Complexity, Metastability And Nonextensivity (Science and Culture: Physics)

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COMPLEXITY, METASTABILITY AND NONEXTENSIVITY 31st Workshop of the international School of Solid State Physics

THE SCIENCE AND CULTURE SERIES

- PHYSICS

Series Editor: A. Zichichi, European Physical Society, Geneva, Switzerland Series Editorial Board: P. G. Bergmann, J. Collinge, V. Hughes, N. Kurti, T. D. Lee, K. M. B. Siegbahn, G. 't Hooft, P. Toubert, E. Velikhov, G. Veneziano, G. Zhou

1. Perspectives for New Detectors in Future Supercolliders, 1991 2. Data Structures for Particle Physics Experiments: Evolution or Revolution?, 1991 3. Image Processing for Future High-Energy Physics Detectors, 1992 4. GaAs Detectors and Electronics for High-Energy Physics, 1992 5. Supercolliders and Superdetectors, 1993 6. Properties of SUSY Particles, 1993 7. From Superstrings to Supergravity, 1994 8. Probing the Nuclear Paradigm with Heavy Ion Reactions, 1994 9. Quantum-Like Models and Coherent Effects, 1995 10. Quantum Gravity, 1996 11. Crystalline Beams and Related Issues, 1996 12. The Spin Structure of the Nucleon, 1997 13. Hadron Colliders at the Highest Energy and Luminosity, 1998 14. Universality Features in Multihadron Production and the Leading Effect, 1998 15. Exotic Nuclei, 1998 16. Spin in Gravity: Is It Possible to Give an Experimental Basis to Torsion?, 1998 17. New Detectors, 1999 18. Classical and Quantum Nonlocality, 2000 19. Silicides: Fundamentals and Applications, 2000 20. Superconducting Materials for High Energy Colliders, 2001 21. Deep Inelastic Scattering, 2001 22. Electromagnetic Probes of Fundamental Physics, 2003 23. Epioptics-7, 2004 24. Symmetries in Nuclear Structure, 2004 25. Innovative Detectors for Supercolliders, 2003 26. Complexity, Metastability and Nonextensivity, 2004

COMPLEXITY, METASTABILITY AND NONEXTENSIVITY 31st Workshop of the International School of Solid State Physics

20 - 26 July 2004

Erice, Sicily, Italy

Editors

C Beck, G Benedek, A Rapisarda and C Tsallis

Series Editor

A. Zichichi

v

World Scientific

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COMPLEXITY, METASTABILITY AND NONEXTENSIVITY Copyright 0 2005 by World Scientific Publishing Co. Re. Ltd

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PREFACE

The International Workshop on Complexity, Metastability, and Nonextensivity was held at the International School on Solid State Physics, International Centre for Scientific Culture “Ettore Majorana” in Erice (Sicily) from the 20th to the 26th of July 2004. The main focus of the workshop was the dynamics of complex systems and the theoretical approaches developed so far to describe them, with particular attention to nonextensive statistical mechanics. There were 29 invited keynote speakers, among them well-known scientists in the fields of statistical mechanics, turbulence, glasses, stochastic dynamics, econophysics, biophysics and networks. In particular, it was a great honour to welcome as a speaker, the 2004 Boltzmann medallist, Professor Ezechiel G.D. Cohen from Rockefeller University, New York, who came directly from India after receiving the important prize. There were also 24 other contributed talks - selected from the most interesting contributed proposals by young physicists. In total, the workshop had 86 participants who came from Europe (Finland, France, Germany, Greece, Hungary, Italy, Netherlands, Portugal, Russia, Spain, Turkey, United Kingdom), Asia (Japan, Taiwan) and America (Argentina, Brazil, Mexico, USA, Canada). About half of them were young researchers and PhD students. Unfortunately, due to limited logistic and financial limitations, many applications could not be accepted. A poster session with 20 posters was also organized and posters remained visible for discussion during the whole period of the conference. The topics covered at the meeting were, more specifically, complexity in physical systems such as glasses and long-range Hamiltonian systems, dynamics of turbulent fluids and granular materials, general complex behaviour relevant to astrophysics, biophysics, geophysics, econophysics, sociophysics and networks. Talks covered both experimental and theoretical aspects with particular attention to applications. From a scientific point of view the meeting was very stimulating and fruitful. Various new results on the dynamics of complex systems were presented and debated. Of particular interest were the talks on the theoretical foundations of nonextensive statistical mechanics. In general, there was a strong effort to cover as many aspects of complexity as possible, pointing to underlying common features and possible interesting theoretical approaches connecting many disciplines at the same time. From the general positive response of

vii

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Vlll

the participants, we can conclude that the primary goal of the workshop was achieved. The meeting obtained a general consensus and it can be considered as another step forward in the direction of putting the field of complexity onto firmer grounds. The meeting was also helpful from a pedagogical point of view. The level of most of the keynotes talks was accessible to students who participated very actively in the discussions. The quantity and the quality of the results presented at the meeting illustrate that the field of complexity is mature and very active. The main aim of the workshop was finding a unifying approach that can cover different disciplines which at first sight appeared to have little in common. In other words, the goal was to gather scientists with common interests but different backgrounds, to strengthen and increase links between communities of scientists working in related but not identical fields like for example nonlinear dynamics, statistical mechanics, noise and fluctuations, mathematical physics, with special attention to applications in other disciplines. From this point of view the workshop was quite successful: the nice and friendly atmosphere at the E. Majorana Centre favoured exchange of ideas and interdisciplinary debates. It helped tremendously the cross-fertilisation of common interests and would induce future collaborations. The young had the golden opportunity to widen their personal experience and also discuss their work with the main experts in the field. The participants’ response was on the whole quite enthusiastic, not only due to the nice and engaging atmosphere and efficient local organisation, but also due to the high scientific level of talks and discussions. We do believe that this conference, the first of its kind organised at the E. Majorana Centre, and its proceedings will remain a useful point of reference for both the experts in the field as well as for students. In conclusion there was a general feeling that a common viewpoint is slowly emerging, although it is still a long way to go. Nonextensive statististical mechanics, superstatistics, the theory of glasses, self-organized criticality, the stochastic resonance formalism, synchronization and networks already provide promising and complementary theoretical paradigms. These approaches are not disconnected as perceived but their precise interconnections will still have to be further clarified. Establishing further links between these various theoretical approaches is a promising line of research which deserves more attention and investigations in the future. The meeting has provided a stimulating environment in this direction and we can say that it has further advanced the field of complexity, an area of research that will continue to increase its level of importance in the years to come.

ix

Before ending, we would like to thank the Director of the centre, Antonino Zichichi, for his great help and enthusiastic support, without which this workshop would not have been possible. Further thanks are due to Fiorella Ruggiu and her collaborators at the Centre E. Majorana in Erice for their constant and efficient help. Last but not least, we are very grateful to the European Science Foundation (via the STOCHDYN programme), the National Institute for Nuclear Physics (INFN), the University of Palermo and the Faculty of Science of the University of Catania, who sponsored this event and provided funds for the organisation.

Editors C. Beck G. Benedek A. Rapisarda C. Tsallis


CONTENTS

Preface

vii

General Aspects

1

Work and Heat Fluctuations in Systems with Deterministic and Stochastic Forces E. G. D. Cohen and R. Van Zon

3

Is the Entropy SQ Extensive or Nonextensive? C. Tsallis

13

Superstatistics: Recent Developments and Applications C. Beck

33

Two Stories Outside Boltzmann-Gibbs Statistics: Mori’s Q-Phase Transitions and Glassy Dynamics at the Onset of Chaos A . Robledo, F. Baldovin and E. Mayoral

43

Time-averages and the Heat Theorem A. Carati

55

Fundamental Formulae and Numerical Evidences for the Central Limit Theorem in Tsallis Statistics H. Suyari

61

Generalizing the Planck Distribution A . M. C. Soma and C. Tsallis

66

The Physical Roots of Complexity: Renewal or Modulation? P. Grigolini

72

Nonequivalent Ensembles and Metastability H. Touchette and R. S. Ellis

81

xi

xii

Applications in Physics

89

Statistical Physics for Cosmic Structures L. Pietronero and F. Sylos Labini

91

Metastability and Anomalous Behavior in the HMF Model: Connections to Nonextensive Thermodynamics and Glassy Dynamics A. Pluchino, A. Rapisarda and V: Latora

102

Vlasov Analysis of Relaxation and Meta-equilibrium C. Anteneodo and R . 0. Vallejos

113

Weak Chaos in Large Conservative Systems - Infinite-range Coupled Standard Maps L. G. Moyano, A. P. Majtey and C. Tsallis

123

Deterministc Aging E. Barkai

128

Edge of Chaos of the Classical Kicked Top Map: Sensitivity to Initial Conditions S. M. Duarte Queirds and C. Tsallis

135

What Entropy at the Edge of Chaos? M. Lissia, M. Coraddu and R. Tonelli

140

Fractal Growth of Carbon Schwarzites G. Benedek, H. V: Tafreshi, A. Podesth and P. Milani

146

Clustering and Interface Propagation in Interacting Particle Dynamics A. Provata and V: K. Noussiou

156

Resonant Activation and Noise Enhanced Stability in Josephson Junctions A . L. Pankratov and B. Spagnolo

168

Symmetry Breaking Induced Directed Motions C.-H. Chang and T. Y. Tsong

178

xiii

Granular Media, Glasses and lhrbulence

183

General Theory of Galilean-invariant Entropic Lattic Boltzmann Models B. M. Boghosian

185

Unifying Approach to the Jamming Transition in Granular Media and the Glass Transition in Thermal Systems A. Coniglio, A. De Candia, A. Fierro, M. Nicodemi, M. Pica Ciamarru and M. Tarzia

194

Supersymmetry and Metastability in Disordered Systems I. Giardinu, A. Cavagna and G. Parisi

204

The Metastable Liquid-liquid Phase Transition: From Water to Colloids and Liquid Metals G. Franzese and H . E. Stanley

210

Optimization by Thermal Cycling A. Mobius, K. H. Hoffmann and C. Schon

215

Ultra-thin Magnetic Films and the Structural Glass Transition: A Modelling Analogy S. A. Cannas, F. A. Tamarit, P. M. Gleiser and D. A. Stariolo

220

Non-extensivity of Inhomogeneous Magnetic Systems M. S. Reis, V . S. Amaral, J. P. Araujo and I. S. Oliveira

230

Multifractal Analysis of Turbulence and Granular Flow 2: Arimitsu and N. Arimitsu

236

Application of Superstatistics to Atmospheric Turbulence S. Rizzo and A. Rapisarda

246

Applications in Other Sciences

253

Complexity of Perceptual Processes F. 2: Arecchi

255

xiv

Energetic Model of Tumor Growth P. Castorina and D. Zappalh

272

Active Brownian Motion - Stochastic Dynamics of Swarms W. Ebeling and U. Erdmann

277

Complexity in the Collective Behaviour of Humans T. Vicsek

287

Monte Carlo Simulations of Opinion Dynamics S. Fortunato

301

A Merton-like Approach to Pricing Debt Based on a Non-Gaussian Asset Model L. Borland, J. Evnine and B. Pochart

306

The Subtle Nature of Market Efficiency J.-P. Bouchaud

315

Correlation Based Hierarchical Clustering in Financial Time Series S. Miccicht, F. Lillo and R. N . Mantegna

327

Path Integrals and Exotic Options: Methods and Numerical Results G. Bormetti, G. Montagna, N. Moreni and 0. Nicrosini

336

Aging of Event-event Correlation of Earthquake Aftershocks S. Abe and N. Suzuki

341

Aging i n Earthquakes Model U. Tirnakli

350

The Olami-Feder-Christensen Model on a Small-world Topology F. Caruso, V Latora, A. Rapisarda and B. Tadii

355

Networks

361

Networks as Renormalized Models for Emergent Behavior in Physical Systems M. Paczuski

363

xv

Energy Landscapes, Scale-free Networks and Apollonian Packings J. P. K. Doye and C. P. Massen

375

Epidemic Modeling and Complex Realities M . Barthe'lemy, A . Barrat, V. Colizza and A. Vespignani

385

The Importance of Being Central P. Crucitti and V. Latora

397


General Aspects


WORK AND HEAT FLUCTUATIONS IN SYSTEMS WITH DETERMINISTIC AND STOCHASTIC FORCES

E. G . D. COHEN AND R. VAN ZON The Rockefeller University, 1230 York Avenue, New York, NY 10021

This is a brief survey of recent progress on fluctuations of performed work and produced heat in non-equilibrium systems in a stationary state, After introducing the Conventional Fluctuation Theorem for systems with only deterministic forces, we review the results for systems with both deterministic and stochastic forces. In at least two such systems (a Brownian particle in a moving harmonic potential and an electric circuit with Nyquist noise and a current source) the work fluctuations satisfy the Conventional Fluctuation Theorem, whereas the heat fluctuations satisfy an Extended Fluctuation Relation, in which large negative fluctuations are much more likely than expected on the basis of the Conventional Fluctuation Theorem. To what extent this new Fluctuation Relation holds for a wider class of systems remains an open question.

1. Introduction Although Statistical Mechanics was originally, in the 19th century, mainly practiced as kinetic theory, which is restricted to dilute gases, in the 20th century it developed into Equilibrium Statistical Mechanics - the molecular understanding of the properties of systems in thermal equilibrium - and non-equilibrium Statistical Mechanics - the molecular understanding of the properties of systems not in thermal equilibrium. The present status of the former is very different from that of the latter. For, while for Equilibrium Statistical Mechanics a general starting point has been provided by the Gibbs ensembles, no such general starting point exists for Non-equilibrium Statistical Mechanics. In the last century major advances have been made in Equilibrium Statistical Mechanics by explaining a host of phenomena on a molecular basis of systems in equilibrium: from the calculation of thermodynamics properties to an understanding of many types of phase transitions in matter, based on scaling and the Renormalization Group theory. Both systems with short as well as long range forces, possibly subjected to electromagnet fields, have been studied. Also dilute and dense systems have been successfully dealt with. In addition, classical and quantum mechanical systems have been considered, where a basic understanding of superfluidity and superconductivity could be especially mentioned. No such dramatic progress has been made in the last century in Non-equilibrium Statistical Mechanics, which has to a large extent continued the kinetic approach, indicated in the 19th century, although replacing mean free path methods by the Boltzmann, Vlasov and related equations, depending on the nature of the non-

3

4 equilibrium systems. A real difficulty here has been that a systematic useful generalization of the Boltzmann and similar equations to higher densities has proved to be exceedingly difficult. On the other hand, the progress in non-linear science - chaos theory and the theory of dynamical systems - has profoundly influenced new developments in Non-equilibrium Statistical Mechanics, which have led to new approaches and insights. Clearly, the understanding of non-equilibrium systems is no less important than that of equilibrium systems, only much more difficult. (See Ruelle1i2.) The lack of a universal starting point here is related to the plethora of characteristic relaxation times, which differ from system to system and are often difficult to identify. This paper deals with a fundamental problem in classical non-equilibrium statis tical mechanics for systems with short range forces, in which some recent progress has been made, although there are still very many questions unanswered and the precise place of the results obtained in the realm of Non-equilibrium Statistical Mechanics is not clear at present. The problem concerns the fluctuations of the external work done on the system and the heat produced by the system in non-equilibrium stationary states not near equilibrium. Near equilibrium the theory of Irreversible Thermodynamics together with hydrodynamics provided a very useful macroscopic description of non-equilibrium phenomena, but beyond this “linear regime”, where the constitutive equations for the fluxes are linear in the gradients of the hydrodynamic quantities (the “forces”), nothing certain is really known. The work on fluctuations is an attempt to penetrate this unknown part of Non-equilibrium Statistical Mechanics. Two different types of fluctuation behavior have so far been identified: one leading to what we will call a Conventional Fluctuation Theorem (CFT) (see Evans et al.4), which for a certain class of (internally thermostatted) dynamical systems has been rigorously mathematically proved by Gallavotti and Cohen5i6; the other, based on a Langevin Equation, leading to what we will call an Extended Fluctuation Relation (EFR) was considered by Van Zon et. where no mathematical proofs have been given so far, but where a very different behavior from the CFT has been found for large heat fluctuations, while work fluctuations follow the CFT. The results of both the CFT and the EFR have been confirmed numerically by Evans et a14;Bonetto et all2; Lepri et all3; Ayton and Evans14; Bonetto and Lebowitz15; Mittag et all6 and Van Zon and Cohen” as well as by laboratory experiments by Ciliberto and Laroche”; Ciliberto et all’; Feitosa and Menon” and Gamier and CilibertoZ0. The latter will be discussed in some detail below. al’7g~10~11,

2. Non-equilibrium Fluctuation Theorems and Relations

2.1. Conventional Fluctuation Theorems In 1993 Evans, Cohen and Morriss4, driven by theoretical considerations, discovered numerically a new Fluctuation Theorem for non-equilibrium stationary states possibly far from thermal equilibrium, which can be stated in general (Gallavotti

5 and Cohen5v6),as follows:

lim T‘OO

1 TT(P) In U+T

7r7(-p)

lim FT(p)= p T+cc

Here FT(p)is the fluctuation function which measures, loosely speaking, the relative probability to observe a positive value of p for the (scaled) entropy production rate to that of an equal in absolute value negative value - p of p . FT(p)is an odd function of p , so that the behavior for p < 0 can be deduced from that for p > 0. Evans et al. considered the statistics of the magnitude of the fluctuations of the xy-component Pxyof the pressure tensor in a strongly sheared fluid in a nonequilibrium stationary state and determined - via a histogram - the ratio of the probabilities that on a very long phase space trajectory, divided into segments on all of which the system spends a time T , Pzywould have a value a or its opposite value -a when integrated over a time segment T . Converting Pxyinto an entropy production rate u by multiplying with the (constant) velocity gradient in the fluid and dividing by the temperature and introducing a scaled u by p = u/u+,where u+ is the average of n over all positive times, their results can be put in the form of Eq. (1). This equality is more precise than the usual qualitative inequality of the Second Law of Thermodynamics, viz. u+ > 0 (see De Groot and Mazur3). In that sense Eq. (1)is a real advance and an extension of the Second Law. It was rigorously proved by Gallavotti and Cohen in 1995 for extremely chaotic, Anosov systems5v6. Another Fluctuation Theorem, for transient, non-stationary, states was derived and proved in 1994 by Evans and SearlesZ1;(see also Cohen and GallavottizZ),but the present paper is mainly concerned with stationary state fluctuations. The Gallavotti-Cohen proof relied heavily on a number of Theorems proved by Sinai for strictly Anosov systems in the Theory of Dynamical Systems. The proof by Gallavotti and Cohen assumes the system t o be Anosov-like i.e. extremely chaotic, (“the Chaotic Hypothesis”). The proof shows that there is also a limited range of validity for the CFT: Ipl < p * , based on an explicit formula for p* (see GallavottiZ3.) However, there is no restriction for the system t o be near equilibrium (e.g. no restriction is imposed on the external field F, in Eq. (2)). Physically one would expect the CFT t o be widely valid for fluid systems, but a precise criterion when the Chaotic Hypothesis is valid is lacking. On the other hand, this C F T for non-equilibrium stationary states has been verified in many numerical simulations mentioned above and also in a number of laboratory experiments by Ciliberto and c o - w o r k e r ~ ~ ~and ~ ~ for ’ ; granular media by Feitosa and Menonlg. In the latter case the systems for which the CFT Eq. (1) was experimentally established did not satisfy some of the conditions used in the Gallavotti-Cohen proof of the CFT. This raises the question again under what precise conditions Eq. (1) holds and it may well be more general than initially surmised. G a l l a ~ o t t and i~~ Gallavotti and RuelleZ5 derived from Eq. (1) key results of Irreversible Thermodynamics, involving only linear deviations of the dissipative fluxes (e.g.Pzy) from

6 equilibrium. Thus Onsager’s reciprocal relations, the Green-Kubo relations and the Fluctuation-Dissipation Theorem were all derived under the condition that the system was near equilibrium (see also Ref. 26). This suggested that the CFT (l), which is not restricted to the linear regime, incorporates something new about the far from equilibrium behavior, which, however, has not been discovered yet. The class of finite sized systems for which Eq. (1) holds can be written in the following form (q = {qi},p = {pi})27

Here i = 1,2, ....,N , N is the number of particles in the system, F, is an external field acting on the i-th of the N particles via Ci and Di, while Fi is the force on particle i due to the other particles in the system and a(q,p)pi is a thermostat term, which is determined from the condition that the total energy (isoenergetic case) or alternatively the total kinetic energy (isokinetic case) of the system remain constant during the stationary state. The thermostats are necessary to avoid a constant heating up of the system, due to the continuous work the field F, does in the stationary state. This thermostat term is the dynamical representation of the thermostatting that occurs in Nature via the walls of the system. The system is finite and closed and thermostatting takes place internally via the -a(q,p)pi term in the equations of motion. Many numerical and also a few laboratory experiments represented by different Fi, Ci and Di have confirmed the correctness of these internal thermostats a ( q , p ) p i as a replacement for physical thermostats. The CFT Eq. (1) holds in the isoenergetic case originally considered by Evans et al. not only for the heat Q , produced in a dissipative system in a stationary state during a time T,but also for the work W, done on the system by an external force in a time T. In fact one has that:

where the bar indicates the average of W,, or Q,, respectively over a trajectory in the non-equilibrium stationary state. In addition, the fluctuations of W, and Q , are equal and both satisfy Eq. (l), or p~ = IQ, respectively. when p is replaced there by pw = Q, While the dynamics in the systems considered so far was classical and deterministic, as given by Hamilton’s equations of motion (apart from the non-Hamilton thermostatting a(q,p)pi term), also systems with stochastic dynamics have been considered, which turned out also to obey Eq. (l),but did not need the Chaotic Hypothesis for a proof, apparently being sufficiently chaotic “by definition” (Kurchan2*; Lebowitz and SpohnZg;Searles and Evans3’).

7 2.2. Extended Fluctuation Relations Inspired by a laboratory experiment by Wang et a131; Van Zon and very recently studied stationary state FT’s (as well as transient FT’s) for a different class of systems than discussed in Sec. 2.1. Not only are these systems of interest by themselves, but combined with what is known about the systems mentioned in Sec. 2.1, they lead to a number of new fundamental questions. The systems considered by Van Zon and Cohen are described below. The relevant aspects of these experiments and their theoretical interpretation are described here under 2.2.1 and 2.2.2 in some more detail. 2.2.1. Brownian Particle The theory of the Brownian particle experiment by Wang et a131 where fluctuations of W, were measured, was discussed by Van Zon and Cohen7~l0,based on an overdamped Langevin Equation which reads: 0 = -a xt - k(xt - v*t)

+ ct

(4)

Here xt is the position of the Brownian particle at time t , a its (Stokes) friction, k the strength of the confining harmonic potential V ( x t )= +$Ixt - x;l2, x; = v*t, where v* is the constant velocity with which the harmonic oscillator “cage” is pulled through the fluid and Ct the fluctuating force acting on the Brownian particle due to the thermal motion of the fluid (water) molecules (see Figure l ) , which is white noise.

Figure 1. Brownian particle in a moving harmonic potential (cf. Van Zon and Cohen lo). In this the average position of the particle in the stationary state and the figure, we chose xt = potential is pulled in the z-direction.

8

An important difference of this system with those considered in Sec. 2.1 is that contrary to either the deterministic or the stochastic dynamics there, leading to the CFT Eq. (l),here a mixed dynamics occurs, in the sense that it is deterministic due to the harmonic force -k(xt - v't) on the particle and stochastic due to the density fluctuations in the water, causing the Brownian motion of the particle. For this system, energy conservation reads as the first law of thermodynamics:

Here, W, is the total work done on the system during time r , i.e,, pulling it with a constant velocity v* through the fluid over a time r; Q, is the total heat developed in the water due to the friction of the Brownian particle during time r an AU, = Ut+,(xt+,) - Ut(xt) is the potential energy difference of the particle in time r. Physically, the pulling of the harmonic potential drags the Brownian particle along, but with a delay, because of its friction with the water, while at the same time this delay necessitates a change in its potential energy from its initial position xt to its final position xt+,. A non-equilibrium stationary state will be reached when the friction force of the water cancels the attractive force on the particle due to the harmonic potential (cf. Fig. 1) and it are the fluctuations around this stationary state which will be studied. stationary state the averages and g, are equal, since -In the-non-equilibrium AU, = Ut+,- U, = 0 then. However, unlike in the cases of only deterministic or only stochastic dynamics (Sec. Zl), P(Q,) # P(W,). The details of the theory to determine the probability distribution functions P(W,) and P(Q,) can be found in Refs. 8, 9. Two theoretical approaches have been used: the theory of large deviations') and the saddle point methodg>''), which give identical results. They also give results consistent with two numerical methods used, one of which is a sampling method and the other a Fast Fourier Inversiong. We only summarize here the most relevant results.

w,

Figure 2. Qualitative difference in the distribution functions of work and heat fluctuations (Van Zon and Cohen") .

9

The distribution function P(W,) is Gaussian, and satisfies a CFT like in Eq. (1) i.e. lim,+m F,(pw) = pw, for all pw. For finite T the solution of the Langevin equation shows that the asymptotic CFT behavior for T + 00 goes as l / ~ . To the contrary, the P(Q,) is not Gaussian, because Q, = W, - AU, depends nonlinearly on x, since AU, is quadratic in x,. To find the P(Q,), its Fourier transform is determined exactly, but the inverse Fourier transform cannot be found exactly. Using the saddle point method for large T and taking into account the singularities in the Fourier transform, one obtains a P(Q,) which has exponential instead of Gaussian tails as a function of Q, (see Fig. 2). Its form is due to an interplay of the Gaussian behavior of P(W,) and the exponential behavior e-pArrT of P(AU,), where 3!, = l / k ~ Twith , ks Boltzmann's constant and T the temperature of the water. This distribution arises physically because it is the stationary distribution for a particle in a potential in contact with a heat bath at a fixed temperature. While the steepest descent method is an asymptotic expansion for T + co, corrections for finite T can be obtained, giving an EFR for non-equilibrium states also for finite T . We emphasize that these finite T corrections a,re very important especially for a comparison with simulations and experiments. Some features of the EFR in these Langevin based models are (cf. figure 4): a) F,(PQ) = p~ O ( ~ / Tfor ) 0 < p~ < 1. Thus the Extended Fluctuation Theorem coincides with the conventional one up to O ( ~ / Tfor ) ( p ~ 1 (i.e., extensive). The total potential energy of this particular model has a logarithmic N-dependance (i.e., nonextensive) at the limiting value a / d = 1. The Lennard-Jones model for gases corresponds to ( a ,d) = ( 6 , 3 ) , and has therefore an extensive total energy. In contrast, if we assume a cluster of

15 stars gravitationally interacting (together with some physical mechanism effectively generating repulsion at short distances), we have ( a ,d ) = (1,3), hence nonextensivity for the total potential energy. The physical nonextensivity which naturally emerges in such anomalous systems is, in some theoretical approaches, desguised by artificially dividing the two-body coupling constant (which has in fact no means of “knowing” the total number of particles of the entire system) by For the particular case a = 0 this yields the widely (and wildly!) used division by N of the coupling constant, typical for a variety of mean field approaches. See for more details. Boltzmann-Gibbs (BG) statistical mechanics is based on the entropy W

SBG = -k)pilnpi,

(5)

i=l

with W

cpi=1, i=l

where pi is the probability associated with the ith microscopic state of the system, and k is Boltzmann constant. In the particular case of equiprobability, i.e., pi = 1 / W (Vi), Eq. ( 5 ) yields the celebrated Boltzmann principle (as named by Einstein 3) :

SBG= kln W

.

(7)

From now on, and without loss of generality, we shall take k equal to unity. Nonextensive statistical mechanics, first introduced in 1988 (see for reviews), is based on the so-called “nonextensive” entropy S, defined as follows: 41516

7,8,9,10,11,12113,14,15

For equiprobability (i.e., pi = 1 / W , V i ) , Eq. ( 8 ) yields

S, = In, W , with the q-logarithm function defined

l6

(9)

as

The inverse function, the q-exponential, is given by

+

if the argument 1 ( 1 - q ) z is positive, and equals zero otherwise. The present paper is entirely dedicated to the analysis of the additivity or nonadditivity of SBG and of its generalization S,. However, following a common (and

16 sometimes dangerous) practice, we shall from now on cease distinguishing between additive and extensive, and use exclusively the word extensive in the sense of strictly additive. 2. The case of two subsystems

Consider two systems A and B having respectively W, and WBpossible microstates. The total number of possible microstates for the system A B is then in principle W = WA+B= WAWB. We emphasized the expression “in principle” because, as we shall see, a more or less severe reduction of the full phase space might occur in the presence of strong correlations between A and B. We shall use the notation p F B (i = 1 , 2 , ...,WA;j = 1,2,...,W B )for the joint probabilities, hence

+

The marginal probabilities are defined as follows:

hence

and

hence W B

cpj”=1. j=1

These quantities are indicated in the following Table.

17 We shall next illustrate the importance of the specification of the composition law. Let us consider two cases, namely independent and (specially) correlated subsystems.

2.1. Two independent subsystems Consider a system composed by two independent subsystems A and B , i.e., such that the joint probabilities are given by A+B = A B

Pi P j

Pi3

(V(i,j)).

(17)

With the definitions

cc p p B WA WB

SBG(A+ B ) = -

,

(18)

and

3=1

we immediately verify that

+

SBG(A+ B) = SBG(A) SBG(B)

(21)

and, analogously, that Sq(A + B ) = Sq(A)

+ Sq(B) + (1 - q)Sq(A)Sq(B).

(22)

Therefore, SBG is extensive. Consistently, S, is, unless q = 1, nonextensive. It is in fact from property (22) that the q # 1 statistical mechanics we are referring to has been named nonextensiue.

2.2. Two specially correlated subsystems Consider now that A and B are correlated, i.e.,

PFB# PfPj”

7

Assume moreover, for simplicity, that both A and B systems are equal, and that WA= WB = 2. Assume finally that the joint probabilities are given by the following Table (with 1/2 < p < 1):

18

It can be trivially verified that Eq. (21) is not satisfied. Therefore, for this special correlation, SBG is nonebensive. It can also be verified that, for q = 0 and only for q = 0, the following additivity is satisfied: So(A

+ B ) = So(A)+ S o ( B ) ,

(24)

therefore So is extensive. Indeed S o ( A + B ) = 2So(A) = 2. We immediately see that, depending on the type of correlation (or lack of it) between A and B , the entropy which is extensive (reminder: as previously announced, we are using here and in the rest of the paper “extensive” to strictly mean “additive”) can be SBC or a different one. Before going on, let us introduce right away the distinction between a T oLet ri possible states (in number W ) and allowed or effective states (in number W e l ). us consider the above case of two equal binary subsystems A and B and consequently W = 4. If they are independent (i.e., the q = 1 case), their generic case corresponds to 0 < p < 1 , hence W e f f= 4. But if they have the above special correlation (i.e., the q = 0 case), their generic case corresponds to 1 / 2 < p < 1 , hence Weff = 3 (indeed, the state ( 2 , 2 ) , although possible a priori, has zero probability). This type of distinction is at the basis of this entire paper. Notice also that the q = 1and q = 0 cases can be unified through We-@= [2l-Q 2l-q - l]’/(’-Q)= [22-Q- l ] ’ / ( ’ - Q ) . This specific unification will be commented later on. Let us further construct on the above observations. Is it possible to unify, at the level of the joint probabilities, the case of independence (which corresponds to q = 1) with the specially correlated case that we just analyzed (which corresponds to q = O)? Yes, it is possible. Consider the following Table:

+

where f q ( p ) is given by the following relation: 2pQ

+ 2(1 -

p)q

- ( f q ) Q - 2 ( p - f q ) Q - ( 1 - 21,

+

fq)Q =

1,

(25)

with f q ( l ) = 1, and 0 5 q 5 1 (later on we shall comment on values outside this interval). Typical curves f q ( p ) are indicated in Fig. 1. Since Eq. (25) is an implicit one, they have been calculated numerically. It can be checked, for instance, that f q ( 1 / 2 ) smoothly increases from zero to 1/4 when q increases from zero to unity, being very flat in the neighborhood of q = 0, and rather steep in the neighborhood of q = 1 . The interesting point, however, is that it can be straightforwardly verified that, for the value of q chosen in f q ( p ) defined through Eq. (25) (and only for that

19 4 )9

+

S q ( A B ) = 2Sq(A)= 2

1-pq-

(1 - p ) q

(26)

9

q-1

where we have used the fact that A = B. In other words, we are facing a whole family of entropies that are extensive for the respective special correlations indicated in the Table just above. 1

0.75

f4 0.5

0.25

n 0

0.25

O5

P

0.75

1

Figure 1. The function f q ( p ) , corresponding to the two-system A = B case (with W A = W B = 2), for typical values of q E [0,1]. A few typical nontrivial (q, fq(1/2)) points are (0.4,0.043295),(0.5,0.064765),(0.6,0.087262), (0.7,0.111289), (0.8,0.138255), (0.9,0.171838),(0.99,0.225630). It can be easily verified that these values satisfy the relation 21-4 - [fq(1/2)Iq - [(1/2) - fq(1/2)Iq = 1/2, which is the simple form that takes Eq. (25) for the p = 1/2 particular case. We also remind the trivial values fo(ll2) = 0 and fi(l/2) = 1/4.

Let us proceed and generalize the previous examples to two-state systems A and B that are not necessarily equal. The case of independence is trivial, and is indicated in the following Table:

20

1

2

1

P?pf

P?P§

PA

2

PAP?

PAPB

PA

A\B ||

pf Of course, Eq. (21) is satisfied. Let us consider now the following Table (with pA + pf > 1): A\B 1 1

2

1

P?+pf-l A

l~P

2

1-pf

Pf

0

1-rf

We verify that Eq. (24) is satisfied. Is it possible to unify the above anisotropic 9 = 1 and q = 0 cases? Yes, it is. The special correlations for these cases are indicated in the following Table:

pf-/,(p?,pf) 3

1-pf ) = /,(pf ,P ), /,(p,l) = p. /,(p,p) = /,(p), /l&tf.pf) = A

where /,( and fo(pf,P?) = P\ + pf — 1- For any value of q in the interval [0, 1], and for any probabilistic pair (p^,pf ), the function /q(pf ,pf ) is (implicitly) defined through

-bf - f,(rt,p? )]' - bf - /,(pf ,pf )]' -[i-pi 1 -pf + /9(pi1,pf)]9 = i

(27)

(We remind that, for the q = 0 particular case, it must be pf +pf > 1). We notice that the special correlations we are addressing here make that all joint probabilities can be expressed as functions of only one of them, say pn+ , which is determined once for ever. More explicitly, we have that pAfB = pA - pA{*~B, pA^B = pf A+B A+B _ A B nA+B Pi —nPi Pn P22 — Pn Eq. (27) recovers Eq. (25) as the particular instance pA = pf. And we can easily verify that, for 0 < q < 1,

21

So, we still have extensivity for the appropriate value of q, i.e., the value of q which has been chosen in Eq. (27) to define the function f g ( x , y ) reflecting the special type of correlations assumed to exist between A and B. In other words, when the marginal probabilities have all the information, then the appropriate entropy is SBG- But this happens only when A and B are independent. In all the other cases addressed within the above Table, the important information is by no means contained in the marginal probabilities, and we have to rely on the full set of joint probabilities. In such cases, SBG is nonextensive, whereas 5, is extensive. Before closing this section dedicated to the case of two systems, let us indicate the Table associated to the q = 0 entropy for arbitrary systems A and B:

B

A\

1

1

WB

1

rf+pf-1

$

PwB

rt

2

P$

0

0

rf

WA

PwA

0

0

r> Pw A

p?

r Pi

1

A

PwB

We easily verify that Eq. (24) is satisfied. For example, the generic case corresponds to all probabilities in the Table being nonzero, excepting those explicitly indicated in the Table. For this case we have S0(A) = WA - 1, S0(B) = WB - 1, and So(A + B) = WA + WB — 2. This is a neat illustration of the fact that, although the full space admits in principle W = W^Wg microstates, the strong correlations reflected in the Table make that the system uses appreciably less, namely, in this example, We" — WA + WB — 1- It is tempting to conjecture the generalization of this expression into Weff = [WA~g + Wg~9 -IjVU-g) for 0 < q < 1. It is clear that Weff < W^WB, the equality holding only for q = 1. Since, strictly speaking, WA, WB and We" are integer numbers, this expression for Weff can only be generically valid for real q ^ 0,1 in some appropriate asymptotic sense. This sense has to be for WA, WB » 1, which however are not fully addressed in the present paper for q ^ 0,1. For the particular instance A = B, we have Weff = [2WA~q - l] 1 /(i-«). We also verify another interesting aspect. If A and B are independent, equal values in the marginal probabilities are perfectly compatible with equal values in the joint probabilities. In the most general independent two-system case, we can simultaneously have p 0 for small P. An example is a x2 distribution of n degrees of f r e e d ~ m

N

PY,

(PO2 0, n > 1) which behaves for P + 0 as

f ( P ) Pn’2-1,

(13)

i.e.

n y=--l (14) 2 . Other examples exhibiting this power-law form are F-distributionslJO. With the above formalism one obtains from eq. (10) PE =

and

7

(15)

-

B ( E ) E-Y-’.

(16)

These types of f(P) form the basis for power-law generalized Boltzmann factors (q-exponentials) B ( E ) ,with the relation 26327128*29

Y + l Z 1

q-1’

Another example would be an c > 0. In this case one obtains

f(P)which for small ,B behaves as f(P)

N

,-‘/PI

37

The above example can be generalized to stretched exponentials: For form e-CP6 one obtains after a short calculation

f(P)of the

f(P)

N

where a i s some factor depending on 6 and c. Of course which type of f(P)is relevant depends on the physical system under consideration. For many problems in hydrodynamic turbulence, log-normal superstatistics seems to be working as a rather good approximation. In this case f(p) is given by

where s and m are parameters

1914,15*16*17135

4. Superstatistical correlation functions

To obtain statements on correlation functions, one has to postulate a concrete dynamics that generates the superstatistical distributions. The simplest dynamical model of this kind is a Langevin equation with parameters that vary on a long time scale, as introduced in 32. Let us consider a Brownian particle of mass m and a Langevin equation of the form .ir = -yv

+ aL(t),

(23)

where v denotes the velocity of the particle, and L ( t ) is normalized Gaussian white noise with the following expectations:

We assume that the parameters (T and y are constant for a sufficiently long time scale T , and then change to new values, either by an explicit time dependence, or by a change of the environment through which the Brownian particle maves. Formal identification with local equilibrium states in the cells (ordinary statistical mechanics at temperature @-I) yields during the time scale T the relation36

38 or

Again, we emphasize that after the time scale T , y and u will take on new values. During the time interval T , the probability density P(v, t ) obeys the Fokker-Planck equation

with the local stationary solution

In the adiabatic approximation, valid for large T , one asumes that the local equilibrium state is reached very fast so that relaxation processes can be neglected. Within a cell in local equilibrium the correlation function is given by 36

Clearly, for t = t’ and setting m = 1 we have

in agreement with eq. (5). It is now interesting to see that the long-term invariant distribution P(v),given bY

$5

depends only on the probability distribution of p = and not on that of the single quantities y and u2. This means, one can obtain the same stationary distribution from different dynamical models based on a Langevin equation with fluctuating parameters. Either y may fluctuate, and u2 is constant, or the other way round. On the other hand, the superstatistical correlation function

can distinguish between these two cases. The study of correlation functions thus yields more information for any superstatistical model. Let illustrate this with a simple example. Assume that u fluctuates and y is constant such that ,f3 = $5 is X2-distributed. Since y is constant, we can get the exponential e-Ylt-t’l out of the integral in eq. (33), meaning that the superstatistical correlation function still decays in an exponential way:

C(t - t’)

e-~lt-t’l.

(34)

39 On the other hand, if u is constant and y fluctuates and @ is still X2-distributed with degree n, we get a completely different answer. In this case, in the adiabatic approximation, the integration over @ yields a power-law decay of C(t - t’):

C ( t - t’)

N

It - t y - 7 ,

(35)

where

Note that this decay rate is different from the asymptotic power law decay rate of the invariant density P(w),which, using (29) and (32), is given by P(w) w-’/(q-l), with 1 n 1 -(37) q-1 2 2 N

--+-.

In general, we may generate many different types of correlation functions for general choices off(@). By letting both u and y fluctuate we can also construct intermediate cases between the exponential decay (34) and the power law decay (35), so that strictly speaking we only have the inequality n 02--1, (38) 2 depending on the type of parameter fluctuations considered. One may also proceed to the position x(t)= ltn(t‘)dt’

(39)

of the test particle. One has

Thus asymptotic power-law velocity correlations with an exponent 7 < 1 are expected to imply asymptotically anomalous diffusion of the form

(x’(t))

N

t”

(41)

with

a=2-0.

(42)

This relation simply results from the two time integrations. It is interesting to compare our model with other dynamical models generating Tsallis statistics. Plastino and Plastino3’ and Tsallis and BukmanrP study a generalized Fokker-Planck equation of the form

40 with a linear force F ( z ) = Icl - kzx and u # 1. Basically this model means that the diffusion constant becomes dependent on the probability density. The probability densities generated by eq. (43) are q-exponentials with the exponent

q=2-u.

(44)

The model generates anomalous diffusion with a = 2/(3- q). Assuming the validity of a = 2 - f j , i.e. the generation of anomalous diffusion by slowly decaying velocity correlations with exponent f j , one obtains

On the other hand, for the X2-superstatistical Langevin model one obtains by combining eq. (36) and (37) the different relation q=-

5 - 39 2q - 2 '

Interesting enough, there is a distinguished q-value where both models yield the same answer: q = 1.453 + f j = q = 0.707

(47)

These values of q and 71 correspond to realistic, experimentally observed numbers, for example in defect turbulence''. So far we mainly studied correlation functions with power law behaviour. But in fact one can construct superstatistical Langevin models that exhibit more complicated types of asymptotic behaviour of the correlation functions. To see this we notice that the asymptotic analysis of section 3 applies t o correlation functions as well, by formally defining

-

1

E := -u2m)t - t') 2

(48) (49)

and writing

To obtain statements on the symptotic decay rate of the superstatistical correlation function, we may just use the same techniques described in section 3 with the replacement E + and f + f. In this way one can construct models that have, for example, stretched exponential asymptotic decays of correlations etc. (see also 39). Asymptotic means here that It-t'l is large as compared t o the local equilibrium relaxation time scale, but still smaller than the superstatistical time scale T, such that the adiabatic approximation is valid.

41 5. Some Applications

We end this paper by briefly mentioning some recent applications of the superstatistics concept. Rizzo and RapisardaZ1J2 study experimental data of wind velocities at Florence airport and find that X2-superstatistics does a good job. Jung and S ~ i n n e ystudy ~ ~ velocity differences in a turbulent Taylor-Couette flow, which is well described by lognormal superstatistics. They also find a simple scaling relation between the superstatistical parameter p and the fluctuating energy dissipation E . Paczuski et aL40 study data of solar flares on various time scales and embedd this into a superstatistical model based on X2-superstatistics = Tsallis statistics. Human behaviour when sending off print jobs might also stand in connection t o such a supers tat is tic^^^. Bodenschatz et al.42have detailed experimental data on the acceleration of a single test particle in a turbulent flow, which is well described by lognormal superstatistics, with a Reynolds number dependence as derived in a superstatistical Lagrangian turbulence model studied by Reynolds15. The statistics of cosmic rays is well described by X2-superstatistics, with n = 3 due to the three spatial dimensions’’. In mathematical finance superstatistical techniques are well known and come under the heading ‘volatility fluctuations’, see e.g.23 for a nice introduction and for some more recent work. Possible applications also include granular media, which could be described by different types of superstatistics, depending on the boundary condition^^^. The observed generalized Tsallis statistics of solar wind speed fluctuation^^^ is a further candidate for a superstatistical model. Chavanis3’ points out analogies between superstatistics and the theory of violent relaxation for collisionless stellar systems. Most superstatistical models assume that the superstatistical time scale T is very large, so that a quasi-adiabatic approach is valid, but Luczka and Z a b ~ r e khave ~ ~ also studied a simple model of dichotomous fluctuations of where everything can be calculated for finite time scales T as well. 24925

References 1. C. Beck and E.G.D. Cohen, Physica 322A, 267 (2003) 2. E.G.D. Cohen, Physica 193D, 35 (2004) 3. E.G.D. Cohen, Einstein und Bolttmann - Dynamics and Statistics, Boltzmann award lecture at Statphys 22, Bangalore, to appear in Pramana (2005) 4. C. Beck, Cont. Mech. Thermodyn. 16, 293 (2004) 5. H. Touchette, Temperature fluctuations and mixtures of equilibrium states in the canonical ensemble, in M. Gell-Mann, C. Tsallis (Eds.), Nonextensiwe Entropy - Interdisciplinary Applications, Oxford University Press, 2004. 6. F.Sattin, Physica 338A, 437 (2004) 7. C. Beck and E.G.D. Cohen, Physica 344A, 393 (2004) 8. J. Luczka, P. Talkner, and P. Hanggi, Physica 278A, 18 (2000) 9. S. Abe, cond-mat/0211437 10. V.V. Ryazanov, cond-mat/0404357 11. A.K. Aringazin and M.I. Mazhitov, cond-mat/0301245 12. C. Tsallis and A.M.C. Souza, Phys. Rev. 67E, 026106 (2003) 13. C. Tsallis and A.M.C. Souza, Phys. Lett. 319A, 273 (2003)

42 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45.

C. Beck, Europhys. Lett. 64,151 (2003) A. Reynolds, Phys. Rev. Lett. 91,084503 (2003) B. Castaing, Y. Gagne, and E.J. Hopfinger, Physica 46D,177 (1990) C. Beck, Physica 193D,195 (2004) K. E. Daniels, C. Beck, and E. Bodenschatz, Physica 193D,208 (2004) C. Beck, Physica 331A,173 (2004) F. Sattin and L. Salasnich, Phys. Rev. 65E,035106(R) (2002) S. Rizzo and A. Rapisarda, in Proceedings of the 8th Experimental Chaos Conference, Florence, AIP Conf. Proc. 742,176 (2004) (cond-mat/0406684) S. Rizzo and A. Rapisarda, cond-mat/0502305 J.-P. Bouchard and M. Potters, Theory of Financial Risk and Derivative Pricing, Cambridge University Press, Cambridge (2003) M. Ausloos and K. Ivanova, Phys. Rev. 68E,046122 (2003) Y . Ohtaki and H.H. Hasegawa, cond-mat/0312568 C. Tsallis, J. Stat. Phys. 52,479 (1988) C. Tsallis, R.S. Mendes and A.R. Plastino, Physica 261A,534 (1998) C. Tsallis, Bmz. J . Phys., 29: 1 (1999) S. Abe, Y. Okamoto (eds.), Nonextensive Statistical Mechanics and Its Applications, Springer, Berlin (2001) P.-H. Chavanis, cond-mat/0409511 H. Touchette and C. Beck, Phys. Rev. 71E,016131 (2005) C. Beck, Phys. Rev. Lett. 87,180601 (2001) B.L. Sawford, Phys. Fluids A3, 1577 (1991) G. Wilk and Z. Wlodarczyk, Phys. Rev. Lett. 84,2770 (2000) S.Jung and H.L. Swinney, Velocity difference statistics in turbulence, Preprint University of Austin (2005) N.G. van Kampen, Stochastic Processes in Physics and Chemistry, North Holland, Amsterdam (1981) A.R. Plastino and A. Plastino, Physica 222A,347 (1995) C. Tsallis and D.J. Bukmann, Phys. Rev. 54E,R2197 (1996) R.G. Palmer et al., Phys. Rev. Lett. 53,958 (1984) M. Baiesi, M. Paczuski, and A.L. Stella, cond-mat/0411342 U. Harder and M. Paczuski, cs/PF/0412027 N. Mordant, A.M. Crawford, E. Bodenschatz, Physica 193D,245 (2004) J.S. van Zon et al., cond-mat/0405044 L.F. Burlaga and A.F. Vinas, J. Geophys. Res. 109,A12107 (2004) J. Luczka and B. Zaborek, Acta Phys. Polon. B 35,2151 (2004)

TWO STORIES OUTSIDE BOLTZMANN-GIBBS STATISTICS: MORI'S Q-PHASE TRANSITIONS AND GLASSY DYNAMICS AT THE ONSET OF CHAOS

A. ROBLEDO': F.BALDOVIN'S~~AND E.

'Instituto de Fisica,

MAYORAL'^

Universidad Nacional Auto'norna de Mixico, Apartado Postal 20-364, Mixico 01000 D.F., Mexico Dipartamento di Fisica, Uniuersitd di Padoua, Via Marzolo 8, I-35131 Padoua, Italy

First, we analyze trajectories inside the Feigenbaum attractor and obtain the atypical weak sensitivity to initial conditions and loss of information associated to their dynamics. We identify the Mori singularities in its Lyapunov spectrum with the appearance of a special value for the entropic index q of the Tsallis statistics. Secondly, the dynamics of iterates at the noise-perturbed transition to chaos is shown to exhibit the characteristic elements of the glass transition, e.g. two-step relaxation, aging, subdiffusion and arrest. The properties of the bifurcation gap induced by the noise are seen to be comparable to those of a supercooled liquid above a glass transition temperature.

Key words: Edge of chaos, q-phase transitions, nonextensive statistics, external noise, glassy dynamics PACS: 05.45.Ac, 64.60.Ak, 05.40.Ca, 64.70.Pf 1. Introduction

Evidence for the incidence of nonextensive dynamical properties at critical attractors in low dimensional nonlinear maps has accumulated and advanced over the last few years; specially with regards to the onset of chaos in logistic maps - the Feigenbaum attractor,'-' and at the accompanying pitchfork and tangent bifurcation^.^^^ The more general chaotic attractors with positive Lyapunov coefficients have fullgrown phase-space ergodic and mixing properties, and their dynamics is compatible with the Boltzmann-Gibbs (BG) statistics. As a difference, critical attractors have * email: rob1edoOfisica.unam.m temail: baldovinOpd.infn.it *email: estelaOeros.pquim.unam.mx

43

44

vanishing Lyapunov coefficients, exhibit memory-retentive nonmixing properties, and are therefore to be considered outside BG statistics. Naturally, some basic questions about the understanding of the dynamics at critical attractors are of current interest. We mention the following: Why do the anomalous sensitivity to initial conditions & and its matching Pesin identity obey the expressions suggested by the nonextensive formalism? How does the value of the entropic index q arise? Or is there a preferred set of q values? Does this index, or indexes, point to some specific observable properties at the critical attractor? From a broader point of view it is of interest to know if the anomalous dynamics found for critical attractors bears some correlation with the dynamical behavior at extremal or transitional states in systems with many degrees of freedom. Two specific suggestions have been recently advanced, in one case the dynamics at the onset of chaos has been demonstrated to be closely analogous to the glassy dynamics observed in supercooled molecular liquids," and in the second case the dynamics at the tangent bifurcation has been shown to be related to that at thermal critical states.l' With regard to the above comments here we briefly recount the following developments:

'

(i) The finding that the dynamics a t the onset of chaos is made up of an infinite family of Mori's q-phase t r a n ~ i t i o n s , ' ~each > ~ ~associated to orbits that have common starting and finishing positions located at specific regions of the attractor. Every one of these transitions is related to a discontinuity in the u function of 'diameter ratios',14 and this in turn implies a q-exponential & and a spectrum of q-Lyapunov coefficientsequal to the Tsallis rate of entropy production for each set of attractor regions. The transitions come in pairs with conjugate indexes q and Q = 2 - q, as these correspond to switching starting and finishing orbital positions. The amplitude of the discontinuities in u diminishes rapidly and consideration only of its dominant one, associated to the most crowded and sparse regions of the attractor, provides a very reasonable description of the dynamics, consistent with that found in earlier studies.lP4 (ii) The realization lo that the dynamics at the noise-perturbed edge of chaos in logistic maps is analogous to that observed in supercooled liquids close to vitrification. Four major features of glassy dynamics in structural glass formers, two-step relaxation, aging, a relationship between relaxation time and configurational entropy, and evolution from diffusive to subdiffusive behavior and finally arrest, are shown to be displayed by the properties of orbits with vanishing Lyapunov coefficient. The previously known properties in control-parameter space of the noise-induced bifurcation gap play a central role in determining the characteristics of dynamical relaxation a t the chaos threshold. 14315

45 2. Mori's q-phase transitions at onset of chaos

The dynamics at the chaos threshold p = pc of the z-logistic map fM(X)= 1 - p 1x1=, z

> 1,-1 5 x 5 1,

(1)

has been analyzed r e ~ e n t l y . ~The - ~ orbit with initial condition xo = 0 (or equivalently, xo = 1) consists of positions ordered as intertwined power laws that asymptotically reproduce the entire period-doubling cascade that occurs for p < pe. This orbit is the last of the so-called 'superstable' periodic orbits at ji,, < pc, n = 1,2, ...,I4 a superstable orbit of period 2O0. There, the ordinary Lyapunov coefficient A1 vanishes and instead a spectrum of q-Lyapunov coefficients A?) develops. This spectrum originally studied in Refs. 13 when z = 2, has been shown 4*7 to be associated to a sensitivity to initial conditions & (defined as &(xo) f lhaz,-ro(Axt/Axo) where Ax0 is the initial separation of two orbits and Axt that at time t ) that obeys the q-exponential form Et(x0) = e.p,[~,(xo)t]

= [I - ( q - 1)A,(xo)

(2)

t]-'/q-'

suggested by the Tsallis statistics. Notably, the appearance of a specific value for the q index (and actually also that for its conjugate value Q = 2 - q ) works out to be due to the occurrence of Mori's 'q-phase transitions' l 2 between 'local attractor structures' at pc. As shown in Fig. 1,the absolute values for the positions x7 of the trajectory with xt=O = 0 at time-shifted T = t + 1 have a structure consisting of subsequences with a common power-law decay of the form ~ - ' / ~ - - 9 . with q = 1 - In 2/(2 - 1) lna(z)? where a(.) is the Feigenbaum universal constant that measures the period-doubling amplification of iterate positions. That is, the attractor can be decomposed into position subsequences generated by the time subsequences 7 = (2k 1)2", each obtained by proceeding through n = 0 , 1 , 2 , ... for a fixed value of k = O , l , 2, .... See Fig. 1. The k = 0 subsequence can be written as xt = exp,-,(-Af)t) with A?) = ( z - 1) In a ( z ) /In 2. q-lyapunou coeficients. The sensitivity &(xo) can be obtained from &(m)N I.n(m-l)/Qn(m)l", t = 2" - 1, n large, where u,,(m) = dn+l,m/dn,m and where dn,m are the diameters that measure adjacent position distances that form the period-doubling cascade sequence.14 Above, the choices Ax0 = dn,m and Ax, = dn,m+t, t = 2" - 1,have been made for the initial and the final separation of the trajectories, respectively. In the large n limit a,(m) develops discontinuities at each rational m/2n+1,14 and according to our expression for &(m) the sensitivity is determined by these discontinuities. For each discontinuity of a,(m) the sensitivity can be written in the forms & = exp,[A,t] and & = exp2-,[A2-,t], A, > 0 and X Z - ~ < 0.7 This result reflects the multi-region nature of the multifractal attractor and the memory retention of these regions in the dynamics. The pair of q-exponentials correspond to a departing position in one region and arrival at a different region and vice versa, the trajectories expand in one sense and contract in

+

46

the other. The largest discontinuity of o,(m) at m = 0 is associated to trajectories that start and finish at the most crowded (z N 1) and the most sparse (z N 0) regions of the attractor. In this case one obtains

the positive branch of the Lyapunov spectrum, when the trajectories start at z and finish at 2 N 0. By inverting the situation one obtains A$) = -

2(2 - 1) In a(.) (2k 1) In 2

+

is infinite , and thus, since U is instead finite, one has in general U #< E >. In other terms, denoting by ~j the value of the internal energy in the cell Zj,one has the condition 1

-X n j E j = U

N

with

U #< E >

1

This indeed is a condition on the initial data or equivalently on the variables nj. So, one is confronted with a large deviation problem, i.e. with the problem that one should compute not the expected value < f >, but rather the conditional expectation < f >(I of f, given the mean energy U . To this end it is sufficient to compute the mean occupation number (which we denote by fij) when the mean energy is U , because one obviously has

59

Note that

fij

satisfies the conditions

N =

x

fij

1

, U = -C~jfij.

j

N

(1)

j

This problem can be solved, under suitable hypotheses (see Ref. [l]),in the limit of large systems. In other terms, one can provide an asymptotic expansion for the mean occupation number fij, the remainder of which tends to zero in the thermodynamic limit. If one assumes that the quantities n j , for different values of j , are independent random variables, one can give a simple expression for the principal term of the expansion. In fact in such a case one has, neglecting the remainder,

0 fij=-x’.(-&.+a) 3 N 3

’

(2)

where the prime denotes derivative, and the function xj(z) is the logarithm of the moment function, i.e. is defined by exp(xj ( z ) )

efSfme--nr dFj 0

The parameters 0 and a are determined by imposing the conditions (l),i.e. by requiring

We can now state the main result of the theory. If one defines the exchanged heat as the difference 6Q = dU - SW, where SW is the mean work performed by the system when an external parameter is changed, then one finds

One then finds that this expression admits BIN as an integrating factor (where 0 is the same quantity entering formula (2)). In fact, introducing vj -x>(z) as an independent variable and the Legendre transform hj(v) of the function xj(z), one indeed has S Q = -Nd ( E 1x h j ( f i j ) ) 0 As a consequence the quantity S = C j h j ( f i j ) / N can be identified with the entropy, and p = O/N with the inverse temperature. It is easy to verify that if the p.d.f. of the occupation number corresponds to a Poisson process (i.e. if F j ( n ) = CkSn e-Ppk/k!,to which there corresponds xj(z) = pe-’ - p ) one gets

ef

h=-

(vj logvj

1

- vj logp ,

60

i.e. the Gibbs distribution for the energy and (obviously) the Boltzmann formula for the entropy. Different p.d.f.'s will give rise to different expressions for both the entropy and the energy distribution. In particular, Figure 2 suggests that xj(z) could decrease more slowly than an exponential for increasing z , for example as an inverse power. As an illustration, one can consider the function

x ( 4 =pe,(-z) -P ,

ef +

where e,(z) (1 (1 - q)z)l/('-q) is the Tsallis q-deformation of the exponential, and one obtains 03

= C(&)(l+ Dq(q - 1 ) E j ) *

ef

+

where C(&) is a suitable normalizing constant, and p, p/(1 (q - 1)a).This distribution coincides with the Tsallis q-distribution(see Refs. [3]) for the energy, while the expressions for the entropy h also coincides with Tsallis q-entropy S, if we express h not in terms of V j , but in terms of the quantities p j p'/qq'/q-lvi'q.

ef

References 1. A. Carati, Physica A 348, 110-120 (2005). 2. M. Baranger,V. Latora, A. Rapisarda, Chaos, Solitons and Fractals 13, 471 (2002). 3. C. Tsallis, J . Stat. Phys. 52, 479 (1988); C. Tsallis, An. Acad. Br. Cienc. 74, no. 3, 393-414 (2002).

FUNDAMENTAL FORMULAE AND NUMERICAL EVIDENCES FOR THE CENTRAL LIMIT THEOREM IN TSALLIS STATISTICS *

HIROKI SUYARI Department of Information and Image Sciences, Chiba University, 263-8522, Japan E-mail: [email protected]

On the way to finding the mathematical structure behind Tsallii statistics, the rigorous formulation of the q-central limit theorem and its proof are expected to play important roles in mathematical physics. This short paper reports some numerical evidences revealing the existence of the central limit theorem in Tsallis statistics reviewing some fundamental formulas such as law of error and q-Stirling’s formula.

1. Q-product uniquely determined by Tsallis entropy Since the birth of Tsallis entropy S, := (1 - Cy=,p y ) / (q - 1), the main approach to the generalization of the traditional Boltzmann-Gibbs statistics has been the maximum entropy principle (MEP) along the same lines of Jaynes’ original ideas, which leads us to a variety of successful theoretical foundations and their applications to unify power-law behaviors in nature 23. Nowadays, the generalized statistical physics is called Tsallis statistics including the Boltzmann-Gibbs statistics as a special case. Through the history of sciences, we have learned an important lesson that there always exists a beautiful mathematical structure behind a new physics. This lesson stimulates us to finding it in Tsallis statistics4. On the way to the goal, we obtain some fundamental theoretical results in Tsallis statistics567. The key concept leading to our results is “q-product” uniquely determined by Tsallis entropy, which is independently introduced by Nivanen et al and Borges g. The q-product @q is defined as follows:

The definition of the q-product originates from the requirement of the following satisfactions:

1% .(

@q

Y) = In, z + In, Y, exp,

@q

expq (Y) = exp, (z + Y)

(2)

*This work was partially supported by the ministry of education, science, sports and culture, grant-in-aid for encouragement of young scientists(b), 14780259, 2004.

61

62 1-s-1

where In, z is the q-logarithm function In, z := 5(z > 0, q E 1-2

EX) and exp, (z)

is the q-exponential function exp, (z) := [1+ (1 - q) x]? with the notation [z]+ := max(0, z } . These functions, In, z and exp, (x), are originally determined by Tsallis entropy and its maximization 23. Moreover, exp, (z) is rewritten by means of the q-product. (3) This representation (3) is a natural generalization of the famous definition of the usual exponential function: exp (x) = lim (1 E)n. This fundamental property n+w

+

(3) reveals the conclusive validity of the q-product in Tsallis statistics. In the following sections, we briefly review the applications of q-product to the fundamental formulations in Tsallis statistics. 2. Law of error, q-Stirling’s formula, and q-multinomial coefficient in Tsallis statistics

Gaussian distribution, the most important distribution, is known to be mathematically characterized by three ways: (i) MEP for Shannon entropy under the second moment constraint, (ii) Gauss’ law of error, and (iii) central limit theorem. MEP for Tsallis entropy under the second moment constraint yields a q-Gaussian as a generalization of a Gaussian distribution. Therefore, we expect the law of error and the central limit theorem in Tsallis statistics. The law of error in Tsallis statistics has been already obtained using q-product 8,‘.

2.1. Law of e r r o r in Tsallis statistics Consider the following situation. We obtain n observed values X I , x2, . . ’ ,xn E R as a result of n measurements for certain observations, where we do not necessarily assume independency. However the infinitesimal probability that the value ( X I , .. . ,X n ) lies in the infinitesimal cube around (XI,.. . , z n ) is assumed t o be proportional to a function L, (8) defined by

L, (8) := f (51- 8) 8,f (22 - 8 ) 8,. . . 8,f (zn - 8)

(4)

for some 8.

Theorem 2.1. If the function L, (8) of 8 for any fied z1,22,. . . ,zn attains the maximum value at 9 = 8’ := CZl xi, then the probability density function f must be a q-Gaussian:

where /3, is a q-dependent positive constant. This q-Gaussian coincides with the probability distribution derived from MEP for Tsallis entropy under the second moment constraint, which recovers a Gaussian distribution when q -+ 1. See for the proof.

‘

63

2.2. Q-Stirling’s formula in Tsallis statistics Using the q-product, we naturally obtain the q-Stirling’s formula. For the q-factorial n!, for n E N and q > 0 defined by n!, := 1 8, . . . 8, n,

(6)

the q-Stirling’s formula (q # 1) is In, (n!,)=

and

q>O

( & + ~1) ~nl-q-1 +(-&)+(&-6,)if if

4#1,2

q=2

(7) where 6, is a q-dependent parameter which does not depend on n. Slightly rough expression of the q-Stirling’s formula ( q # 1) is In, (n!,)

N

{-2nq

(In, n - 1) if q > 0 and q # 1 , 2 n-Inn if q = 2

These q-Stirling’s formulas recover the famous Stirling’s formula when q -+ 1. See for the proof.

2.3. Q-multinomial coeficient in Tsallis statistics The q-multinomial Coefficientin Tsallis statistics is defined by

k where n = Ci=l ni, ni E N (i = 1 , . . . ,k) . 0, is the inverse operation to @,I which is defined by

x0,y:=

{

[.I-,

- yl-9

+ 13 i+t , if x > 0, y > 0,

0,

z1-q

- yl-Q

+ 1 > 0,

otherwise.

(10)

0, is also introduced by the following satisfactions as similarly as 8,. In, x 0, y = In, x - In, y,

exp, (x) 0, exp, (Y)= exp, (x - Y)

.

(11)

Applying the definitions of 8, and 0, t o (9), the q-multinomial coefficient is explicitly written as

From the definition (9), the q-multinomial coefficient clearly recovers the usual multinomial coefficient when q + 1. When n goes infinity, the q-multinomial coefficient (9) has a surprising relation to Tsallis entropy as follows:

64 The present relation (13) tells us some significant mathematical structures: (i) There always exists a one-to-one correspondence between Tsallis entropy and the q-multinomial coefficient. In particular, (13) reveals the following (%, . . . , %) is equivalent to important equivalence: “Maximization of S Z - ~ n that of the q-multinomial coefficient n1 . . . n k ] when n is large.”

[

(ii) The relation (13) reveals a surprising symmetry: (13) is equivalent to

for q > 0 and q # 2. This expression represents that behind Tsallis statistics there exists a symmetry with a factor 1 - q around q = 1. Substitution of some concrete values of q into (14) helps us understand this symmetry. 3. Numerical computations revealing the existence of the central limit theorem in Tsallis statistics

It is well known that any binomial distribution converge to a Gaussian distribution when n goes infinity. This is a typical example of the central limit theorem in the usual probability theory. By analogy with this famous result, each set of normalized q-binomial coefficients is expected to converge to each q-Gaussian distribution with the same q when n goes infinity. As shown in this section, the present numerical results come up to our expectations. In Fig.1 Fig.3, each set of bars and solid line represent each set of normalized q-binomial coefficients and q-Gaussian distribution with normalized q-mean 0 and normalized q-variance 1 for each n when q = 0.1,0.5,0.9, respectively. Each of the three graphs on the first row of each Fig represents two kinds of probability distributions stated above, and the three graphs on the second row of each Fig represent the corresponding cumulative probability distributions, respectively. From these 3 figures, we expect the convergence of a set of normalized q-binomial coefficients to a q-Gaussian distribution when n goes infinity. Other cases with different q represent the similar convergences as these cases. In order to confirm these convergences required for the proof of the central limit theorwm in Tsallis statistics, we compute the maximal difference Aq+ among the values of two cumulative probabilities (a set of normalized q-binomial coefficients and q-Gaussian distribution) for each q = 0.1, 0.2, . . . , 0.9 and n. Aq+ is defined max by Aq,n := .j=o ... n IFq-bino (2) - Fq-Gauss (z)I where Fq-bin0 (2) and Fq-Gauss (2) are cumulative probability distributions of a set of normalized q-binomial coefficients and its corresponding q-Gaussian distribution, respectively. Fig.4 results in convergences of Aq,nto 0 when n -+ co for q = 0.1, 0.2, . . . , 0.9. This result indicates that the limit of every convergence is a q-Gaussian distribution with the same q E (0,1] as that of a given set of normalized q-binomial coefficients. N

65

.

.

&

Fig

Fig.1-Fig.4.

Convergences of a set of normalized q-binomial coefficients to a q -Gaussian distribution

The present convergences reveal a possibility of the existence of the central limit theorem in Tsallis statistics. The central limit theorem in Tsallis statistics provides not only a mathematical result in Tsallis statistics but also the physical reason why there exist universally power-law behaviors in many physical systems. In other words, the central limit theorem in Tsallis statistics mathematically explains the reason of ubiquitous existence of power-law behaviors in nature.

References 1. C. Tsallis, J. Stat. Phys. 52, 479-487 (1988). 2. C. Tsallis et al., Nonextensive Statistical Mechanics and Its Applications, edited by S. Abe and Y . Okamoto (Springer-Verlag, Heidelberg, 2001). 3. C. Tsallii et al., Nonextensive Entropy: Interdisciplinary Applications, edited by M. Gell-Mann and C. Tsallis (Oxford Univ. Press, New York, 2004). 4. H. Suyari, IEEE Trans. Inform. Theory., 50, 1783 (2004). 5. H. Suyari and M. Tsukada, Law of error in Tsallis statistics, to appear in IEEE. Trans. Infcrm. Theory. 6. H. Suyari, q-Stirling's formula in Tsallii statistics, LANL e-print cond-mat/0401541. 7. H. Suyari, Mathematical structure derived from the q-multinomial coefficient in Tsallis statistics, LANL e-print cond-mat/0401546. 8. L. Nivanen, A. Le Mehaute, Q.A. Wang, Rep.Math.Phys. 52, 437 (2003). 9. E.P. Borges, Physica A 340, 95 (2004).

GENERALIZING THE PLANCK DISTRIBUTION

ANDRE M. C. SOUZA Departamento de Fisica, Universidade Federal de Sergipe 49100-000, Sao Cristovao-SE, Brazil E-mail: amcsouzaOufs.br CONSTANTINO TSALLIS Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM, 87501 USA E-mail: [email protected] and Centro Brasileiro de Pesquisas Fisicas Rua Xavier Sigaud 150, 28290-180 Rio de Janeiro-RJ, Brazil Along the lines of nonextensive statistical mechanics, based on the entropy Sq = k(1 C i p 5 ) / ( q - 1) ( S l = - k C i p i lnpi), and Beck-Cohen superstatistics, we heuristically generalize Planck’s statistical law for the black-body radiation. T h e procedure is based yq (with y(0) = l), on the discussion of the differential equation dy/dx = -aly-(a,-al) whose q = 2 particular case leads t o the celebrated law, as originally shown by Planck himself in his October 1900 paper. Although the present generalization is mathematically simple and elegant, we have unfortunately no physical application of it at the present moment. It opens nevertheless the door t o a type of approach that might be of some interest in more complex, possibly out-of-equilibrium, phenomena.

We normally obtain the statistical mechanical equilibrium distribution by optimizing, under appropriate constraints, an entropic functional, namely the Boltzmann-Gibbs (BG) entropy SBG = -k pi lnpi. The success and elegance of this variational method are unquestioned. But at least one more possibility exists, namely through differential equations. Such a path is virtually never followed. Indeed, such an approach might seem quite bizarre at first sight. But we should by no means overlook that it has at least one distinguished predecessor: Planck’ s law for the black-body radiation. Indeed, Planck published two papers on the subject in 1900. The first one in October, the second one in December The bases of both of them were considered at the time as totally heuristic ones, although kind of different in nature. The second paper might be considered as a primitive form of what has now become the standard approach to statistical mechanics, based on the optimization of an entropy functional, the connection with Bose-Einstein statistics, and, ultimately, with the Boltzmann-Gibbs thermal theory for a quantum harmonic oscillator The first paper ’, however, is totally based on simple arguments regarding an ordinary differential equation. It is along this line that the present paper

xi

’.

’.

66

67 is constructed. If SBG is extremized under appropriate constraints, we obtain the famous BG weight p ( E ) = p ( 0 ) e-PE. This distribution can be seen as the solution of the differential equation dp/dE = - p p . Since more than one decade, a lot of effort is being dedicated to the study of the so called “nonextensive statistical mechanics”, based on the generalized entropy S, = k(1- c i p : ) / ( q - 1) (S1= SBG) (for a review, see 5 ) . The extremization of this entropy under appropriate constraints yields p ( E ) = p ( 0 ) e;PE, where e; = [l (1 - q ) z ] ’ / ( l - Q )(eT = e z ) . This distribution, which has been shown to emerge in many natural and artificial systems 5 , can be seen as the solution of the differential equation d[p/p(O)]/dE= -pb/p(O)]‘J. As a next step, we may consider even more complex systems, namely those which exhibit, for increasing E , a crossover from nonextensive to BG statistics. Such appears to be the case of cosmic rays ‘. Such situations can be handled with a differential equation which unifiesthe previous two ones, as follows:

+

Excepting for the fact that here q may be noninteger, this differential equation is a particular case of Bernoulli’ s differential equation. Its solution is given by

P(E) =

P(0)

[1+ Lk (e(q-l)PlE - I)]

1

,

(2)

Fi

P1

which precisely exhibits the desired crossover for q > 1 and 0 < ,& 1, we have p 0: e-PIE. In the limit p,/p1 --t 00 and p(O)p1/& + C , where C is a constant, Eq. (2) becomes N

which, for q = 2, becomes

If we multiply this statistical weight by the photon density of states g ( E ) a E2 and by the energy E , we have the celebrated frequency spectral density

where we have identified + l/kBT and E hu. It is in this precise sense that Eq. (3) (hence Eq. (2)) can be seen as a generalization of Planck statistics. For q > 1, Eq. (3) can be written as --f

m

’ 0 = C

n=O

d(n,q ) e-OIEn ,

68 10 9 -

0

1.o

I

I

I

1.5

2.0

2.5

3.0

9 Figure 1. Degeneracy d ( n , q ) as function of q ( n = 0,1,2,3); q = 2 corresponds to Planck law.

where

En = [(q - 1).

1 + 11 E 0: n + q-1’

(7)

and

r(z)being the Gamma function. We may now follow Planck’ s path in his December 1900 paper, where he introduced the discretization of energy that eventually led to the formulation of quantum mechanics. Consistently, we may interpret En as a discretized energy and d ( n , q ) as its degeneracy. We see that, Vq > 1, the spectrum is made of equidistant levels, like that of the quantum one-dimensional harmonic oscillator. The situation is definitively different in what concerns the degeneracy (see Fig. 1). Only for q = 2 we have the remarkable property d ( n , 2 ) = 1 (Vn), which recovers the harmonic oscillator problem. At this point, let us emphasize that any thermostatistical weight (that of thermal equilibrium for instance) reflects the microscopic dynamics of the system. This

69 fact was addressed by Einstein in 1910 ', and was recently revisited by several authors (see 8 , for instance). It was shown also, on quite general grounds, in '. In the same vein, a dynamical theory of weakly coupled harmonic oscillators system was recently used for deducing the functional relation between energy variance and mean energy that was conjectured by Einstein in connection with Planck' s formula, thus exhibiting that it is a consequence of pure dynamics It is within this dynamical interpretation that Beck and Cohen introduced their superstatistics'l. Indeed, nonequilibrium systems might exhibit spatio-temporal fluctuations of intensive quantities, e.g., the temperature. They assumed then that the inverse temperature P might itself be a stochastic variable, such that the generalized distribution of energy is expressed as

'.

where the distribution f(P)satisfies dPf(P) = 1. The effective statistical mechanics of such systems depends on the statistical properties of the fluctuations of the temperature and similar intensive quantities. Naturally, if there are no fluctuations of intensive quantities at all, the system must obey BG distribution (i.e., f(P) = S(P - l/kBT)). They also showed that, if f(P) is the y-distribution (see also 12), one obtains the q-exponential weight of nonextensive statistical mechanics. Moreover, for small variance of the fluctuations, the nonextensive statistical distribution is once again reobtained. See l3 for an entropic functional which, extremized under appropriate constraints, recovers the distribution of superstatistics. We straightforwardly obtain, through Laplace transform, that the superstatis tical distribution f(P) corresponding t o the p(E)/p(O)given by Eq. (2) is

Moreover, we define

sow

dPf(P)(...). The notation qBc (BC stands for Beck-Cohen) has where (...) = been introduced to avoid confusion with the present q. Only when f (P) equals the y-distribution we have qBC = q. Using Eq. (10) and integrating we obtain

Replacing( 12) into (11)we obtain

-

It is worthy remarking that, for all admissible f(f?), we can write the asymptotic expression p(E)/p(O)= (e-oE) e-(o)E(l + where o = d m= (qBC -

O'

e),

70 I0

a

0.1 0.8

la1

-

so.-

0.8 02

0.0

00

-

@I

(4

01-

0.8

-

0.-

02-

0

2

1

3

Id

4

E

-

2

00

B

Figure 2. Functions

[#

$#( l e f t ) and f(0)(right) for (p) = 1. (a) Boltzmann-Gibbs distribution

= e P E ;f(0)= 6(p-1)]; (b) q = QBC = 1.8 distribution

f ( ~ )=

;[6(0-

&,-l.z50]; 0.8 . r ( i . 2 5 )

(c) ( q , Q B C ) = (2,3/2) distribution

=

(l+o,~E~E)'.25;

[% = (*;

f(p) =

[s

i)+ ;6(p - 1) + $6(p - 5 ) + ...I]; (d) ( q , q B C ) = (3/2,5/4) distribution = - f(0)= $[6(0- 4) + 6(p - a) + $6(0 - 1) + ...I]. In the cases (a,c,d), what is

w; 4 1-In2

[$

represented is not

f(0) strictly speaking, but rather the weights of the Dirac delta's.

Finally, we may rewrite distribution (2) as follows:

hence, through Laplace transform,

(15) Observe that, for all q, if qBc -+ 1 we obtain the BG distribution. In addition, we see that p generically assumes discrete values in f(P). If we focus on the limit of continuous values for p, we must have (using Eq. (10)) Ap 3 @(n 1) - P(n) = pI(q - 1) --+ 0, and this is obtained (see Eq. (13)) when (p) -+ 0 (i.e., high temperature) or qBc ---t q (i.e., q-statistics) . In Fig. 2 we present typical examples of pairs (P(E)/P(O), f(P)>. Summarizing, we obtained the distribution corresponding to the differential equation (l),expected to characterize a class of physical stationary states where a

+

71 crossover occurs between nonextensive and BG statistics. This led us to a possible generalization of Planck law. We obtained also the Beck-Cohen superstatistical distribution f(p) associated with such type of crossovers between statistics. Along similar lines, it is possible t o study crossovers between q and q’ statistics, with eventual applications in turbulence and other complex phenomena.

Acknowledgments Partial support from PCI/MCT, CNPq, PRONEX, FAPERJ and FAP-SE (Brazilian agencies) is acknowledged.

References 1. L.J. Boya, physics/O402064. 2. R. Balian, From Microphysics to Macrophysics, Vol. I, 140 and Vol. 11, 218 (SpringerVerlag, Berlin, 1991/1992). 3. M. Planck, Verhandlungen der Deutschen Physikalischen Gessellschaft 2, 202 and 237 (1900) [English translation: D. ter haar, S. G. Brush, Planck’s Original Papers in Quantum Physics (Taylor and Francis, London, 1972)]. In his “Ueber eine Verbessemng der Wien’schen Spectralgleichung” 19 October 1900 paper, Planck writes the following equations: d2S/dU2 = a / [ U ( p U] ( S and U being the entropy and internal energy respectively; a and p are constants), and dS/dU = 1/T (T being Kelvin’s absolute temperature). Replacing the latter into the former leads to d U / d ( l / T ) = @ / a )U ( l / a ) U 2 . From this differential equation, he eventually obtains his famous law, namely E = CA-5/(eC’XT - 1) (A being the wavelength; C and c are constants). If, as usually done nowadays, we express this spectral density in terms of the frequency u cc 1/X, we obtain the familiar expression, proportional to ~ ~ / ( e ~ ‘-” 1) / ~(c’ > 0 being a constant). 4. C. Tsallis, J. Stat. Phys. 52, 479 (1988); E.M.F. Curado and C. Tsallis, J. Phys. A24, L69 (1991) [Corrigenda: A24,3187 (1991) and A25, 1019 (1992)l; C. Tsallis, R.S. Mendes and A.R. Plastino, Physica A261,534 (1998). For a regularly updated bibliography of the subject see http://tsallis.cat.cbpf.br/biblio.htm. 5. M. Gell-Mann and C. Tsallis, Nonextensive entropy - Interdisciplinary Applications (Oxford University Press, New York, 2004). 6. C. Tsallis, J.C. Anjos and E.P. Borges, Phys. Lett. A310,372 (2003). 7. A. Einstein, Annalen der Physik 33,1275 (1910). 8. E.G.D. Cohen, Physica A305, 19 (2002); E.G.D. Cohen, Boltzmann and Einstein: Statistics and dynamics - A n unsolved problem, Boltzmann Award Communication at Statphys-Bangalore-2004, Pramana (2005), in press. 9. A. Carati, Physica A348, 110 (2005). 10. A. Carati and L. Galgani, Phys. Rev. E61,4791 (2000). 11. C. Beck and E.G.D. Cohen, Physica A321,267 (2003). 12. G . Wilk and Z. Wlodarczyk, Phys. Rev. Lett. 84,2770 (2000); C. Beck, Phys. Rev. Lett. 87,180601 (2001). 13. C. Tsallis and A.M.C. Souza, Phys. Rev. E67,026106 (2003).

+

+

THE PHYSICAL ROOTS OF COMPLEXITY: RENEWAL OR MODULATION?

PAOLO GRIGOLINI* Center for Nonlinear Science, University of North Texas, P.O. Box 311427, Denton, Texas 76203-1427 E-mail: grigodunt.edu Dipartimento di Fisica E. Fermi, Via Bonarroti, 2 I 56127, Pisa, Italy Istituto dei Processi Chimico Fisici del CNR Area della Ricerca di Pisa, Via G. Moruzzi 1,56124 Pisa, Italy

We show that the emergence of a non-Poisson distribution might have different physical origins. We study two distinct ways to generate a non-Poisson distribution, the first from within the renewal theory, and the second based on infinitely slow modulation, a condition that makes this second perspective equivalent to superstatistics. We prove that these different origins yield different physical effects, aging in the former case, and no aging in the latter.

1. Introduction

Here we adopt the simple minded definition of complexity science, as the field of investigation of multi-component systems characterized by non-Poisson statistics. On intuitive ground, this means that we trace back the deviation from the canonical form of equilibrium and relaxation, to the breakdown of the conditions on which Boltzmann’s view is based: short-range interaction, no memory and no cooperation. Thus, the deviation from the canonical form, which implies total randomness, is a measure of the system complexity. However, this definition of complexity does not touch the delicate problem of the origin of the departure from Poisson statistics. Here we limit ourselves to considering two different proposals, which we shall refer to as renewal and modulation, both generating non-Poisson distributions. Thus, to a first sight, one might be tempted to conclude that they are indistinguishable, leaving no motivation whatsoever to prefer the one to the other. We shall prove that it is not so, and that an aging experiment can be done, to distinguish modulation from renewal. *Work supported by grant 70525 of the Welch Foundation

72

.

73 2. An example of complex system: the blinking quantum dots

The physical process that we adopt here as a paradigm of complexity is the phenomenon of non-Poisson intermittent fluorescence, producing a sequence of ”light on” and ”light off’ states. The well known experiment by Dehmelt on a single ion, studied by Cook and Kimble ’, is an example of non-complex intermittence, given the fact that the distribution of sojourn times is exponential. An example of fluorescence intermittency to be termed complex is given instead by the blinking phenomenon in semiconductor nanocrystallytes 3. In fact, in this case the waiting time distributions are found to fit an inverse power law for some time decades. In this paper, for simplicity sake, we assume that the ”light on” and ”light oil” time distributions are identical, and are thus described by the same waiting time distribution $J(r). Throughout this paper we adopt the form

with p > 1. This distribution is properly normalized, and the parameter T , making this normalization possible, gives information on the lapse of time necessary to reach the time asymptotic condition where +(r)becomes identical to an inverse power law. We shall see that the main conclusions of this paper are not confined to the inverse power law form of Eq. (l),being valid for any form of non-Poisson distribution. The choice of the form of Eq. (1)is dictated by the simplicity criterion. This form has been known for many years 4, see for instance Ref. ’, and following Metzler and Nonnenmacher and Metzler and Klafter we shall be referring to it as Nutting law. This form is also obtained by means of entropy maximization from a non-extensive form of entropy and, for this reason, is referred to by an increasing number of researchers as Tsallis distribution. The theoretical discussion of this paper rests on a time series { ~ i } , created in such a way as to correspond to the distribution of Eq. (1). After creating this time series, according to either the renewal or modulation prescription, we use it to generate a sequel of events in time. The first event occurs at time t = 71, the second at time t = 71 72, and so on. The time intervals between two consecutive events are called laminar regions, and the reason for this name will be made clear by the discussion of Section 3.

’

+

3. Renewal

As a prototype of renewal model we shall refer to the following dynamic process. Let us consider a particle moving within the interval I = [0,1]driven by the following equation of motion d

-&Y

= ay=,

74

with 221,

(3)

and O 0), the system exhibits a ferromagnetic transition at the critical energy E, = 0.75JN. Here we will focus on the out-of-equilibrium behavior of the ferromagnetic HMF ( J > 0), when the system is prepared in a fully magnetized configuration, at an energy close below E,, with uniformly distributed momenta (“water-bag” initial conditions). Under these initial conditions the system evolves to a spatially homogeneous state with well defined macroscopic characteristics and whose lifetime increases with the system size, eventually reaching equilibrium. Numerical experiments have shown the disappearance of the family of homogeneous MESs below a certain energy close to 0.68JN. 334

2. Equations of motion

It is convenient to write the Hamiltonian (1) in the simplified form:

where we have introduced the magnetization per particle .

N

and for simplicity we have taken J = 1. The equations of motion read

for i = 1 , . . . ,N , with Oi = (cosOi, -sin&). Without loss of generality, we can set the axes such that m,(t = 0) = 0. If, additionally, the distribution of momenta is symmetrical, then m Z ( t )= 0, W. In that case, the equations of motion become

115

Notice that these equations can be seen as the equations for a pendulum with a time-dependent length. 3. First stage of relaxation Fully magnetized states violently relax t o a state of vanishing magnetization, within finite size corrections. The most elementary approach to describing the relaxation of m, from a given initial condition, is to perform a series expansion around t = 0, i.e.,

m(t) =

C k!1

-ck

tk.

k10

In our case, the initial condition is such that m = 1 (with m, = 0), then one obtains the following coefficients for mv(t)

Q=l c2

= -(P2)0

c4 =

( P ~+)4(P2)o ~

cs = - ((p6)o+ 26(p4)0 + 1 6 ( p 2 ) o + 1 8 ( p 2 ) i )

and C,dd = 0, where averages are calculated with the initid distribution of momenta h ( p ) . If h ( p ) at t = 0 is symmetrical around p = 0, then m(t) is an even function of t. In particular, if the initial condition is water-bag, i.e., Bi = 0,Vi and additionally pi are uniformly distributed in the interval [-p,,p,], then (from Hamiltonian (I), p, = 6 , with E the energy per particle), one obtains

;:(

;)

+-

t4

+.

,

The convergence of this series is very slow and, given that a general expression is not available, only the very short time of the relaxation can be described. 4. Vlasov equation

On the other hand, the evolution equation of the reduced probability density function (PDF) in p-space isformally equivalent to the Vlasov-Poisson system'

where V = - m . i ( B ) andm=JdBF(O)Jdpf(B,p,t). I f m = r n i j , then

df + p -af at

ae

-

msine-af = 0 , aP

(4)

116 with

The Vlasov equation (4) can be cast in the form

-af_ - [Lo+ Ll(t)l f, at

where Lo = -pa@ and & ( t ) = m(t)sinOap. We will consider states (for instance, with vanishing magnetization) for which the term L l ( t ) can be treated as a perturbation. It is convenient to switch t o the interaction representation, i.e., to define T(t) = e-Lotf((t), then

-

where L1 = eVLotLleLot. The equation for the propagator therefore,

6(t)=

1

6,such that F(t)= 6(t)T(O),is a6/at = zl6,

+

Jd

t

dt%l(t~)6(t~),

and recursively, one has

The solution at order k of the Vlasov Eq. (4)is

where the index (k) indicates the order at which the expansion (6) is truncated. Fk-om here on, we will deal with continuous distributions, hence our treatment is valid in the thermodynamic limit.

4.1. Lowest-order truncation

6

At zeroth-order, the propagator is approximated by N G ( O ) = 1. This is equivalent to neglecting the magnetization. Thus, if m = 0, the truncation is exact. For the initial distribution f(O,p,O) = g(O)h(p), where g(0) is uniform in [-7r,n] (hence, m = 0) and h(p) is an arbitrary even function, both distributions remain unaltered in time, consistently with the numerical simulations in Fig. 7 of '. In fact, if m = 0 for any time, there are no forces t o drive the system out of the macroscopic state. If the initial condition is f(0,p, 0) = 6(0)h(p),where 6 is the Dirac delta function and h(p) an arbitrary even function of p (as in our case of interest), although

117 m # 0 , L1 is small (it is null at t = 0 and remains small for later times), allowing a perturbative treatment. Then, we have f(O)(e,p,t) = e-Pta”(e,p,o)

= h(p)

W

1

q e -pt)

= %h(p)

C

eik(’-Pt).

k=-w Therefore,

h(o)(p,t) = S _ : w ( O ) ( e , p , t ) =

w,

that is, at zeroth-order, the distribution of momenta, whatever it is, does not change in time. However the angular distribution does indeed change. For instance, in the particular case of the water-bag distribution, where h(p) is a uniform distribution in [ - p O , p , ] , we obtain

Notice that the distribution of angles becomes uniform in the long time limit. It gets uniform through a mechanism of phase mixing, where particles do not interact (remember that magnetization has been neglected). From Eqs. (5) and (8), the zeroth-order magnetization is

whose expansion in powers of time yields

+ -103E 2 t 4 + . . .

m(O)(t)= 1 - &t2

Observe that this expansion up to second-order coincides with the exact one, given by Eq. (2), for any E . 4.2. First-order truncation

Now, recalling that $1

=1

+

d t l E l ( t l ) , from (7), at first-order, we have

f(l)(e,p,t ) = e-PtaOW(t)f(e,p,

+

= fO)(e,p,t)

where

El ( t )= ePtae

0)

1‘

dtlEl(tl)f(e,p,o),

m(t)sin 0 8, e-Ptae. Then

rt

118 1.o E

= 0.69

10.0

t

m2 ...... Om order -1st order

0.5

0.0 0.1

1.o

Figure 1. Squared magnetization as a function of time. The initial state is fully magnetized with uniformly distributed momenta for E = 0.69. Symbols correspond to numerical simulations with N = 1000, the distribution of momenta is regular. Dashed lines correspond to the zeroth-order approximation obtained from Eq. (9). Full lines correspond to the first-order approximation given by Eq. (10)

After some algebra, for the case h(p) uniform in [-p,,p,], we obtain

Substituting m(t)by m(')(t)one obtains the magnetization at first-order. Moreover,

+ h(p)

s' 0

dtl tlm(tl) cos(pt1)

+

.

We recall that, for the uniform distribution, h'(p) 0: [b(p p,) - b(p - p,)]. This explains why h(p, t ) presents two spikes at p = fp,. Fig. 1 shows the first stage of the relaxation of the magnetization. Numerical simulations were performed for N = 1000. Increasing the system size does not change the numerical curve in the time interval considered (t 5 50). Of course, for longer times the curve becomes size d e ~ e n d e n tThe ~ > ~squared magnetization rapidly decreases from its initial value m2 = 1 down t o zero at t 2: 2. Then, it remains very close to zero up to t = 20. From then on, one observes bursts of small amplitude. Since Vlasov equation is exact in the thermodynamic limit, it describes the exact N = 1000 evolution up to time t N 50. The zero order approximation describes correctly m2 vs t for a very short time (t 2: 0.4). The first-order approximation describes satisfactorily the violent initial relaxation (up t o t 1: 2), but it does not reproduce the structure appearing later. Higher order corrections are required to describe that behavior. Extrapolation of numerical sir nu la ti on^^?^ shows that m + 0 in the thermodynamic limit. This regime settles for times beyond the scope of our approximation.

119 4.3. Equilibrium

For completeness, let us discuss the distributions at thermal equilibrium'l. If the system has already attained equilibrium, then at = 0. Let also assume that the equilibrium distribution can be factorized, i.e., f ( 0 , p ) = g(0) h(p). Then, from (4)7 as ah pas h(p) = msinOg(0)-. aP Assuming h(p) = Aexp(-Pp2/[2fi]), Eq. (11) reduces to ag/a0 = -msinBg(0). Thus g(e) = Ceomcos',

(12)

with the normalization constant C = 1/[27rI0(/3m)],where I0 is the modified Bessel function of zeroth-order. The equilibrium magnetization can be obtained from the consistency condition (5):

thus recovering the results of canonical calculations

'.

4.4. Meta-equilibrium

Although we have not found the long-time solution of Vlasov equation, starting from fully magnetized initial conditions, numerical simulations3 indicate that in the thermodynamic limit the system tends to a spatially homogeneous state. We have seen in Sect. 4.1 that, once reached a homogeneous state, the distribution of momenta, whatever it is, does not change in time. But, the question is whether the homogeneous solutions are stable or not under perturbations. One one hand, the Vlasov approach is a good approximation to the discrete dynamics, on the other finite-size effects may be the source of perturbations that may take the system out of a Vlasov steady state. Therefore, we will perform a stability test (valid in the thermodynamic limit) and discuss the results under the light of the discrete dynamics. There is the well known Landau analysis which concerns linear stability. A more powerful stability criterion for homogeneous equilibria has been proposed by Yamaguchi et al. '. This is a nonlinear criterion specific to the HMF. It states that f(p) is stable if and only if the quantity

is positive (it is assumed that f is an even function of p ) . This condition is equivalent to the zero frequency case of Landau's recipe ','. Yamaguchi et al. showed that a distribution which is spatially homogeneous and Gaussian in momentum becomes unstable below the transition energy E,, = 3/4 (see also v,1)' in agreement

120 with analytical and numerical results for finite N systems. They also showed that homogeneous states with zero-mean uniform f(p) are stable above E = 7/12 = 0.58... (see also In the same spirit, it is instructive to analyze the stability of the family of q-Gaussian distributions ‘1’).

f(P) 0: exP,(-aPZ) = [I- 4 1 - q)P2]

VO-9) 7

(14)

which allows to scan a wide spectrum of PDFs, from finite-support to power-law tailed ones, containing as particular cases the Gaussian (q = 1) and the water bag (q = -m). In Eq. (14), the normalization constant has been omitted and the parameter a > 0 is related to the second moment (p2),which is finite only for q < 5/3. In the homogeneous states of the HMF one has (p2) = 2~ - 1, as can be easily derived from Eq. (1). Then, the stability indicator I as a function of the energy for the q-exponential family reads

I=1-

3-9 2(5 - 3q)(2~- 1)

’

Therefore, stability occurs for energies above E~

3 =

Eq

q-1

4 -t 2(5

-

3q) ’

(16)

The stability diagram is exhibited in Fig. 2. It is easy to verify that one recovers the known stability thresholds for the uniform and Gaussian distributions. We remark that Eq. (16)states that only finite-support distributions, corresponding to q < 1, are stable below E , ~ . This agrees with numerical studies in the meta-equilibrium regimes of the HMF. . .................................................... ................i.................. .:...............’ : i

513

0.5

7112

?

314

Figure 2. Stability diagram of the q-Gaussian ansatz for the momentum PDFs.

We have also shown recently l2 that a similar analysis can be performed for a very simple family of functions exhibiting the basic structure of the observed f(p), basically, a uniform distribution plus cosine. Fitting of numerical distributions leads

121 to points in parameter space that fall close t o the boundary of Vlasov stability, and exit the stability region for energies below the limiting value E N 0.68. This result is confirmed when the stability criterion is applied t o the discrete distributions arising from numerical simulations 12, although for the discrete dynamics the magnetization is not strictly zero. The stability index I is positive for energies above E N 0.68. The fact that the stability indicator becomes negative below E N 0.68 signals the disappearance of the homogeneous metastable phase at that energy. In fact, extrapolation of numerical simulations to the thermodynamic limit confirm this result. The present stability test only applies t o homogeneous states. Strictly speaking, m = 0 does not imply that the states are inhomogeneous. However, the sudden relaxation that leads to the present MESs mixes particles in such a way that m = 0 and spatial homogeneity are expected to be synonimous. Below E = 0.68, the measured distributions are evidently inhomogeneous ( m # 0). In these cases, negative stability refers to hypothetical homogeneous states having the measured f(p).

5. Final remarks We have seen that, although our approach is valid in the continuum limit, it gives useful hints on the finite size dynamics. Of course, it can not predict complex details of the discrete dynamics. However, the present approach gives information on the violent initial relaxation from fully magnetized states, for sufficiently large system. It also explains the dissapearance of homogeneous MESs below a certain energy observed by extrapolation of numerical simulations to the thermodynamic limit. Moreover, the identification of MESs with Vlasov solutions is also consistent with the fact that when the thermodynamic limit is taken before the limit t + oc), the system never relaxes t o true equilibrium, remaining forever in a disordered state.

Acknowledgements C.A. is very grateful to the organizers for the opportunity of participating of the nice meeting at the Ettore Majorana Foundation and Centre for Scientific Culture in Erice.

References 1. M. Antoni and S. Ruffo, Phys. Rev. E 53, 2361 (1995). 2. T. Dauxois, V. Latora, A. Rapisarda, S. Ruffo and A. Torcini, in Dynamics and Thermodynamics of Systems with Long Range Interactions, edited by T. Dauxois, S. Ruffo, E. Arimondo and M. Wilkens, Lecture Notes in Physics Vol. 602, Springer (2002). 3. A. Torcini and M. Antoni, Phys. Rev. E 59, 2746 (1999). V. Latora, A. Rapisarda, and S. Ruffo, Physica A 280, 81 (2000); V. Latora, A. Rapisarda, and C. Tsallis, Phys. Rev. E 64, 056134 (2001); V. Latora and A. Rapisarda, Chaos, Solitons and Ractals 13, 401 (2002); A. Giansanti, D. Moroni, and A. Campa, ibid., p. 407; V.

122 Latora, A. Rapisarda, and C. Tsallis, Physica A 305,129 (2002);M. Montemurro, F. A. Tamarit and C. Anteneodo, Phys. Rev. E 67,031106 (2003). 4. Pluchino, V. Latora and A. Rapisarda, Physica D 193,315 (2003). 5. C. Tsallis, J. Stat. Phys. 52, 479 (1988);C. Tsallis, in Nonextensive Statistical Mechanics and ats Applications, edited by S. Abe and Y. Okamoto, Lecture Notes in Physics Vol. 560 (Springer-Verlag, Heidelberg, 2001); Chaos, Solitons and Fractals 13, 371 (2002); Non Extensive Thermodynamics and Physical Applications, edited by G. Kaniadakis, M. Lissia, and A. Rapisarda, Physica A 305 (Elsevier, Amsterdam, 2002). See http://tsallis.cat.cbpf.br/biblio.htm for further bibliography on the subject. 6. M. Antoni, H. Hinrichsen, and S. Ruffo, Cham, Solitons and Ractals 13,393(2002). 7. R. Balescu, Statistical Dynamics (Imperial College Press, London, 2000). 8. Y.Y.Yamaguchi, J. Bar& F. Bouchet, T. Dauxois and S. Ruffo, Physica A 337 , 36 (2004). 9. M.Y. ‘Choi and J. Choi, Phys. Rev. Lett. 91, 124101 (2003). 10. S. Inagaki, Prog. Theo. Phys. 90,577 (1993). 11. V. Latora, A. Rapisarda and S. Ruffo, Physica D 131,38 (1999). 12. C.Anteneodo and R.O. Vallejos, Physica A 344,383 (2004).

WEAK CHAOS IN LARGE CONSERVATIVE SYSTEM INFINITE-RANGE COUPLED STANDARD MAPS

LUIS G . MOYANO Centro Brasileiro de Pesquisas Fisicas Rua Xawier Sigaud 150, Urca 22290-180 Rio de Janeiro, Brazil E-mail: moyanoOcbpf.br ANA P. MAJTEY Facultad de Matemcitica, Astronomia y Fisica Universidad Nacional de Co'rdoba, Ciudad Universitaria 5000 Co'nloba, Argentina E-mail: [email protected] CONSTANTINO TSALLIS Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA E-mail: tsallisOsantafe.edu and Centro Bmsileiro de Pesquisas Fisicas, Rua Xavier Sigaud 150, Urca 22290-1 80, Rzo de Janeiro, Brazil We study, through a new perspective, a globally coupled map system that essentially interpolates between simple discrete-time nonlinear dynamics and certain long-range many-body Hamiltonian models. In particular, we exhibit relevant similarities, namely (i) the existence of long-standing quasistationary states (QSS), and (ii) the emergence of weak chaos in the thermodynamic limit, between the present model and the Hamiltonian Mean Field model, a strong candidate for a nonxtensive statistical mechanical approach.

PACS numbers: 05.10.-a, 05.20.Gg, 05.45.-a, 05.90.+m 1. Introduction

In the last years, considerable effort has been made in order to clarify the role that nonextensive statistical mechanics' plays in physics. In this context, there has been significantly growing evidence relating several, physically motivated, nonlinear dynamical systems. It has been repeatedly put forward that the statistical behaviour of a physical system descends from its microscopic dynamics2. Consequently, the study of paradigmatic nonlinear dynamical systems is important in order to describe and

123

124 understand anomalies and deviations from the well known Boltzmann-Gibbs (BG) statistical mechanics. The scenario within which we are working tries to capture the most relevant features of nonextensive statistical mechanics in the complete range of dynamical systems: from extremely simple dissipative low-dimensional maps to complex conservative Hamiltonian dynamics3. In the present paper we will make specific progress along these lines by focusing on a model which illustrates the deep similarities that can exist between nonlinear coupled maps and many-body Hamiltonian dynamics. Let us first recall a paradigmatic and intensively studied many-body infiniterange coupled conservative system, namely the Hamiltonian Mean Field (HMF) model4s5:

The HMF model may be thought of as N globally coupled classical spins (inertial version of the XY ferromagnetic model). Its molecular dynamics exhibit a remarkably rich behaviour. When the initial conditions are out of equilibrium (for example, the so called waterbag initial conditions6), it can present an anomalous temperature evolution (we consider T = 2 K / N , being K the total kinetic energy). These states are characterized by a first stage (quasistationary state, Q S S ) whose temperature is different from that predicted by the BG theory, followed by a crossover t o the expected final temperature. These QSS appear to be a consequence of the longrange coupling. They are important because their duration diverges with N , thus becoming the only relevant state for a macroscopic system'. At the other end of the range of dynamical systems we may consider a dissipative, one-dimensional model, such as the logistic map (and its universality class): 2t+l

= 1 - px:

(t = 0 , 1 , 2...; 2 E [-1,1]; p E [0,2]).

(2)

Because of its physical importance, the logistic map is one of the most studied lowdimensional maps. Despite its apparently simple form, it exhibits a quite complex behaviour. Important progress has recently been made which places this model as an important example of the applicability of nonextensive statistical mechanical concepts. Indeed, Baldovin and Robledo8 rigorously proved that, at the edge of chaos (as well as at the doubling-period and tangent bifucations), the sensitivity t o initial conditions is given by a q-exponential function Furthermore, they proved8 the q-generalization of a Pesin-like identity concerning the entropy production per unit time. For the stationary state at the edge of chaos, the entropic index q can be obtained analytically. Moreover, when a small external noise is added, a twestep relaxation evolution is foundg, similarly to what occurs for the HMF case. At this point, a natural question may arise. Is it possible to relate the results found for such simple maps to the anomalies found in the HMF model? Further-

'.

125 more, can we treat various nonlinear dynamical systems within the nonextensive statistical mechanics theory? Many studies are presently addressing such questions. A first step that can be done in this direction is to move closer to a Hamiltonian dynamics by considering a symplectic, conservative map. This is the case of the widely investigated Taylor-Chirikov standard maplo:

+ 1) + q t ) + 1) = p ( t ) + f sin[2d(t)]

e(t + 1) = p ( t p(t

(mod 11, (mod 1 ) .

(3)

This map may be obtained, for instance, by approximating the differential equation of a simple pendulum by a centered difference equation, and converting a secondorder equation into two first-order equations. The standard map was studied along the present lines by Baldovin et all’. For symplectic maps, what plays a role analogous to the temperature is the variance , ( ) denotes the ensemble of the angular momentum: T = 0; = ( p 2 ) - ( P ) ~where average. Beginning with the same type of initial conditions as before (waterbag), we observe once again a two-plateaux relaxation process, suggesting a connection with the phenomena already described for the HMF model. A step forward to capture the behaviour of the HMF system of rotors is to consider N standard maps, coupled in such way as to mantain their symplectic (hence conservative) structure. However, there are several ways to achieve this. A particular coupling in the m o m e n t a has been recently addressedg with quite interesting results such as QSS relaxation and nonergodic occupation of phase space. A different type of coupling is addressed in the next Section. 2. Symplectic coupling in the coordinates

As before, we consider N standard maps but, this time, with a global, symplectic coupling in the coordinates :

This coupling arises as a natural choice. In fact, applying to the HMF model the difference procedure mentioned above for the standard map, we obtain precisely the a = 0 particular instance of model (4). This model has already been addressed in the literature12, but in a quite different context, related to the study of the Lyapunov exponents in the completely chaotic regime. We present next numerical simulations of the map system (4).We calculated the evolution of the variance of the momenta 0;. Our results show that, for waterbag initial conditions and appropriate ranges for the parameters a and b, two-step relaxation processes are again found. In Fig. 1 we show these results for different sizes of the system. It can be seen that the crossover time t , grows as t , N N1.07* thus never reaching BG equilibrium when N 00. In other words, the N + co and t + co limits do not commute. --f

126

0.02

t

Il '

8

Figure 1. Temperature evolution illustrating the presence of tw+step relaxation (QSS) for typical system sizes. We have used a = 0.05, b = 2, and waterbag initial conditions within po = 0.3f0.01. Ensemble averages were done, typically over 100 realizations. Only much longer simulations could confirm, or exclude, the possibility that all curves, i.e. VN, saturate at the equal-probabilityvalue 1/12 N 0.08. Inset: The crossover time t , corresponds to the inflexion point of T versm log t.

Finally, we calculated the largest Lyapunov ezponent (LLE) XL (we recall that Lyapunov exponents measure the instability of dynamicd trajectories, and provide a quantitative idea of the sensitivity t o the initial conditions of the system). Indeed, for the (a, b)-parameters in the range illustrated in Figs. 1 and 2, we found that the dependence of the LLE with the sistem size is consistent with XL N-0.40*0 i.e., a clear indication of weak chaos in the thermodynamic limit. N

0' 10'

' ",,...'

lo2

''sl.*.*'

lo3

''*ll..ll

lo4

''....I.'

lo5

''......I

lo6

".,.A t

Figure 2. Time dependence of the eflective largest Lyapunov exponent Xr. for typical sizes (same parameters as in Fig. 1). Inset: N-dependence of the asymptotic value of XL, consistent with weak chaos in the thermodynamic limit.

Summarizing, we presented a conservative model consisting in N standard maps

127 symplectically coupled through the coordinates. We have found results suggestively similar to those obtained for other nonlinear dynamical systems including the HMF model. More specifically, we found the double plateaux in the time evolution of the temperature, and a LLE which approaches zero for increasing size. We are currently studying several other quantities (e.g., correlation functions and momenta probability distribution functions), as well as the influence of (a, b) on the present ones. These results place naturally the present system within a series of nonlinear dynamical systems which starts with one-dimensional dissipative maps, follows with low-dimensional conservative maps, then many symplectically coupled maps, and ends with long-range many-body Hamiltonians. They all share important phenomena, typically related, in one way or another, to weak cham and long-standing nonergodic occupation of phase space. These features precisely constitute the scenario within which nonextensive statistical mechanics appears t o be the adequate thermostatistical theory, in analogy to Boltzmann-Gibbs statistical mechanicis, successfully used since more than one century for strongly chaotic and ergodic systems. LGM thanks the organizers for warm hospitality at the meeting in Erice, Italy, in particular A. Rapisarda. Partial finantial support from CNPq, Faperj and Pronex/MCT (Brazilian agencies) is acknowledged as well.

References 1. C. Tsallis, J. Stat. Phys. 52, 479 (1988). For a review see M. Gell-Mann and C. Tsallis, eds., Nonextensive Entropy - Interdisciplinary Applications, (Oxford University Press, New York, 2004). For bibliography see http://tsallis.cat.cbpf.br/biblio.htm 2. E.G.D. Cohen, Physica A305, 19 (2002); C. Tsallis, Physica A340, 1 (2004); E.G.D. Cohen, Boltzmann Award Communication at Statphys-Bangalore-2004, Pramana (2005), in press. 3. C. Tsallis, A. Rapisarda, V. Latora and F. Baldovin, in Dynamics and Thermodynamics of Systems with Long-Range Interactions, eds. T. Dauxois, S. Ruffo, E. Arimondo and M. Wilkens, Lecture Notes in Physics 602 (Springer, Berlin, 2002), p. 140. 4. M. Antoni and S. Ruffo, Phys. Rev. E52, 2361 (1995). 5. T. Dauxois, V. Latora, A. Rapisarda, S. Ruffo and A. Torcini, in Dynamics and Thermodynamics of Systems with Long-Range Interactions, eds. T. Dauxois, S. Ruffo, E. Arimondo and M. Wilkens, Lecture Notes in Physics 602 (Springer, Berlin, 2002), p. 458. 6 . By waterbag initial conditions we mean totally aligned spins, with momenta taken from

a uniform distribution. See, for instance, A. Pluchino, V. Latora and A. Rapisarda, Physica A338, 60 (2004). 7. A. Pluchino, V. Latora and A. Rapisarda, Physica D193, 315 (2004). 8. F. Baldovin and A. Robledo, Phys. Rev. E66, 045104(R) (2002); F. Baldovin and A. Robledo, Phys. Rev. E69, 045202(R) (2004). 9. F. Baldovin, L.G. Moyano, A. P. Majtey, A. Robledo and C. Tsallis, Physica A340, 205 (2004). 10. E. Ott, Chaos in Dynamical Systems, (Cambridge University Press, Cambridge, 1993). 11. F. Baldovin, E. Brigatti and C. Tsallis, Phys. Lett. A320, 254 (2004). 12. V. Ahlers, R. Zillmer and A. Pikovsky, Phys. Rev. E63, 036213 (2001).

DETERMINISTIC AGING *

ELI BARKAI Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel E-mail: barkaieamail. biu.ac.il We investigate aging behavior in a dynamical system: a non-linear map which generates sub-diffusion deterministically. Behaviors of the diffusion process are described using aging continuous time random walks. We briefly relate the aging behavior to other anomalous features of the map: q exponential sensitivity of trajectories to initial conditions, divergence of escape times from unstable fixed points, anomalous diffusion, breaking of ergodicity, and the absence of an invariant measure.

There is growing interest in physical systems which exhibit aging behavior. Aging is found in glasses, polymers, and in random walks in random environments. These disordered complex systems are composed of many interacting units and stochastic forces govern their dynamics. In contrast we recently showed that a low dimensional model, a deterministic non-linear map which has no element of disorder built into it, exhibits aging behavior The aging behavior is related to diverging of average waiting time in vicinity of unstable fixed point of the map under investigation (see details below) The diverging waiting time is also responsible for a non-stationary evolution which leads t o anomalous diffusion and ergodicity breaking which are behaviors related t o aging. The dynamics of the map in vicinity of the unstable fixed point, is governed by q exponential sensitivity of the trajectories on initial conditions, i.e. weak chaos (see details below). The relation of such q exponential behavior in models related t o ours and Tsallis statistics is a subject of ongoing research Probably the simplest theoretical tool which generates normal and anomalous diffusion deterministically are one dimensional maps

'.

'.

39495.

Zt+l = xt

+F(zt)

(1)

with the following symmetry properties of F ( x ) : (i) F ( z ) is periodic with a periodicity interval set t o l, F ( z ) = F ( z N ) , where N is an integer. (ii) F ( z ) has inversion anti-symmetry; namely, F ( z ) = - F ( - z ) , while t in Eq. (1)is the discrete time. Geisel and Thomae considered a rather wide family of such maps which behave like

+

F ( z ) = axz for z + +0, *This work is supported by the center of complexity-jerusalem

128

(2)

129 where z > 1. Eq. (2) defines the property of the map close to its unstable fixed point. In numerical experiments soon to be discussed I will use the map 1

F ( z ) = (2z)=, 0 5 z 5 2

(3)

which together with the symmetry properties of the map define the mapping for all z. In Fig. 1 I show the map for three unit cells. It is important to emphasize that the main properties of the aging behavior I investigate will not be sensitive to the detailed shape of the map, besides its behavior in vicinity of the fixed points Eq. (2). To investigate aging, e.g. numerically, I choose an ensemble of initial conditions x-t, which is chosen randomly and uniformly in the interval -112 < x - ~ ,< 112. The quantity of interest is the displacement in the interval (0, t ) ,z = zt -20 which is obtained using the map Eq. (1). Previous work ' 6 considered the non-aging regime, namely t, = 0. In numerical simulations averages like (z2(t,,t ) ) ,are averages over the set of initial conditions, which generally depend both on t and on t,. In an ongoing process a walker following the iteration rules may get stuck close to the vicinity of unstable fixed points of the map (see Fig. 1). It has been shown, both analytically and numerically, that probability density function (PDF) of escape times of trajectories from the vicinity of the fixed points decays like a power law '. To see this, one considers the dynamics in half a unit cell, say 0 < 2 < 112. Assume that at time t = 0 the particle is on z* residing in vicinity of the fixed point z = 0. Close to the fixed point we may approximate the map Eq. (1) with the differential equation dzldt = F ( z ) = axz. This equation is reminiscent of the equation defining the q generalized Lyaponov exponent '. The solution is written in terms of the q-exponential function, ezpq(y) = [l (1 - q)y]'/@-q) where q = 2 and

+

(4) We invert Eq. (4)and obtain the escape time from z* to a boundary on b (z* < b < b z*-z+l b-'+l 112) is t N S,.[F(z)]-'da: using Eq. (2) t N a - x] , a In q behavior. The PDF of escape times $(t) is related to the unknown PDF of injection points q(z*),through the chain rule $(t) = v(z*)ldz*/dtl. Expanding q(z*)around the unstable fixed point z* = 0 one finds that for large escape times

' T[

- r(--o

$(t)

A

t-l-a

1

, a = - (2 - 1) '

(5)

where A depends on the PDF of injection points, namely on how trajectories are injected from one cell to the other. The parameter A will be sensitive to the detailed shape of the map, which implies that it is non-universal. In contrast the parameter z depends only on the behavior of the map close to the unstable fixed points. When z > 2 corresponding to a < 1 the average escape time diverges. The consequence of this is a non-stationary evolution which leads to anomalous diffusion, ergodicity

130

*

breaking, and aging. In turn these behaviors are related to the observation that the invariant time independent density is never reached (the latter is defined only on 0 < x < 1/2 with suitable boundary conditions and see also '). Since in our problem q = z > 1 the relation between q and the anomalous diffusion exponent is Ly = l/(q - 1).

-0.5

0

0.5

1 xl

I 1.5

2

2.5

Figure 1. The map zt+l = zt f F ( z t ) ,defined by Eq. (3) with z = 3. The linear dash-dot curve is zt+l = zt. The unstable fixed points are on zt = 0,1,2.

To consider stochastic properties of the aging dynamics I now investigate aging continuous time random walks ACTRW loill, deriving an explicit expression for the asymptotic behavior of the Green function. ACTRW describes the aging properties of the well known CTRW, and was introduced in the context of aging of the trap model by Monthus and Bouchaud lo. ACTRW considers a one-dimensional nearest neighbor lattice random walk, where lattice points corresponds to the cells of the iterated maps. Waiting times on each lattice point are assumed to be described by $(t). Note that after each jumping event it is assumed that the process is renewed, namely, we neglect correlation between motions in neighboring cells. This assumption will be justified later using numerical simulations. As mentioned start of the ACTRW process is at t = -t, and our goal is to find the ACTRW Green function P ( x ,t,, t ) , were 2 is the random displacement in the interval (0, t ) after the random walk was aged for a period t,. In ACTRW we must introduce the distribution of the first waiting time tl: the time elapsing between start of observation at t = 0 and the first jump event in the interval (0,t). Let ht,(tl) be the PDF of tl. Let hS(u)be the double Laplace

131 transform of hta(tl)

11,12

when z > 2 corresponding to a

< 1 in Eq. (5) sin (7ra)

A

hta(tl)

t,*

(7)

t y (tl f t a ) '

which is valid in the long aging time limit. Note that Eq. (7) is independent of the exact form of $ ( t ) , besides the exponent a. When a + 1 the mass of the PDF ht,(tl) is concentrated in the vicinity of tl + 0, as expected from a 'normal process'. I have checked numerically the predictions of Eq. (7) for z = 3, analyzing trajectories generated by the map Eq. (3) with three different aging times. In Fig. 2 I show the probability of making at-least one step in the interval (0,t ) :$ hta( t ) d t = l - p o ( t , , t ) ,where po(t,, t ) is the probability of making no steps, i.e. the persistence probability. The results show a good agreement between numerical results and the theoretical prediction Eq. (7) without fitting. Fig. 2 clearly demonstrates that as the aging time becomes larger the time for the first jumping event, from one cell to its neighbor, becomes larger in statistical sense (i.e. the older the particle gets its tendency to make a jump decreases). The aging behavior is clearly related to the slow escape times from the vicinity of fixed points (when z > 2), as the age of the process is increased there is more likelihood of finding the particles very close to the

09080706-

-a o s I

-

cp 04

-

030201

-

00

1

2

3 I

4

5 x lo'

Figure 2. The probability of making at least one step in a time interval (0,t ) for different aging times specified in the figure. The solid curve is the theoretical prediction EQ. (7),the dotted, dashed, and dot dashed curves are obtained from numerical solution of the map with z = 3.

132 unstable fixed points, which in turn means that they become more localized. The interesting observation is that this aging behavior is captured by the limit theorem Eq. (7). We now investigate the ACTRW Green function. Let P ( k ,s, u ) be the doubleLaplace -Fourier transform (z + k , t, + s, t + u ) of P ( x ,t,, t ) , then showed lill

1

P ( k , s, u ) = su

+

[Ict (4- Ict ( S ) l P - cos I)@ 2L

(u - s) [l- 11(s)] [l - ? (u) /l cos ( k ) ].

Eq. (8) is a generalization of the well known Montroll-Weiss equation describing the non-equilibrium CTRW process l31I4. Note that only if the underlying process is a Poisson process, the Green function P ( z ,t,, t ) is independent of the age of the process t,. Before considering the behavior of the Green function P ( x ,t,, t ) let me consider the second moment. By differentiating Eq. (8) with respect to k twice and setting k = 0, and using Tauberian theorem, I obtain the mean square displacement of the random walk for t ,t , >> All"

For times t >> t, I recover the standard CTRW behavior, (z2(t,,t)) 0: t" 14. For t 12913

Physical requirements l4 on the resulting entropy select l5 0 5 a 5 1and 0 5 p < 1. All the entropies of this class: (i) are concawe 12, (ii) are Lesche stable 16, and (iii) yield nonnalizable distributions 15; in addition, we shall show that they (iv) yield a finite non-zero asymptotic rate of entropy production for the logistic map with the appropriate choice of a. We have considered the whole class, but we shall here report results for three interesting one-parameter cases: (1) the original Tsallis proposal (a = 1 - q, /3 = 0):

(2) Abe’s logarithm

+

+

where PA = 1/(1 a ) and p = a/(l a ) , which has the same quantum-group symmetry of and is related to the entropy introduced in Ref. 17; (3) and Kaniadakis’ logarithm, a = /3 = IC, which shares the same symmetry group of the relativistic momentum transformation l8

The sensitivity to initial conditions and the entropy production has been studied in the logistic map xi+l = 1 - pxq at the infinite-bifurcation point pm = 1.401155189. The generalized logarithm G(E) of the sensitivity, E(t) = ( 2 p ) t lzil ~ ~for~1 ~5 t 5 80, has been uniformly averaged by randomly choosing 4 x lo’ initial conditions -1 < xo < 1. Analogously to the chaotic regime, the deformed logarithm of E should yield a straight line g(E(t)) =G(G(At)= ) At. Following Ref. 8, where the exponent obtained with this averaging procedure, indicated by (...), was denoted q:lns for Tsallis’ entropy, each of the generalized logarithms, (G( %K. when ri = d2, as found from DF calculations, nanotubes are more likely to occur. However the local values of i? and K either at the surface termination into vacuum, where the growth takes place by cluster addition, or at the contact with a catalyst, are likely to be quite different from the above values (which have been fitted to regular structures) and should be obtained from ab-initio calculations. One should consider that the local change in the electronic structure, e.g., a IT bond-charge depletion or accretion, can substantially modify F . The charge redistribution produced by a catalyst depends on the actual size of catalyst nanoparticles, which may explain why the growth of schwarzites supersedes that of nanotubes when metallorganic precursors are used. In this case the metallic particles are in general very small and highly dispersed. The ordinary configurationalentropy of a schwarzitemade of fs 6-fold rings and fi 7-fold rings can be derived from number of possible tiling combinationsforfa +f7=fand is given by

Thus the free energy at temperature Tcan be obtained from Eqs. (10) and (12) and the minimal conditionH = 0 as

where As is the area of a 6-fold ring. The total area A is related to the connectivity and the total number of atoms N. by the equation

where A* = 6A7- 7As = 0.598 a*,with AT the area of the 7-fold ring and d the average interatomic distance. For a periodic schwarzite [A is proportional to A and both are proportional to No. Thus also F(A,Q, Equation 15, is proportional to A, and can be written as

153 where ?? is the average Gauss curvature defmed through the Gauss-Bonnet theorem, Equation (12) as = 2xx / A. This ensures an exact extensivity for the thermodynamic functions of periodic schwarzites. For a fxed No

and T there are equilibrium values for the connectivity and the pore average size R ,which are obtained from Equation (15) by setting a F ( A , T ) l a x = O . Itis found

with the activation potential @ = Zm-+yA* g 29.8 eV . For SCBD experiments with an average deposition energy E. = 0.15 eV per atom it may be assumed kT = 2E. = 0.3 eV, which gives R=880nrn. The present calculated mesoscopic size of the pores has the right order of magnitude of, though somewhat larger than those observed in TEM images for the same deposition energy. 2.3 Serf-aflnily and Non-extemivily

The observed random schwanites, however, are far from being periodic. Probably a better model is the self-affine construction described by Equation (5). For this model the total area A is no longer simply proportional to x. It is found

where the film thickness t (also in units of ao)is also dependent on x. This clearly makes, for p> 0, the free energy, Equation (1 5), non-extensive, and yields a correcting factor in the expression of the equilibrium connectivity:

and similarly for the average pore size. There is not enough information about the actual value of a0 (something of the order of the initial pore size at the catalyst surface) for a quantitative comparison with the experiment, though Equation (21) introduces some interesting dependence of the ratio I x I eq/Na(constant for the periodic case) on the film thickness through the (non-zero) growth exponent fl These aspects would deserve further investigation. The non-extensivity of the thermodynamic functions for the self-affine structures suggests an analysis in terms of Tsallis non-extensive entropy [13,14], which in the present case is written as

with the parameter q # 1 andfa +f7=f: It is easy to show that for q + 1 the q-entropy S, tends to the ordinary entropy S , Equation (14). By constructing Fq = E - TS, with E given by Equation (lo), H = 0 and the integration made with the Gauss-Bonnet theorem, and by minimizing F, with respect to x at a constant No, the equilibrium connectivity and average pore size can be obtained in a rather involved algebraic form. However in the limit of small q - 1 it is found

154

The comparison of Equation (23) with Equation (21) shows some link between the deviation from extensivity q - 1 and the emergence of self-affinity, as argued from Equation (20).

Acknowledgement

One of us (G.B.) acknowledges a partial support by MIUR, Italy, under the program PP.IN03.

References

1. H. W. Kroto, J. R. Heath, S. C. O'Brien, R. F. Curl and R. E. Smalley,Nature 318, 162 (1985). 2. S. Iijima, Nature 324, 56 (1991). 3. L. D. Ratter, Z. Schlesinger, J. P. McCauley, N. Coustel, J. E. Fisher and A. B. Smith, Nature 355,532 (1992). 4. H. Wang, A. A. Setlur, J. M. Lauerhaas, J. W. Dai, E. W. Seelig and R. P. H. Chang, Appl. Phys. Lett. 72,2912 (1998). 5. C. Niu, E. K. Sichel, R. Hoch, D. May and H. Tennent, Appl. Phys. Lett. 70, 1480 (1997). 6. G. Benedek and M. Bernasconi, in Encyclopaedia of Nanoscience and Nanotechnology (Marcel Dekker, Inc., New York 2004) p. 1235. 7. E. Barborini, P. Piseri, P. Milani, G. Benedek, C. Ducati and J. Robertson, Appl. Phys. Lett. 81, 3359 (2002) and E. Gerstner, Nature, Materials Update, 7 Nov 2002. 8. P. Milani e S. Iannotta, Cluster Beam Synthesis ofNanoshrctured Materials (Springer, Berlin 1999). 9. G. Benedek, H. Vahedi-Tafieshi, E. Barborini, P. Piseri, P. Milani, C. Ducati and J. Robertson, Diamond and Rel. Muter. 12,768 (2003). 10. D. Donadio, L. Colombo, P. Milani and G. Benedek, Phys. Rev. Lett., 84,776 (1999). 1 1 . T. Lenosky, X. Gonze, M. Teter and V. Elser, Noture 355,333 (1992). 12. M. Bogana, D. Donadio, G. Benedek and L. Colombo, Europhys.Lett., 54,72 (2001). 13. C. Tsallis, J. Stat. Phys.. 52,479 (1988). 14. C. Tsallis, in Non-extensive Entropy - Interdisciplinary Applications, M. Gell-Mann and C. Tsallis Eds. (Oxford University Press, 2004) p.1, and present volume. 15. C. Tsallis, present Volume. 16. A. L. McKay, Nature 314,604 (1985). 17. A. L. McKay andH. Terrones, Nature 352,762 (1991). 18. H. Terrones and A. L. McKay, in The Fullerenes, H. W. Kroto, J. E. Fisher and D. E. Cox, Eds. (Pergamon Press, Oxford 1993) p. 113. 19. D. Vanderbilt and J. Tersoff, Phys. Rev. Lett. 68,511 (1992). 20. S. Gaito, L. Colombo and G. Benedek, Europhys. Left. 44,525 (1998). 21. H. A. Schwarz, Gesammelte Mathematische Abhandlungen (Springer, Berlin 1890). 22. M. O'Keeffe, G. B. Adam and 0. F. Sankey, Phys. Rev. Len. 68,2325 (1992). 23. G. Benedek, L. Colombo, S. Gaito, E. Galvani and S. Serra, J. Chem. Phys. 106,23 11 (1997). 24. M. CotC, J. C. Grossman,M. L. Cohen and S. G. Louie, Phys. Rev. 858,664 (1998). 25. D. Hilbert and S. Cohn-Vossen, Amchauliche Geornefrie (Springer, Berlin 1932). 26. S . T. Hyde, in Sponges, F o a m andErnulsions, J. F. Sadoc and N. Rivier, Eds. (Kluver, Dordrecht 1999) p. 437 27. D. Hoffman, Nature 384,28 (1996). 28. R. Buzio, E. Gnecco, C. Boragno, U.Valbusa, P.Piseri, E. Barborini and P. Milani, SurJ Sci. 444, LI (2000). 29. P. Milani, A. Podesta, P. Piseri, E. Barborini, C. Lenardi, C. Castelnova, DiamondandRel. Muter. 10,240 (2001). 30. C. Castelnova, A. PodestA, P. Piseri, P. Milani, Phys. Rev. E 65,21601 (2001). 31. A. L. Barabasi and H. E. Stanley, Fractal Concepts in Suflace Growth (Cambridge University Press, Cambridge 1983). 32. R. Osserman, A Survey ofMinirnal Suflaces (Dover, New York 1986). 33. W. Helfrich, Z. Naturforsch. 28,768 (1973). 34. S. T. Hyde, in Foams andEmulsions, J. F. Sadoc and N. Rivier (Kluwer, Dordrecht, 1999) p. 437. 35. C. Oguey, in F o a m andEmulsions, J. F. Sadoc andN. Rivier (Kluwer, Dordrecht, 1999) p. 471.

155 36. J. M. Sullivan, in Foams andEmulsions, J. F. Sadoc and N. Rivier (Kluwer, Dordrecht, 1999) p. 379. 37. C . T. White eta!, in BuckminsterfuNerenes,W. E. Billups and M. A. Ciufolini (VCH, New York, 1993) p. 125. 38. J. F. Sadoc, in Foams andEmulsions, J. F. Sadoc and N. Rivier, Eds. (Kluver, Dordrecht 1997) p. 51 1.

CLUSTERING AND INTERFACE PROPAGATION IN INTERACTING PARTICLE DYNAMICS

A. PROVATA Institute of Physical Chemistry, National Center for Scientific Research “Demokritos” 15310 Athens, Greece E-mail: aprovataO1imnos.chem.demokritos.gr

V. K. NOUSSIOU Institute of Physical Chemistry, National Center for Scientific Research “Demokritos” 25320 Athens, Greece and Department of Chemistry, University of Athens 10679 Athens, Greece We study the development of rough interfaces in lattice models with multispecies nearestneighbour interactions. In particular, we study a bimolecular and a quadrimolecular interacting particle model and the Ziff - Gulari - Barshad model which involves both spontaneous (single particle) and cooperative (multiparticle) reactive steps. We show that interface roughening follows a scaling function in all models and the critical exponents depend on the particular type of interactions and the number of species involved.

1. Introduction

In recent studies considerable interest is devoted to the development of models which dcscribe reactive processes taking place on low dimensional lattices.’-’’ As it has been shown,” when a process takes place on a low dimensional support, the vacancies of the support must also be taken into account to properly describe the steady state properties and the dynamics. The lattice models which consider the support vacancies as an independent species and can thus be directly implemented on lattice are called ”lattice compatible models”. As an example of a lattice compatible model we have studied the Lattice LotkaVolterra (LLV) model” which is described by the following scheme:

XI -k x2 5 2x2 x2 f

s 4 2s

156

157 where X1 and Xz are the reactive species while S represents the empty lattice sites. In the reactive scheme (l),Eq.(la) corresponds to reaction between X1 and X;?, provided they reside in neighbouring sites, while Eq.( lb) corresponds to desorption of X z from the lattice leaving an empty site S, provided that a neighbouring empty site S already exists. Similarly, Eq.(lc) corresponds to desorption of XI. In this, lattice compatible, LLV mechanism a lattice site is allocated for every species, XI, X 2 , or S . Equivalently, the total number of sites of the lattice that are empty ( S ) ,or covered by X1 or Xz is constant, N.

N

= x1 +x2

+s

(2) a counter example, the original Lotka-Volterra (LV) model: [XI Xz 2 x 2 , X2 3 0, X1 3 2x11 is not lattice compatible. The fact that the LV model does not take vacant sites into account makes it inappropriate for the direct implementation of on lattice interactions. In the third step, for example, X1 gives 2x1. If this step was to be implemented on a lattice, there would be no available site for the extra X1 produced to adsorb. In order to use the LV model on lattice, modifications of the mechanism are necessary and introduction of empty sites. The LLV model is therefore a special modification of the LV model that is applicable on lattice. Except for the LLV model, many other lattice compatible models have been used in literature, such as the Lattice Limit Cycle (LLC) model,13 the epidemic mode1,l4- l6 the Ziff-Gulari-Barshad (ZGB) model.' , 2 Some of these models, e.g. the LLV and the LLC, have been designed to study basic pattern formation mechanisms, while others have been designed to simulate specific reactive processes; e.g. the ZGB model was used to simulate the catalytic CO oxidation on a Pt surface. Heterogeneous catalytic reactions are important examples of processes that are best described by lattice compatible models. Clustering and pattern formation (including oscillations) are observed as a result of molecule interactions on catalysts. Various examples of patterns arise in experiment, such as in the CO oxidation on Pt,17 the NO reduction on Fth or Pt,17,18the NO CO reaction on Pt,l79l9 etc. (The NO H2 reaction has also been studied on substrates with different proper tie^).'^ In the case of reactions on catalysts, patterns are concentration gradients on the surface which evolve in time. Stripes, target patterns and spiral waves are the usual patterns that appear in the above experimental processes. By far the richest variety of spatiotemporal patterns has been observed in catalytic CO oxidation on P t ( l l 0 ) - target patterns, rotating spiral waves, solitary oxygen pulses, standing waves and chemical turbulence -.17 Simulating realistic processes like the ones above demands more complicated models than the LLV, the LLC, or even the ZGB model. In the meantime, both approaches, that is studying basic mechanisms as well as simulating simplified specific processes, are necessary in the effort to find the true mechanistic pathways of real processes. In particular, the ZGB model is very important in that it predicts spatiotemporal phenomena such as kinetic phase

As

+

+

2

+

158 transitions and interface propagation through a minimal mechanism and simultaneously it corresponds to a real chemical system. On the other hand, the LLV and the LLC model have been very successful in producing patterns, thus shedding light in the direction of identifying the mechanisms that are responsible for pattern formation in general. In the current study we will focus on the interface propagation between the different species in bi-molecular and quadri-molecular reactive schemes and on determining the scaling of the interface width. We will show that the characteristic exponents depend on the number of species and also on the parameter values. In the next section we will study the bimolecular reactive scheme, while in section 3 we study the quadrimolecular reactive scheme. Section 4 we devote to pattern formation and interface propagation in the ZGB model while in the concluding section we draw our main results and we discuss open problems. 2. Bimolecular Reactive Schemes

In the case of bimolecular reactions, as in the LLV model, previous studies have demonstrated the formation of fractal clusters in free LLV systems and stripes and spiral patterns in LLV systems with specific initial condition^.^ An important element for the creation of such patterns was the autocatalytic nature of all the steps involved in the LLV scheme (Eqs.(la),(lb),(lc)), which are of the form

A + B ~ B + B

(3)

This model, known as the epidemic model, has been extensively studied in 1 i t e r a t ~ r e .The l ~ ~bimolecular ~~ character of this kind of reactions results in competition of the domains of species A and B and intrusion of the B domains within the A domains. This particular type of interaction gives rise to a rough interface when two phases A and B interact, even if we start from a completely linear interface. In this section we will study the roughening of a 1-dimensional linear interface, when the system (3) is realised in a 2-d square lattice via Kinetic Monte Carlo (KMC) simulations. The realisation of system (3) is as follows. (1) Start with a 2-d square lattice of size LxL filled with particles A or B and with given initial conditions and concentrations. (2) At every Elementary Time Step (ETS) choose one lattice site at random. (3) If the lattice site chosen contains a B particle then disregard the site and go to algorithm step 5. (4) If the lattice site chosen contains an A particle then select one of the four neighbours at random. If the selected neighbour is A go to algorithm step 5. If the selected neighbour is B then with probability p change A to B. ( 5 ) CONTINUE. One ETS is completed and the algorithm returns to step 2 starting a new ETS.

159 The time unit we use is the Monte Carlo Step (MCS) which is equivalent to LxL ETS; that is, one MCS is the time required for a number of trials equal to the total number of lattice sites (LxL) to be completed.

Figure 1. Interface roughening in the bimolecular model (3) (a) Initial stages of evolution (after 2 MCS), (b) snapshot after 20 MCS, (c) 50 MCS, (d) 100 MCS. Parameter values are L = 28, p = 1.0.

To study the interface roughening in model (3) we start the algorithm with initial conditions in which the whole system is covered by A particles (coloured gray) except for a linear band which is covered by B (black) and thus the interface between the A and B phases is linear (see Fig. 1). In Fig. 1 we present some representative stages of the interface roughening as the KMC algorithm proceeds. Figure l(a) represents initial stages of evolution (after only 2 MCS), while in Figs. l(b), l(c) and l(d) we observe how the surface has evolved after 20 MCS, 50 MCS and 100 MCS respectively. The system size is L = 28, while p = 1.0. To describe the interface roughening we calculate the average height < h(t) > and the width of the interface w ( t ) which are defined as

c L

< h(t) >=

i=l

h(i,t )

(4)

160 and

l N w2(t)= - C(h(i, t)-

< h(t) > ) 2

(5)

i=l

where h(i,t ) is the height of the i-th column a t time t.

Figure 2. The scaling of the width as a function of time for the bimolecular reaction scheme. The straight line represents a power law with an exponent of 2/3.

In Fig. 2 we present w2 as a function of time for p l = 1.0. For both values in a double logarithmic scale, the function w 2 shows a linear increase, then a second linear region, while for larger times w2 reaches a plateau. The behaviour of w 2 after the initial transitory phase can be described by a scaling function

w(t)

t L”

= Laf(-)

where (Y is called the roughness exponent and z is called the dynamic exponent. The scaling function w(t) behaves as

while

161 where fl is the growth exponent. Calculating the exponents after an initial transitory regime where d ( t )cc t, the growth enters a roughening phase and W 2 ( t ) = t2P

(9)

with 2p = 0.65 f 0.05

(10)

The exponent fl values are very close to 0.33, u-ich is the p value for t-5 Eden model, the LLV model and the Kardar - Parisi - Zhang (KPZ) equation.'O This is a strong indication that the epidemic model is in the same universality class described by the KPZ equation. 3. Quadrimolecular Reactive Schemes We will now consider a more complicated quadrimolecular reactive scheme in which two A particles and two B particles are involved as follows 2A+ 2B 3 4B

(11)

that is, when two A particles are found to be neighbours with two B particles, both A particles change into B particles. The KMC scheme which simulates the process (11) has the following form (1) We start with a 2-d square lattice of size LxL filled with particles A or B and with given initial conditions and concentrations. (2) At every ETS one lattice site is chosen at random. (3) If the lattice site contains a B particle the algorithm jumps to step 5 . (4) If the lattice site contains an A particle and amongst the 4 first nearest neighbours there is another A particle and 2 B particles then with probability p both A particles change into B particles. (5) CONTINUE. One ETS is completed and the algorithm returns to step 2 starting a new ETS. Following the above algorithm we have realised the quadrimolecular scheme on a 2-d square lattice with the following initial condition: All lattice sites are covered by A particles (gray) except for a linear band which is covered by €3 (black) (see Fig. 3). In Fig. 3 we present some representative stages of the interface roughening as the KMC algorithm proceeds. Fig. 3(a) represents initial stages of evolution (after only 10 MCS), while in Figs. 3(b), 3(c) and 3(d) we observe the surface evolution after 100 MCS, 200 MCS and 300 MCS respectively. The system size is L = 2 ' , while p = 1.0. This initial configuration will initiate reaction while having almost zero roughness. Note that a perfectly linear interface cannot initiate reaction. For this reason

162

Figure 3. Interface roughening in the quadrimolecular reaction scheme (11) (a) Initial stages of evolution (after 10 MCS), (b) snapshot after 100 MCS, (c) 200 MCS, (d) 300 MCS. Parameter values are L = 28, p = 1.0.

the initial state contains a toothy interface between the A and B phases which slowly develops considerable roughness. In Fig. 4 we present the evolution of the width w 2 as a function of time in this process. In a double logarithmic plot, the scaling follows a power law of the form of Eq.(9) with /? = 0.5 f 0.05. This exponent is distinctly different from the one calculated for the bimolecular model indicating that these two models do not belong to the same universality class. We can then conclude that the number of interacting species is important in defining the roughening exponents in growth models. 4. Interface Formation in the ZGB model

The ZGB model, which is a lattice compatible model as well, has been introduced for the simulation of the Langmuir - Hinschelwood (LH) mechanism of the CO oxidation that takes place on the surface of various metals e.g. P t , Pd, Rd.'-' Our realisation of the LH mechanism is inspired by the ZGB model, yet the implementation is slightly different. The lattice compatible LH mechanism has the following form

163

100

I

I

I 1

I 10

I

?

I 100

time (MCS)

Figure 4. The scaling of the width as a function of time for the quadrimolecular reaction scheme. The straight line represents a power law with an exponent of 1

Co(ad8) + O(ads)-’COZ(g) -k 2 s

(124

where S are the vacant lattice sites, and the subscripts g and ads imply that the molecules are in the gaseous and adsorbed state respectively. The particles in the gas phase affect the on lattice interactions only through the adsorption probabilities of CO and 0 2 which express the mole fractions of CO, and OZ(,). The KMC algorithm we used for the LH simulation on a 2-d lattice is the following (1) We start with a 2-d square lattice of size LxL either vacant (S)or filled with CO or 0 particles with given initial concentrations and conditions. (2) At every Elementary Time Step (ETS) one lattice site is chosen at random. (3) If the lattice site chosen is vacant (S) then: a) with probability yco, CO adsorbs, b)with probability 1 - yco two 0 adsorb, one on this site and one on a randomly selected neighbor (provided it is vacant (S)). (yco is the ”mole fraction” of CO in the ”gas phase”). (4) If the lattice site chosen contains a CO particle then one of the four neighbours is selected at random. If this neighbour contains an 0 then both CO and 0 change to S. If the selected neighbour is CO or S then the site is disregarded and the algorithm goes to step 6 .

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(5) If the lattice site chosen contains an 0 particle then one of the four neighbours is selected at random. If this neighbour is CO then both 0 and CO change to S. If the selected neighbour is 0 or S then the site is disregarded and the algorithm goes t o step 6. (6) CONTINUE. One ETS is completed and the algorithm returns to step 2 starting a new ETS.

(C)

(d)

Figure 5. Interface roughening in the ZGB reaction scheme (a) Initial condition of the system for t = OMCS, (b) snapshots after t = SOMCS, (c) t = 270MCS and (d) t = 39OMCS. Parameter values are L = 200, p = 0.3.

More generally we can denote A = CO, B = 0, S = (vacant lattice site) and yco = p . Using this mechanism we have simulated systems of various sizes ranging from L = 30 to L = 500 and with parameter values ranging from p=O to 1. As can be seen in Figs. 5 and 6, phase separation and wave propagation take place in this model for specific parameter regions. In particular, in Fig. 5 we have started with the specific initial conditions presented in Fig. 5(a), i.e. a circular area (disc) of B (coloured gray) surrounded by A (black). Due to the reactive step 12c, reaction between A and B takes place and vacant sites (coloured white) arise at the disc circumference. Reactions take place only on these regions of vacant sites which travel both towards the center of the disc and the outside. However for small p such as p = 0.3 these areas have a tendency to propagate within black areas (towards the outside of the disc that is covered by A particles) while the originally circular circumference of the disc roughens. In Figs. 5(b), 5(c) and 5(d) we observe various

165

Figure 6. Interface roughening and propagation in the ZGB reaction scheme (a) Initial condition of the system for t = OMCS, (b) Snapshots a t = 2 0 M C S , (c) t = GOMCS, and (d) t = 12OMCS. Parameter values are L = 200, p = 0.5.

stages of roughening where the effect of the bimolecular step 12c is predominant. The intrusion of the A domains towards the center of the disc (covered by B) happens more slowly due to the small adsorption probability of A. Also as soon as an A is deposited next to one or more B particles it is very likely to disappear, since the reaction step 12c happens with high probability, r=l. Thus the area of the disc that contains only B particles grows with time. For larger values of p the system goes through a critical point (pc 0.4) where propagation towards the center and the outside of the disc happens at the same rate. In this case we have a reactive steady state of the system and so it never reaches a poisoned state. Thus for values of p near the critical point stable, fractal clusters are observed. For still larger values of p , intrusion of phase A within phase B prevails. In Fig. 6 we also present the diagram of phase separation for different initial conditions shown in Fig. 6(a), that is a random deposition on a band enclosed in a bulk region of B particles. The boundary conditions are periodic in both the x and y directions, the system size is L = 200 and the adsorption probability p of A is 0.5. For these parameter values the phase A gradually dominates the initially random area while vacant sites (S) arise at the interface between A and B regions because of the reaction step 12c between them (as in the previous case of Fig. 5). The adsorption of A particles is favoured on the vacant sites since the adsorption of B (12b) demands the simultaneous selection of two neighbouring vacant sites. Thus when p = 0.5 the A regions grow in expense of the B regions. The roughness of the interface between the A and B regions increases N

166 with time at first (see Fig. 6(b)) until the roughness reaches a plateau (Figs. 6(c) and 6(d)). The steady state roughness depends on the system size L . Thus, as in the above cases, when cooperative phenomena are involved in reactive dynamics clustering, interface propagation and roughening are observed. More detailed study is expected to shed light on the roughening transition away from the critical point as well as on the cluster structure at the critical point.

Conclusions In the current study we examine the interface propagation and roughening on a surface between phases in competition. We present three models of interacting particle systems with a variable degree of interactions, that is a) the bimolecular model A B+B B, b) the quadrimolecular reactive model 2A 2B+4B and the more complex ZGB model which involves spontaneous and cooperative reaction steps. In all these models we observe clustering of homologous species due to the cooperative character of the interactions. The various clusters compete and we observe interface roughening between the different clusters. The roughness of the surface follows a scaling function and the roughening exponents depend on the type of the interactions and the number of species involved. Namely, for the bimolecular interaction model, the dynamic scaling exponent is p = 0.32 f 0.03, while for the quadrimolecular interaction model is p = 0.5 f 0.03. It is therefore clear that the dynamic scaling exponent (and the universality class) depends crucially on the degree of interactions. More detailed study needs to be carried out in order to investigate the values of the a exponent for all the models and to determine the clustering characteristics of the ZGB model at the critical point of p and far from it.

+

+

+

Acknowledgments The authors would like to thank Dr. G. A. Tsekouras, and Profs. V. Havredaki and A. A. Tsekouras for helpful discussions.

References 1. R. M. Ziff, E. Gulari and Y. Barshad, Phys. Rev. Lett., 56, 2553 (1986). 2. B. J. Brosilow, E. Gulari and R. M. Ziff, J. Chem. Phys., 98,674 (1993); C.A. Voigt and R. M. Ziff, Phys. Rev. E, 56, R6241 (1997). 3. J. W. Evans and M. S. Miesch, Phys. Rev. Lett., 66, 833 (1991); M. Tammaro and J. W. Evans, Phys. Rev. E,52,2310 (1995); M. Tammaro and J. W. Evans, J. Chem. Phys., 108,762 (1998). 4. D. J. Liu and J. W. Evans, Phys. Rev. Lett., 84,955 (2000). 5. V. P. Zhdanov, Phys. Rev. E 59, 6292, (1999); V. P. Zhdanov, Surf. Sci. Rep., 45, 231, (2002). 6. E. V. Albano and J. Marro, J. Chem. Phys., 113,10279 (2000). 7. H. Rose, H. Hempel and L. Schimanksy-Geier, Physica A 206, 421, (1994).

167 8. A. Provata, J. W. Turner and G . Nicolis, J. Stat. Phys. 70,1195 (1993). 9. A. Tretyakov, A. Provata and G . Nicolis, J. Phys. Chem. 99,2770 (1995). 10. A. Provata, G. A. Tsekouras, Phys. Rev. E, 67,art. no 056602 (2003). 11. A. Provata, G. Nicolis and F. Baras,'J. Chem. Phys. 110,8361 (1999). 12. L. Frachebourg, P. L. Krapivsky and E. Ben-Naim, Phys. Rev. E, 54,6186 (1996). 13. . V. Shabunin, F. Baras and A. Provata, Phys. Rev. E,66,art. no 036219 (2002). 14. W. Wang and X. Q. Zhao Math.Biosci., 190,97 (2004). 15. 0. Alves, C.E. Ferreira, F. P. Machado Math. Comput. Simulat., 64,609 (2004). 16. J. D. Murray, Mathematical Biology, Springer, Verlag 2002. 17. R. Imbihl and G. Ertl, Chem. Rev., 95 697 (1995). 18. Y . De Decker, F. Baras, N. Kruse, G. Nicolis J. Chem. Phys., 117,22 (2002). 19. N. Hartman, Y . Kevrekides and R. Imbihl J. Chem. Phys., 112,15 (2000). 20. M. Kardar, G. Parisi and Y. -C. Zhang Phys. Rev. Lett., 56,889 (1986).

RESONANT ACTIVATION AND NOISE ENHANCED STABILITY IN JOSEPHSON JUNCTIONS

A. L. PANKRATOV Institute for Physics of Microstructures of Russian Accademy of Science, GSP-105, Nizhny Novgorod 603950, Russia E-mail: alp9apm.sci-nn0v.m

B. SPAGNOLO Dipartimento d i Fisica e Tecnologie Relative and INFM, Group of Interdisciplinary Physics* Universitb da Palenno, Viale delle Scienze pad. 18, I-90128 Palenno, Italy E-mail: spagnoloOunipa.it

We investigate the interplay of two noise-induced effects on the temporal characteristics of short overdamped Josephson junctions in the presence of a periodic driving. We find that: (i) the mean life time of superconductive state has a minimum as a function of driving frequency, and near the minimum it actually does not depend on the noise intensity (resonant activation phenomenon ); (ii) the noise enhanced stability phenomenon increases the switching time from superconductive to the resistive state. As a consequence there is a suitable frequency range of clock pulses, at which the noise has a minimal effect on pulse propagation in RSFQ electronic devices.

1. Introduction and Basic Formulas

The investigation of thermal fluctuations and nonlinear properties of Josephson junctions (JJs) is very important owing to their broad applications in logic devices. Superconducting devices in fact are natural qubit candidates for quantum computing because they exhibit robust, macroscopic quantum behavior l. Recently, a lot of attention was devoted to Josephson logic devices with high damping because of their high-speed switching 2*3. The rapid single flux quantum logic (RSFQ), for example, is a superconductive digital technique in which the data are represented by the presence or absence of a flux quantum @po = h/2e in a cell which comprises Josephson junctions. The voltage pulse from a moving single flux quantum is the unit of information. The short voltage pulse corresponds to a single flux quantum moving across a Josephson junction, that is a 277 phase flip. However the operating temperatures of the high-Tc superconductors lead to higher noise levels by increasing the probability of thermally-induced switching errors. Moreover during *electronic address: http://gip.dft.unipa.it

168

169 the propagation within the Josephson transmission line fluxon accumulates a time jitter. These noise-induced errors are one of the main constraints to obtain higher clock frequncies in RSFQ microprocessors 2 . In this work after a short introduction with the basic formulas of the Josephson devices, the model used to study the dynamics of a short overdapmed Josephsonn junction is described. In the next section two main noise-induced phenomena observed in metastable states, namely the resonant activation and the noise enhanced stability, are shortly presented. Finally in the last section the results and the interplay of these noise-induced phenomena on the temporal characteristics of the Josephson devices are discussed. The role played by these noise-induced effects in the accumulation of timing errors in RSFQ logic devices is analyzed. The Josephson tunneling junction is made up of two superconductors separated from each other by a thin layer of oxide 4 . The phase difference cp between the wave function for the left and right superconductors is given by the Josephson equation

where V ( t )is the potential difference across the junction, e is the electron charge, and ti = h/27r is the Planck’s constant. A small junction can be modelled by a resistance R in parallel with a capacitance C across which is connected a bias generator and a phase-dependent current generator, Isincp, representing the Josephson supercurrent due to the Cooper pairs tunnelling through the junction. Since the junction operates at a temperature above absolute zero, there will be a white Gaussian noise current superimposed on the bias current. Therefore the dynamics of a short overdamped J J , widely used in logic elements with high-speed switching and corresponding to a negligible capacitance C, is obtained from Eq. (1) and from the current continuity equation of the equivalent circuit of the Josephson junction. The resulting equation is the following Langevin equation

valid for p 1, switches the junction into the resistive state. An output voltage pulse will appear after a random switching time. We will calculate the mean value and the standard deviation of this quantity for two different periodic driving signals: (i) a dichotomous signal, and (ii) a sinusoidal one. We will consider different values of the bias current i, and of signal amplitude A. Depending on the values of i, and A as well as values of signal frequency and noise intensity, two noiseinduced effects may be observed, namely the resonant activation (RA) (see Refs.[&S,ll] and the noise enhanced stability (NES) (see Refs.[5,10-12]. These effects have different role on the temporal characteristics of the Josephson junction and occur because of the presence of metastable states in the periodic potential profile of the Josephson tunnel junction and the thermal noise. Specifically the RA phenomenon minimizes the switching time and therefore also the timing errors in RSFQ logic devices, while the NES phenomenon increases the mean switching time producing a negative effect

+

’.

2. Noise induced effects 2.1. Resonant Activation

The escape of a Brownian particle moving in a fluctuating metastable potential shows resonant activation phenomenon, that is the average escape time has a minimum as a function of the oscillating frequency of the potential barrier. This effect was theoretically predicted in Ref.[6], where random fluctuations of the potential were considered, and experimentally observed in tunnel diodes and in underdamped Josephson tunnel junctions ’. The fluctuations of the potential barrier can be random or periodic between two limiting configurations of the potential, upper and lower positions respectively. The average frequency of fluctuations must be less than the natural frequency of the system at the metastable state. Recently the RA effect was obtained theoretically in a piece-wise linear dichotomously fluctuating potential with metastable state If the potential fluctuations are very slow, the average escape time is equal to the average of the crossing times over upper and lower configurations of the barrier, and particularly in this case the slowest process determines the value of the average escape time. In the limit of very fast fluctuations, the Brownian particle ”see” the average barrier and the average escape time is equal to the crossing time over the average barrier. In the intermediate regime, the crossing is strongly correlated with the potential fluctuations and the average escape time exhibits a minimum at a resonant fluctuation rate. In the following Fig. 2 we show a typical picture of RA phenomenon observed in a metastable fluctuating potential l l .

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-6

-4

-2

0

2

4

0

Figure 2. Semilogarithmic plot of the average escape time as a function of the mean switching rate of the piecewise linear metastable potential profile for seven different values of the noise intensity D.

2.2. Noise Enhanced Stability

The noise-enhanced stability (NES) phenomenon was observed experimentally and and, as arecent review, numerically in various physical systems (see Refs. [3,5,10,11] Ref. [12]). The investigated systems were subjected to the action of two forces: additive white noise and driving force. The driving force was futed, periodical or random. The noise enhanced stability effect implies that, under the action of additive noise, a system remains in the metastable state for a longer time then in the deterministic case, and the escape time has a maximum as a function of noise intensity. We can lengthen or shorten the mean lifetime of the metastable state of our physical system, by acting on the white noise intensity. The noise-induced stabilization, the noise induced slowing down in a periodical potential, the noise induced order in one-dimensional map of the Belousov-Zhabotinsky reaction, and the transient properties of a bistable kinetic system driven by two correlated noises, are akin to the NES phenomenon 12. In the next Fig. 3 we report the behavior of the average escape time as a function of the noise intensity for a piece-wise linear metastable potential subjected to dichotomous random fluctuations ll.

1.4.

1.3

D Figure 3. Semilogarithmic plot of the normalized average escape time as a function of the white noise intensity D for three values of the dimensionless mean switching rate of the piece-wise linear metastable potential profile.

173 3. Temporal characteristics Now we investigate the following temporal characteristics: the mean switching time (MST) and its standard deviation (SD) of the Josephson junction described by Eq. (2). These quantities may be introduced as characteristic scales of the evolution

s W(p, t)dp, to find the phase within one period of the

9 2

of the probability P ( t ) =

9 1

potential profile of Eq. (3). We choose therefore p2 = T , (PI = -T and we put the initial distribution on the bottom of a potential well: po = arcsin(i0). A widely used definition of such characteristic time scales is the integral relaxation time '. The mean switching time T = ( t )may be introduced in the form

where w(t) =

,*

and the SD of the switching time is c =

d< t 2 > - < t >2.

Let us focus on the case of dichotomous driving, f ( t ) = Asign(sin(wt)). The results of computer simulations are shown in Fig. 4. Both MST and its SD does not depend on the driving frequency below a certain cut-off frequency, above which the characteristics degrade. In the frequency range from 0 t o 0 . 2 therefore ~ ~ we can describe the effect of dichotomous driving by time characteristics in a constant potential. The exact analytical expression, as well as asymptotic representation of the MST has been obtained in R,ef.[5]. For an arbitrary y we have

and for y

(
^. The location of the glass transition, T K ,corresponds to a cusp in the function @(Tcmf). The dynamical crossover point To might correspond to the position of a characteristic shaking amplitude I?’ found in experiments and simulations where the “irreversible” and “reversible” regimes approximately meet.

Fluid

I

i

2

I

I

I

I

I , ,

2.5 Tconf

Figure 3. The density, = N , / ( 2 ( 2 ) - l), for N s = 0.6 as a function of T,,,f. amazis the maximum density reached by the system in the crystal phase.

5. Monte Carlo tap dynamics

The model, Eq. (5), simulated in 3d by means of Monte Carlo tap dynamics” presents a transition from a fluid to a crystal as predicted by the mean field approximation, density profiles in good agreement with the mean field ones, and in the fluid phase a large increase of the relaxation time as a function of the inverse tap amplitude. In the following section we study a more complex model for hard spheres, where an internal degree of freedom allows to avoid cry~tallization~’ In the following the tap duration is fixed, TO = lOMCsteps/particle, and

200

0

0

0

0.7

Figure 4. The bulk density, iP E N/L2(2(z) - l), is plotted as function of Tr for TO = 10 MCsteps/partzcZe. The empty circles correspond to stationary states, and the black stars to out of stationarity ones. iPmax is the maximum density reached by the system in the crystal phase, QmaX = 6/7.

different tap amplitudes, Tr, are considered. In Fig. 4 the bulk density, N / L 2 ( 2 ( z )- l),is plotted as a function of Tr: @(Tr)has a shape resembling that found in the “reversible regime’’ of tap experiments314, and moreover very similar to that obtained in the mean field calculations and shown in Fig. 3. At low shaking amplitudes (corresponding to high bulk densities) a strong growth of the equilibration time (i.e. the time necessary to reach stationarity) is observed, and for the lowest values here considered (the black stars in Fig. 4) the system remains out of stationarity. In conclusions the system here studied presents a jamming transition at low tap amplitudes as found in real granular media. In order to test the predictions of the mean field calculations, in the following we measure quantities usually important in the study of glass transition: The relaxation functions, the relaxation time and the dynamical susceptibility, connected to the presence a dynamical correlation length. The autocorrelation function has a behavior very similar to that found in usual glass-formers: At low values of the tap amplitudes, Tr, two-step decays appear, well fitted in the intermediate time region, by the 0-correlator predicted by the mode coupling theory for supercooled liquidsz5 and at long time by stretched exponentids’l . In Fig. 5 the relaxation time, 7,is plotted as a function of the tap amplitude, Tr: A clear crossover from a power law to a different regime is observed around a tap amplitude To. The power law divergence can be interpreted as a mean field behavior, followed by a hopping regime. We note that a similar behavior is found inz6 where the escape time of a system in a sub-diffusive medium has a similar

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Figure 5. The relaxation time, T , as function of the t a p amplitude inverse, T;’. The dashed line is a power law, (Tr - T ~ ) - 7 2with , TD = 0.40 f 0.01 and 7 2 = 1.52 f0.10. The continuous line is an Arrhenius fit, eAITr, with A = 17.4f 0.5 (the data in this region are also well fitted by both a super-Arrhenius and Vogel-Fulcher laws).

shape as a function of the inverse diffusion coefficient (i.e. l / T ) . In this case the escape time obeys a generalized Arrhenius law. The divergence of the relaxation time at vanishing tap amplitude is consistent with the experimental data of Philippe and Bideau4 and D’Anna et aLZ7. Their findings are in fact consistent with an Arrhenius behavior as function of the experimental tap amplitude intensity. However a direct comparison with our data is not possible since we do not know the relation between the experimental tap amplitude and the tap amplitude in our simulations. In Fig. 6 we plot the dynamical non linear susceptibility, x(t)’l at different Tr, which exhibits a maximum at a time, t*(Tr).The presence of a maximum in the dynamical non linear susceptibility is typical of glassy system^^^,^'. In particular the value of the maximum, x ( t * ) ,diverges in the pspin model as the dynamical transition is approached from above, signaling the presence of a diverging dynamical correlation length. In the present case the value of the maximum increases as Tr decreases (except at very low Tr where the maximum seems t o decrease30). The growth of X ( t * ) in our model suggests the presence of a growing dynamical length also in granular media.

6. Conclusions In conclusions using standard methods of statistical mechanics we have investigated the jamming transition in a model for granular media. We have shown a deep connection between the jamming transition in granular media and the glass

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Figure 6. The dynamical non linear susceptibility, X ( t ) , (normalized by X ( t o ) , the value at t o = 1) as a function of t , for tap amplitudes T y = 0.60, 0.50, 0.425,0.41,0.40,0.385,0.3825(from left to right).

transition in usual glass formers. As in usual glass formers the mean field calculations obtained using a statistical mechanics approach t o granular media predict a dynamical transition a t a finite temperature, T D ,and, at a lower temperature, T K , a thermodynamics discontinuous phase transition t o a glass phase. In finite dimensions 1) t h e dynamical transition becomes only a dynamical crossover as also found in usual glass formers15~20~23 (here the relaxation time, 7 ,as a function of both the density and the t a p amplitude, presents a crossover from a power law t o a different regime); and 2) the thermodynamics transition temperature, T K ,seems t o go t o zero (the relaxation time, 7,seems t o diverge only at Tr P 0, even if a very low value of the transition temperature is consistent with the data).

References 1. S. F. Edwards and R. B. S. Oakeshott, Physica A 157 (1989) 1080. A. Mehta and S. F. Edwards, Physica A 157 (1989) 1091; S.F. Edwards, in (Disorder in Condensed Matter Physics” p. 148, Oxford Science Pubs (1991); and in Granular Matter: a n interdisciplinary approach, (Springer-Verlag, New York, 1994), A. Mehta ed. 2. “Unifying concepts in granular media and glasses”, (Elsevier Amsterdam, 2004), Edt.s A. Coniglio, A. Fierro, H.J. Herrmann, M. Nicodemi. 3. J. B. Knight, C. G . Fandrich, C. N. Lau, H. M. Jaeger and S. R. Nagel, Phys. Rev. E 51 (1995) 3957; E. R. Nowak, J. B. Knight, E. Ben-Naim, H. M. Jaeger and S. R. Nagel, Phys. Rev. E 57 (1998) 1971; E. R. Nowak, J B. Knight, M. Povinelli, H. M. Jaeger and S. R. Nagel, Powder Technology 94 (1997) 79. 4. P. Philippe and D. Bideau, Europhys. Lett. 60 (2002) 677. 5. H. A. Makse and J. Kurchan, Nature 415 (2002) 614. 6. A. Fierro, M. Nicodemi and A. Coniglio, Europhys. Lett. 59 (2002) 642; Phys. Rev. E

203 66 (2002) 061301; Europhys. Lett. 60 (2002) 684. 7. J . J . Brey, A. Prados and B. Shchez-Rey, Physica A 275 (2000) 310. 8. D. S. Dean and A. Lefevre, Phys. Rev. Lett. 86 (2001) 5639. 9. G. Tarjus and P. Viot, Phys. Rev. E, 69:011307, 2004. 10. J . Berg, S. Franz and M. Sellitto, Eur. Phys. J. B 26 (2002) 349. 11. M. Taraia, A. de Candia, A. Fierro, M. Nicodemi, A. Coniglio, Europhys. Lett. 66, 531 (2004). 12. M. Tarzia, A. Fierro, M. Nicodemi, A. Coniglio, Phys. Rev. Lett. 93, 198002 (2004). 13. M. Nicodemi, Phys. Rev. Lett. 82 (1999) 3734. 14. A. Barrat et at., Phys. Rev. Lett. 85 (2000) 5034. 15. L. F . Cugliandolo and J. Kurchan. Phys. Rev. Lett., 71:173-176, 1993. 16. L. F. Cugliandolo, J. Kurchan, and L. Peliti. Phys. Rev. E, 55:3898-3914, 1997. 17. A. J . Liu and S. R. Nagel, Nature 396 (1998) 21. 18. C. S. O’Hern, S. A. Langer, A. J. Liu and S. R. Nagel, Phys. Rev. Lett. 86 (2001) 111. C. S. O’Hern, L. E. Silbert, A. J . Liu and S. R. Nagel, Phys. Rev. E 68, 011306 (2003). 19. M. MCzard and G. Parisi, Eur. Phys. J. B 20, 217 (2001). 20. G. Biroli and M. MBzard, Phys. Rev. Lett. 88,025501 (2002). 21. A. Fierro, M. Nicodemi, M. Tarzia, A. de Candia and A. Coniglio, cond-mat/O~l2l20. 22. In the case of uniform density profile, i.e. u ( z ) = const., we have u ( z ) = @ (where = N/L2(2(z) - 1)) below the maximum height and zero above. 23. C. Toninelli, G. Biroli, D. S. Fisher, Phys. Rev. Lett. 92, 185504 (2004). 24. M. PicaCiamarra, M. Tarzia, A. de Candia, and A. Coniglio, Phys. Rev. E 67,057105 (2003); Phys. Rev. E 68,066111 (2003). 25. W. Gotze, in Liquids, Freezing and Glass Transition, eds. J.P. Hansen, D. Levesque, and Zinn-Justin, Elsevier (1991). T . Franosch, M. Fuchs, W. Gotze, M.R. Mayr and A.P. Singh, Phys. Rev. E 55,7153 (1997). M. Fuchs, W. Gotze and M. R. Mayr, Phys. Rev. E 58,3384 (1998). 26. E. K. Lenzi, C. Anteneodo and L. Borland, Phys. Rev. E 63,051109 (2001). 27. G. D’Anna and G. Gremaud, Nature 413, 407 (2001); G. D’Anna, P. Mayor, A. Barrat, V. Loreto, and F. Nori, Nature 424,909 (2003). 28. S. Franz, C. Donati, G. Parisi and S. C. Glotzer, Philos. Mag. B 79,1827 (1999). C. Donati, S. Franz, S. C. Glotzer and G. Parisi, J. Non-cryst. Solids, 307,215 (2002) 29. S. C. Glotzer, V. N. Novikov, and T . B. S c h r ~ d e rJ, . Chem. Phys. 112,509 (2000). 30. Interestingly this anomalous behavior seems to occur around the crossover temperature TD previously calculated. The origin of this behavior, also observed in molecular dynamics simulations of a usual glass former2’, is still unclear.

SUPERSYMMETRY AND METASTABILITY IN DISORDERED SYSTEMS

IRENE GIARDINA, ANDREA CAVAGNA AND GIORGIO PARIS1 Institute for Complex Systems INFM-CNR and Department of Physics, University of Rome La Sapienza, P.le A . Moro 2, 00185 Rome, Italy E-mail:irene.giardinaQromal.infn.it The presence of metastable states is a well known feature of disordered systems and plays a crucial role in the slowing down of the dynamics and the occurrence of the glass transition. A deep understanding of the geometric structure of these states and its implications on the dynamical behaviour therefore represents a very important issue. We will show that when analyzing the properties of metastable states, and in particular their entropic contribution, a supersymmetry is revealed at a formal level, which has a clear physical interpretation. Systems that have different structures of metastable states seem t o behave differently in terms of this supersymmetry: for some of them the supersymmetry is obeyed, for others it is spontaneously broken. We will discuss the physical meaning of the supersymmetry breaking and its connection with the cavity method.

1. Introduction Disordered systems are often said to have a complex landscape, which may be thought as the fundamental reason of the non trivial behaviour exhibited at low temperature. This statement can actually be made more precise, and a whole series of analytical and numerical analysis have been performed t o show the deep connections between the topological structure of the energy (or, when it is possible, free energy) function and thermodynamical properties. One feature that seems ubiquitous is the presence of a huge number of metastable states, i.e. states that are locally stable or quasi-stable but have energy density higher than the equilibrium one. These states have been studied in details via analytical techniques and numerical simulations for a great variety of models. The general scenario that emerges is the following. There is an exponential number of metastable states, i.e. the number of metastable states at a given free energy density f scales as N ( f )= expNC(f), where the entropy C has a finite support f E [ f o , f t h ] , and is an increasing function being zero at fo (the ground-states) and maximal at f t h (the threshold highest energy states). This situation is quite generic, however there are cases where these metastable states do have a crucial role, and some others where they seem much less relevant. One can then distinguish two classes of systems: i) A first class, where metastable states influence the asymptotic dynamics. In

204

205 this case if the system is started at low temperature with a high energy (random) initial condition, with very high probability at long times it will remain trapped at the threshold level: the dynamics is consequently slowed down, until activation processes enable to cross barriers and explore lower energy regions. This activation time-scale may however be quite large, and for mean-field systems is infinite, meaning that the system never reaches the equilibrium energy. Another important feature, much related to the one above, concerns the behaviour in temperature: these systems exhibit a dynamical crossover (which becomes a true transition in mean-field) t o an activated Arrhenius-like behaviour at a certain temperature T d . Interestingly, Td is higher than the critical temperature where thermodynamical anomalies occur, and is therefore a purely dynamical phenomenon. Examples of systems belonging to this class, are pspin models glass forming super-cooled liquids of the fragile kind *, K-satisfiability problems for certain ranges of the control parameter ‘. ii) For systems in the second class instead, metastable states, despite having a finite entropy, are not relevant at all. They can sometimes be computed by analytical or numerical means, but do not influence the dynamical behaviour. In this case then, also at low temperature, the system is always able to asymptotically reach the equilibrium level at the bottom of the metastable states energy band. Dynamically, anomalies (i.e. divergence of the relaxation time) only occurr at the static transition, as one would naturally expect. Examples of systems in this class are mean-fiels models of spin-glass (e.g. the Sherrington-Kirkpatrick model 16); certain disordered problems on random graphs and satisfiability problems for K = 3 and low values of the control parameter ‘. An intriguing question is then why metastable states have such a different role in these two classes. What we have understood in the last years through a series of analytical works on mean-field models and numerical studies on finite dimensional ones, is that what really matters is the nature and the stability properties of the metastable states. For systems of the first class, one can show that metastable states are indeed locally stable, and therefore correspond to confining regions of the configuration space. Besides, their global structure is very robust and does not change much when external perturbation are applied to the system. On the other hand, the situation is completely different for models of the second class. In this case, metastable states are not “truly” stable: when a description in terms of some free-energy functional is possible (for example for mean-field systems) one can show that they have an almost soft-mode l 2 9 l 3 determining a quasi-instability. As a consequence, they are not completely confining. Also, some arguments indicate that they have small attraction basins 14. Finally, as we shall better discuss, their whole structure is very fragile to external perturbations. Interestingly, for mean-field models of spin-glasses, where analytical computations can be performed, these features of metastable states and the difference in their physical role are captured in an elegant formal description where the model

’,

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belongs t o one class or to the other according to whether a certain supersymmetry is obeyed or not. To understand how this supersymmetry comes into play we have to explain more in details how to deal with metastable states and how t o investigate their properties. 2. Metastable states in mean-field spin glass models

Metastable states in mean-field spin-glasses can be identified with the local minima of a mean-field free energy F (also known as the TAP free energy 5 ) , that is a function of the local magnetizations mi of the system. The number of local minima of F(m) can be written as

where the delta-functions enforce the stationarity of the free-energy, &F(m) = 0, and the determinant of the second derivative (the Hessian) gives the appropriate normalization. By using an integral representation for the delta-functions and the determinant, the number N can be expressed in terms of an effective action S given by lo S = z&F(rn)

+ &$JjL@jF(rn) ,

(2)

where xi is a standard commuting variable, while &,~)i are anticommuting Grassmann variables, and sums over repeated indices are understood. This action S is invariant under a generalized form of the Becchi-Rouet-StoraTyutin supersymmetry 6i7i8 namely, under the transformation bmi = E $Ji; 66 = -E z i , where E is an infinitesimal Grassmann parameter. The physical meaning of this supersymmetry becomes clearer if we look at the Ward identities generated by it. One of them, in particular, reads (mixj) (&$Jj) = 0, which, with some simple algebra, can be rewritten as lo,ll

+

Here the brackets indicate an average over all metastable states, i.e. (...) = l/NC,. . . with a a state label. This relation is nothing else than the static average fluctuation-dissipation theorem (FDT) and shows that this formal supersymmetry encodes an important physical property of the system. 3. Supersyrnmetry breaking and its physical interpretation

It turns out that, while for mean-field models belonging to the first class described in the Introduction this supersymmetry is always obeyed ', for models of the second class it is in fact spontaneously broken when metastable states with high enough free-energy are considered However, as we have seen, this supersymmetry is intimately connected t o a fluctuation-dissipation relation that we expect on physical 12913315.

207 grounds to hold. How is it then possible that supersymmetry is broken and what is the interpretation of this breaking ? To answer this question we must understand under what circumstances such a general relation as the FDT of Eq. (3) may be violated. We have said that for models of the second class metastable states (at least most of them) have an almost softmode in one direction. Indeed recent studies show that at low temperatures metastable states, i.e. minima of the free-energy functional, are organized into minimum-saddle pairs. The minimum and the saddle are connected along a mode that is softer the larger the system size N , corresponding then t o marginal directions in the thermodynamic limit. Also, their free energy difference vanishes as N + 00. One may wonder whether this peculiar structure and the presence of marginal modes may be related to FDT violations. Let us then start to look at the fluctuation-dissipation relation for a single metastable state (we remind that in (3) we have an average over all the states). A metastable state is defined as a particular solution m of the stationarity equations a F ( m ) / d m i = 0. Let us add an external field h, the equation that identifies the metastable state then becomes a F ( m ) / a m i - hi = 0. If we now differentiate with respect to hi we get dmildhj = [aaF(m)];l, i.e. the FDT. Thus, for each state, the FDT is a very natural relationship between the susceptibility and the curvature of the minima of the free energy. Note that this relation also holds when marginal modes are present, since both sides diverge. Thus clearly, if a connection exists between features of metastable states and validity of the average FDT, it must not concern the states considered individually, but rather their global structure: FDT is always valid inside one state, even if marginal, but something goes wrong when we consider averages over all of them, such as in Eq. (3). Indeed another important consequence of marginality is for these systems the extreme fragility towards external perturbations. Since minima and saddles of the free-energy come into pairs and are connected via an almost zero mode, it is clear that even an infinitesimal (i.e. of order 1 / N ) external field may destabilize some states: if the applied field is opposite to local magnetization of the state and its intensity is of the same order as the free-energy difference with the nearby saddle, the minimum-saddle pair will “merge” and the state will disappear. On the other hand, virtual states, i.e. inflection points of the free energy with a very small second derivative, may be stabilized by the field, giving rise to pairs of new states. Thus, the number of states may change dramatically when an external field is applied. In such a situation we can then understand what may go wrong when doing averages. If we want t o compute the average FDT, as in (3), we must start from the average magnetization: 12,13115

Then, to get the 1.h.s. of equation (3) we must differentiate with respect to a magnetic field. The problem is that, due to marginality, some elements in the sum

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defining the average mangnetization may disappear or appear as the field goes to zero. More precisely, we have,

The key point is that the elements in the two sums at the 1.h.s. of the relation above may be different, because of the action of the field. Therefore, even though the static fluctuation-dissipation relation holds for each individual state, when we sum over all states, an anomalous contribution arises due to the instability of the whole structure with respect to the field and relation (3) is thus violated. Supersymmetry breaking is thus the mathematical expression of a great instability in the structure of metastable states. It has also important consequences in a well-known and widely applied approch to disordered system known as the cavity method l S , l . The basic assumption of this method is that by adding one extra degree of freedom (for example, a spin) to a large system, its physical properties do not change dramatically, and that it is therefore possible to write some recursive equations connecting the system with one extra spin to the old one. In the thermodynamic limit, these relations become self-consistent equations for the physical observables, for example for the distribution of the local magnetization P(mi). Unfortunately, the main assumption of this method, that is the stability under the addition of one new degree of freedom, is no longer valid if the supersymmetry is broken. In fact, in this case, even the small field produced by the new spin and acting on the rest of the system, may, as we have discussed, completely change the structure of metastable states of the system. When this happens, it becomes harder to write equations connecting the old and new properties of the system. For example this cannot be done any longer for the distribution P(mi). However, the cavity approach can be modified to deal with this problem. The crucial point is that the effect of a new spin is somehow analogous to an external field. Thus, in a way, a system with one more spin can be compared with a system that has one spin less and an appropriate field acting on it. Or, in other words, if one considers a system in presence of an external magnetic field, the adding of one spin can be balanced by appropriately tuning the field. As a result, self-consistence equations can be written for P(mi(hi),i.e. for the conditional distribution of the local magnetizations at given external magnetic field 19.

References 1. M.Mezard, G.Parisi and M.A. Virasoro, S p i n Glass T h e o r y and beyond, World Scientific, Singapore (1987). 2. See, e.g. M. MBzard, Physica A 306 25 (2002). 3. See, e.g., O.C. Martin, R. Monasson, R. Zecchina, Theoretical Computer Science 265, 3 (2001).

209 4. A.Montanari, G.Parisi, F. Ricci-Tersenghi, Preprint cond-mat/0308147 (2003). 5. D.J. Thouless, P.W. Anderson and R.G. Palmer, Phil. Mag., 35,593 (1977). 6. C. Becchi, R. Rouet and A. Stora, Comm. Math. Phys. 42,127 (1975); I.V. Tyutin Lebedev preprznt FIAN 39 (1975). 7. G. Parisi and N. Sourlas, Phys. Rev. Lett. 43,744 (1979). 8. Kurchan J 1991 J. Phys. A: Math. Gen. 24 4969 9. A. Cavagna, J.P. Garrahan and I. Giardina, J. Phys. A 32,711 (1998). 10. A. Cavagna, I. Giardina, G. Parisi and M. Mezard, J. Phys. A 36,1175 (2003). 11. A. Crisanti, L. Leuzzi, G. Parisi, and T. Rizzo, Phys. Rev. B 68,174401 (2003). 12. T. Aspelmeier, A. J. Bray, M. A. Moore, Phys. Rev. Lett. 92,087203 (2004) 13. A. Cavagna, I. Giardina and G. Parisi, Phys. Rev. Lett. 92,120603 (2004) 14. M. Moore, private communication. 15. G. Parisi and T. Rizzo, Preprint cond-mat/0401509. 16. D. Sherrington and S. Kirkpatrick, Phys. Rev. Lett. 32,1792 (1975). 17. A.J. Bray and M.A. Moore, J. Phys. C 13,L469 (1980). 18. M.Mezard, G.Parisi and M.A. Virasoro, Europhys. Lett. 1,77 (1986). 19. A. Cavagna, I. Giardina and G. Parisi, Phys. Rev. B, to be published (February 2005).

THE METASTABLE LIQUID-LIQUID PHASE TRANSITION: FROM WATER TO COLLOIDS AND LIQUID METALS.

GIANCARLO FRANZESE Departament de Fisica Fonamental, Universitat de Barcelona Diagonal 64 7,08028 Barcelona, Spain, E-mail: gfranzesedub. edu H. EUGENE STANLEY Center for Polymer Studies and Department of Physics, Boston University, Boston, MA 02215 USA E-mail: [email protected] The possibility of a liquid-liquid (LL) phase transition has been proposed as an interpretation of the anomalous behavior of liquids such as water, whose isobaric density has a maximum for decreasing temperature. By using theoretical models and numerical simulations, we show (i) that this property for molecular liquids implies the existence of a LL critical point, (zi) that the existence of a LL critical point does not imply the anomalous density behavior in systems whose effective potential resembles an isotropic soft-core attractive potential, such as protein solutions, colloids, star-polymers and, to some extent, liquid metals.

1. Water Anomalies and Their Interpretations

We all know that ice floats on water, while solid forms of normal substances are denser than their liquid form. This anomalous property of water is a manifestation of the density maximum at 4 "C at ambient pressure. Many other anomalies are known for water, especially in the supercooled liquid region where the liquid is metastable with respect to the crystal. For example, the absolute magnitude of isobaric heat capacity1, isothermal compressibility', and thermal expansivity3, appear as if they might diverge to infinity at a temperature of about -45"C, while in normal liquids all three response functions decrease as temperature decreases. These anomalies have been interpreted in different ways. (i) The stability-limit interpretation4, assumes that in the pressuretemperature (P-T) plane the limits of stability of the supercooled, superheated and stretched liquid water form a single retracing spinodal line. However, no experimental or numerical evidence of a retracing spinodal has thus far been found5. (ii) The singularity-free interpretation6i7 predicts no retracing spinodal and envisages that the experimental data represent apparent singularities, due to anticorrelated fluctuations of volume and entropy, responsible for the anomalies. (iii) Finally, the liquid-liquid (LL) phase transition hypothesis' proposes the presence of a first order line of phase transitions, possibly ending in a critical point,

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separating two liquids differing in density, the high density liquid (HDL) and the low density liquid (LDL), and responsible for the anomalies. The last two interpretations can be recovered within the same model by smoothly tuning a parameter, i.e. they could be complementary, describing different physical s i t u a t i ~ n s ~ ~ 'The ~ ~ 'two ~ . interpretations seem t o suggest a one-way implication going from the LL phase transition to the density anomaly: the LL phase transition hypothesis apparently implies the presence of density anomaly, but the singularityfree interpretation suggests that the density anomaly does not imply the occurrence of a LL phase transition. Understanding the relation between the density anomaly and the LL phase transition is relevant to understand this transition, and to predict in which systems a LL phase transition can be found experimentally. The interest in this issue has been renewed by recent experimental evidences of LL phase transitions in one-component systems such as phosphorous'2, triphenyl phosphite13 and Yz03-A120314. 2. Does the Density Anomaly Imply the LL Phase Transition?

To answer this question, we consider a model for a water-like molecular fluid with density anomaly and with intermolecular and intramolecular interactions". To mimic the density anomaly we assume, motivated by experimental observations", that the formation of a HB leads to an expansion in local volume7, V = VOf N H B V H B . Here VOis the volume of the liquid with no HBs, N H B = C(i,j) ninjboij,oji is the total number of HBs in the system, and U H B is the specific volume per HB. We partition the fluid into N cells of equal size and associate a variable ni with each cell i = 1,.. . ,N , where ni = 1 if a molecule occupies the cell and ni = 0 otherwise. The cells have the size comparable to that of a water molecule, and the molecules have four arms, one per possible HB. Experiments" show that the relative orientations of the arms of a water molecule are correlated, suggesting an orientational intramolecular interaction between the arms, with a finite interaction energy. Hence, we introduce the Hamiltonian"

The first two terms describe the isotropic and orientational contribution, respectively, of the intermolecular interaction, where E > 0 is the van der Waals attraction energy for molecules in nearest neighbor (NN) cells summed over all the possible NN cells, and J > 0 is the energy gain per each HB formed between molecules in NN cells. The Potts variable uij = 1,.. . ,q, with a finite number q of possible orientations, represents, for the molecule in cell i, the orientation of the arm facing the cell j , with molecules in NN cells forming a HB only if they are correctly oriented, i.e. if 6,ij,,,ji = 1 (6=,b = 1 if a = b and 6a,b = 0 otherwise). The third term accounts for the intramolecular interaction, with an energy gain J , > 0 for each of the 4Cz = 6 different pairs (k,l ) i of arms of the same molecule

212

i with the appropriate orientation (buiLlCil= 1). For J , = 0 we recover the model introduced in Ref.7 that predicts the singularity-free scenario. We perform analytic calculations in mean field approximation" and off-lattice Monte Carlo (MC) simulation", finding for J , > 0 a LL phase transition, ending in a critical point whose temperature decreases to zero and vanishes with J, (Fig. l)ll. General considerations suggest that the liquid-liquid phase transition for water occurs below the glass temperature, i.e. outside the accessible experimental range". Therefore, this model predicts that the singularity-free scenario for anomalous liquids is strictly valid only for molecular liquids with vanishing intramolecular interaction, while for a finite intramolecular interaction the density anomaly implies a LL phase transition. Once this implication is proved, one can explore the implication in the other direction, in order to clarify if experimental investigation of a possible LL phase transition should be limited only to anomalous liquids.

3. Does a LL Phase Transition Imply the Density Anomaly? To answer this question we inve~tigate'~, by molecular dynamics (MD) simulation^^^^^^, integral equation^'^!^^ and modified van der Waals approach16, the phase behavior of an isotropic soft-core attractive potential for a single-component system in 3 dimensions (inset Fig. 2), similar to potentials used to describe systems such as colloids, protein solutions or, to some extent, liquid metals. For largeenough attractive range we find a gas-LDL phase transition and, at lower T , higher p and higher P, a LDL-HDL phase transition16. For short attractive range and small repulsive shoulder (Fig. 2), we predict a phase diagram with a gas-LDL phase transition and a gas-HDL phase transition, both ending in critical points and both metastable with respect to the crystal phase. The latter phase diagram is reminiscent of the experimental phase diagram for fluid phosphorous, showing gas-LDL phase transition and gas-HDL phase transition12. In all the cases we studied, this potential does not show density anomaly, so our work shows that, at least theoretically, the occurrence of a LL critical point, or more generally the coexistence of two liquids, does n o t necessarily imply the presence of a density anomaly. This result suggests the possibility of finding a LL phase transition in a class of systems wider than liquids with density anomalies. 4. Conclusions

Our results for a water-like model predict that a liquid with a density anomaly and non-zero intramolecular interaction has a LL phase transition, which may be pre-empted by inevitable freezing. This conclusion applies for network-forming systems, such as water or silicals. Recently an analogous conclusion has been reached by other authors with a different approachlg. On the other hand, by studying an isotropic soft-core attractive potential, we show15i16,17that a system with a LL phase transition, and possibly a LL critical

213

,,TMD (Temperature of Maximum Density: ‘a\.-.

4

‘9

0

40

C’ 0

0.5

1

Temperature T/E Figure 1. The mean field P-T phase diagram for the water model in Sec.2 with J O / e = 0.05, showing the gas-liquid first-order phase transition (black) line ending in the critical point C,the (dashed blue) line of temperatures of maximum density (TMD) and the LDLHDL first-order phase transition (red) line ending in the critical point C‘.

point, does not necessarily have a density anomaly. Since this kind of potential describes systems like colloids, solutions of biomolecules and liquid metals, within specific approximations, it is reasonable t o predict t h a t a LL phase transition could be experimentally investigated in these systems. In particular, we found results reminiscent of t h e recently-investigated phosphorus experimental phase diagram12. We thank our collaborators, S. V. Buldyrev, G. Malescio, M. I. Marquks, F. Sciortino, A. Skibinsky, and M. Yamada. We thank Ministerio de Ciencia y Tecnologia (Spain) and NSF Chemistry Program C H E 0096892 and CHE0404673 for support.

References C.A. Angell, M. Oguni, and W.J. Sichina, J . Phys. Chem. 8 6 , 998 (1982). R.J. Speedy, C.A. and Angell, J . Chem. Phys. 6 5 , 851 (1976). D.E. Hare, and C.M. Sorensen, J. Chem. Phys. 87, 4840 (1987). R. J. Speedy, J. Phys. Chem. 86,3002 (1982); M. C. D’Antonio and P. G. Debenedetti, J. Chem. Phys. 86, 2229 (1987). 5. S. Sastry, F. Sciortino, and H. E. Stanley, J. Chem. Phys. 98, 9863 (1993). 6. H. E. Stanley and J. Teixeira, J. Chem. Phys. 73, 3404 (1980). 7. S. Sastry et al., Phys. Rev. E 53, 6144 (1996); L. P. N. Rebelo et al., J. Chem. Phys. 109, 626 (1998); E. La Nave et al., Phys. Rev. E 5 9 , 6348 (1999). 8. P. H. Poole et al., Nature (London) 360, 324 (1992). 9. P. H. Poole et al., Phys. Rev. Lett. 73, 1632 (1994); S. S. Borick et al., J. Phys. Chem. 99, 3781 (1995); C. J. Roberts et al., Phys. Rev. Lett. 77, 4386 (1996); C. J. Roberts and P. G. Debenedetti, J. Chem. Phys. 105, 658 (1996); T. M. Truskett et al., ibid. 111, 2647 (1999). 10. G . Franzese, and H.E. Stanley, J . Phys. Condens. Matter 14, 2201 (2002);

1. 2. 3. 4.

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Number Density p ( aJ )

0

0.b7

o.ii 0.i3 Number Density p ( a-3)

0.69

0.i5

Figure 2. The MD P-p phase diagram for the potential in the inset (with U R / V A= 0.5, w R / a = 1, w A / a = 0.2), showing the gas phase, the LDL phase and the HDL phase, with a gas-LDL critical point (CI) and a gas-HDL critical point (Cz) with the corresponding coexistence regions. Panel (b) is a blow up of panel (a) in the vicinity of C1.

11. G . Franzese, M. I. MarquBs, and H.E. Stanley, Phys. Rev. E67, 011103 (2003). 12. Y . Katayama et al., Nature (London) 403, 170 (2000); Science 306, 848 (2004); G. Monaco et al. Phys. Rev. Lett. 90, 255701 (2003). 13. R. Kurita and H. Tanaka, Science 306, 845 (2004). 14. S. Aasland and P. F. McMillan, Nature 369, 633 (1994); M. C. Wilding et al., J. Non-Cryst. Solids 297, 143 (2002). 15. G. Franzese et al., Nature 409, 692 (2001); Phys. Rev. E 66 051206 (2002). 16. A. Skibinsky et al., Phys. Rev. E 69 061206 (2004). 17. G. Malescio et al., cond-mat/0412159 (2004). 18. ISaika-Voivod et al., FSciortino and P.H.Poole, Phys. Rev. E 63, 011202-1 (2001). 19. F. Sciortino, E. La Nave, and P. Tartaglia, Phys. Rev. Lett. 91, 155701 (2003).

OPTIMIZATION BY THERMAL CYCLING

A. MOBIUS Leibniz Institute for Solid State and Materials Research Dresden, PF 2701 16, 0-01171 Dresden, Germany E-mail: [email protected]

K.H. HOFFMANN T U Chemnitz, Institute of Physics, 0-09107 Chemnitz, Germany E-mail: [email protected] c . SCHON Max Planck Institute for Solid State Research, 0-70569 Stuttgart, Germany E-mail: [email protected]

Thermal cycling is an heuristic optimization algorithm which consists of cyclically heating and quenching by Metropolis and local search procedures, respectively, where the amplitude slowly decreases. In recent years, it has been successfully applied to two combinatorial optimization tasks, the traveling salesman problem and the search for low-energy states of the Coulomb glass. In these cases, the algorithm is far more efficient than usual simulated annealing. In its original form the algorithm was designed only for the case of discrete variables. Its basic ideas are applicable also to a problem with continuous variables, the search for low-energy states of Lennard-Jones clusters.

1. Introduction

Optimization problems with large numbers of local minima occur in many fields of physics, engineering, and economics. They are closely related to statistical physics, see e.g. Ref. [l].In the case of discrete variables, such problems often arise from combinatorial optimization tasks. Many of them are difficult to solve since they are NP-hard, i.e., there is no algorithm known which finds the exact solution with an effort proportional to any power of the problem size. One of the most popular such tasks is the traveling salesman problem: how to find the shortest roundtrip through a given set of cities 2 . Many combinatorial optimization problems are of considerable practical importance. Thus, algorithms are needed which yield good approximations of the exact solution within a reasonable computing time, and which require only a modest effort in programming. Various deterministic and probabilistic approaches, so-called search heuristics, have been proposed to construct such approximation algorithms. A considerable part of them borrows ideas from physics and biology. Thus simu-

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lated annealing and relatives such as threshold accepting as well as various genetic algorithms have successfully been applied to many problems. Particularly effective seem to be genetic algorithms in which the individuals are local minima For recent physically motivated heuristic approaches we refer to thermal cycling ', optimization by renormalization *, and extremal optimization g. For problems with continuous variables, approaches which combine Monte-Carlo procedures for global search with deterministic local search by standard numerical methods, for example the basin-hopping algorithm, have proved to be particularly efficient They can be considered as relatives of the genetic local search approaches for the case of discrete variables. Here we focus on the thermal cycling algorithm and illuminate the reasons for its efficiency. 516.

loill.

2. Thermal cycling algorithm Simulated annealing can be understood as a random journey of the sample (i.e. the approximate solution) through a hilly landscape formed by the states of its configuration space. The altitude, in the sense of a potential energy, corresponds to the quantity to be optimized. In the course of the journey, the altitude region accessible with a certain probability within a given number of steps shrinks gradually due to the decrease of the temperature in the Metropolis simulation involved. The accessible area, i.e., the corresponding configuration space volume, thus shrinks until the sample gets trapped in one of the local minima.

t Figure 1. Time dependence of the energy E (quantity t o be optimized) of the sample currently treated in the cyclic process. Gaps in the curve refer t o cycles where the final state has a higher energy than the initial state, so that the latter is used as initial state of the next cycle too.

Deep valleys attract the sample mainly by their area. However, it is tempting to make use of their depth. For that, we substitute the slow cooling down by a cyclic process: First, starting from the lowest state obtained so far, we randomly deposit energy into the sample by means of a Metropolis process with a certain temperature T, which is terminated, however, after a small number of steps. This

217 part is referred t o as heating. Then we quench the sample by means of a local search algorithm. Heating and quenching are cyclically repeated where the amount of energy deposited in a cycle decreases gradually, see Fig. 1. This process continues until, within a ‘reasonable’ CPU time, no further improvement can be found. It is an essential feature of the thermal cycling algorithm that two contradicting demands are met in heating: the gains of the previous cycles have to be retained, but the modifications must be sufficiently large, so that another valley can be reached. Thus the heating process has to be terminated in an early stage of the equilibration. An effective method is to stop it after a fixed number of successful Metropolis steps. The efficiency of the proposed algorithm depends to a large extent on the move class considered in the local search procedure. For discrete optimization problems, it is a great advantage of our approach that far more complex moves can be taken into account than in simulated annealing so that the number of local minima is considerably reduced. The local search concerning complex moves can be enormously sped up by use of branch-and-bound type algorithms. Their basic idea is to construct new trial states following a decision tree: At each branching point, a lower bound of the energy of the trial state is calculated. The search within the current branch is terminated as soon as this bound exceeds the energy of the initial state. The basic thermal cycling procedure can be easily accelerated in three ways: (i) partition of the computational effort into several search processes in order to minimize the failure risk 14, (ii) restricting the moves in heating to the ‘sensible sample regions’ by analyzing previous cycles, or by comparing with samples considered in parallel, and (iii) combining parts of different states 7912.

3. Applications The thermal cycling algorithm was first tested on the traveling salesman problem ’. For that, we considered problems of various size from the TSPLIB95 l5 for which the exact solutions, or at least related bounds, are known. Fig. 2 gives a comparison of thermal cycling data with results from simulated annealing and from repeated local searches starting from random states. For a meaningful characterization of the algorithms, it relates mean deviations from the optimum tour length to the CPU-time effort for various parameter values. The diagram includes data for two move classes: (a) cutting a roundtrip twice, reversing the direction of one of its parts, and connecting the parts then again, or shifting a city from one to another position in the roundtrip; (b) same as (a) and additionally rearrangements by up to four simultaneous cuts as well as Lin-Kernighan realignments 16. Fig. 2 shows that, for the traveling salesman problem, thermal cycling is clearly superior t o simulated annealing, already if the same move class is considered in both procedures - the simulated annealing code had been carefully tuned too -. However, when taking advantage of the possibility to incorporate more complex moves, thermal cycling beats simulated annealing by orders of magnitude in CPU time. Applied to an archive of samples instead of t o a single one, it can compete

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Figure 2. Relation between CPU time, TCPU (in seconds for one PA8000 180 MHz processor of an HP K460), and deviation, 6L = L,,,, - 27686, of the obtained mean approximate solution from the optimum tour length for the Padberg-Rinaldi 532 city problem 0: repeated quench t o stability with respect t o move class (a) defined in the text; A: simulated annealing; x and m: thermal cycling with ensembles of various size, and local search concerning move classes (a) and (b), respectively. In all cases, averages were taken from 20 runs. Errors (lu-region) are presented if they exceed the symbol size. The lines are guides t o the eye only.

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1o4

7 CPU t=cI

Figure 3. Mean deviation of the energy of the lowest state found from the ground state energy, 6E = Em,,, - Eground state, related t o the CPU time TCPU (180 MHz PA8000 processor of HP K460) for one realization of the threedimensional Coulomb glass lattice model with 1000 sites, half filling, and medium disorder strength 13. X : simulated annealing; A: multistart local search considering simultaneous occupation changes of up t o four sites; m: thermal cycling. For simulated annealing and multistart local search, averages were taken from 20 runs, for thermal cycling from 100 runs. In thermal cycling, the ground state was always found within 500 seconds.

219 with leading genetic local search algorithms '112. For several years, we have used thermal cycling as standard approach in numerical investigations of the Coulomb glass, which is basically an Ising model with long-range interactions. Also in this case, thermal cycling proved to be a very efficient tool. Fig. 3 presents data from an investigation comparing several algorithms 13. In simulated annealing, we could efficiently treat only particle exchange with a reservoir and one-particle hops inside the sample, that is occupation changes of one or two sites. However, in the deterministic local search, the simultaneous occupation modification of up to four sites could be considered by means of branch-and-bound approaches. Therefore, the corresponding multistart local search yields significantly better results than simulated annealing. Thermal cycling of the low-energy states proves to be still far more efficient than the local search repeatedly starting from random states. It is tempting to apply the thermal cycling approach also to problems with continuous variables. Thus we have considered Lennard-Jones cluster of various size because the energy landscapes of this system are known to have large numbers of local minima. The heating consisted of simulateneously shifting all atoms by small distances a few times according to a thermal rejection rule, and the quench combined the Powell algorithm with a systematic consideration of symmetry positions. The ground states of several clusters of up t o 150 atoms could be reproduced within 'reasonable' CPU times. Further related investigations should be promising.

References 1. Y. Usami and M. Kitaoka, Int. J. Mod. Phys. B 11, 1519 (1997). 2. D. S. Johnson and L. A. McGeoch, in Local Search in Combinatorial Optim.ization, eds. E. Aarts and J. K. Lenstra (Wiley, Chichester, 1997), p. 215. 3. S. Kirkpatrick, C. D. Gelatt, Jr., and M. P. Vecchi, Science 220, 671 (1983). 4. J. H. Holland, Adaptation in Natural and Artificial Systems: an Introductory Analysis with Applications to Biology, Control, and Artificial Intelligence, (University of Michigan Press, Ann Arbor, 1975). 5. R. M. Brady, Nature 317, 804 (1985). 6. P. Merz and B. Reisleben, in Proc. 1997 IEEE Int. Conj. on Evolutionary Computation, Indianapolis, (IEEE Press,1997), p. 159. 7. A. Mobius, A. Neklioudov, A. Diaz-Shchez, K.H. Hoffmann, A. Fachat and M. Schreiber, Phys. Rev. Lett. 79, 4297 (1997). 8. J. Houdayer and 0. C. Martin, Phys. Rev. Lett. 83, 1030 (1999). 9. S. Boettcher and A. G. Percus, Phys. Rev. Lett. 86, 5211 (2001). 10. D. J. Wales and J. P. K. Doye, J. Phys. Chem. A 101 (1997), 5111. 11. M. Iwamatsu and Y . Okabe, Chem. Phys. Lett. 399,396 (2004). 12. A. Mobius, B. F'reisleben, P. Merz and M. Schreiber, Phys. Rev. E 5 9 , 4667 (1999). 13. A. Diaz-SAnchez, A. Mobius, M. Ortuiio, A. Neklioudov and M. Schreiber, Phys. Rev. B 62, 8030 (2000). 14. B. A. Huberman, R. M. Lukose and T. Hogg, it Science 275, 51 (1997). 15. www.iwr.uni-heidelberg.de/groups/comopt/so~ware/~SPLIB95 16. S. Lin and B. Kernighan, Operations Research 21, 498 (1973).

ULTRA-THIN MAGNETIC FILMS AND THE STRUCTURAL GLASS TRANSITION: A MODELLING ANALOGY

S. A. CANNAS AND F. A. TAMARIT Facultad de Matemcitica, Astronomia y Fisica Universidad Nacional de Cdrdoba Ciudad Universitaria, 5000 Cdrdoba, Argentina E-mail: [email protected]. edu.ar and [email protected]. ar P. M. GLEISER Centro Atdmico Bariloche, Sun Carlos de Bariloche, 8400 Rio Negro, Argentina E-mail: gleiser0cab.cnea.gov.ar

D . A. STARIOLO Departamento de Fisica, Universidade Federal do Rio Grande do Su1, C P 15051, 91501-979, Porto Alegre, Brazil E-mail: starioloOif.uj?gs. br In this work we study a two dimensional king model for ultrathin magnetic films which presents competition between short range ferromagnetic interactions and long-range antiferromagnetic dipolar interactions. We present evidence that the dynamical and thermodynamical properties of the model allow for an alternative interpretation, in terms of glass forming liquids. In particular, the existence of a first order phase transition between a low temperature crystal-like ordered phase a high temperature liquid-like disordered phase, which can be supercooled below the melting point, together with a drastic slowing down after a quench t o low temperatures suggest that these materials could present a phenomenology similar t o that observed in glass forming liquids.

1. Introduction The physics of glass forming liquids and structural glasses in general, appears today as a great challenge in statistical mechanics and chemical physics. Despite the huge effort devoted to the field and the enormous improvements obtained in the comprehension of these complex systems, due both t o theoretical and experimental studies, there are still many open questions concerning their phenomenology. Most of the theoretical knowledge in the field resides today in two different approaches. On one side, there are different phenomenological and first principles microscopic theories (for a recent review see Ref.l). Among the last ones,

220

221 perhaps the most successful one is the mode coupling theory ', in the sense of having more experimentally verified predictions. However, up to now no one of the existing theories can account for a complete description of the observed phenomenology. On the other hand, the constant improvement in the computational capacity allowed the implementation of very accurate molecular dynamics simulations. Most of these simulations are based on small binary systems of particles interacting through Lennard-Jones like potentials. While the existing microscopic theoretical approaches seem to be very limited due to the complexity of the analytical treatment, the numerical approach is limited by the small number of particles that can be considered and the extremely small time span that a simulation can cover, specially for modelling systems that have an astonishing slow relaxation dynamics '. In a completely different scheme, the statistical physics community has been looking, since many decdes ago, for a simple lattice model able to catch (independently of the degree of accuracy of its microscopic description) those few relevant ingredients which are responsible of the rich dynamical and thermodynamical phenomenology of these materials. There is a general consensus in the community in the sense that a relevant element in the description of structural glasses is the appearance of frustration at the level of microscopic interactions between molecules. And, unlike what happens in many other complex systems, as for instance spin glasses, this frustration is not due to the existence of randomness, but to the emergence of competition between attractive and repulsive interactions acting on each particle. Among the many different approaches presented in the literature in order to introduce a lattice model for structural glasses, we want to mention in first place that presented by Shore, Holzer and Sethna 3, since, as will become clear in short, it is intimately related to the scope of our paper. They consider the magnetic Ising system on a square and on a cubic regular lattices, with ferromagnetic interactions between nearest-neighbors plus antiferromagnetic interactions between next-nearest neighbors spins, and without taking into account any kind of randomness in the Hamiltonian of the model. For the two dimensional case, they actually found a relatively simple dynamical and thermodynamical behavior, which is far from being glassy. But the situation was completely different for the three dimensional case, where at very low temperatures they found a drastic slowing down of the relaxation, ruled by a logarithmic domain growth law, proper of glassy systems. This simple model, without imposed disorder and with competition, showed to be able to reproduce at least partially and qualitatively the complex phenomenology of a glassy processes. Nevertheless, its main limitation was the existence of a second order phase transition between the high temperature disordered phase (analog to a liquid state) and the ferromagnetic ordered phase (analog to a crystalline state). This continuous transition without coexistence of phases can not give account of the process of supercooling a liquid, which is intimately related to the process of

222 forming a structural glass. Since then, many other attempts have been done in order to improve that first intent. In a series of papers, Lipowski and co-workers considered the same model introduced by Shore et al. plus a four spin plaquette ferromagnetic term in the Hamiltonian. This model captures most of the complex dynamics of the original model, and presents a first order phase transition, as desired. Nevertheless, its ground state is infinitely degenerate, a fact that can be hardly associated to the crystalline ordering of a solid. Later on, Cavagna, Giardina and Grigera considered a two dimensional model with a two terms Hamiltonian: the previously described four spin ferromagnetic plaquette plus a five term ferromagnetic plaquette. This is actually an excellent model which describes most of the expected features of a twedimensional structural glass. Another interesting approach considered a system with nearest-neighbors coupling and antiferromagnetic coulomb interactions, which also proved t o be an adequate model for describing three dimensional glass forming liquids. Nevertheless, all these systems, besides their great value as statistical mechanics prototypes for modelling a structural glass without imposing disorder in the Hamiltonian, are not really inspired by any physical realization. In this work instead, we will present a model which has been vastly analyzed during the last ten years and is considered to be the proper tool for describing the physics of real ultra thin magnetic films, but instead of paying attention to the magnetic behavior, we will show that the model, and consequently perhaps ultra thin films, present evidence of displaying a structural glass-like state. 2. A metal on metal ultra thin film model

The physics of ultra-thin magnetic films has deserved a great interest during the last years, not only because of their multiple technological applications, such as data storage and catalysis, but also because their study has opened novel and nontrivial questions related t o the role of microscopic competitive interactions in the overall behavior of a system. In particular, under suitable thermal and magnetic conditions, ultra thin magnetic films form unusual complex patterns of magnetization 7,8. And precisely, most of the potential technological applications of these materials reside in the ability of controlling these patterns, both in time and space, with a high degree of accuracy. For instance, in the case of data storage the stabilization of very small metastable magnetic domains could eventually increase the compression obtained nowadays in recording devices. The model we will analyze in this paper has been used mainly in the study of metal films on metal substrates, as for example Fe on Cu or Co on Au lo. In these cases, an adequate theoretical description of the system must take into account, at least, a three terms Heisenberg Hamiltonian, including: i) the usual ferromagnetic exchange interactions between nearest-neighbors spins, ii) the dipoledipole interactions which, despite their considerable small strength (when compared with the exchange interactions) become relevant due to their long range, and finally iii) a sur-

223 face anisotropy term that takes into account the magnetic influence of the substrate on the spins of the film. The anisotropy induced by the dipolar interaction tends to align the spins parallel to the film, but, as the thickness of the film is reduced (usually around approximately five monolayers) the surface anisotropy overcomes the anisotropy of the dipolar interaction and the system suffers a reorientation transition at which the spins suddenly align perpendicular to the plane defined by the proper film. Under these particular conditions, the physics of the material can be appropriately described by replacing the Heisenberg spin variables by the much simpler Ising magnetic moments located at the nodes of a square lattice, and the Hamiltonian takes the form:

where Si = f l and 6 is the ratio between the exchange JOand dipolar Jd interactions strengths (6 = Jo/Jd). Here the first term represents the ferromagnetic exchange interaction and the sum runs only over nearest-neighbors spins, while the second one represents the dipole-dipole interaction once the spins have aligned perpendicular to the plane. In this last case, the sum runs over all pairs of spins of the lattice and rij is the distance, measured in crystal units, between the sites i and j. Then, the system is ruled only by two variables: the usual temperature T and the parameter 6, which depends on the composition and preparation of the sample. We will restrict ourselves to consider the case Jd > 0, in such a way that 6 > 0 corresponds to ferromagnetic exchange interactions and 6 < 0 to antiferromagnetic ones. Note that the model introduced by Shore et al. can be considered as truncated version of the model defined by Hamiltonian (1). In 1995 MacIsaac and coauthors l1 presented the first study of the thermodynamics of the model (for a complete review of the subject see 12). Concerning the order of the ground state, it is antiferromagnetic when 6 < 6, M 0.425, but when 6 > 6, the system orders forming stripes whose width h depends on the value of 6. In particular, h increases as 6 increases and, surprisingly, the ferromagnetic state is always metastable respect to a striped one. In other words, irrespectively of the strength of the dipole-dipole interactions, the frustration induced by the antiferromagnetic term avoids the usual ferromagnetic ordering. In the same paper l1 they also presented a phase diagram of the model, obtained through Monte Carlo simulations on a relatively small system of N = 16 x 16 spins. They observed that, for fixed values of 6, the system suffers an order-disorder phase transition between a low temperature striped phase and a high temperature tetragonal phase. The last one consists of extended magnetic domains characterized by predominantly square corners, which induces a four fold rotational symmetry (as can be clearly observed in numerical simulations of the structure factor). The existence of this tetragonal phase has been recently verified experimentally in a fcc Fe on Cu(100) films and had already been predicted by Abanov et al. by means of a continuous approximation 13. '7'

224

disordered T

Figure 1. The phase diagram for intermediate values of 6 obtained in L = 24 lattices. Triangles: critical temperature obtained from the maximum in the specific heat; circles: stability line of the h l stripe phase; open squares: stability line of the h2 stripe phase; diamonds: first order transition lines between low temperature ordered phases. TP indicates a triple point.

i

t

00

Figure 2. Specific heat 2)s. T for 6 = 3 (corresponding to an h = 4 ground state) and three different system sizes. Some typical equilibrium configuration a t the indicated temperatures for L = 48 are shown below. Note the sequence of transitions h4 tetragonal --t paramagnetic.

-

In Fig. 1 we present a detail of the phase diagram obtained in l4 corresponding to the intermediate values of 6. Here h l and h2 indicates the regions where the ground states have widths h = 1 and h = 2, respectively, and AF indicates the antiferromagnetic phase. The gray region indicates the presence of metastable states. The lines (diamonds) separating the low temperature phases are all first order ones, and TP indicates the existence of a triple point (as will be become clear in the next section).

225

.84

-

.83

-

.a. -81

-

.m .70

T,-0.776

.TI 0 . W

.wO5

,0010

,0015

,0020

,0025

.o030

.OM5

.I040

. a 5

L-2 Figure 3. Pseudo critical temperatures T: (maximum of the specific heat) and T," (minimum of the Binder cumulant) vs. L-* for 6 = 2.

3. Super cooled tetragonal liquid state

In this section we will present some recent and preliminary evidence that strongly suggests that metal on metal ultra thin magnetic films could have a glassy transition. Supercooled glass forming liquids have the property of getting trapped into a liquid metastable state (with respect to the crystalline state), when cooled below the melting temperature under suitable conditions. But, if the cooling is done suddenly enough at sufficiently low temperatures, the characteristic relaxation time attains macroscopic scales, the supercooled state is structurally arrested and the material behaves as a solid without any pattern of long range order. Under these conditions the systems becomes a glass. Then, a basic ingredient for a glass transition model is the existence of a first order phase transition between a high temperature liquid-like disordered phase and a low temperature crystal-like ordered phase. We will now show that the order-disorder phase transition observed in the model described by Hamiltonian (l),is actually a weak first order one, at least for intermediate values of b (though some analytical approximate results l6 suggest that this could be valid for any finite value of 6). In Fig. 2 we plot the specific heat CL as a function of the temperature T for b = 3, and three different system sizes. We can clearly identify two peaks. The low

226 temperature one increases with the size L and coincides with the temperature at which the tetragonal phase appears. Then it is associated with the stripe-tetragonal transition. The second broader peak does not manifest any dependence on the system size and indicates the continuous decay of the tetragonal phase into the paramagnetic phase. We also present some snapshots of typical configurations of the system for different temperatures. What about the order of the transition? One way of determining it is through the analysis of the finite size scaling behavior of different moments of the energy, like the specific heat

and the Binder fourth order cumulant

which permit to distinguish between continuous and discontinuous transitions. In a first order phase transition the specific heat presents a maximum at a pseudo critical temperature T,'(L) and the Binder cumulant a minimum at another different pseudo critical temperature T,"(15).Those temperatures present the following finite size scaling behavior T;(L) T, AL-2 and T,"(L) T, BL-2, with B > A , T, being the transition temperature of the infinite system 15. In Fig. 3 we plot T,' and T," vs. 1/L2 for 6 = 2, identifying clearly the finite size scaling behavior expected in a first order phase transition 16. Moreover, numerical simulations of the energy histogram around the transition point for 6 = 2 show a two peak structure, proper of this kind of transitions 16. Next we will show that it is possible to get a supercooled metastable state below the melting temperature, another significant feature of glass forming liquids. In Fig. 4 we plot (full circles) the average internal energy per particle u(T) as a function of the temperature in a quasi-static cooling from a high temperature. Each point corresponds t o an average over many different initial conditions and sequences of random numbers. In the same plot (empty triangles), we also display the result of a quasi-static heating from the ground state. We clearly observe the emergence of hysteresis, proper of a first order phase transition. From these energy curves we obtained the free energy per spin on cooling and heating, finding that both curves intersect at T, = 0.805 f 0.005, a temperature that we identify with the melting point. Finally, let us describe the behavior of the system when it is suddenly quenched into a very low temperature. In Fig. 5 we plot the time evolution of the internal energy per spin along a single Monte Carlo run for 6 = 2 , L = 32 and T = 0.2. We observe that the system is stuck into a disorder configuration, with a very slow relaxation rate (almost logarithmic, as can be observed in the inset). The dashed line indicates the energy per spin of the ground state, and we see that the system is magnetically arrested in this out of equilibrium disordered state. We also present, at different times, snapshots of the corresponding pattern of magnetization, where N

+

N

+

227 -0.9

-1 .o

3 -1 .l

n-J

I I I

-1.2

0.5

0.6

0.7

0.8

0.9

1 .o

T Figure 4. Internal energy per spin u(T)obtained by quasistatic cooling from infinite temperature and quasistatic heating from the ground state.

-1.2

!

Figure 5. Time evolution of the energy per spin in a single MC run. Snapshots of the spin configurations are shown below the figure. The inset presents the same results for the time evolution of the energy per spin in a log-normal plot.

one can recognize and almost tetragonal phase. This behavior, characterized by a slow relaxation of a disordered liquid-like phase well below the melting temperature,

228 which can neither be associated with nucleation nor with coarsening, can be clearly interpreted in terms of the appearance of a glassy phase. 4. Conclusions

In this paper we have revisited the two dimensional Ising model with competing nearest-neighbors ferromagnetic interactions and long range antiferromagnetic dipoldipole interactions from a new point of view. Instead of concentrating on the magnetic properties of the model, we have investigated a possible characterization of the tetragonal phase as a lattice version of a liquid that can give place to a glass state a very low temperatures. It has been well established in our simulations that the order-disorder temperature driven phase transition between the tetragonal and striped phases is a weak discontinuous one, as revealed by the scaling law of the pseudo critical temperatures obtained from the specific heat and the fourth order Binder cumulant. Analitical results on a continuous version of the model give further support to this conclusions 16. We have also shown, by simulating a quasistatic cooling from infinite temperature, that the tetragonal phase can be supercooled well below the melting temperature. Furthermore, when the system is suddenly cooled down into a low enough final temperature, it gets stuck in a glass like state, characterized by an extremely slow relaxation process. It is important here to mention that previous papers had already reported the existence of some glassy like phenomena, as for instance, aging l7ll8 and logarithmic domain growth law but all of them were connected to the existence of metastable stripe states, which it is well known, modify the landscape of the free energy function. Instead, the results presented in this paper refer to a different and novel observation, namely, the existence of a first order phase transition between the tetragonal (liquid) and the striped phases (crystal) and the supercooling of the former in a long standing metastable state well below the melting temperature, indicating the emergence of a two dimensional glass. Finally, let us stress that the present model, unlike all the other lattice models cited in this papers, did not arise as a statistical mechanics toy model able to catch the main features of a glass forming liquid. On the contrary, the model is widely accepted to be the proper one for describing the physics of real metal on metal ultra thin magnetic films. In other words, our results strongly suggest that these materials present a glass transition, a fact that would have many relevant technological consequences. As much as we know, this is the first example of a physical realization of a two dimensional magnetic system without imposed disorder that can be considered as a glass forming liquid. Nevertheless, this point requires further investigation. In that sense, a careful study of the behavior of the relaxation time as the temperature decreases, as well as an adequate characterization of the nucleation process, will not only be a clear confirmation of the existence of a glass state but will also bring light into its

229 nature. Works along these lines are in progress and will be published elsewhere. Experimental checks of our predictions will b e also very helpful.

Acknowledgments This work was partially supported by grants from Consejo Nacional de Investigaciones Cientificas y TBcnicas CONICET (Argentina), Agencia CQdoba Ciencia (Cbrdoba, Argentina), Secretaria de Ciencia y Tecnologia de la Universidad Nacional de C6rdoba (Argentina), CNPq (Brazil) and ICTP grant NET-61 (Italy). P.M.G. acknowledges financial support from Fundaci6n Antorchas (Argentina).

References 1. R. Schilling, Theories of the strzlctural glass transition, appears in ”Collective Dynamics of Nonlinear and Disordered Systems”, eds. G. Radons, W. Just and P. Hwussler, Springer (2003) - also in cond-mat/0305565. 2. W. Kob, J. Phys.: Condens. Matter 11, R85-Rl15 (1999). 3. J. D. Shore and J. P. Sethna, Phys. Rev. B 433782 (1991); J. D. Shore, M. Holzer and J. P. Sethna, Phys. Rev. B 46, 11376 (1992). 4. A. Lipowski, J. Phys. A 30, 7365 (1997); Lipowski and D. Johnston, J. Phys. A 33, 4451 (2000); Phys. Rev. E 61, 6375 (2000); Phys. Rev. E 64, 041605 (2001). 5. A. Cavagna, I. Giardina T . S. Grigera, J. Chem. Phys. 118, 6974 (2003); Europhys. Lett. 61, 74 (2003). 6. P. Viot and G . Tarjus, Europhys. Lett. 44, 423 (1998); G. Tarjus, D. Kivelson, and P. Viot, J. Phys : Cond. Matter. Special issue: Unifying concepts in Glass Physics 12, 6497 (2000); M. Grousson, G. Tarjus and P. Viot, Phys. Rev. E 62, 7781 (2000); Phys. Rev. E 64, 036109 (2001); J. Phys: Cond. Matter 14, 1617 (2002). Phys. Rev. E 65, 065103(R) (2002) 7. A. Vaterlaus, C. Stamm, U. Maier, M. G. Pini, P. Politi and D. Pescia, Phys. Rev. Lett. 84, 2247 (2000). 8. 0. Portmann, A. Vaterlaus and D. Pecia, Nature 444, 701 (2003). 9. D. P. Pappas, K. P. Kamper and H. Hopster, Phys. Rev. Lett 64, 3179 (1990). 10. R. Allenspach, M. Stampanoni and A. Bischof, Phys. Rev. Lett 65, 3344 (1990). 11. A. B. MacIsaac,J. P. Whitehead, M. C. Robinson and K. De Bell, Phys. Rev. B 51, 16033 (1995). 12. K. De’Bell, A. B. MacIsaac and J. P. Whitehead, Rev. Mod. Phys. 72, 225 (2000). 13. A. Abanov, V. Kalatsky, V. L. Pokrovsky. and W. M. Saslow, Phys. Rev. B 51, 1023 (1995). 14. P.M. Gleiser,, F.A. Tamarit and S.A. Cannas, Physica D 168-169, 73 (2002). 15. J. Lee and J. M. Kosterlitz, Phys. Rev. B 43, 3265 (1991). 16. S.A. Cannas, D. A. Stariolo and F. A. Tamarit, Physical review B 69, 092409 (2004). 17. J. H. Toloza, F. A. Tamarit and S. A. Cannas Phys. Rev. B 58, R8885 (1998). 18. D. A. Stariolo and S. A. Cannas, Phys. Rev. B, 60, 3013 (1999). 19. P. M. Gleiser, F. A. Tamarit, S. A. Cannas and M. A. Montemurro, Phys. Rev. B 68, 134401 (2003).

NON-EXTENSIVITY OF INHOMOGENEOUS MAGNETIC SYSTEMS

M. S. REIS* AND V. S. AMARAL Departamento de Fisica and CICECO Universidade de Aveiro 3810-193 Aveiro, Portugal J. P. ARAUJO IFIMUP, Departamento de Fisica Universidade do Porto 4150 Porto, Portugal I. S. OLIVEIRA Centro Brasileiro de Pesquisas Fisicas Rua Dr. Xavier Sigaud 150, Urca 22290-180 Rio de Janeiro-RJ, Brasil

In recent publications we developed the main features of a generalized magnetic system, in the sense of the non-extensive Tsallis thermostatistics. Our mean-field-non-extensive models predict phase transitions of first and second order, as well as various magnetic anomalies, as a direct consequence of non-extensivity. These theoretical features are in agreement with the unusual magnetic properties of manganites, materials which are intrinsically inhomogeneous. In the present work, we consider an inhomogeneous magnetic system composed by many homogeneous subsystems, and show that applying the usual Maxwell-Boltzmann statistics to each homogeneous bit and averaging over the whole s y s tem is equivalent of using the non-extensive approach. An analytical expression for the Tsallis entropic parameter q was obtained, and showed to be related t o the moments of the distribution of the inhomogeneous quantity. Finally, it is shown that the description of manganites using Griffiths phase can be recovered with the use of the non-extensive formalism.

1. Introduction

Tsallis statistics is applicable to systems which present non-extensivity. Broadly speaking, in order to be non-extensive, a system must present at least one of the following properties: (i) long-range interactions, (ii) long-time memory, (iii) fractality and (iv) intrinsic inhomogeneity '. Manganese oxides, or simply manganites, seems to embody three out of these four ingredients: they present Coulomb long'e-mail: mariorOfis.ua.pt

230

231 range interactions ' v 3 v 4 , they are formed by grains with fractal shapes and they are intrinsically inhomogeneous 5,6,7. In a sequence of previous publications we have shown that the magnetic properties of manganites, some of them very unusual, can be properly described within a mean-field approach using Tsallis statistics. In Ref. lo it is pointed out that the value of the entropic parameter q of a system is related to its magnetic susceptibility. In the present work, through an analogy to the paper of Beck 11, Beck and Cohen l2 and Wilk and Wlodarczyk 13,we consider an inhomogeneous magnetic system composed by many homogeneous parts, each one of them described by the Maxwell-Boltzmann statistics. By averaging the magnetization over the whole system, we recover the Tsallis non-extensivity. We obtain an analytical expression for the entropic parameter q and its dependence with temperature T . Finally, we show that the description of manganites using Griffiths phase 14,15can be incorporated to the non-extensive approach. 8y9,10,

2. Model Description

Homogeneous and Non-Extensive (HNE) case: In Ref. lo an expression for the classical non-extensive magnetization (Generalized Langevin Function) was obtained:

where q E 8 is the Tsallis entropic parameter and

x = - PH kT In this model", the non-extensive correlations lie inside each cluster, whereas the interactions inter-clusters remain extensive. Thus, pne means the magnetic moment of each non-extensive cluster. Inhomogeneous and Extensive (IE) case: Consider an inhomogeneous system formed by smaller Maxwell-Boltzmann homogeneous bits, each of them with magnetization M . We can write the average magnetization considering two types of distribution: A. Distribution of magnetic moments:

B. Distribution of temperature:

where

M ( p , T ,H ) =

[

coth x

I:

--

232

is the usual Langevin function (which yields the magnetization of each small Maxwell-Boltzmann cluster), f and g represent the respective distribution functions and z is the same of Eq. 2. Suppose we have the situation described in (A). Equaling the magnetic saturation value of the HNE case (Eq. 1) and IE case (Eq. 3), we obtain:

where ( p ) is the first moment of the f(p) distribution. Equaling the susceptibilities x4 = (x), and using Eq. 6, we found an expression that connects the q parameter to the moments of the f(p) distribution:

This general result, valid for any f(p), is analogous to that obtained by Beck and Cohen l2 for the case of a Brownian particle travelling through a distribution of temperatures. Proceeding in analogy with case (B), one finds:

where p is the single value of magnetic moment present in the system. Equaling the susceptibilities xq = ( X ) T and using Eq. 8, we obtain:

3. Connections with Griffiths Singularity Salamon and co-workers I41l5 have used the idea of Griffiths singularity t o study manganites. The authors considered a distribution of the inverse magnetic suscep tibility A:

to explain the sharp downturn in the (x)-'(T) curve (behavior usually observed in manganites). In the expression above, c is a parameter of the distribution, I'(--, -) stands for the Incomplete Gamma Function and

where a is a free parameter, /3 =0.38 is a critical exponent for the pure system, assumed to be a 3D Heisenberg-like, and TG is the Griffiths temperature 14,15. From Eq. 10 one can find the average susceptibility: PT

233 and, consequently, its inverse:

(13) that fits the strong downturn usually found on manganites. However, since the Curie Law tells us that

x=-

PZ

3kT ' inhomogeneities in x can arise from distributions of either p or T (or both). From the above equation we can obtain, for instance, a corresponding distribution of magnetic moments:

and, consequently, the q parameter:

The dependence of q with temperature T arises since the first and second moments of the distribution f ( p ) have such dependence. In other words, q(T)is consequence of ( P W ) and ( P 2 ) ( T ) . From the above, one can write the non-extensive magnetic susceptibility:

that is equal to the average one. It is important t o note that, once we know f(p), the value of q can be direct obtained, and, consequently, xq. The present model does not consider the entropic index q as a fitting parameter, but as a known quantity, previously determined and direct related to the inhomogeneity of the system. Finally, an analogous reasoning applies to the case of a distribution of T . 4. Connections with Experimental Results

Pro.o5C~.g5Mn03(T, =110 K) is an example of manganite that presents a strong downturn, around the Curie temperature, in the curve of the inverse susceptibility as a function of temperature. A sample of this manganite was prepared by the starting from the stoichiometric amount of PrzO3 sol-gel method with urea l6l1', (99.9 % pure), CaC03 (99 %) and MnOz (99 %). The final crushed powder was compressed and sintered in air at 1300 "C during 60 hours, with a subsequent fast freezing of the sample. X-ray diffraction pattern confirmed that the sample lies in the Pbnm space group, without vestige of spurious phases. The temperature dependence of the susceptibility x (= M / H at low magnetic field) was carried out using a commercial SQUID magnetometer.

234

Figure 1 presents the experimental data and the model above described (Eq. 17). To obtain this result, the distribution of magnetic moments f(p) presented in Eq. 15 has the following parameters: c =-0.04, a =0.002 K and TG =510 K. The entropic index q (inset of figure l ) , is not a free parameter and could be obtained a priori, using Eqs. 7 and 15,

I0

Experimental data Present model (Eq. 17)

2 P

0 rl

X

n

E' W

01

a 1 0

v

*

'x

0

Figure 1. Experimental (open circles) and theoretical (solid line - Eq. 17) temperature dependence of the inverse susceptibility for the Pro.o5Ca,095MnOsmanganite. See text for details concerning the theoretical description.

5. Conclusion

In the present work, we considered an inhomogeneous magnetic system composed by many homogeneous subsystems, and show that applying the usual MaxwellBoltzmann statistics to each homogeneous bit and averaging over the whole system is equivalent of using the non-extensive approach. An analytical expression for the Tsallis entropic parameter q was obtained, being related to the moments of the distribution of the inhomogeneous quantity. Finally, it is shown that the description of manganites using Griffiths phase can be recovered with the use of the non-extensive formalism. 6. Acknowledgments We thank CAPES-Brasil and GRICES-Portugal for financial support concerning the Brasil-Portugal bilateral cooperation. M.S.R thanks CNPq-Brasil.

235

Bibliography 1. For a complete and updated list of references, see the web site: tsallis.cat.cbpf.br/biblio.htm. 2. LORENZANA J., CASTELLANI C. AND CASTROC.D. Phys. Rev. B 64 (2001) 235127. 3. LORENZANA J., CASTELLANI C. AND CASTROC.D. Phys. Rev. B 64 (2001) 235128. 4. MOREOA., YUNOKIS. A N D DAGOTTOE. Science 283 (1999) 2034. 5. DAGOTTOE., HOTTAT. AND MOREOA. Phys. Rep. 344 (2001) 1. 6. DAGOTTOE. Nanoscale phase separation and colossal magnetoresistance: The physics of manganites and related compounds. (Springer-Verlag, Heidelberg, 2003) . 7. BECKERT., STRENGC., Luo Y., MOSHNYAGA V., DAMASCHKE B., SHANNON N. AND SAMWER K. Phys. Rev. Lett. 89 (2002) 237203. 8. REIS M.S., FREITAS J.C.C., ORLANDOM.T.D., LENZI E.K. A N D OLIVEIRAI.S. Europhys. Lett. 58 (2002) 42. O AMARAL V.S., LENZIE.K. A N D OLIVEIRAI.S. Phys. Rev. 9. REIS M.S., A R A ~ J J.P., B 66 (2002) 134417. V.S., A R A ~ J J.P. O AND OLIVEIRA I.S. Phys. Rev. B 68 (2003) 10. REIS M.S., AMARAL 014404. 11. BECK C. Phys. Rev. Lett. 87 (2001) 180601. 12. BECK c. AND COHENE.G.D. Physica A 322 (2003) 267. 13. WILK G. AND WLODARCZYK z. Phys. Rev. Lett. 84 (2000) 2770. 14. SALAMON M.B., LIN P. AND CHUNS.H. Phys. Rev. Lett. 88 (2002) 197203. 15. SALAMON M. AND CHUNS. Phys. Rev. B 68 (2003) 014411. 16. VAZQUEZ-VAZQUEZ C., BLANCOM.C., LOPEZ-QUINTELA M., SANCHEZ R.D., RIVAS J. AND OSEROFF S.B. J . Mat. Chem. 8 (1998) 991. J.P., TAVARES P.B., GOMES A.M. A N D 17. REIS M.S., AMARALV.S., A R A ~ J O OLIVEIRAI.S. Submatted to Phys. Rev. B (2004).

MULTIFRACTAL ANALYSIS OF TURBULENCE AND GRANULAR FLOW T. ARJMITSU Graduate School of Pure and Applied Sciences, University of Tsukuba, Ibaraki 305-8571,Japan E-mail: [email protected]

N. ARIMITSU Graduate School of Environment and Information Sciences, Yokohama National University, Yokohama 240-8501,Japan E-mail: [email protected]

Abstract The probability density function of velocity fluctuations of granular turbulence (granulence) observed by Fladjai and Roux in their twdimensional simulation of a slow granular flow under homogeneous quasi-static shearing is studied by multifractal analysis (MFA) proposed by the authors. MFA is a unified self-consistent approach for the systems with large deviations, which has been constructed based on the Tsallis-type distribution function that provides an extremum of the extensive RBny or the non-eztensive Tsallis entropy under appropriate constraints. It is shown by the present precise analysis that the system of granulence and of turbulence indeed have common scaling characteristics as was pointed out by Radjai and Roux. Keywords: multifractal analysis, velocity fluctuation, turbulence, granulence

1

Introduction

There have been reported that granular materials in the rapid flow regime present non-Gaussian velocity distributions in various situations, e.g., in a vibrated bed [l,2, 31, in a fluidized beds [4], in a fluidized granular medium between two walls 151, in homogeneous granular fluids 16, 71, in granular gases IS] and so on. Radjai and Roux [9] observed non-Gaussian distribution function in their two-dimensional simulation of a slow granular flow subject to homogeneous quasi-static shearing. They reported that there is an evident analogy between the scaling features of turbulence and of granular turbulence (granulence) in spite of the fundamentally different origins of fluctuations in these systems.

236

237 In this paper, we apply the multifractal analysis (MFA) [lo, 11, 12, 13, 14, 15, 16, 17, 18, 19, 201 of fluid turbulence to granulence in order to see how far MFA works in the study of the data observed by Radjai and Roux [9]. MFA is a unified selfconsistent approach for the systems with large deviations constructed by following the assumption [21] that the strengths of the singularities distribute in a multifractal way in real physical space. The appearance of singularities originates from the invariance of the Navier-Stokes (N-S) equation under a scale transformation. The distribution function of singularities is assumed to be given by the Tsallis-type distribution function [22] that provides an extremum of the extensive RQny [23] or the non-extensive Tsallis entropy (22, 241 under appropriate constraints. This distribution of singularities determines the tail part of the probability density function (PDF). The parameters appeared in the theory are determined, uniquely, by the intermittency exponent representing the strength of intermittency. On the other hand, observed PDF should include the effect resulted from the term in the N-S equation violating the invariance under the scale transformation (the dissipative term). However, there has been no ensemble theory of turbulence including this effect, and the situation remained at the stage where almost all the theories are just trying to explain observed scaling exponents of the mth order velocity structure function, i.e., the mth moment of velocity fluctuations. MFA counts this effect as something determining the central part of PDF narrower than its standard deviation. We are assuming that the fat-tail part, which the PDF of intermittent systems took on, is determined by the global characteristics of the system, and that the central part of PDF is a reflection of the local nature of constituting eddies.

2

Basic Equations for Granular Flow

A set of basic equations for the flow of granular media is given [25] by the equation of continuity for the mass density p: ap/at

with the notation granular media:

aj =

+ ai ( p u i ) = 0

(1)

a / d x j , the equation of motion for the velocity field ii of

pauipt

+ p ( i i . $)tii = -aip + a j U $

(2)

with the fluid pressure p and the dissipative stress tensor a’.. = ij(d (30,. - D . . - b . . D k k 3% 2 23 3%

-

+

(3)

4Wji)

where Dji = ajui, and the equation for the angular velocity field media: Iawjilat I ( i i . t ) W j i = a k p k j i 2uji

wji

of granular

+

(4)

with I = ma2/2 being the momentum inertia for 2-dimensional disks with radius a and mass m, the moment Pkji

= a2ij(g) (bkjatwei - Skiaewej

+2akWji +a j w k i - a i w k j ) ,

(5)

238 and u3< = (u(ii- cij) 12. Here, the generalized viscosity f j ( g ) is defined by fj(g)

C(P) = 2J;;aPf

= C(P);/g,

G I 7

(6)

with pf being the friction coefficient between granular particles (disks), y the filling factor of the disks, and

2

=

dEjiEji/4

+ RjiRji/2 + a2 ( R j j k a j j k + a j & j i k + Q j i k a i j k ) /8

(7)

4

where Eji = D ; - b j i D k k / 2 , Rji = wji-D3;, R j i k = a j w i k with D: = ( D j i fD i j ) . The energy of granular media dissipating per unit mass and per unit time is given by the dissipation function @ = C(p)p b. We confine ourselves in this paper to the case of an incompressible granular flow where the mass density p is constant in time and space. Then, (1) reduces to V . G = 0, and ( 2 ) to aUi/dt

+ (G.G ) u =~ -&p + U ( ~ ) V '+U(~3 0 . .- D . . - 4 w . . ) 3 32

23

(8)

32

with p = p / p and the generalized kinematic viscosity u ( g ) defined by v ( g ) = f j ( g ) / p . When the angular velocity is induced by velocity field 77, it is given by wji = (Dji - D i j ) 12, i.e., w' = x G. Then, the stress tensor (3) reduces to u;; = @) (Dji D i j ) . Note that, for an ordinary fluid, f j ( g ) becomes a constant viscosity f j representing the frictional characteristics of the fluid. In this case, (8) reduces to the N-S equation for incompressible fluid

+

a77/at + (72. G ) G = - G p

+UV2C

(9)

+

with the kinematic viscosity u = f j / p , and the equation of motion aw'/at x (w'x G)= vV2w' for w' is no more an independent equation but is derived from (9).

Multifractal Analysis

3

MFA rests on the invariance of the basic equation of the type aG/at

+ (GI G)IZ= - 9 p + [disspative term(s)]

(10)

under the scale transformation [21, 261

z

--$

d = A?,

77 + GI= ~

4

3

~ t

-+ , ti = ~

1 - ~ / 3 ~ ,-+

=~

2

4

3

(11) ~

for arbitrary real number a when the effect of the dissipative term(s) is negligible in certain region, and on the assumption that the singularities due to the invariance distribute themselves in a multifractal way in physical space. The scaling invariance leads the scaling law

239 with 6, = &/lo = 6-n ( n = 0, 1 , 2 , . . .) for the velocity fluctuation 6u, = Iu(o+e,)u(0)l of the nth multifractal step where u represents a component of the velocity field 5. We will put 6 = 2 in the following in this paper that is consistent with the energy cascade model of turbulence.' The singularity appears in the velocity derivative Iu'I = limn-+a,uk c( limn+m 6, a/3-1 which diverges for a < 3. Here, we introduced the nth velocity difference uk = Sun/& for the characteristic length en. Note that a is a measure of the strength of singularities. It is assumed, in A&A model within MFA, that the singularities due to the scale invariance distribute themselves in a multifractal way in physical space with the Tsallis-type distribution function, i.e., the probability P(n)(a)dato find in real space a singularity with the strength a within the range a a da is given by [ll,12, 131 P(n)(a) = (ZC))-'{l[((Y- (YO) /Ao]'}~/('-~) (13) N

+

with (Aa)' = 2X/[(1- q ) ln21. Here, q is the entropy index introduced in the definitions of the Rknyi and the Tsallis entropies.' This distribution function provides us with the multifractal spectrum f ( a )= 1 (1 - q)-l log,[l - (a- ao)'/(Aa)'] which, then, produces the mass exponent

+

.(a)

= 1 - aoQ+ 2X@(1+

G)-' + (1

- q)-l[l

- lOg,(l

+ A)]

(14)

with C, = 1+2ij2(1-q)X In 2. The multifractal spectrum and the mass exponent are related with each other through the Legendre transformation [26]: f ( a )= aQ+~(?j) with a = -dr(Q)/dg and 4 = df (a)/da. The formula of the PDF, II(")(un),of velocity fluctuations is assumed to consists of two parts, i.e., II(n)(u,) = rIg)(un) ArI(n)(un) where the first term is with the transformation of the related to P(,)(a) by II$'(Iunl)dun 0: P(n)(a)da variables (12), and the second term is responsible to the contributions coming from the dissipative term(s) in (10) violating the invariance under the scale transformation (11). Then, we have the velocity structure function in the form ( ( 1 ~ ~ 1 " ) ) = Jdun1unlmII(")(Un) = 272'+(l-27F')am6$with 272) = Jdunlu,lmAII(n)(un), a, = {2/[-(1+ -)]}'I' and the scaling exponent

+

~have > ~ peculiar features regarding the non-stationarity character of the wind data and the high turbulence intensity a aThe turbulence intensity of the wind speed are typically expressed in terms of standard deviation, u,,,of velocity fluctuations measured over 10 t o 60 minutes normalized by the mean wind speed V. [Iv = 91.Typical values for complex terrain are I,, 2 0.2 while, on the other hand for microscale turbulence one usually has Iv 0(10-2) N

246

247 many similarities with microscopic turbulence exists. For a detailed comparison between the two, one can see the recent paper by Bottcher et al.’. In this short paper we discuss a study of a turbulent wind data series recently measured at Florence airport for a period of six months. We show by means of a statistical analysis that we can describe this example of atmospheric turbulence within the by means of the nonextensive approach adopted in refs. more general superstatistics formalism introduced in ref. 13. The latter justifies the successful application of Tsallis statistics in different fields, and more specifically in turbulence experiment^",^^^^^,^^,^^,^^. We will show that such an approach is meaningful and can reveal very interesting features which could have also a very practical utility for safety reasons when applied to air traffic control services. Part of this study has just been published l7 and a longer paper with a complete and exhaustive analysis is in preparation 18. 8p10111912914115,

2. Statistical analysis of wind measurements

The wind velocity measurements, were taken at Florence airport and were done for a time interval of six months, from October 2002 to March 2003. Data were recorded by using two 3-cups runway heads anemometers, each one mounted on a 10 m high pole, located at a distance of 900 m and with a sampling frequency of one measure every 5 minutes. Although in our experiment we actually could not control the Reynolds number, as usually done in microscopic turbulence, and despite our low sampling frequency (3.3 . 10-3Hz) and the high intermittency of our wind data, we found several features of canonical turbulence as we will discuss in the following. We performed, on our time series, a statistical analysis using conventional mathematical tools which are normally adopted in small scale physical turbulence studied in laboratory. In particular we investigated correlations, spectral distributions as well as probability density functions of velocity components of returns and differences see refs. for more details. In this short contribution for simplicity and lack of space we discuss only returns of the longitudinal velocity components measured by one of the two anemometers (in the present case the one closest to the runway head 05 and labeled RWYUS) defined by the following expression 17118

z(qT = V,RWY05(t + 7)- V,RWY05(t) , Vz(t)being the longitudinal velocity component at time t and 7 being a fixed time interval ’. The same analysis was done also for the transversal components and for velocity difference between the two anemometers with similar results 17,18.

bReturns are here defined in a slight different way from those used in econophysics, i.e.: dt+r)-s(t)

z(t)

’

248

2.1.

Correlations and power spectra

Our data show very strong correlations and power spectra with the characteristic -5/3 law in the high-mid portion of the entire spectrum, see fig.1 in ref.”. However the dissipation branch in the high-range frequency, well known in micro-scale (or is here missing due to the low-frequency high-frequency) turbulence analysis Correlation functions also show an initial exponential decay, sampling used followed by a power law-decay modulated by the day-night wind periodicity which is a well known phenomenon. No significant difference was found for day and night periods, when air traffic is almost absent. For more details please see refs. ‘i6,

17318.

17118.

2.2. The superstatistics approach for wind velocity pdfs

The superstatistics formalism proposed recently by C. Beck and E.G.D. Cohen is a general and effective description for nonequilibrium systems. For more details see the original article and the paper by Beck in this volume13. In the superstatistics approach one considers fluctuations of an intensive quantity, for example the temperature, by introducing an effective Boltzmann factor

where f(P)is the probability distribution of the fluctuating variable have for the probability distribution 1 P(E)=-B(E) ,

p, so that we

z

(3)

Z = l o B(E)dE .

(4)

with the normalization given by

One can imagine a collection of many cells, each of one with a defined intensive quantity, in which a test particle is moving. In our atmospheric turbulence studies, the time series of the wind velocity recordings, are characterized by a fluctuating variance, so the returns (l),cannot be assumed to be by a ”simple” Gaussian process. They show a very high intermittent behavior stronger than that one usually found in small-scale fluid turbulence experiments. In our analysis we considered the following quantities: (a) the wind velocity returns x defined by eq.(l), (b) the corresponding variance of the returns x which we indicate with u , (c) the fluctuations of u, whose variance we indicate with the symbol C. We extracted from the experimental data, using an fixed time interval 7,the distribution for the fluctuations of the longitudinal wind component variance. The aim is to slice the time series in ”small” pieces in which the signal is almost Gaussian and apply superstatistics theory. This fluctuating behavior of F is plotted in Fig.1 for a time interval 7 = 1 hour. In Figs. 2 and 3 we then plot the probability

249 cs fluctuations of the RWYOS longitudinal wind component 10

Time window

9

:t

2=

I hi

I

Figure 1. Variance fluctuations of the longitudinal wind velocity component for the anemometer RWYOS obtained with a moving time window T of one hour.

c=2.70 q = 1.37

-1

Figure 2. Standardized pdf of the fluctuating variance corresponding t o the previous figure (open points) are compared with a Gamma distribution (full line) and with a Log-normal distribution (dashed line) sharing the same mean (ao) and variance (C) e x t r x t e d from experimental data. The Log-normal is not able t o reproduce the experimental data.

distribution of the variance o for 7 = 1 and 3 hours respectively. In the same figures we plot for comparison a Gamma (full curve) and a Log-normal (dashed curve) distribution6 characterized by the same average and variance extracted from the experimental data. In this sense, the curves are not fitted to the data. The comparison clearly shows that the Gamma distribution is able to reproduce very nicely the experimental distribution of the u fluctuations and that this type of distribution show robustness for different time windows choices. This is at variance with the Log-normal distribution which is usually adopted in microscopic turbulence

250

Figure 3. Standardized pdf of the fluctuating variance similar to t h e previous figure but c o r r e sponding t o a windowing of three hours in our longitudinal velocity components (open points). We show for comparison also a Gamma distribution (full line) and a Log-normal distribution (dashed line) sharing the same mean (uo)and variance (C). Also in this case the Gamma distribution reproduces very well the experimental data at variance with the Log-normal one.

and which in this case is not able to reproduce the experimental data. In general using Beck and Cohen notation13 we have for the Gamma distribution

with

where 2c is the actual number of effective degrees of freedom and b is a related parameter. Inserting this distribution into the generalized Boltzmann factor ( 2 ) one gets the q-exponential curve 14,

P ( z ) = (1 - (1 - q)PoE(z))&

.

(7)

iz2

In our analysis we have E = with z defined by eq.(l)11>17>18. Considering the fluctuations of the variance a of the returns z, we get the following correspondence with the original superstatistics formalism @=ar

,

a(P)=C(a,)

,

Po=<ar

>=a0

.

(8)

In the present case, we get for the Gamma distributions which describe the experimental variance fluctuations reported in Figs.2 and 3 the characteristic values c = 2.70 and c = 3.22, for a time interval of 1 and 3 hours, from which, using eq. (6), we get the corresponding q-values q = 1.37 and q = 1.31. In Fig.4 we plot the probability density function P ( z ) of the experimental longitudinal returns for different time intervals, i.e. 1 hour (full circles), 3 hours (open diamonds) and 24 hours (open

25 1 squares). For comparison we plot a Gaussian distribution (dashed curve) and the q-exponential curves (7) characterized by the q-values extracted from the Gamma distributions of Figs.2 and 3 for a time interval T corresponding to 1 and 3 hours. The q-exponential curves reproduce very well the experimental data which, on the other hand, are very different from the Gaussian pdf. However one can notice that for a very long time interval, i.e. T =24 hours, the data are not so far from being completely decorrelated and therefore the corresponding experimental pdf is closer to the Gaussian curve. Notice that the theoretical curves are not fitted, and that the superstatistic approach, in a self-consistent and elegant way, is able to explain and characterize in a quantitative way the wind data. In a similar way one can extract theoretical curves which reproduce the wind velocity differences pdfs with similar entropic q-values, although in that case an asymmetry correction has t o be considered t o better reproduce the tails of the p d f ~ ~ ~ ~ ' ' .

Figure 4. Comparison between standardized longitudinal velocity returns pdfs for three different time intervals (T = 1,3,24 hours)and the q-exponential curves with the q-value extracted from the c parameter of the Gamma distribution shown in the previous figures. A Gaussian pdf is also shown as dashed curve. See text.

In our analysis the large and intermittent wind velocity variance fluctuations y e reproduced very well by a Gamma type superstatistics excluding the Log-normal one and this gives exactly the Tsallis q-exponential for the velocity returns pdfs. However one has to say that the situation is much more difficult for the less fluctuating velocity flow of the microscale fluid turbulence. We add as a final remark that, very recently a similar method has been adopted by a research group at the NASA Goddard Space Flight Center to analyze the solar wind speed fluctuations16.

252 3.

Conclusions

We have studied a temporal series of wind velocity measurements recorded at F1e rence airport for a period of six months. T h e statistical analysis for the velocity components shows intermittent fluctuations which exhibit power-law pdfs. Applying the superstatistics formalism, i t is possible to extract a Gamma distribution from the probability distributions of the variance fluctuations of wind data. T h e characteristic parameter c of this Gamma distribution gives t h e entropic index q of the Tsallis q-exponential which is then able to reproduce very well the velocity returns and differences pdfs. Beyond the successful application of superstatistics and Tsallis thermostatistics for turbulent phenomena and the corresponding theoretical implications, we think t h a t this work shows a useful and interesting method t o characterize and study in a rigorous and quantitative way atmospheric wind d a t a for safety flight conditions in civil and military aviation.

Acknowledgements T h e authors are indebted t o C. Beck, E.G.D. Cohen, S. Ruffo, H.L. Swinney and C. Tsallis for suggestions and discussions.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

16. 17. 18.

A.N. Kolmogorov, J . Fluid Mech. 13,82 (1962). A.M. Obukhov, J. Fluid Mech. 13,77 (1962). R.H. Kraichnan,J. Fluid Mech. 62,305 (1974). B. Castaing, Y . Gagne, E.J. Hopfinger, Physica D 46, 177 (1990). S.B. Pope, Turbulent Flows, Cambridge University Press (2000). U. Frisch, Turbulence, Cambridge University Press (1995). K.R. Sreenivasan, Rev. Mod. Phys. 71,S383 (1999). F.M. Ramos, M.J. A. Bolzan, L.D.A. SB, R.R. Rosa, PhysicaD 193,278 (2004). F. Bottcher, St. Barth and J. Peinke, eprint [nlin.A0/0408005]. C. Beck, Physica A 277,115 (2000); C. Beck, Phys. Rev. Lett. 87,18060 (2001); C. Beck, Physica A 306,189 (2002). C. Beck, G.S. Lewis and H.L. Swinney Phys. Rev. E 63,035303 (2001). C.N. Baroud and H.L. Swinney Physica D 284,21 (2003). C. Beck and E.G.D. Cohen Physica A 322,267 (2003) and C.Beck in this volume [cond-mat/0502306]. C. Tsallis J . Stat. Phys. 52,479 (1988). See for example C. Tsallis, Physica D 193,3(2004) and the other papers published in the same volume. For un updated list of references on the generalized thermostatistics and its applications, see also http://tsallis.cat.cbpf.br/biblio.htm. L.F. Burlaga and A.F. Vixias Geophysical Research Letters, 31,L16807 (2004); L.F. Burlaga and A.F. Viiias Journal of Geophysical Research, 109,A12107 (2004). S. Rizzo and A. Rapisarda, 8th Experimental Chaos Conference,l4-17 June 2004, Florence, Italy, AIP Conference proceedings Vol. 742, p. 176, [cond-mat/0406684] S. Rizzo and A. Rapisarda (2005) to be submitted.

Applications in Other Sciences


COMPLEXITY OF PERCEPTUAL PROCESSES F. TIT0 ARECCHI Department of Physics University of Firenze, Italy

At the borderline between neuroscience and physics of complex phenomena, a new paradigm is under investigation ,namely feature binding. This terminology denotes how a large collection of coupled neurons combines external signals with internal memories into new coherent patterns of meaning. An external stimulus spreads over an assembly of coupled neurons, building up a corresponding collective state. Thus, the synchronization of spike trains of many individual neurons is the basis of a coherent perception. Based on recent investigations, a novel conjecture for the dynamics of single neurons and, consequently, for neuron assemblies has been formulated. Homoclinic chaos is proposed as the most suitable way to code information in time by trains of equal spikes occurring at apparently erratic times; a new quantitative indicator, called propensity ,is introduced to select the most appropriate neuron model. In order to classify the set of different perceptions, the percept space is given a metric structure by introducing a distance measure between distinct percepts. The distance in percept space is conjugate to the duration of the perception in the sense that an uncertainty relation in percept space is associated with time limited perceptions. Thus coding of different percepts by synchronized spike trains entails fundamental quantum features . It is conjectured that they are related to the details of the perceptual chain rather than depending on Planck‘s action.

1

Feature binding

1.1. Neuron synchronization

It is by now established that a holistic perception emerges, out of separate stimuli entering different receptive fields, by synchronizing the corresponding spike trains of neural action potentials [Von der Malsburg, Singer]. Action potentials play a crucial role for communication between neurons [Izhikevich]. They are steep variations in the electric potential across a cell’s membrane, and they propagate in essentially constant shape from the soma (neuron’s body) along axons toward synaptic connections with other neurons. At the synapses they release an amount of neurotransmitter molecules depending upon the temporal sequences of spikes, thus transforming the electrical into a chemical carrier. As a fact, neural communication is based on a temporal code whereby different cortical areas which have to contribute to the same percept P synchronize their spikes. Limiting for convenience the discussion to the visual system, spike emission in a single neuron of the higher cortical regions results as a trade off between bottom-up stimuli arriving through the LGN (lateral geniculate nucleus) from the retinal detectors and threshold modulation due to top-down signals sent as conjectures by the semantic

255

256

memory. This is the core of ART (adaptive resonance theory [Grossberg]) or other computational models of perception [Edelman and Tononi] which assume that a stable cortical pattern is the result of a Darwinian competition among different percepts with different strength. The winning pattern must be codirmed by some matching procedure between bottom-up and top-down signals. 1.2. Perceptions, feature binding and Qualia

The role of elementary feature detectors has been extensively studied in the past decades [Hubel]. By now we know that some neurons are specialized in detecting exclusively vertical or horizontal bars, or a specific luminance contrast, etc. However the problem arises: how elementary detectors contribute to a holistic (Gestalt) perception? A hint is provided by [Singer]. Suppose we are exposed to a visual field containing two separate objects. Both objects are made of the same visual elements, horizontal and vertical contour bars, different degrees of luminance, etc. What are then the neural correlates of the identification of the two objects? We have one million fibers connecting the retina to the visual cortex, through the LGN. Each fiber results from the merging of approximately 100 retinal detectors (rods and cones) and as a result it has its own receptive field. Each receptive field isolates a specific detail of an object (e.g. a vertical bar). We thus split an image into a mosaic of adjacent receptive fields. Now the “feature binding” hypothesis consists of assuming that all the cortical neurons whose receptive fields are pointing to a specific object synchronize the corresponding spikes, and as a consequence the visual cortex organizes into separate neuron groups oscillating on two distinct spike trains for the two objects(fig.1) Direct experimental evidence of this synchronization is obtained by insertion of microelectrodes in the cortical tissue of animals just sensing the single neuron [Singer]. Indirect evidence of synchronization has been reached for human beings as well, by processing the EEG (electro-encephalo-gram) data [Rodriguez et al.]. Based on the neurodynamical facts reported above, we can understand how this occurs [Grossberg]. The higher cortical stages where synchronization takes place have two inputs. One (bottom-up) comes from the sensory detectors via the early stages which classify elementary features. This single input is insufficient, because it would provide the same signal for e.g. horizontal bars belonging indifferently to either one of the two objects. However, as we said already, each neuron is a nonlinear system passing close to a saddle point, and the application of a suitable perturbation can stretch or shrink the interval of time spent around the saddle, and thus lengthen or shorten the interspike interval. The perturbation consists of top-down signals corresponding to conjectures made by the semantic memory (fig.2).

257

Fig. 1: Feature binding: the lady and the cat are respectively represented by the mosaic of empty and tilled circles, each one representing the receptive field of a neuron group in the visual cortex. Within each circle the processing refers to a specific detail (e.g. contour orientation). The relations between details are coded by the temporal correlation among neurons, as shown by the same sequences of electrical pulses for two filled circles or two empty circles. Neurons referring to the same individual (e.g. the cat) have synchronous discharges, whereas their spikes are uncorrelated with those referring to another individual (the lady) [from Singer].

}Top -Down

]--up

Fig.2 ART = Adaptive Resonance Theory. Role of bottom-up stimuli kom the early visual stages an top-down signals due to expectations formulated by the semantic memory. The focal attention assures the matching (resonance) between the two streams [from Julesz].

In other words, the perception process is not like the passive imprinting of a camera film,but it is an active process whereby the external stimuli are interpreted in terms of past memories. A focal attention mechanism assures that a matching is eventually

258

reached. This matching consists of resonant or coherent behavior between bottom-up and top-down signals. If matching does not occur, different memories are tried, until the matching is realized. In presence of a fully new image without memorized correlates, then the brain has to accept the fact that it is exposed to a new experience. Notice the advantage of this time dependent use of neurons, which become available to be active in different perceptions at different times, as compared to the computer paradigm of fixed memory elements which store a specific object and are not available for others (the so called “grandmother neuron” hypothesis). We have above presented qualitative reasons why the degree of synchronization represents the perceptual salience of an object. Synchronization of neurons located even far away from each other yields a space pattern on the sensory cortex, which can be as wide as a few square millimeters, involving millions of neurons. The winning pattern is determined by dynamic competition (the so-called “winner takes all” dynamics). This model has an early formulation in ART and has been later substantiated by the synchronization mechanisms. Perceptual knowledge appears as a complex selforganizing process. Naively, one might expect that a given “qualia”, that is, a private sensation as e.g. the red of a Titian painting, is always coded by the same sequence of spikes. If so, in a near future the corresponding information could be retrieved by a high resolution detector, and hence a Rosetta stone could be established between the spike sequences and the qualia. Such a naive expectation which would lead to a world without privacy, is altogether wrong for the following reasons. After the initial experience of that qualia, the first time one has seen that Titian, any further repetition of that experience, either by memory recollection or by re-watching the painting occurs in presence of new experiential elements (one has become older, hisher store of memories has drastically mutated) and these novelties contribute to feature binding by a modified synchronization pattern. Evidence of such a fact has been established by Freeman [Freeman] reporting the synchronization pattern of the olfactory bulb of a rabbit, recorded by a large number of electrodes; as the same odor is presented twice, with an intermediate odor in between, the two patterns are all together different, even though the animal behavior hints at the same reaction. Freeman’s experiment is contrasted by the fact that some olfactory neurons of the locust yield the same bursts of spikes for the same odor [Rabinovich et al.]. Presumably, lower animals as locusts have a much smaller semantic repertoire than rabbits or humans, and hence for them the dream of the Rosetta stone has some validity. 2

Homoclinic chaos, synchronization and propensity

Let us model the neurodynamics of spike formation As for the dynamics of the single neuron, a saddle point instability separates in parameter space an excitable region, where axons are silent, from a periodic region, where the spike train is periodic (equal interspike intervals). If a control parameter is tuned at the saddle

259

point, the corresponding dynamical behavior (homoclinic chaos) consists of a frequent return to the instability [Allaria]. This manifests as a train of geometrically identical spikes, which however occur at erratic times (chaotic interspike intervals). Around the saddle point the system displays a large susceptibility to an external stimulus, hence it is easily adjustable and prone to respond to an input, provided this is at sufficiently low frequencies; this means that such a system is robust against high frequency noise as discussed later. HOMOCLINIC CHAOS through saddle connection in 3D

Chaos a < y Periodic stable oscillations a = y

Susceptibility

x

x= response/stimulus

-easy synchronization : external forcing,

DSS,NIS

-bursting

Telecomunicaiion neuronal dynamics

Fig.3 Schematic view of the phase space trajectory approaching the saddle S and escaping from it. Chaos is due to the shorter or longer permanence around S; fiom a geometrical point of view most of the orbit P provides a regular spike.

Such a type of dynamics has been recently dealt with in a series of reports that here I recapitulate as the following chain of l i e d facts. 1) A single spike in a 3D dynamics corresponds to a quasi-homoclinic trajectory around a saddle focus SF (fured point with 1 (2) stable direction and 2 (1) unstable ones); the trajectory leaves the saddle and returns to it (Fig.3).We say “quasi-homoclinic”because, in order to stabilize the trajectory away from SF, a second fixed point, namely a saddle node SN, is necessary to assure a heteroclinic connection. The experiment on a C02 laser confrms this behavior(Fig.4)

260

80

82

84

88

time (ms)

a8

so

Fig.4 Experimental time series of the laser intensity for a C02 laser with feedback in the regime of homoclinic chaos. (b) Time expansion of a single orbit. (c) Phase space trajectory built by an embedding technique with appropriatedelays [from Allaria et al.].

A train of spikes corresponds to the sequential return to, and escape from, the SF. A control parameter can be set at a value BC for which this return is erratic (chaotic interspike interval). As the control parameter is set above or below BC, the system moves from excitable (single spike triggered by an input signal) to periodic (yielding a regular sequence of spikes without need for an input), with a frequency monotonically increasing with the separation OB from BC [Meucci]. Around SF , any tiny disturbance provides a large response. Thus the homoclinic spike trains can be synchronized by a periodic sequence of small disturbances (Fig. 5). However each disturbance has to be applied for a minimal time, below which it is no longer effective; this means that the system is insensitive to broadband noise, which is a random collection of fast positive and negative signals[Zhou et all. The above considerations lay the floor for the use of mutual synchronization as the most convenient way to let different neurons respond coherently to the same stimulus, organizing as a space pattern. In the case of a single dynamical system, it can be fed back by its own delayed signal. As the delay is long enough the system is decorrelated with itself and this is equivalent to feeding an

26 1 independent system. This process allows to store meaningful sequences of spikes as necessary for a short term memory [Arecchi et a1.20021.

I 0

1

2

3

4

77

78 time <ms)

78

I

5

time (ms)

Fig. 5. Left: experimental time series for different synchronization ratios induced by periodic changes of the control parameter. (a) I : 1 locking, (b) 1 :2, ( c ) I :3, (d) 2 :I . Right: when the system is not able to spike for each period of the driver, a phase slip (one spike less or more) occurs, it is a jump of +/- 271 if the interspike interval is normalized to 2a. The rate of +/- phase slips increases with the offset of the driving frequency from the natural frequency (associated with the average interspike interval of the free system).

Several neuron models (integrate-and-fire, Hodgkin-Huxley, FitzHughNagumo, Hindmarsh-Rose ) have been used by different investigators. We have introduced the propensity to synchronization as a quantitative indicator of how easy is for a chaotic system to recognize an external input (Fig. 6) [Arecchi et a1 20031 In presence of localized stimuli over a few neurons, the corresponding disturbances propagate by inter-neuron coupling ( either excitatory or inhibitory); a synchronized pattern is uniquely associated with each stimulus; the degree of mutual synchronization is measured by the disappearance of phase slips, or defects in a space-time fabric [Leyva et al.].

262 5 1 .

I

.

I

.

,

. *.

, .

I

.

,

.

I

.

,

. ..

-0

Fig.6 The coherence parameter R is defined as the ratio between the average IS1 (interspike interval) and its r.m.s. fluctuation. R is unity for a fully chaotic system and tends to infinity for a periodic system. Here we plot in log scale the ratio between R for a driving periodic disturbance of 1% to a control parameter and Rfiee for the free system, at different frequencies o away from the natural one 00 (average of the chaotic spiking in the unperturbed system), For HC (circles) the ratio is 30 at w0 and it goes up to 104for higher frequencies; for the Lorenz system(squares) the ratio stays flat to 1. We thus take this ratio as a quantitative indicator of the propensity to synchronization

These facts have been established experimentally and confirmed by a convenient model in the case of a class B laser with a feedback loop which readjusts the amount of losses depending on the value of the light intensity output[Arecchi 1987 b]. I here recall the classification widely accepted in laser physics. Class A lasers are ruled by a single order parameter, the amplitude of the laser field, which obeys a closed dynamical equation; all the other variables having much faster decay rate, thus adjusting almost instantly to the local field value. Class B lasers are ruled by two order parameters, the laser field and the material energy storage providing gain; the two degrees of freedom having comparable characteristic times and behaving as activator and inhibitor in chemical dynamics [Areccbi 1987al The above listed facts hold in general for any dynamical system which has a 3dimensional sub-manifold separating a region of excitability from a region of periodic oscillations: indeed, this separatrix has to be a saddle focus. 3

Time code in neural information exchange

How does a synchronized pattern of neuronal action potentials become a relevant perception?

263 Not only the different receptive fields of the visual system, but also other sensory channels as auditory, olfactory, etc. integrate via feature binding into a holistic perception. Its meaning is “decided” in the PFC (pre-frontal cortex) which is a kind of arrival station from the sensory areas and departure for signals going to the motor areas. On the basis of the perceived information, motor actions are started, including linguistic utterances [Rodriguez et al.]. Sticking to the neurodynamical level, and leaving to psychophysics the investigation of what goes on at higher levels of organization, we stress here a fundamental temporal limitation. Taking into account that each spike lasts about 1 msec, that the minimal interspike separation is 3 msec, and that the decision time at the PCF level is estimated to be = 200ms , we can split Tinto 200/3 -66 bins of 3 msec duration, which are designated by 1 or 0 depending on whether they have a spike or not. Thus the a priori total number of different messages which can be transmitted is

2 6 6 = 6I~019. However we must account also for the average rate at which spikes proceed in our brain, which is t= 40 Hz (so called y band, (average ISI) = 25 ms). When we account for

S

this rate we can evaluate a reduction factor a = Y = 0.54 where S is an entropy

T

[Rieke et all, thus there are roughly 2‘ = 10” words with significant probability. Even though this number is large, we are still within a finitistic realm. Provided we have time enough to ascertain which one of the different messages we are dealing with, we can classify it with the accuracy of a digital processor, without residual error. But suppose we expose the cognitive agent to fast changing scenes, for instance by presenting in sequence unrelated video frames with a time separation less than 200 msec. While small gradual changes induce the sense of motion as in movies, big differences imply completely different subsequent spike trains. Here any spike train gets interrupted after a duration AT less than the canonical This means that the brain cannot decide among all coded perceptions having the same structure up to AT, but different afterwards. Whenever we stop the perceptual task at AT shorter than the total time T , then the bin stretch.T-AT (we measure the times in bin units) is not explored. This means that all stimuli which provide equal spike sequences up to AT, and differ afterwards by at least one spike will cover an uncertainty region AP whose size is given by

r.

hp = 2d2-4T = p M e - d T l n 2

(1) -

where PM=lO” is the maximum perceptual size available with the chosen T c 66.6 bins per perceptual session and rate r=40 Hz.Relation (1) is very different from the standard uncertainty relation

AP.AT=C

(2)

264 that we would expect in a word-bin space ruled by Fourier transform relations. Indeed, the trascendental equation (1) is more rapidly converging at short and long AT than the hyperbola (2). We fit (1) by (2) in the neighborhood of a small uncertainty AP = I0 words, which corresponds to AT = 62 bins. Around AT = 62 bins the local uncertainty (2) yields a quantum constant

C = 10.62 = 620words x bins

(3)

To convert C into Js as Planck’s h, consider that: i)l bin = 3 ms ii)in order to jump from an attractor corresponding to one perception to a nearby one, a minimal amount of energy is needed, corresponding to one spike; but one spike requires the energy corresponding to about lo7 transitions ATP-+ADP+ P [Laughlin et all each one taking 0.3 eV; thus the total energy quantum is about joules .The conversion factor is then:

Ic ~ o - ~=~i oJZ.0 ~ l %

Quantum limitations were also put forward by Penrose [Penrose] but on a completely different basis. In his proposal, the quantum character was attributed to the physical behavior of the “microtubules” which are microscopic components of the neurons playing a central role in the synaptic activity. However, speaking of quantum coherence at the h-bar level in biological processes is not plausible, if one accounts for the extreme vulnerability of any quantum system to decoherence processes, which make quantum superposition effects observable only in extremely controlled laboratory situations, and at sub-picosecond time ranges ,not relevant for synchronization purposes in the 10-100 msec range. Our tenet is that the quantum C-level in a living being emerges from the limited time available in order to take vital decisions ;it is logically based on a non-commutative set of relevant variables and hence it requires the logical machinery built for the h-bar quantum description of the microscopic world where non-commutativity emerges from use of variables coming from macroscopic experience ,as coordinate and momenta, to account for new facts. A more precise consideration consists in classifying the spike trains. Precisely, if we have a sequence of identical spikes of unit area localized at erratic time positions TIthen the wholesequence is represented by

f(o=Cw-r/)

(3)

where {q} is the set of position of the spikes. A temporal code, based on the mutual position of successive spikes, depends on the moments of the interspike interval distributions

m/

= {TI - r/-l} Different ISIs encode different sensory information.

(4)

265

A time ordering within the sequence (3) is established by comparing the overlap of two signals as (3) mutually shifted in time. Weighting all shifts with a phase factor and summing up, this amounts to constructing a Wigner function [Wigner] -t-ou

W(t, co) = \f(t + T 12)f(t - T 12) exp(icoT)dT

(5)

If now/is the sum of two packets f=fi+f2 as in Fig. 8, the frequency-time plot displays an intermediate interference. Eq. (5) would provide interference whatever is the time separation between/; and f2. In fact, we know that the decision time T truncates a perceptual task, thus we must introduce a cutoff function g(r) ~ CXp(—T2 IT1) which transforms the Wigner function as -t-cu

W(t, co) = J/0 + T 12)f(t - T / 2) exp(i aw)g(r)c/r

(5')

In the quantum jargon, the brain physiology breaks the quantum interference for t > T (decoherence phenomenon ).

4

The role of the Wigner function in brain operations.

We have seen that feature binding in perceptual tasks implies the mutual synchronization of axonal spike trains in neurons which can be even far away and yet contribute to a well defined perception by sharing the same pattern of spike sequence.The synchronization conjecture was given experimental evidence by inserting several micro-electrodes probing each one single neuron in the cortex of cats and then studying the temporal correlation in response to specific visual inputs[Singer et al]. In the human case, indirect evidence is acquired by exposing a subject to transient patterns and reporting the time-frequency plots of the EEG signals [Rodriguez et al.]. Even though the space resolution is poor, phase relations among EEG signals coming from different cerebral areas at different times provide an indirect evidence of the synchronization mechanism. The dynamics of homoclinic chaos (HC) was motivated by phenomena observed in lasers and then explored in its mathematical aspects, which display strong analogies with the dynamics of many biological clocks ,in particular that of a model . HC provides almost equal spikes occurring at variable time positions and presents a region of high sensitivity to external stimuli; perturbations arriving within the sensitivity window induce easily a synchronization, either to an external stimulus or to each other (mutual synchronization ) in the case of an array of coupled HC individuals (from now on called neurons).

266

But who reads temporal information contained across synchronized and oscillatory spike trains?[MacLeod et all. In view of the above facts, we can model the encoding of external information on a sensory cortical area (e.g. V1 in the visual case) as a particular spike train assumed by an input neuron directly exposed to the arriving signal and then propagated by coupling through the array. As shown by the experiments on feature binding [Singer], we must transform the local time information provided by Eqs (3) and (4) into a spatial information which tells the amount of a cortical area which is synchronized. If many sites have synchronized by mutual coupling, then the read out problem consists in tracking the pattern of values (3), one for each site. Let us take for simplicity a continuous site coordinate r. In case of two or more different signals applied at different sites a competition starts and we conjecture that the winning information (that is, the one channeled to a decision center) corresponds to a “muioritv rule”. Precisely, if the encoding layer is a 1dimensional chain of N coupled sites activated by external stimuli at the two ends (i=l and i=N),the majority rule says that the prevailing signal is that which has synchronized more sites. The crucial question is then :who reads that information in order to decide upon? We can not recur to some homunculus who reads the synchronization state. Indeed, in order to be made of physical components, the homunculus itself should have some interpreter which would be a new homunculus, and so on with a “regressio ad infinitum ”. On the other hand, it is well known that ,as we map the interconnections in the vision system,V1 exits through the Vertical stream and the Dorsal stream toward the Inferotemporal Cortex and Parietal Cortex respectively. The two streams contain a series of intermediate layers characterized by increasing receptive fields ; hence they are cascades of layers where each one receives converging signals from two or more neurons of the previous layer. Let us take for the time being this feed-forward architecture as a network enabled to extract relevant information upon which to drive consequent actions . We show how this cascade of layers can localize the interface between two domains corresponding to different synchronization. It is well known that ON/OFF cells with a center-surround configuration perform a first and second mace derivative[Hubel]. Suppose this operation was done at certain layer , At the successive one, as the converging process goes on, two signals will converge on a simple cells which then performs a hieher order derivative, and so on. This way ,we build a power series of space derivatives. A translated Eunction as f(ri-4 is then reconstructed by adding up many layers, as can be checked by a Taylor expansion. Notice that the alternative of exploring different neighborhoods {of r by varying { would imply a moving pointer to be set sequentially at different positions, and there is nothing like that in our physiology.

267 The next step consists in comparing the function f ( r + g with a suitable standard, to decide upon its value. Since there are no metrological standards embedded in a living brain, such a comparison must be done by comparing f with a shifted version of itself, something like the product f(r+W f(r-4 Such a product can be naturally be performed by converging the two signalsf(rY@ onto the same neuron, exploiting the nonlinear (Hebbian )response characteristic limited to the lowest quadratic nonlinearity, thus taking the square of the sum of the two converging inputs and isolating the double product. This operation is completed by summing up the different contributions corresponding to different 6, with a kernel which keeps track of the scanning over different 6, keeping information on different domain sizes. If this kernel were just a constant, then we would retrieve a trivial average which cancels the 4 information. Without loosing in generality, we adopt a Fourier kernel exp(i kg and hence have built the quantity

It contains information on both the space position r around which we are exploring the local behavior, as well as the frequency k which is associated with the space resolution . As well known, it contains the most complete information compatible with the Fourier uncertainty

Notice that building a joint information on locality@) and resolution (k) by physical measuring operations implies such an intrinsic limitation.

In summary ,it appears that the Wigner function is the best read-out of a synchronized layer that can be done by exploiting natural machinery, rather than recurring to a homunculus. The local value of the Wigner function represents a decision to be sent to motor areas triggering a suitable action. In order to have a suitable description of the feature binding mechanism in terms of a Wigner function in time (at a single site) and space (over an array of sites) we need an evolution equation. But which are the physical objects whose evolution describes the phenomenology? I conjecture that the transition from de-coupled to fully synchronized neurons is controlled by the dynamics of defects (see Fig. 7). A preliminary treatment of defects as harmonic oscillators is summarized in Fig.9 . The total Hamiltonian then rules the evolution equation for the Wigner function and shou1.d provide a complete quantum description.

268

Synchronizatian patterns in arrays of homoclinic chaotic systems

20

20

40

20

40

20

40

40

Site index

& = ( a ) 0 . 0 ; ( b ) 0 . 0 5 ; ( c ) O . l ; (d)0.12; ( e ) 0 . 2 5

28

Fig. 7. Space -time representation of spike positions for a linear array of 40 neurons and for different amounts of nearest neighbor coupling. At ~ 0 . 2 5percolation has been reached, in the sense that spikes at all sites are connected (besides a mutual time lag to the transfer operation); at FO , no correlation at all among different sites; in between, a partial synchronization with the evidence of defects as phase slips (one spike more, or less, with respect to the neighbor site)

The oscillating interference of Fig.8 is crucial for quantum computution.The reason why we don’t see it in ordinary life is “decoherence”. Let us explain what we mean. If the system under observation is affected by the environment, then the two states of the superposition have a finite lifetime; however, even worse, the interference decays on a much shorter time (the smaller, the bigger is the separation between the two states): so while the separation is still manageable for the two polarization states of a photon, it becomes too big for the two states of a macroscopic quantum system. Precisely if we call the intrinsic decay of each one of the two states of the superposition, then the mutual interference in the Wigner function decays with the so-called decoherence time

where D2is the square of the separation D in phase space between the centers of the Wigner functions of the two separate states. Notice that, in microscopic physics, 02 is measured in units of fi . Usually

o2/ f i is much bigger than unity for a macroscopic

269

system and hence T,,, is so short that any reasonable observation time is too long to detect a coherent superposition. Instead, in neurodynamics, we perform measurement operations which are limited by the intrinsic sizes of space and time resolutions peculiar of brain processes. The associated uncertainty constant C is such that it is very easy to have a relation as (9) with d / C comparable to unity, and hence superposition lifetimes comparable to times of standard neural processes. This implies the conjecture that for short times or close cortical domains a massive parallelism typical of quantum computation should be possible. The threshold readjustment due to expectations arising from past memory acts as an environmental disturbance, which orthogonalizes different neural states, destroying parallelism.

Fig.8: Wiper distribution of two localized sinusoidal packets shown at the top. Bottom :frequency-time

representationof the levels of the (real but not always positive!) Wiper function. The oscillating interference is centered at the middle time-frequencylocation

270

Dynamical model for a synchronized HC array ax

-=

at

f ( x )+E.V?X

linearize around the saddle focus

Fig. 9. A summary of the strategy to characterize the quantum aspects of the time code. Here x is an Ndimensional (N=6 for HC) dynamical field depending on position r. Its local dynamics is given by the equation Rx). Furthermore, the space coupling of strength E is given by the Laplacian. Linearizing the equation at parameter values below the percolation threshold, Fourier transforming and diagonalizing the linearized equation, we have harmonic oscillators. Their coordinate and momenta do not commute, because of the fundamental energy-time uncertainty with the quantum constant C. From the k-dependence of the oscillator amplitudes a(k) one reconstructs the space features of the defects

References

1. Allaria E., Arecchi F.T., Di Garbo A., Meucci R. 2001 “Synchronization of

homoclinic chaos” Phys. Rev. Lett 36, 791. 2. Arecchi F.T., 1987a “Instabilities & chaos in single mode homogeneous line lasers“, in: Instabilities and chaos in quantum optics, (eds. F.T. Arecchi & R.G. Harrison), Springer Series Synergetics, Vol. 34 ,pp. 9-48. 3. Arecchi F.T ., MeucciR., Gadomski W. 1987b “Laser dynamics with competing instabilities”,Phys. Rev. Lett., 58, 2205 4. Arecchi F.T., Meucci R., Allaria E., Di Garbo A., Tsimring L.S., 2002 “Delayed self-synchronization in homoclinic chaos” Phys. Rev. E, 65,046237

271 5. Arecchi F.T.,Allaria E., Leyva I. ,2003 “A propensity criterion for networkingin an arrayof coupled chaotic systems” Phys.Rev.Lett. 91,234101 6. Edelman, G.M., and G. Tononi 1995 “Neural Darwinism: The brain as a selectional system” in “Nature’s Imagination: The frontiers of scientific vision”, J. Cornwell, ed., pp.78-100, Oxford University Press, New York. 7. Freeman, W.J. 1991.” The Physiology of Perception” Sci Am. 264(2),78-85. 8. Grossberg S., 1995 “The attentive brain” The American Scientist, 83,439. 9. Hubel D.H., 1995 “Eye, brain and vision”, Scientific American Library, n. 22, W.H. Freeman, New York. 10. Izhikevich E.M., 2000 “Neural Excitability, Spiking, and Bursting” Int. J. of Bifurcation and Chaos. 10, 1171 11. Laughlin S.B., de Ruyter van Steveninck,R..,Anderson J. 1998 “The metabolic cost of neural information” Nature Neuroscience 1,36 12. Leyva I. , Allaria E., Boccaletti S. and Arecchi F. T. ,2003, “Competition of synchronization patterns in arrays of homoclinic chaotic systems”, (eprint:nlin.PS/0302008). 13. MacLeod, K., Backer, A. and Laurent, G. 1998. “Who reads temporal information contained across synchronized and oscillatory spike trains?”, Nature 395: 693-698. 14. Meucci R., Di Garbo A., Allaria E., Arecchi F.T. 2002 “Autonomous Bursting in a Homoclinic System” ”,Phys. Rev. Lett. 88, 144101 15. Penrose R. 1994 “Shadows of the Mind” Oxford University Press New York. 16. Rabinovich M.,Volkovskii A.,Lecanda P.,Huerta R.,Abarbanel H. and Laurent G. 2001”Dynamical encoding by networks of competing neuron groups”, Phys. Rev. Lett. 87,068 102 17. Rieke, F., Warland, D., de Ruyter van Steveninck, R. and Bialek, W. 1996. “Spikes: Exploring the neural code”, MIT Press, Cambridge Mass.. 18. Rodriguez E., George N., Lachaux J.P., Martinerie J., Renault B.and Varela F. 1999, “Perception’s shadow:Long-distance synchronization in the human brain”, Nature 397:340-343. 19. Singer W. and E Gray C.M., 1995, “Visual feature integration and the temporal correlation hypothesis” Annu.Rev.Neurosci. 18,555. 20. Von der Malsburg C., 1981 “The correlation theory of brain function”, reprinted in E. Domani, J.L. Van Hemmen and K. Schulten (Eds.), “Models of neural networks 11”, Springer, Berlin. 21. W i p e r E.,1932 “ On the Quantum Correction For Thermodynamic Equilibrium” Phys.Rev,70,749 22. Zhou C.S. , Allaria E. , Boccaletti S. , Meucci R., Kurths J.,and.Arecchi F.T .,2003, “Noise induced synchronization and coherence resonance of homoclinic chaos”, Phys. Rev. E 67,015205

ENERGETIC MODEL OF TUMOR GROWTH

P. CASTORINA AND D.ZAPPALA INFN, Sezione di Catania and Dept. of Physics, University of Catania, Via 5 '. Sofia 64, I-95123, Catania, Italg E-mail: [email protected] A macroscopic model of the tumor Gompertsian growth is proposed. This approach is based on the energetic balance among the different cell activities, described by methods of statistical mechanics and related to the growth inhibitor factors. The model is successfully applied to the multicellular tumor spheroid data.

1. Introduction

A microscopic model of tumor growth in vivo is still an open problem. However, in spite of the large set of potential parameters due to the variety of the in situ conditions, tumors have a peculiar growth pattern that is generally described by a Gompertzian curve ', often considered as a pure phenomenological fit of the data. More precisely there is an initial exponential growth (until 1-3 mm in diameter) followed by the vascular Gompertzian phase2. Then it seems reasonable to think that cancer growth follows a general pattern that one can hope to describe by macroscopic variables and following this line of research, for example, the universal model proposed in has been recently applied to cancer4. In this talk we present a macroscopic model of tumor growth, proposed in 5, that: i) gives an energetic basis to the Gompertzian law; ii) clearly distinguishes between the general evolution patterns, which include the internal feedback effects, and the external constraints; iii) can give indications on the different tumor phases during its evolution. The proposed macroscopic approach is not in competition with microscopic models 6, but it is a complementary instrument for the description of the tumor growth. 2. Cellular energetic balance and Gompertzian growth

The Gompertzian curve is solution of the equation

d N = Nyln dt

(%)

where N ( t ) is the cell number at time t, y is a constant and Nmasis the theoretical saturation value for t + 00.

272

273

a. (9)

It is quite natural to identify the right hand side of (1) as the number of proliferating cells at time t and then to consider f p ( N )= 71n as the fraction of proliferating cells and 1 - f p ( N ) = fnp the fraction of non proliferating cells. Since f p ( N ) depends on N ( t ) ,there is a feedback mechanism usually described by introducing some growth inhibitor factors which increase with the number of nonproliferating cells and are responsible for the saturation of the tumor size. The concentration of inhibitor factors should be proportional to the number of nonproliferating cells which is maximum at N = N,,, 7. If one considers that, during the growth, each cell shares out its available energy at time t , in the average, among its metabolic activities, the mechanical work ( associated with the change of the tumor size and shape) and the increase of the number of cells, it is conceivable to translate the previous cellular feedback effect in terms of energy content. Indeed, as shown in 5 , the specific energy for the growth should be proportional to fnp and the average metabolic plus mechanical energy per cell, M e , is proportional to f,. As we shall see this reproduces the observed cellular feedback. The model 5 , based on an analogy with statistical mechanics, assumes that in a larger system B , the body, there is a subsystem A, the tumor, made of N ( t ) cells at time t, with total energy U ,which has specific distributive mechanisms for providing, in the average, the amount of energy U / N to each cell. Then we indicate with EM the energy needed for the metabolic activities of A , with R the energy associated with the mechanical work required for any change of size and shape of A , and with p the specific energy (i.e. per cell) correlated to the change in the number of cells N , and, by assuming that these three processes summarize the whole cellular activity, we have U = EM R p N . Let us assume that the system A slowly evolves through states of equilibrium with the system B , defined by macroscopic variables, analogous for instance to the inverse temperature 8, that have the same value for the two systems, although it should be clear that in our case we do not have red thermodynamical equilibrium because the system B supplies the global energy for the slow evolution of the subsystem A. Within this scheme, there are many microscopic states of the system A, compatible with the macroscopic state, defined by 8, p and V , which are built by a large number of states of each single cell. These microscopic states of each cell have minimum total energy e and in an extremely simplified picture, we assume an energy spectrum of the form = e 16, where 1 is an integer and 6 is the minimum energy gap between two states. With this spectrum the grand partition function 2 is given by the following product Z(B, V ,p ) = lT&exp ( e - O ( Q - p ) ) and the corresponding grand potential, which is natural to associate to the energy R related to the mechanical work in our problem, is given by R(/?,V,p) = -(R/p)exp[-p(e - p ) ] where R = 1/(1- e-pa). The average value of N , defined for constant V and 8, turns out to be N = -80. According to the basic rules of statistical mechanics, the product of the “entropy times the temperature”, which in our system corresponds to EM introduced above,

+ +

+

2 74

is

where C is given by C = Rp6 exp(-pd). Fkom the previous equations it is straightforward to express p in terms of N : p = e (1//3) ln(N/R). To find the evolution of the system A which takes into account the internal feedback mechanism we recall some results obtained in :

+

the energetic balance requires that the growth with cellular feedback starts at a minimum number of cells N , and saturates at a maximum value, Nmaz related by N,,, = N,exp(l +,Be). For N , >> 1, Me = E M / N R/N = (1/p) ln(N,,,/N) is a decreasing function of N > Nm and there is a simultaneous reduction of the total metabolic energy per cell and an increase of specific energy required for the growth: there is an energetic balance between Me and p = (l/p)(l pe 1n(N/Nma,)). Moreover Me is proportional t o fp(t).

+

+ +

Then it is possible to derive the Gompertz equation for the growth, A N in an interval At. A N is proportional to the number of proliferating cells and then one can write A N = c1 At fp(t)N = c2 At Me N where c1 and c2 are constants. This gives in the continuum limit Eq. (1) with y = c2/,4.

3. Application to Multicellular Tumor Spheroids The first step to analyze the phenomenological implications of the model and t o describe the dependence of the growth on the external conditions is t o consider the multicellular tumor spheroids (MTS). A minimal MTS description consists of a spherical growth where a) the thickness, k, of the layer where the nutrient and oxygen is delivered (the crust) is independent on the spheroid radius R; b) the cell density is constant; c) the cells in the crust, receive a constant supply of nutrient for cell; d) at time t the cells are non proliferating if they are at distances d < R - k if R > k from the center of the spheroid. For R < k all cells are proliferating. To separate the effects of the external constraints due t o energy supply from those related t o biomechanical conditions, it is better t o consider first the MTS growth without external and internal stress and to introduce later these effects.

3.1. Energetic MTS growth In this case the external conditions are experimentally modified by changing the oxygen and nutrient concentration in the environment. At fixed value of these concentrations, the maximum allowed number of cells in the MTS is N,,, . For R < k all cells receive the nutrient and oxygen supply while for R > k there is a fraction of non proliferating cells and the feedback effect starts. The growth of the MTS is due to the proliferating cells in the crust and one obtains that the MTS radius R, for

275

R >> k, follows a Gompertzian law as the one in Eq. (l),with N replaced by R and N,,, by La,, where La, is the maximum radius of the spheroid corresponding to the maximum number of tumor cells Nmaz. The experimental results show that after 3-4days of initial exponential growth the spheroids essentially follow the Gompertzian pattern'. According to our model, at time t > t*,such that R(t*)= k, a fraction of the total cells becomes non proliferating, the feedback effect starts and the growth rate decreases according to the Gompertz law. The number of cells at time t' is fixed by the condition N ( t * )= N , 5 . On the other hand, the variation of the concentration of nutrient and/or of oxygen modifies the total energy supply, that is the value of N,,, and, since N , = N,,,exp(-ep - 2), there is a clear correlation among the external energetic "boundary conditions", the value N,,, and the thickness of the viable cell rim which corresponds to the radius of the onset of necrosis. It can be shown that (G, is the glucose concentration):

k(G,) = cy ( N h t , - N&z1'3)

+ ko

(3)

where cy and ko are constants depending on the supplied oxygen. From Eq. (3) one obtains the correlation among N,,,, G, and k. In Fig.1 and Fig. 2 the previous behaviors are compared with data without optimization of the parameters.

Figure 1. Thickness (pm) vs. glucose concentration (mM). Figure (a) is for an oxygen concentration of 0.28 mM and Figure (b) is for an oxygen concentration of 0.07 mM.

2k

Figure 2. Spheroid saturation cells number vs. diameter (pm) at which necrosis first develops. Circles refer to culture in 0.28 mM of oxygen Tkiangles refer to culture in 0.07 mM of oxygen.

276 Table 1. Comparison with the experimental data as discussed in the text. The experimental error is about 240%

cg(percent)

3.2.

2Rmax(Q)

[m] exper.

2Rmax(Q) [ ~ m fit]

0.3

450

0.5

414

452 429

0.7

370

404

0.8

363

394

Biomechanical eflects

The experimental data indicate that when MTS are under a solid stress, obtained for example by a gel, the cellular density p is not constant and depends on the external gel concentration C,. In particular the results in show that: 1) an increase of the gel concentration inhibits the growth of MTS; 2) the cellular density at saturation increases with the gel concentration. In the model the mechanical energy is included in the energetic balance of the system by the term R = -N/B = -PV where the pressure is P ( t ) = p ( t ) / p . The introduction of this term decreases the value of N,,, with respect t o the case in Sect. 3.1 and this reduction should also imply a decrease of the maximum size of the spheroids, i.e. R,,,(P) by increasing the pressure. The comparison with the data is reported in Table I for C, in the range 0.3 - 0.8 % (see ti for details).

References 1. B. Gompertz, Phyl. Trans. R. SOC., 115,513 (1825). 2. G.G. Steel, “Growth Kinetic of tumors”, Oxford Clarendon Press, 1977; Cell tissue Kinet., 13,451 (1980); T.E. Weldon, “Mathematical models in cancer research”, Adam Hilger Publisher, 1988 and refs. therein. 3. G.B. West et al., Nature 413,628 (2001). 4. C. Guiot et al., J. Theor. Biol. 25, 147 (2003). 5. P. Castorina and D. ZappalA, “Tumor Gompertzian growth by cellular energetic balance”, q-bio.CB/0407018. 6. M. Marusic et al., Cell Prolif. 27 73 (1994); Z. Bajzer et 06. in: “Survey of model for tumor-immune system dynamics”, J.A. Adams and N. Bellomo eds., Birkhauser 1997; A. Bru et al., Phys. Rev. Lett. 81, 4008 (1998); Z. Bajzer, Growth Dev. Aging, 63, 3 (1999); N. Bellomo et al., “Mathematical topics on the modelling complex mulicellular systems and tumor immune cells competition”, Preprint: Politecnico di Torino, 2004. 7. J. P. Freyer, R.M. Sutherland, Cancer Research, 46,3504 (1986). 8. L. Norton et al., Nature 264, 542 (1976); L. Norton Cancer Research, 48,7067 (1988). 9. G. Helmlinger et al., Nature Biotechnology ,15,778 (1997).

ACTIVE BROWNIAN MOTION - STOCHASTIC DYNAMICS OF SWARMS

WERNER EBELING AND UDO ERDMANN Institut fur Physik, Humboldt- Universitat zu Berlin Newtonstrafle 15, 12489 Berlin, Germany Email: [email protected], [email protected]. de We summarize the essential features of the new model of wtive Brownian dynamics and applications ranging from the dynamics of molecular clusters in non-equilibrium to moving swarms of animals.

1. Characteristics of the dynamics of clusters and swarms

This paper covers a wide range of related phenomena reaching from the dynamics of molecular clusters in non-equilibrium t o moving swarms of animals. We introduce the general notation of “swarms” for confined systems of particles (or more general objects) in non-equilibrium. The study of non-equilibrium clusters of molecules begins more than 70 years ago with the pioneering papers of Farkas, Becker and Doring. However most of these studies are restricted to near equilibrium phenomena, as moving over a threshold of the free energy. Compared t o the theory of clusters, the study of objects like swarms of animals is a rather young field of physical studies (see e.g. Refs. 1, 2, 3, 4). Since the dynamics of swarms of driven particles has captured the interest of theorists, many interesting effects have been revealed and in part already explained. We mention the comprehensive survey of Okubo and Levin’ on swarm dynamics in biophysical and ecological respect. Further we mention the survey of Helbing’ covering traffic and related self-driven many-particle systems and the comprehensive books of Vicsek’ Mikhailov and Calenbuhr3 and of Schweitzer4. In the book of Okubo and Levin5 we find a classification of the modes of collective motions of swarms of animals. It is discussed that animal groups have three typical modes of motion: (i) translational motions, (ii) rotational excitations and (iii) amoeba-like motions. For example, Ordemann, Balazsi and Moss6>’studied the modes of motion of Duphniu. Depending on the existence of a external light source a whole swarm of these

277

278

animals switches from a uncorrelated type of motion to a very correlated type. The whole swarm starts to rotate then. At present it seems to be impossible to describe all the complex collective motions observed in nature. Instead we study in the following the collective modes and the distribution functions of a simple model. We investigate finite systems of particles confined by attracting forces which are self-propelled by active friction and have some interactions. This is considered as a rough model for the collective motion of non-equilibrium clusters and of swarms of cells and organisms as well8. For alternative models based on velocity-velocity interactions see Refs. 2, 9, 10. From the point of view of statistical mechanics the main purpose of this work is the study of the dynamics of active Brownian particles including interactions. The self-propelling of the particles is modeled by active friction as introduced in earlier work”. The interaction between the particles is modeled by harmonic (linear) forces or by Morse potentials. The consideration is restricted to 2 - d models. Driving the system by negative friction we may bring the system to far from equilibrium states. Studies of one-dimensional models have shown that driven interacting systems may have many a t t r a ~ t o r s ~Noise ~ i ~ ~may . lead to transitions between the deterministic attractors. In the case of two-dimensional motion of interacting particles, positive or negative angular momenta may be generated. This may lead to left/right rotations of pairs, clusters and swarms. We will show that the collective motion of large clusters of driven Brownian particles reminds very much the typical modes of parallel motions in swarms of living entities. 2. Dynamics in external fields in rigid body approximation

We introduce interactions described by the potential U(r1,. . . ,r N ) and postulate a dynamics of Brownian particles determined by the Langevin equation: where [ ( t )is a stochastic force with strength D, and a &correlated time dependence. The dissipative forces are expressed in the form

The coefficient y denotes a velocity-dependent friction, which possibly has a negative part. This way the dynamics of our Brownian particles is determined by the Langevin equation with dissipative contributions. In the case of thermal equilibrium systems we have y(v) = 70 = const.. In the general case where the friction is velocity dependent we will assume that the friction is monotonically increasing with the velocity and converges to 70at large velocities. In the following we will use the following ansatz based on the depot model for the energy supply 11,14

(4)

279 where c, d, q are certain positive constants characterizing the energy flows from the depot to the particle. Dependent on the parameters yo,c, d, and q the dissipative force function may have one zero at v = 0 or two more zeros with

vz - -6; d O - C

6=qd - 1.

(5)

CYO

Here 6 is a bifurcation parameter. In the case 6 > 0 a finite characteristic velocity vo exists. Then we speak about active particles. For IvI < VO, the dissipative force is positive, i.e. the particle is provided with additional free energy. Hence, slow particles are accelerated, while the motion of fast particles is damped (see Fig. 1). The asymptotics for large velocities is passive. Now we will discuss the motion of 1

.

0.5

p -

..................

6=0

-6=1

'

............... 6=1.2

-----

6=2

,,

Figure 1. The typical form of a friction function with active (negative) part a t small velocities (parameter 6 = C 1).

+

active particles in a two-dimensional space, r = {q,q}.The case of constant external forces was already treated by Schienbein et al.15316. Symmetric parabolic external forces were studied in Refs. 11, 17 and the non-symmetric case is being investigated in Ref. 18. Here we will study Pd-systems of N 2 2 particles. Let us imagine s swarm of active Brownian particles which are pairwise bound to a cluster which is rigid. Then the problem is restricted to the motion of the center of mass 1 I XI = XZ = - c x i z . (6) Nxzi1; We consider here only the free motion of the center of mass and the motion in an external field. The relative motion under the influence of the interaction is neglected so far. The free motion of the center of mass M is described by the equations XI = Vl x 2

=

vz

MVl = - M y (Vl, V2)Vl MV2

= - M y (Vl, V2) v 2

+ &G(t) +J2D,E2(t)

(74 (7b)

280 The stationary solutions of the corresponding Fokker-Planck equation reads’’

where D, = D/m2. For simplification we specify now the potential U as 1

U(X1,X,) = -a (Xf -tX,”) . 2

(9)

First, we discuss the deterministic motion, which is described by four coupled firstorder differential equations. The motion of the center of mass corresponds to the motion of 1 particle in an external field:

XI = ~1 X2

= VZ

- ax1

(104

rnV2 = -my ( ~ 1 ~, 2 ) - ax2

(lob)

mlil = -my

( ~ 1vZ) ,

For this case we have shown earlier“ that a limit cycle in the four-dimensional space is developed, which corresponds to leftlright rotations with the frequency W O . The projection of this periodic motion to the planes (21, q )and (u1, v2) are circles

xf + x,”= r$

V:

+ hz= u:.

(11)

The trajectories converge to limit cycles and the energy to

H

-+

2 EO= muO

(12)

This corresponds to an equal distribution between kinetic and potential energy. In explicite form we may represent the motion on the limit cycle in the four-dimensional space by the four equations”

+

X1 = ro cos(w0t a) Xz = T O sin(w0t + 3)

V1 = -row0 sin(w0t

+ a)

Vz = row0 cos(wt + a)

(134 P3b)

The frequency is given by the time the particle need for one period moving on the circle with radius ro with constant speed UO. This leads to wo = ro/uo and means that the particles oscillate with the frequency given by the linear oscillator frequency. The trajectory on the limit cycle defined by equations (13) is like a hula hoop in the four-dimensional space. The projections to the ( z 1 , ~space ) as well as ‘the projections to the (u1,uZ) space are circles. The projections to the subspaces (21, u2) and (22,ul} are like a rod. In the four-dimensional space the attractor has therefore the form of a hula hoop. A second limit cycle is obtained by reversal of the velocity. This second limit cycle forms also a hula hoop which is different from the first one, however both limit cycles have the same projections to the (21, 22) and to the (ul, u2) plane. The motion in the ( 2 1 , ~ plane ) has the opposite sense of rotation in comparison with the first limit cycle. Therefore both limit cycles correspond to opposite angular momenta. L3 = +Mrouo and L3 = -Mrouo. Applying similar arguments to the stochastic problem we find that the two hooprings are converted into a distribution looking like two embracing hoops with finite

28 1

size, which for strong noise converts into two embracing tires in the four-dimensional space. In order to get the explicite form of the distribution we may introduce amplitude-phase representations". The probability crater is located above the two deterministic limit cycles on the sphere T O = vo/w0. Strictly speaking not the whole spherical set is filled with probability but only two circle-shaped subsets on it, which correspond to a narrow region around the limit sets, The full stationary probability has the form of two hula hoop distributions in the four-dimensional space. This was confirmed by simulations". The projections of the distribution to the (21, 2 2 ) plane and to the {q,v2) plane are smoothed two-dimensional rings. The distributions intersect perpendicular the (21, v2) plane and the {x2,wl} plane. Due t o the noise the Brownian particles may switch between the two limit cycles, this means inversion of the angular momentum (direction of rotation)ll,ls. As a result rotating clusters are getting unstable similar to asymmetric driven ascillators18. 3. Dynamics of self-confined Morse clusters of driven particles The study of the full many-body dynamics of interacting driven particles including drift is an extremely difficult task. Therefore we will present here first the result of some simulations. In particular we studied Morse interactions described by the interaction potential @(T)

= A [exp(-m)

-

112

-

A

(14)

Our simulations for swarms with Morse interactions (Fig. 2). show rotating clusters. We observe rotations changing from time to time the sense of rotations due to

Figure 2. The two possible stationary states of a rotating cluster of 20 particles. The arrows correspond to the velocity of the single particle. In the presence of noise the cluster changes from time to time the direction of rotation.

stochastic effects. Further we see a slow drift of the clusters. The rotating swarms simulated in our numerical experiments remind very much the dynamics of swarms

282

studied in papers of Viscek and c o l l a b ~ r a t o r sand ~ ~ ~in other recent w0rks91111g. The translation mode corresponds to a driven motion of a free particle located in the center of mass supplemented by a small oscillatory relative motion against the center of mass. The solutions for the rotational model are similar to what we have found in the rigid approximation for the case of external fields. The probability is distributed around two limit cycles corresponding to left or right rotations. The result of several simulations which show cluster configurations and amoeba-like configurations is shown in Fig. 3. lllW

70 60 50

40

30 20

10

0

10

20

30

40

so

I

60

-

D

70

10

20

30

,=lOW

40

50

60

70

so

64

70

1IIwo

70

70

60

60 SO

40 30

20 10

0 0

10

20

30

40

50

60

70

0

10

20

30

40

Figure 3. Rotating and drifting clusters as well as amoeba-like configurations of 625 particles with Morse interactions.

4. The model of harmonic swarms w i t h global coupling

This section is devoted to some analytical estimates. Due to the great complexity of the dynamics of swarms we need further simplifications. In the following we will reduce all interactions to a global coupling of the particles. We consider twodimensional systems of N point masses m with the numbers 1 , 2 , . . . ,i, . . . ,N . We assume that the masses m are connected by linear pair forces mw; (ri - rj). The dynamics of the system is given by the following equations of motion r.- v . . '$

$ ,

miri +mu: (ri - R(t)) = Fi(vi)- a2(vi- V) + m&(t) (15)

The term proportional to a2 denotes a small force tending to parallelize the velocities of the particles in the swarm. Again we start with an investigation of the

283 translational mode of this system. For the mean velocity we find by summation and expanding around V in a symbolic representation V = F(V)

+ -21 (bv)* F”(V)* (bv)+ . . .

(16)

For the relative motion 6vi = v i- V we get in first order approximation

siri

+ wi6ri = -r * 6vi + J2D,Ei(t)

(17)

In the translational mode of this system all the particles form a noisy flock which moves with nearly constant velocity modulus

V(t) = R(t) = won;

ri(t) - R(t) = 0 i = 1,. . . ,N

(18)

The direction n may change from time to time due to stochastic influences. The distribution function of the flock is Boltzmann-like. However this distribution is stable only in the region where the friction tensor I? has only positive eigenvalues. This is for sure if V 2 = vi. Our solution breaks down if the dispersion bv2 is so large that the linearization around V is no more possible. With increasing noise we find a bifurcation. This corresponds to the findings of Mikhailov and Zanette for equivalent one-dimensional systems”. We note that in the two-dimensional system the dispersion of the relative velocity 6v is not isotropic. The dispersion in the direction of the flight V is smaller than perpendicular to it. We introduce here an isotropic approximation which allows for explicite solutions of the bifurcation problem. Beyond the region of stability of the translational mode the swarm converges to a rotating swarm at rest (see Fig. 4). This second stationary state of the 6Z.7

.

233.817.5

.I,.*

,

,

, .- . .

. .. .. , . . . .

1

.111

Figure 4. Possible stable states of a system of N = 100 globally coupled active particles. In the left picture we see a typical elliptical configuration of a swarm for a noise strength below the critical value. The right picture shows the spherical configuration corresponding to a rotational state. The noise strength is beyond the critical one.

swarm corresponds to left/right rotating ring configurations with a center at rest 8,19. In order to describe the numerical results semi-quantitatively, we introduce

284 further approximations. At first we simplify the equation for the mean momentum V similar as in Ref. 20 assuming

v = (a- pv2) (V) - P C ( 6 V i ) 2 - 2-P N

C (VbVi) svi + . .. + & Z [ ( t )

(19)

N i

i

In order to find explicite solutions we decouple the center of mass motion from the relative motion. By averaging with respect to Svi and neglecting the tensor character of the coupling to the relative motion we get d -v dt

= (01

-

Here the effective driving strength proximated by

PV2)V a1

a1 = a

+ ... +

(20)

(which strictly speaking is a tensor) is a p

P

- s-

N

C(6Vi)2

i

The factor s is between 1 (corresponding to strictly perpendicular fluctuations) and 3 (corresponding to only parallel fluctuations). As some reasonable average we will assume s N 2. The corresponding velocity distribution is

This way we find the most probable velocity

The most probable velocity of the swarm is shifted to values smaller than for the free motion. The shift with respect to the free mode VOis proportional to the noise strength D. For the fluctuations around the center om mass of the swarm we find

6vi

+ d 6 r i = -mi+ &Zti(t)

(24)

Here I? = 2PV: - a follows from a diagonal approximation of the tensor I?. In this way the relative distribution can be approximated as

Now we get a quadratic equation for the dispersion Dw

(sv2) = a!

with the solution

- 2sp (6w2)

285 The corresponding effective friction reads

r=2

[ + 1J1

At the critical noise strength D, = a2/8sP the dispersion has its maximal and the effective friction its minimal value, for larger noise strength the dispersion and the effective friction get complex. In simulations we found for Q = ,B = 1 a critical noise strength D P M 0.06721. This is very close to our theoretical estimate with s = 2 which gives DYit = 1/16. This way we gave a simple explanation for the transition from translational to rotational modes. A more advanced theory will be developed elsewhere".

5. Conclusions We studied here the active Brownian dynamics of swarms of confined particles with velocity-dependent friction and attracting interactions. Confinement was created either by pair-wise linear attracting forces, or by attracting Morse interactions. The basic results of our observations may be summarized as follows:

cluster drift: We see in the simulations clusters drifting clusters rotating very slowly and clusters without rotations which move rather fast (see Fig. 3). The latter state corresponds to the translational mode studied in the previous section for N = 2. Here most of the energy is concentrated in the kinetic energy of translational movement. generation of rotations: As we see from the simulations, small Morse clusters up to N 2: 20 generate left/right rotations around their center of mass. The angular momentum distribution is bistable. This corresponds to the rotational mode studied above for N = 1,2. breakdown of rotations: The rotation of clusters may come to a stop due to several reasons. The first is the anharmonicity of clusters. As we have shown in our previous work18, strong anharmonicity destroys the rotational mode. Another reason which was investigated here are noise induced transitions. shape distribution: Under special conditions the shape of the clusters is amoebalike and is getting more and more complicated". A theoretical interpretations of the shape dynamics is still missing. cluster composition: With increasing noise we observe a distribution of clusters of different size. Again a theory of clustering in the two-dimensional case is still missing. For the case of one-dimensional rings with Morse interactions several theoretical results are a ~ a i l a b l e l ~ . ~ ~ . We have given here first an analysis of several simple cases. This way we could identify several qualitative modes of movement. f i r t h e r we have made a numerical study of special N particle systems. In particular we investigated the rotational and translational modes, the clustering phenomena and noise induced transitions.

286 We did not intend here t o model any particular problem of biological or social collective movement. We note however t h a t t h e study of dynamic modes of collective movement of swarms may be of some importance for t h e understanding of many biological and social collective motions. To support this view we refer again t o t h e book of Okubo and Levin5 where t h e modes of collective motions of swarms of animals are classified i n way which reminds very much the theoretical finding for t h e model investigated here. In particular we mention also t h e motion of animals in water, for example t h e collective motion of Duphniu6~7~24325.

References

D.Helbing, Rev. Mod. Phys. 73,1067 (2001). T. Vicsek, Fluctuations and Scaling in Biology, Oxford University Press, Oxford, 2001. A. S.Mikhailov and V. Calenbuhr, From Cells to Societies, Springer, Berlin, 2002. F. Schweitzer, Brownian Agents and Active Particles, Springer, Berlin, 2003. A. Okubo and S. A. Levin, Diffusion and Ecological Problems: Modern Perspectives, Springer, New York, 2 edition, 2001. 6. A. Ordemann, G. Balazsi, and F. Moss, Nova Acta Leopoldina NF 88,87 (2003). 7. A. Ordemann, G. Balazsi, and F. Moss, Physica A 325,260 (2003). 8. F.Schweitzer, W. Ebeling, and B. Tilch, Phys. Rev. E 64,021110 (2001). 9. T.Vicsek, A. Czirbk, E. Ben-Jacob, I. Cohen, and 0. Shochet, Phys. Rev. Lett. 75, 1226 (1995). 10. A. Czir6k and T. Vicsek, Physica A 281,17 (2000). 11. U. Erdmann, W. Ebeling, F. Schweitzer, and L. Schimansky-Geier, Eur. Phys. J. B 15,105 (2000). 12. W.Ebeling, U. Erdmann, J. Dunkel, and M. Jenssen, J. Stat. Phys. 101,443 (2000). 13. J. Dunkel, W.Ebeling, U. Erdmann, and V. A. Makarov, Int. J. Bif. Chaos 12,2359 (2002). 14. F.Schweitzer, W.Ebeling, and B. Tilch, Phys. Rev. Lett. 80,5044 (1998). 15. M. Schienbein and H. Gruler, Bull. Math. Biol. 55,585 (1993). 16. M. Schienbein, K.Franke, and H. Gruler, Phys. Rev. E 49,5462 (1994). 17. W.Ebeling, F. Schweitzer, and B. Tilch, BioSystems 49,17 (1999). 18. U. Erdmann, W.Ebeling, and V. S. Anishchenko, Phys. Rev. E 65,061106 (2002). 19. W.Ebeling and F. Schweitzer, Theory in Biosciences 120,207 (2001). 20. A. S. Mikhailov and D. Zanette, Phys. Rev. E 60,4571 (1999). 21. U. Erdmann, W.Ebeling, and A. S. Mikhailov, Noise induced transition from translational to rotational motion of swarms, http: //arxiv. org/abs/physics/04120372004, 2004. 22. W.Ebeling and U. Erdmann, Complexity 8,23 (2003). 23. J. Dunkel, W.Ebeling, and U. Erdmann, Eur. Phys. J. B 24,511 (2001). 24. A. Ordemann, G. Balazsi, E. Caspari, and F. Moss, Daphnia swarms: from single agent dynamics to collective vortex formation, in Fluctuations and Noise in Biological, Biophysical, and Biomedical Systems, edited by S . M. Bezrukov, H. Frauenfelder, and F. Moss, volume 5110 of Proceedings of SPIE, pages 172-179,Bellingham, 2003,SPIE. 25. U. Erdmann, W.Ebeling, L. Schimansky-Geier, A. Ordemann, and F. MOSS,Active brownian particle and random walk theories of the motions of zooplankton: Application to experiments with swarms of Daphnia, http ://arxiv. org/abs/q-bio. PE/0404018,2004. 1. 2. 3. 4. 5.

COMPLEXITY IN THE COLLECTIVE BEHAVIOUR OF HUMANS TAMAS VICSEK Biological Physics Department and Research Group of Hung. Acad. Sci., Eotvos Universiv, Phzmdny p . Stny lA, H-1117 Budapest, Hungav

Can we reliably predict and quantitatively describe how large groups of people behave? Here we discuss an emerging approach to this problem which is based on the quantitative methods of statistical physics. We demonstrate that in cases when the interactions between the members of a group are relatively well defined (e.g, pedestrian traffic, synchronization, panic, etc) the corresponding models reproduce relevant aspects of the observed phenomena. In particular, people moving in the same environment typically develop specific patterns of collective motion including the formation of lanes, flocking or jamming at bottlenecks. We simulate such phenomena assuming realistic interactions between particles representing humans. The two specific cases to be discussed in more detail are waves produced by crowds at large sporting events and the main features of escape panic under various conditions. Our models allow the prediction of crowd behaviour even in cases when experimental methods are obviously not applicable and, thus, are expected to be useful in assessing the level of security in situations involving large groups of excited people

1. Introduction It is becoming increasingly evident that the application of ideas, methods and results of statistical physics to a wide range of phenomena occurring outside of the realm of the non-living world is a fruitful approach leading to numerous exciting discoveries. Among many others, examples include the studies of various group activities of people from the physicist’s viewpoint. Here, I shall give a partial account of some of our new investigations in this direction, involving the interpretation of such collective human activities as group motion and synchronization. On the small scale side of the size/complexity spectrum, in the world of atoms and molecules collective behaviour is also considered to be an important aspect of the observed processes. Furthermore, there are articles on collectively migrating bacteria, insects or birds and additional interesting results are published on phenomena in which groups of various organisms or non-living objects synchronize their signals or motion. This is the natural scientist’s aspect of how many objects behave together. However, if you search for a collective behaviour related item with your web browser most of the texts popping up will be concerned with group activities of humans including riots, fashion or panics. What is common in these seemingly diverse phenomena involving interpretations ranging from social psychology to statistical physics? The answer is that they happen in

287

288

systems consisting of many similar units interacting in a relatively well defined manner. These interactions can be simple (attractiodrepulsion) or more complex (combinations of simple interactions) and may take place between neighbours in space or on a specific underlying network. Under some conditions, in such systems various kinds of transitions occur; during these transitions the objects (particles, organisms or even robots) adopt a pattern of behaviour which is nearly completely determined by the collective effects due to the presence of all of the other objects in the system. What are the motivations for this sort of research? Mankind has been experiencing a long successful period of technological development. This era has been the result of a deeper understanding of the various physical and chemical processes due to the outstanding advances in the related sciences. After these achievements there is now a growing interest in a better, more exact understanding of the mechanisms underlying the main processes in societies as well. There is a clear need for the kind of firm, reliable results produced by natural sciences in the context of the studies of human behaviour. The revolution in information and transportation technology brings together larger and larger masses of people (either physically or through electronic communication). New kinds of communities are formed, including, among many others, internet chat groups or huge crowds showing up at various performances, transportation terminals or demonstrations. Since they represent relatively simple examples, these groups or communities of people provide a good subject from the point of studying the mechanisms determining the phenomena taking place in societies. Below we discuss new quantitative approaches to collective behaviour based on the exact methods of statistical physics. It is clear that the methods developed in natural sciences contain a significantly smaller amount of subjectivity than those used for the interpretation of human behaviour. If more exact approaches could be applied to social situations they could provide the desired objectivity, reproducibility and predictability. We demonstrate that in cases when the interactions between the members of a group are relatively well defined (e.g, pedestrian traffic, rhythmic applause, panic, soccer fans in stadiums, etc) the corresponding numerical models reproduce relevant aspects of the observed phenomena. Simulating models in a computer has the following advantages: i) by changing the parameters different situations can easily be created ii) the results of an intervention can be predicted and iii) more efficient design of the conditions for the optimal outcome can be assisted. In addition to possible applications, our approach is useful in providing a deeper insight into the details of the mechanisms determining collective phenomena occurring in social groups (see Fig. 1 for an observation of a simple kind of collective human behaviour: spontaneous lane formation in crowds of oppositely moving pedestrians).

289

Figure 1. A simple kind of collective human behaviour: spontaneous lane formation in crowds of oppositely moving pedestrians

Most of the results I am discussing next are available through the home page http://angel.elte.hd-vicsek. Many other recent studies have been devoted to the question of how the concepts common in statistical physics (fluctuation, phase transitions, scaling, etc) can be applied to a group of humans. A particularly entertaining and exhausting review of these efforts is given in the very recent book by Philip Ball [ 13 written in a popular science style. There exist additional remarkable works in similar directions by groups working on traffic, evacuation dynamics, econophysics, and on further related topics (see, e.g., Refs 2-4). 2. Mefhods Our central statement is that collective behaviour can be very efficiently studied by the methods developed by statistical physicists. The related theoretical and numerical approaches provide reliable, sometimes exact description of the processes taking place in many particle systems. We assume that under some conditions a large group of humans can be considered as a collection of particles, since there are various situations where the interaction of people is reasonably well defined (e.g., two people heading towards each other in a corridor will avoid each other just as if they had a repulsive physical force acting between them).

For the last two decades perhaps the most fruitful approach to collective phenomena has been the application of computer simulations. In such studies a simple model is

290

constructed which is supposed to grab the most relevant features of the system to be studied. Then, by letting the algorithm run in the computer while monitoring the parameters of the models a great variety of collective phenomena can be observed. The true test of a model is a careful comparison of its predictions with the behaviour of the real system. Examples

The rest of the paper will present examples of group behaviour of people which could be successfully interpreted by computer simulations and the related theoretical concepts. It is hoped that the process of simultaneous investigation of particular examples and the abstraction of their most general features will in time lead to a coherent theoretical description of collective human behaviour.

Collective motion

Here we first address the more general question whether there are some global, perhaps universal features of collective motion [ 5 ] . Such behaviour takes place when many organisms are simultaneously moving and parameters like the level of perturbations or the mean distance between the individuals is changed. The simple and generic model we introduced some time ago to study collective motion assumes two rules: a) Follow the others, or in other words, try to take on the average velocity of your neighbours. b) In addition, an amount of randomness is added to the actual velocity (to account for example for the level of excitement of the pedestrians).

Simulations result in a completely disordered motion if the level of perturbations is large (each particle moves back and forth randomly). However, if the noise is smaller than a critical value (just as in the case of the ordering of ferromagnets), groups of particles are spontaneously formed the groups merge (aggregate) and sooner or later join into a single large group moving in a direction determined in a non-trivial way by the initial conditions (Fig. 2).

291

Figure 2. The simple model of collective motion leads to a globally ordered motion of particles for intermediate noise levels (a), while results in flocks moving in random directions if the level of fluctuations is larger @). The interaction radius is indicated as a horisontal line segment.

3. Applications to situations involving crowds

A) Consider, as a thought experiment, thousands of people standing on a square and trying to look in the same -- however, previously undetermined -- direction, after being asked to do so. A nice example for human collective behaviour would be if all of them managed to face the same direction. Can they do it? Statistical physicists can predict for sure that this cannot be done. They recall a theorem valid for particles with short ranged ferromagnetic interactions stating that in two dimensions no long range ordered phase (all magnets pointing in the same direction) can exist in such a system for any finite temperature and zero external field. So what happens? Locally people are looking almost in the same direction, but on a large scale, e.g., seen from a helicopter -just as the little magnets -- they locally form vortex-like directional patterns due to the small perturbations due to human errors. Curiously enough, if the crowd is allowed to choose from a few discrete directions, the ordering can be realized. Perhaps even more interestingly, our models of flocking (based on the follow the neighbours rule) predict that if the people are asked to move in the same direction they will be able to do it.

B) In the latter models, if the moving particles are confined to move around in a closed circular area stable motion can be maintained only by the simultaneous rotation of all of the objects around the centre. Remarkably enough, under some conditions even humans move in groups in a manner predicted by simple models. Indeed, in Mecca each year thousands of people circle around the Kaba stone as they are trying to both keep on moving and not confronting with others. C) Next we focus on a system of oppositely moving pedestrians in a corridor. Here the corridor is wide enough (its width is several times larger than the diameter of a person). Half of the pedestrians is assumed to move from left to right, the rest in the opposite direction. In the associated model it is assumed that the particles tend to take on a

292 constant speed in their desired direction and are avoiding each other due to a repulsive force.

Figure 3. Results from a simulation of oppositely moving particles in a strip geometry (yellow to the left, red to the right). A simple repulsive force and motion on a continuous plane (there is no underlying grid) have been assumed. An intermediatenumber of particles leads to lane formation, while a large density results in a jamming, turbulent flow (bottom).

Simulations of this simple model based on the solution of the corresponding Newton's equations of motion reproduce the experimentally observed behaviour surprisingly well. A spontaneous formation of lanes of uniform walking directions in "crowds" of oppositely moving particles can be observed (Fig. 3). It is clear that lane formation will maximize the average velocity in the desired walking direction which is a measure of the "efficiency" or "success" of motion. Note, however, that lane formation is not a trivial effect of this model, but eventually arises only due to the smaller relative velocity and interaction rate that pedestrians with the same walking direction have. Once the pedestrians move in uniform lanes, they will have very rare and weak interactions. 4. Panic

One of the most disastrous forms of collective human behaviour is the kind of crowd stampede induced by panic, often leading to fatalities as people are crushed or trampled. Sometimes this behaviour is triggered in life-threatening situations such as fires in crowded buildings; at other times, stampedes can arise from the rush for seats or seemingly without causes. Although engineers are finding ways to alleviate the scale of such disasters, their frequency seems to be increasing due to greater mass events. Next we show that simulations based on a model of pedestrian behaviour can provide valuable insights into the mechanisms of and preconditions for panic and jamming by incoordination [61. The available observations on escape panic have encouraged us to model this kind of collective phenomenon in the spirit of self-driven many-particle systems. We assume, in addition to the earlier considered socio-psychological forces the relevance of physical

293 forces as well since the latter ones become very important in the case of a dense crowd with strong drive to get through a narrow exit. Each pedestrians of mass mi likes to move with a certain desired speed vo(t) into a certain direction eo(t) , and therefore tends to correspondingly adapt his or her actual velocity vi(t) with a certain characteristic time z. Simultaneously, he or she tries to keep a velocity-dependent distance to other pedestrians j and walls W. This can be modelled by “interaction forces” fi/ and fiw, respectively. In mathematical terms, the change of velocity in time is then given by the acceleration equation

The fi/ interaction forces include an exponentially decaying, repelling term expressing socio-psychological effects, and two additional “physical” terms corresponding to elastic repulsion andfviction forces between the bodies of people [6]. The fiwinteraction with the walls is treated analogously. To avoid model artefacts (gridlocks by exactly balanced forces in symmetrical configurations), a small amount of irregularity of almost arbitrary kind is needed. This irregularity was introduced by uniformly distributed pedestrian diameters ri in the interval [0.5m, 0.7m], approximating the distribution of shoulder widths of soccer fans. Based on the above model assumptions, it is possible to simulate several important phenomena of escape panic. The simulated outflow from a room is well-coordinated and regular, if the desired velocities are normal. However, for desired velocities above 1S-m/s, i.e., for people in a rush, we find an irregular succession of arch-like blockings of the exit and avalanche-like bunches of leaving pedestrians, when the arches break. “Faster-is-slower effect’’ due to impatience: Since clogging is connected with delays, trying to move faster can cause a smaller average speed of leaving, if the friction parameter is large. This effect is particularly tragic in the presence of fires, where the fleeing people reduce their own chances of survival. Improved outflows can be reached by columns placed asymmetrically in front of the exits preventing the build up of fatal pressures.

In fact, as it is clear from our visualization of the pressure distribution in a panicking crowd near an exit, the force experienced by the people is quickly changing and is highly fluctuating. The situation is quite similar to that observed and calculated for granular flows. The spots corresponding to high stress form bridge like structures connected in a hierarchical manner. Furthermore, this structure is completely reorganized in a short time after a “discharge”, i.e., after the region close to the exit is for a short time relaxed due to the successful exit of a group of people previously temporarily hindered by others from leaving.

294 More practical versions

After the basic model is given it can be applied to cases of increasing complexity and relevance to practical settings. Thus, we investigated the following additional cases: i) large crowds, ii) complicated “geometry” (parameters of actual rock concerts organized at a square in the downtown of a major city and in a stadium - both in Belgium), iii) effects of impatiencelanxiety and the combinations of these. Here I would like to discuss briefly, how the effects of the level of anxiety of people involved in escape panic can be taken into account in a setting with several exits of varying size (Fig. 4).

Figure 4. Snapshot of a simulation of people trying to escape from a room with 5 exits. The pressure distribution is colour coded (ranging from bright red corresponding to larger values to darker green denoting smaller pressures). There is an intensive traffrc of people changing exits due to impatience.

For this, several new aspects of the dynamics have to be considered. First of all, we have to allow that - in case someone becomes too anxious about not being able to proceed or, in other words, looses hisher patience - people could choose an alternative exit if they become unhappy with the one they chose previously. For this, we constantly have to keep a track of their level of anxiety. We assume, that if a particle cannot proceed quickly enough (the distance it makes in a given time interval is below a previously set anxiety level) it become “frustrated or anxious” and makes a decision about choosing a new exit. This decision is based on a number of parameters but, qualitatively, it is inversely proportional to the distance of the exits and to the number of people which would block this particle from leaving through an exit. Thus, most of the time, a particle within a crowd surrounding an exit, chooses this given exit, however, some other particles, closer to the edge of such a crowd are likely to chose an alternative exit. As a result, in a simulation of this sort there is a permanent redistribution of the crowds

295 around the exits: the larger ones tend to “evaporate” faster and the particles leaving them “condensate” at the exits with smaller number of particles. This process makes the whole system more efficient; in other words, giving to ‘anxiety factor” an increasing weight in the simulations leads to a faster overall escape rate! Although this result is not too surprising, it is somewhat paradoxical, since impatience or anxiety is usually not thought of as source of more optimal choices. Finally, we investigate a situation in which pedestrians are trying to leave a smoky room, but first have to find one of the invisible exits. Each pedestrian may either select an individual direction or follow the average direction of his neighbours in a certain radius or try a mixture of both. We assume that both options are weighted with some parameter (1-p)and p, respectively, where O

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Figure 4. The cumulative degree distributions for the inherent structure and Apollonian networks.

number of nodes (in fact with t = 7 and 4376 nodes and 13 122 edges). One could argue that it would be more appropriate to compare to an Apollonian network with the same spatial dimension. However, the properties of the Apollonian networks are very similar irrespective of dimension, so we chose to use the two-dimensional example simply because the properties of this case have been most comprehensively worked out. To study the size dependence of the network properties, as in Fig. 3 we have to make a further choice. For the inherent structure networks we follow clusters with an increasing number of atoms, and hence an increasing dimension of configuration space. Again it could be argued that we should be comparing to an Apolonian network of fixed number of generations, but increasing dimension, but the useful feature of examining a network of fixed dimension and increasing t instead is that the variable t behaves in a somewhat similar way to the number of atoms. For example, the number of nodes is an exponential function of t , whereas it only increases polynomially with the dimension of the system.2g As already mentioned, the number of minima is an exponential function of the number of atoms. From Fig. 3, one can see that both types of networks have small-world properties. Firstly, for both networks the average separation between nodes scales no more than logarithmically with system size, as for a random graph. The stronger sub-logarithmic behaviour for the inherent structure networks is because the average degree increases with network size (the random graph result is in fact 1, = log N,/ log(k)) whereas it is approximately constant for the Apollonian networks. The increase in ( k ) is simply because the ratio of the number of transition states to minima on a potential energy landscape is a linear function of the number of atoms.31 Secondly, the clustering coefficient, one measure of the local ordering within a network, has values that are significantly larger than for a random network. The size dependence of this property depends on how it is defined. If it is

380

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Figure 5. The dependence of the degree of a node on the potential energy of the corresponding minimum for LJ14. The data points are for each individual minimum and the solid line is a binned average.

as the probability that any pair of nodes with a common neighbour are themselves connected (Cl) then it decreases quite rapidly with size. The second definition (CZ) is as the average of the local clustering coefficient, where the latter is defined as the probability that the neighbours of a particular node are themselves connected. The second definition gives more weight to the low-degree nodes that, BS we shall see later, have a higher local clustering coefficient. That Cz tends to a constant value for the Apollonian network, rather than decaying weakly as for the inherent structure networks, reflects the stronger degree dependence of the local clustering coefficient. Both networks also have a power-law tail to their degree distribution, and so are scale-free networks. The exponent is slightly larger for the inherent structure networks (2.78 compared to 2.59). This heterogeneous degree distribution is easier to understand for the Apollonian network, and reflects the fractal nature of the packing^.^' At each stage in the generation of the network, the degrees of the nodes double, i.e. new nodes preferentially connect to those with higher degree, and so the highest degree nodes correspond to those that are 'oldest' and have larger associated disks. For the inherent structure networks, the high-degree nodes correspond to minima with low potential energy (Fig. 5). Our rationale for this correlation between degree and potential energy is that the lower-energy minima have larger basin areas,33 and hence longer basin boundaries with more transition states on them. The scale-free character of these networks must reflect the hierarchical packing of these basins with larger basins surrounded by smaller basins, which in turn are surrounded by smaller basins, and so on, in a manner somewhat similar to the Apollonian packing. Thus, the comparison of the inherent structure and Apollonian networks can provide some

381

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k/ ~ ~ for the inherent structure networks the degree dependence is somewhat reduced at small k. This similar behaviour partly reflects the common spatial character of the networks. The smaller low-degree nodes have a more localized character and so their neighbours are more likely to be connected, whereas the larger high-degree nodes can connect nodes that are spatially distant from each other and so are less likely to be connected. The behaviour of c ( k ) also partly reflects the correlation^^^ evident in Fig. 6(b). Both networks are disassortative, that is nodes are more likely to be connected to nodes with dissimilar degree. By contrast, for an uncorrelated network, knn(k) would be independent of degree. However, it is well known that disassortativity can arise for networks, as here, in which multiple edges and self-connections are not present.3g Indeed, for the inherent structure networks k,,(k) for a random network with the same degree distribution looks almost identical.” An additional source of disassortativity is present in the Apollonian networks, because, except for the initial disks, there are no edges whatsoever between nodes with the same degree; disks created in the same generation all go in separate interstices in the structure and so cannot be connected. Therefore, that k,,(k) for the two types of networks follow each other quite so closely is probably somewhat accidental. The behaviour seen for most of the network properties discussed so far is fairly common for scale-free networks. Therefore, a better test of the applicability of the Apollonian analogy to the energy landscape is to examine the spatial properties of

382 1

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Figure 7. The degree dependence of the basin areas for the LJ14 energy landscape and disk areas for the Apollonian packing. Both lines represent the average values for a given k.

the two systems directly. For the inherent structure networks, in agreement with the suggestion made earlier, there is a strong correlation between the degree of a node and the hyperarea of the basin of attraction that is similar to the degree dependence of the disk area seen for the Apollonian networks (Fig. 7). This result therefore implies that there is also a strong dependence of the basin area on the energy of a minimum with the low-energy minima having the largest basins. It also provides strong evidence that the scale-free topology of the inherent structure networks reflects the heterogeneous distribution of basin areas. The distribution of disk areas for the Apollonian packing reflects its fractal ~haracter.~’ It is in fact a power-law4’ with an exponent that depends upon the fractal dimension of the packing,41 as illustrated in Fig. 8 For high-dimensional packings this exponent tends to -2.” Preliminary results suggest that there is a similar power-law distribution for the hyperareas of the basins of attraction on an energy landscape, confirming the deep similarity between these two types of system, and suggesting that configuration space is covered by a fractal packing of the basins of attraction. 3. Conclusion

In this chapter we have looked at some of the fundamental organizing principles of complex multi-dimensional energy landscapes. By viewing the landscapes as a network of minima that are linked by transition states, we have found that the topology of this network is scale-free. Unlike most scalefree networks, the origin of this topology must be static. We believe that it is driven by a very heterogeneous size distribution for the basins of attraction associated with the minima, with the large basins having many connections. In this paper, we have explored whether space-filling packings of disks and hyperspheres, such as the Apollonian packings, and their associated contact networks can provide a good model of how the energy

383 t

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in a two-dimensional

landscape is organized. We have shown that these systems share a deep similarity both in t h e topological properties of the networks a n d the spatial properties of the packings. In fact, our results suggest that t h e energy landscape can be viewed as a fractal packing of basins of attraction. Although this conclusion can provide a n explanation for the scale-free topology of t h e inherent structure network, it itself demands an explanation. Why are t h e basins of attraction organized in this fractal manner? We will explore this in future work.

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384 18. F. Liljeros, C. R. Edling, L. A. N. Amaral, H. E. Stanley and Y. Aberg, Nature 411, 907 (2001). 19. D. J. Watts and S. H. Strogatz, Nature 393,440 (1998). 20. A. L. BarabLi and R. Albert, Science 286,509 (1999). 21. J. P. K. Doye, Phys. Rev. Lett. 88,238701 (2002). 22. J. P. K. Doye and C. P. Massen, J. Chem. Phys. 122, in press (2005); condmat/0411144. 23. C. P. Massen and J. P. K. Doye, cond-mat/0412469. 24. A. F. Rozenfeld, R. Cohen, D. ben Avraham and S. Havlin, Phys. Rev. Lett. 89, 218701 (2002). 25. C. P. Warren, L. M. Sander and I. M. Sokolov, Phys. Rev. E 66,056105 (2002). 26. D. ben Avraham, A. F. Rozenfeld, R. Cohen and S. Havlin, Physica A 330, 107 (2003). 27. C. Herrmann, M. Barthelemy and P. Provero, Phys. Rev. E 68,026128 (2003). 28. J. S. Andrade, H. J. Herrmann, R. F. S. Andrade and L. R. d a Silva, condmat/0406295. 29. J. P. K. Doye and C. P. Massen, Phys. Rev. E 71,in press (2005); cond-mat/0407779. 30. T. Aste, T. Di Matteo and S. T. Hyde, Physica A 346,20 (2005). 31. J. P. K. Doye and D. J. Wales, J. Chem. Phys. 116,3777 (2002). 32. B. B. Mandelbrot, The Fractal Geometry of Nature, W. H. Freeman, New York (1983). 33. J. P. K. Doye, D. J. Wales and M. A. Miller, J. Chem. Phys. 109,8143 (1998). 34. S. N. Dorogovtsev, A. V. Goltsev and J. F. F. Mendes, Phys. Rev. E 65, 066122 (2002). 35. E. Ravasz and A. L. BarabLi, Phys. Rev. E 67,026112 (2003). 36. F. Comellas, G. Fertin and A. Raspaud, Phys. Rev. E 69,037104 (2004). 37. A. L. BarabLi, Nature Reviews Genetics 5, 101 (2004). 38. S. N. Soffer and A. Vbquez, cond-mat/0409686. 39. J. Park and M. E. J. Newman, Phys. Rev. E 68,026112 (2003). 40. Z. A. Melzak, Math. Comput. 16,838 (1966). 41. S. S. Manna and H. J. Herrmann, J. Phys. A 24,L481 (1991).

EPIDEMIC MODELING AND COMPLEX REALITIES

MARC BARTHELEMY', ALAIN BAR RAT^, VITTORIA COLIZZA~,ALESSANDRO VESPIGNANI~ School of Informatics and Biocomplexity Institute, Indiana University Bloomington, IN, USA Laboratoire de Physique The'orique (UMR du CNRS 8687) Batiment 210, UniversitC de Paris-Sud 91405 Orsay, France Informatics tools have recently made it possible to achieve an unprecedented static and dynamical picture of our society, providing increasing evidence for the presence of complex features and emerging properties at various levels of description. We present here a brief overview of how epidemic modelling is affected by the complexity characterizing the structure and behavior of real world societies and the new opportunities offered by a coherent inclusion of complex features in the understanding of disease spreading.

1. Introduction

The mathematical modelling of epidemics is a very active field of research that crosses different disciplines. Epidemiologists, computer scientists and social scientists share a common interest in studying spreading phenomena and rely on very similar models for the description of the diffusion of viruses, knowledge and inn* vation. In particular, understanding and predicting an epidemic outbreak requires a detailed knowledge of the contact networks defining the interactions of the p o p ulation at various scale ranging from the individuals interactions to the traveling patterns. Thanks to the development of new computational capabilities, a variety of largescale data on social networks have become available and amenable to scientific analysis. This has lead to the accumulation of ample evidence for the presence of complex and heterogeneous properties of many evolving networks some of them being of great interest for the spreading of epidemics A central result is that some networks are characterized by complex topologies and very heterogeneous structures A striking example of this situation is provided by scale-free networks characterized by large fluctuations in the number of connections (degree) k of each vertex. This feature usually finds its signature in a heavy-tailed degree distribution with power-law behavior of the form P ( k ) k-7, with 2

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