in a ‘super-strong’ sensitivity to initial conditions, i.e., a sensitivity that is fasterthan exponential [11].

4. Nonextensivity and criticality

The implications of joining the results described in the previous two sections areapparent. In the subsystem of in-nite size R → ∞ the dynamics of critical �uctuations

A. Robledo / Physica A 344 (2004) 631–636 635

obey the nonextensive statistics. This is expressed via the time series of the averageorder parameter �n, i.e., trajectories �n with close initial values separate in a super-exponential fashion according to Ref. [11] with q = (2� + 1)=(� + 1)¿ 1 and with aq-Lyapunov coeGcient �q determined by the system parameter values �, g1, g2 and �0

[5]. Also, when considering an ensemble of trajectories {�n} with prescribed distribu-tion of initial conditions, the q-Pesin identity Kq = �q holds with the rate of entropyproduction Kq evaluated according to the nonextensive entropy Sq.It is interesting to comment on the conditions for the incidence of q-statistical prop-

erties at criticality and the manner in which these develop. The order parameter pro-lefor a large but -nite-size domain R�1 has the form [6–8] (r) = Ad(r2 − r20)

(2−d)=2,that, parenthetically, can be rewritten in terms of a q-exponential. There is a singularityin (r) when r = r0, the reference position. We keep in mind the requirement R�r0for the critical clusters (r) to be of relevance to the partition function Z and also themap shift parameter dependence on the domain size � ∼ R−d. The time evolution of �displays laminar episodes of duration 〈n〉 ∼ �−�=(�+1) and the Lyapunov coeGcient inthis regime is �1 ∼ � [7]. Within the -rst laminar episode the dynamical evolution of� obeys q-statistics, but for very large times the occurrence of many di<erent laminarepisodes leads to an increasingly chaotic orbit consistent with the small �1 ¿ 0 andBG statistics is recovered. As R increases (R�r0 always) the time duration of thenonextensive regime increases and in the limit R → ∞ there is only one in-nitely longlaminar nonextensive episode with �1 = 0 and with no crossover to BG statistics. Onthe other hand when R ¿ r0 the clusters (r) are no longer dominant, for the in-nitesubsystem R → ∞ their contribution to Z vanishes [6–8] and no departure from BGstatistics is expected to occur.This study is developed in Ref. [5].

5. Current questions on q-statistics

What is the signi-cance of the connections we have presented? Are there other con-nections between critical phenomena and transitions to chaos? Are all critical states—in-nite correlation length with vanishing Lyapunov coeGcients—outside BG statistics?When does BG statistics stop working? Where, and in that case why, does Tsallis statis-tics apply? Is ergodicity failure the basic playground for applicability of generalizedstatistics? Is there a link between critical dynamics and glassy dynamics?Suggestive results and partial answers to these questions are given here and in the

cited references. In relation to this it is pertinent to mention the following recentdevelopment. In Ref. [19] it is argued that the dynamics at the noise-perturbed edgeof chaos in logistic maps is analogous to that observed in supercooled liquids closeto vitri-cation. That is, the three major features of glassy dynamics in structural glassformers, two-step relaxation, aging, and a relationship between relaxation time andcon-gurational entropy, are displayed by orbits with �1 =0. Time evolution is obtainedfrom the Feigenbaum RG transformation, it is expressed analytically via q-exponentials,and described in terms of nonextensive statistics.

636 A. Robledo / Physica A 344 (2004) 631–636

Acknowledgements

It is a great pleasure to dedicate this work to Constantino Tsallis on the occasion ofhis 60th birthday. I am grateful for hospitality o<ered at Angra dos Reis, Brazil, andfor partial support by DGAPA-UNAM and CONACyT (Mexican Agencies).

References

[1] K.G. Wilson, Physica 73 (1974) 119.[2] M.E. Fisher, Rev. Mod. Phys. 46 (1974) 597.[3] K.G. Wilson, Rev. Mod. Phys. 55 (1983) 583.[4] M.E. Fisher, Rev. Mod. Phys. 70 (1998) 653.[5] A. Robledo, submitted for publication.[6] N.G. Antoniou, Y.F. Contoyiannis, F.K. Diakonos, C.G. Papadoupoulos, Phys. Rev. Lett. 81 (1998)

4289.[7] Y.F. Contoyiannis, F.K. Diakonos, Phys. Lett. A 268 (2000) 286.[8] N.G. Antoniou, Y.F. Contoyiannis, F.K. Diakonos, Phys. Rev. E 62 (2000) 3125.[9] Y.F. Contoyiannis, F.K. Diakonos, A. Malakis, Phys. Rev. Lett. 89 (2002) 035701.[10] A. Robledo, Physica A 314 (2002) 437;

A. Robledo, Physica D 193 (2004) 153.[11] F. Baldovin, A. Robledo, Europhys. Lett. 60 (2002) 518.[12] C. Tsallis, J. Stat. Phys. 52 (1988) 479.[13] For a recent review see, M. Gell-Mann, C. Tsallis (Eds.), Nonextensive Entropy—Interdisciplinary

Applications, Oxford University Press, New York, 2004, in press. See http://tsallis.cat.cbpf.br/biblio.htmfor full bibliography.

[14] F. Baldovin, A. Robledo, Phys. Rev. E 66 (2002) 045104(R);F. Baldovin, A. Robledo, Phys. Rev. E 69 (2004) 045202(R).

[15] E. Mayoral, A. Robledo, Physica A 340 (2004) 219.[16] F. Baldovin, L.G. Moyano, A.P. Majtey, A. Robledo, C. Tsallis, Physica A 340 (2004) 205.[17] F.K. Diakonos, D. Pingel, P. Schmelcher, Phys. Lett. A 264 (1999) 162.[18] See, for example, H.G. Schuster, Deterministic Chaos. An Introduction, 2nd Revised Edition, VCH

Publishers, Weinheim, 1988.[19] A. Robledo, Phys. Lett. A, in press, and cond-mat/0307285;

A. Robledo, Physica A, in press.