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SPECIAL ISSUE & DIRECTORY 36/6 Nonextensive statistical mechanics: 2005 I N 9 n 9 o s
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European Physical Society PUB no Lattice and Boltzmann and superstatistics Atmospheric turbulence eur mechanics andnetworks Nonextensive statistical ᭡ ᭡ ᭡ PAGE 218 PAGE 192 P ne AGE 189 oph xtensive diffusion y sics news NO VEMBER / See the article by B.M. Boghosian and J.P. Boon p.192 andJ.P. Boon Boghosian by B.M. thearticle See C November/December 2005 Volume 6 36Number europhysics landscape of landscape ofa appears whosespatial structure bydetected fluctuations local measuring precursor be phenomenacan beforeBut any pattern fingering becomes visible, between two fluidswithdifferentdestabilization oftheinterface mobilities. DECEMBER o FEATURES NEWS v 9 Noise induced phenomenaand 197 194 Lattice and Boltzmann 192 189 andentropy Extensivity production 186 issueoverview: Special 185 0 Complexity and ofseismicity 206 202 232 Pictures from EPS13 Pictures 232 Conferences in2006 231 er pic tur none interacting system Relaxation andaginginalong-range nonextensive diffusion superstatistics A none nonextensive statistics N oai .Wio Horacio S. F andJean Boon Boghosian Pierre M. Bruce C. and YuzuruSato MurrayGell-Mann Constantino Tsallis, Jean andConstantino Boon Tsallis Pierre ..VarotsosP.A. Ugurand Tirnakli Sumiyoshi Abe, A ne glassy behaviour , acsoA TamaritCelia Anteneodoand A. rancisco ndr tmospheric turbulence and q one e: 2005 B -Gaussian -Gaussian by P. Grosfils). andwells”(simulation “hills IW N ACTIVITIES AND VIEWS w tr F ec ingering is a generic phenomenonthat isageneric resultsingering from the ea R xtensivity x k, x t ends t E.G.D ensiv ensiv apisar new perspectives , . e sta e ther da and Alessandro Pluchino oe n .Rizzo Cohen andS. tistic mo dynamics and dynamics al mechanics: news 208 228 Entropy andstatistical complexity 224 221 Nonextensive statistical mechanics 218 214 ion Nuclear astrophysical plasmas: 211 3 Directory 234 nonextensivity infinancialmarketsnonextensivity sy A.R. S L Rosso andO.A Plastino A. A.K. Stefan Thurner Robledo A. Marcello Lissia Quarati andPiero L in br information implic N and complex scale-free networks C rates andfusion distribution functions ong-r isa B q ritical attractorsritical andq-statistics one st entropy andself-gravitating ems P aaoa n ..Rendell andR.W. Rajagopal ain activity or x last ations to quantum t ange memor land ensive statistical mechanics: ino y and CONTENTS 183
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184 europhysics news NOVEMBER/DECEMBER 2005 a hroyais It does soby connection, advancing aspecific cal thermodynamics. between themechanicaluseful bridge microscopic laws andclassi- 29-8 i eJnioR,Brazil 22290-180 Rio deJaneiro-RJ, USA 87501, Belgium B-1050Bruxelles, Bruxelles, Jean Boon Pierre new perspectives newtrends, mechanics: N Special issueoverview Features andNews wide-ranging next time! the usual mixof referenceeral work on “nonextensive mechanics”. statistical Back to Issue thisSpecial thegen- make It of will volume. inasingle articles tograteful thePublisher for accepting to accommodate the all We are astandardissue. EPN passeslargelymaterial thesize of guestsEditors The were soefficient thatthecollected rewarding. It isalways askitsreaders to EPNwill make aneffort. time, This address arduous developments to cover recent progress inPhysics. to It necessary issometimes tohad face aheavier taskthanusual. ourthanksgo to who theEPNdesigner tions thanusual inEPN: equa- proportion of is difficultandcould ahigher notgo without subject The thecontributions. one of in each aco-author being of Editorial and theintroductory authorsguished but alsoinwriting distin- animpressivedid agreat jobnotonly inselecting of set Europhysics News on “Nonextensive They Mechanics”. Statistical I Foreword eur B 3 2 1 etoBaier ePsussFscs Xavier 150, Sigaud Centro Brasileiro dePesquisas Fisicas, Santa F –CP231Université Campus Plaine Libre de CNLPCS, Tsallis asguestsEditors for thepresent Issue Special of t isourpleasure to welcome Jean Pierre andConstantino Boon oph mnso otmoaypyis It establishes aremarkably physics. contemporary uments of themon- mechanicsoltzmann-Gibbs isone (BG)statistical of one y e I sics news nst tt,19 yePr od at e New Mexico SantaFe, 1399Hyde Park Road, itute, x 1 t and Constantino Tsallis ensive statistical NO VEMBER / DECEMBER 2005 ,3 2, The Editors satisfies concavity only in the interval 0 concavitysatisfies only intheinterval Renyi entropy Indeed, inadequate for thermodynamical purposes. an int fy thesef fi itisnoteasyto because isquite This important state). stationary of microscopic phase space inorder to ultimately approach some kind real situationswhere to thesystem explore isstriving its of ety Boltzmann’s famous 27-year-old thisstate andof deep foundation The of importance. hence itsfundamental has remarkable andubiquitous properties, fthesystem), of par iswt h irsoi ttso h ytm oee,the However, thesystem. themicroscopicsius with states of where frequent inphysical andeconomics, sciences aswell asinbiology isparticularly This accomodate hypothesis. thissimplifying with andeven systems social do not artificial, phenomena innatural, However many important hence ergodicity. hence mixing, chaos, state denominated ary station- addresses special centrally thevery theory That that. about No surprise or evenpredictions slightly canbe inadequate. strongly its Outside thisdomain, as any other human construct. intellectual applicability, It hasadelimited domain of isnotuniversal. theory S ≠ fi index q with none i” n17 i nnnierdnmc,more on specifically sis”) in1871lieon nonlinear dynamics, ty of by its replaced strong chaos istypically at themicroscopic dynamical level, rw o xoetal ihtm,but like rather apower-law. grows time, notexponentially with answ cations are nowadays available thatpoint towards the affirmative ihteetoya nrdcdi lsia hroyais,and theentropywith asintroduced inclassical thermodynamics), the results), L mechanics? BGstatistical of similarto those andmethods concepts with tions -situa- anomalous intheBGsense -though important these of some the t on isbased thegeneraliza- first proposed in1988, approach, This these w io BG niteness of the entropy production per unittime per production theentropy of niteness esc nd entropic thatsimultaneouslysatis- functionals andgenerically 1, all the other three properties mentioned above. The extensivi- The theother three all properties mentionedabove. 1, n o usinte rssntrly namely: naturally, A question then arises r ticular case ticular = - mo he-stab x e t S f e weak weak r non-equlibrium e k r . e ha q the d nsive statistical mechanics nsive statistical ∑ est A the our p ynamical eevsaseilmnin ned fwe compose if Indeed, mention. deserves aspecial i W = S ing f BG ve 1 Many theoretical, experimental and observational indi- andobservational experimental Many theoretical, ilit version, when the sensitivity to conditions theinitial thesensitivity when version, q = p ∈ extensivity extensivity i o c ,andrelevant for reproducibility theexperimental of y, r onca ln e operties. Renyi entropy, for instance, is known to isknown be for instance, Renyi entropy, operties. r r o hrceiigmlircas But itseems formultifractals. orm characterizing q nt y which appearsto play satisfactorily thatrole is experimental robustness experimental = 1(se p and l r y andd i performing a performing lo generalizations oftheusual exponential and reduces for ensemble a isageneralizationdistribution oftheusual escort entropy.The Boltzmann-Gibbs q- ᭣ the in N ln o vity vity nisdsrt eso,o theentropy alaClau- of in itsdiscrete version, py by theexpression garithmic functions whichare retrieved functions garithmic by q e B (e Stosszahlansatz S xp ttoaysae r h omnrl.Then, states are common the rule. stationary o q thermal equilibrium. thermal one 1 xp (r x: -entropy generalizes thestandard e theBox). = (relevant for matchinghaving anatural onen e ynamical q The two basic functions thatThe appear two basicfunctions levant for thethermodynamicalstability levant S xtensive Statistical are Mechanics the x BG ) =e ..the i.e. , tial andthe v q er = 1. | and itss xp 1- aging func l motn rpris Among properties. y important q S q (ln q | (“molecular chaos hypothe- BG <1epnin Similarly << 1expansion. shar q < x ubse theory iscontained asthe theory ) = q- q (technically as known es with es with This macroscopicThis state ≤ lo Is to itpossible address tion t x q . ,advoae,for q and violates, 1, gar They are simple ue (r elevant for avari- elevant nt developments. ithm with o whichit S BG a v FEATURES ariety of ariety strong 185 BG
features FEATURES subsystems that are (explicitly or tacitly) probabilistically indepen- dent, then SBG is extensive whereas Sq is, for q ≠ 1, nonextensive. This fact led to its current denomination as “nonextensive entropy”. Extensivity and entropy However, if what we compose are subsystems that generate a non- trivial (strictly or asymptotically) scale-invariant system (in other words, with important global correlations), then it is generically Sq production for a particular value of q ≠ 1, and not SBG, which is extensive. Ask- ing whether the entropy of a system is or is not extensive without Constantino Tsallis 1, 2, Murray Gell-Mann 1 and Yuzuru Sato 1 * 1 indicating the composition law of its elements, is like asking whether Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico some body is or is not in movement without indicating the referen- 87501, USA tial with regard to which we are observing the velocity. 2 Centro Brasileiro de Pesquisas Fisicas, Xavier Sigaud 150, The overall picture which emerges is that Clausius thermody- 22290-180 Rio de Janeiro-RJ, Brazil namical entropy is a concept which can accomodate with more * [email protected], [email protected], [email protected] than one connection with the set of probabilities of the microscop- ic states. SBG is of course one such possibility, Sq is another one, and it seems plausible that there might be others. The specific one to be n 1865 Clausius introduced the concept of entropy, S, in the used appears to be univocally determined by the microscopic Icontext of classical thermodynamics. This was done, as is well dynamics of the system. This point is quite important in practice. If known, without any reference to the microscopic world. The first the microscopic dynamics of the system is known, we can in prin- connection between these two levels of understanding was pro- ciple determine the corresponding value of q from first principles. posed and initially explored one decade later by Boltzmann and As it happens, this precise dynamics is most frequently unknown then by Gibbs. One of the properties that appear naturally within for many natural systems. In this case, a way out that is currently the Clausius conception of entropy is the extensivity of S, i.e., its used is to check the functional forms of various properties associ- proportionality to the amount of matter involved, which we inter- ated with the system and then determine the appropriate values of pret, in our present microscopic understanding, as being q by fitting. This has been occasionally a point of – understandable proportional to the number N of elements of the system. The W but nevertheless mistaken – criticism against nonextensive theory, Boltzmann-Gibbs entropy SBG ≡ -k∑i=1 pi ln pi (discrete version, but it is in fact common practice in the analysis of many physical where W is the total number of microscopic states, with proba- systems. Consider for instance the determination of the eccentrici- bilities {pi}, and where k is a positive constant, usually taken to be ties of the orbits of the planets. If we knew all the initial conditions kB). SBG satisfies the Clausius prescription under certain condi- of all the masses of the planetary system and had access to a colos- tions. For example, if the N elements (or subsystems) of the sal computer, we could in principle, by using Newtonian mechanics, system are probabilistically independent, i.e., pi1,i2,…,iN = determine a priori the eccentricities of the orbits. Since we lack that pi1pi2…piN , we immediately verify that SBG(N) ∝ NSBG(1). If the (gigantic) knowledge and tool, astronomers determine those eccen- correlations within the system are close to this ideal situation (e.g., tricities through fitting. More explicitly, astronomers adopt the local interactions), extensivity is still verified, in the sense that mathematical form of a Keplerian ellipse as a first approximation, SBG(N) ∝ N in the limit N → ∞. There are however more com- and then determine the radius and eccentricity of the orbit through plex situations (that we illustrate later on) for which SBG is not their observations. Analogously, there are many complex systems extensive. The question then arises: Is it possible, in such complex for which one may reasonably argue that they belong to the class cases, to have an extensive expression for the entropy in terms of the that is addressed by nonextensive statistical concepts, but whose microscopic probabilities? The general answer to this question still microscopic (sometimes even mesoscopic) dynamics is inaccessible. eludes us. However, for an important class of systems (e.g., asymp- For such systems, it appears as a sensible attitude to adopt the math- totically scale-invariant), one such entropic connection is known, ematical forms that emerge in the theory, e.g. q-exponentials, and namely then obtain through fitting the corresponding value of q and of similar characteristic quantities. Coming back to names that are commonly used in the literature, (1) we have seen above that the expression “nonextensive entropy” can be misleading. Not really so the expression “nonextensive statistical mechanics”.Indeed, the many-body mechanical systems that are (N = 0) 1 1 primarily addressed within this theory include long-range interac- (N = 1) π10 π11 1/2 1/2 tions, i.e., interactions that are not integrable at infinity. Such (N = 2) π20 π21 π22 1/3 1/6 1/3 systems clearly have a total energy which increases quicker than N, (N = 3) π30 π31 π32 π33 3/8 5/485/48 0 where N is the number of its microscopic elements. This is to say a (N = 4) π40 π41 π42 π43 π44 2/5 3/40 1/20 0 0 total energy which indeed is nonextensive. Acknowledgements ᭡ Table: Left: Most general set of joint probabilities for N equal and distinguishable binary subsystems for which only the The present special issue of Europhysics News is dedicated to a number of states 1 and of states 2 matters, not their ordering. hopefully pedagogical presentation, to the physics community, of Right: Triangle with ⑀ = 0.5 and d = 2 constructed by the main ideas and results supporting the intensively explored and modifying the Leibnitz-triangle. In general qsen = 1-(1/d). quickly evolving nonextensive statistical mechanics. The subjects For N = 5, 6, … a full triangle emerges (on the right side) all that we have selected, have been chosen in order to provide a gen- the elements of which vanish. For any finite N, the Leibnitz eral picture of its present status in what concerns both its rule is not exactly satisfied, but it becomes asymptotically foundations and applications. It is our pleasure to gratefully satisfied for N → ∞. See details in [3]. acknowledge all invited authors for their enthusiastic participation.
186 europhysics news NOVEMBER/DECEMBER 2005 ( so L whether or nottheentropywhether is. nonextensive istypically theenergy In such atheory views). as g expressionThis wasproposed basisfor in1988[1]asapossible a eur t 1 ∑ relation asthe nFg1a.We that see in Fig.1(a). triangle: t The side asindicated inTable I [3]. on its triangle bility “right” introducing azero then gradually proba- Leibnitz triangle, with have They constructed been by starting probabilities inthetable. such asystem represented canbe asinTable Iwith probabilities of joint The which takes variable values 1 and2. bilistic binary elements those corresponds to a proba- that eachof simplicity and dist p otal meas erizes a typical atypical erizes ,n π eneralization of Boltzmann- Gibbs statistics currently Gibbs Boltzmann- statistics referred to of eneralization r et us impose thescale-invariant usimpose et constraint n N me e o 3 nonextensive statistical mechanics statistical nonextensive L ( =0 0 bab oph n tu lutae for both usillustrate, et , ( π — N-n ,1 …, 1, = 0, 31 xamples [3]. ilit N! y inguishab )! , sics news ( n π ies int π ! ur 32 1 π 0 , e associated is then redistributed on a strip on the e associated isthen redistributed on astrip N,n , π π Leibnitz rule. Leibnitz 33 o e = 11 B N ) = (1/4, 1/12, 1/12, 1/4), etc. By inserting these By inserting etc. 1/4), 1/12, 1/12, ) =(1/4, 12 /) ( 1/2), ) =(1/2, le s 1 ( oltzmannian system. xpression (1)we cancalculate -1; NO Consider asystem composed of π ubsy N,n VEMBER N S q ∈ ,3 ).Hereafter we refer to this . …) 3, 2, = st is q m o lmns.Let usassume for ems (or elements). [0, xesv nyfrq only for extensive Indeed, it is satisfied by itissatisfied the Indeed, = 1and / 1]; DECEMBER π N 2 0 = 1, se[]frasto minire- [2]for(see of aset q , π ≠ L 21 1, 2, et usnowet consider the , 2005 π the extensivity of extensivity the 3, 22 π …; 13 /,1/3), 1/6, ) =(1/3, N,n = 1. This charac- This = 1. n S + q = 0, ( N π N N,n + ) assho 1, id Leibnitz 1 …, e nt = S ical N π q w in N- ). n linear size is ( ue then cube, b p dime + [ “nonextensive isvalid entropy”, culate N D dent. are explicitly or tacitly assumed to probabilistically be indepen- π ly thatof meas no titybtol smttcly .. for i.e., but only asymptotically, strictly It [3]thatthissystem shown canbe theLeibnitz satisfies not rule aitna n,then Hamiltonian one, no dynamics. In particular its value of In particular dynamics. classical deterministic andfollowing nonlinear one, specific “left side” whose width is width whose side” “left sen y r N ( o ( W w N w o 0 i S bability sets, bability N = 1, q > -iesoa hs pc,anddenote by )-dimensional phase space, stands for hl o drs opeeydfeetpolm name- nowe shall address acompletely different problem, nsion. S ur ( > →∞ B q n d S e. )/ π is q ,…, 2, ( N k W entropy production per unittime. production per entropy N e e ] +(1- 1 .I thephase space isa If ). tnieol o q only for xtensive not egttersl hw nFg () We that see 1(b). ) we theresult get in Fig. shown e the > ⑀ W ,thediscrepancies becoming larger as > …, . The total The number . W ed sensitivity ( S N ( n mak N q q ) = is e q )) isg sen )[ xt f S W o e apar D q e ( r ). The property property The ). iven by nsive only value for of aspecial 0 ( A N / )/ easo ⑀ ⑀ ) ~2 k d D ][ = 1/2. iino itinto cells whose small of tition . The distribution issuch distribution The that . ( N ns thatw S ) W only fthe system isaclassical If . q W d ( s B ( N ( N )/ N N D nfc,for alarge class of In fact, fthesubsystems if ( ) k >1o cells (designated ) >> 1of ( N sfie.We consider the is fixed. ∝ ], which led to which led theterm ], N S N →∞ ill soon become soon ill clear )-dimensional hyper- W q ( →∞ The systemThe now isa A +B 0 / S ᭣ S aids lines are visual coloured dashed are inblack.The data the actual All for details). chaos (see[4] and weak (d) (b) c (a) orglobally locally listically dep dependence of for details).Time- sy ⑀ ⑀ ,where ), q q st F D for strong (c) f W fwe now cal- If . or pr ( ig N ems (see[3] )/ endenc . or ) 0 . k w 1: its Lebesgue r elated = [ ith FEATURES obabi- Size- d S A s D N e of q q ≡ ( , ( and → A N space fr 187 )/ o ) ∞ m k ∝ B ] . features FEATURES
systems (q ≠ 1), as might well be the case for long-range- interacting Hamiltonians [7]. Let us illustrate, for a one-dimensional map, an important property associated with Eq. (2). The sensitivity ξ to the initial conditions is defined through ξ≡lim∆x (0)→ 0 ∆x(t)/∆x(0), where ∆x(0) is the discrepancy of two initial conditions. For a wide class of one-dimensional systems we have the upper bound ξ = λqsent x 1/(1-q) x x eqsen , where eq ≡ [1+(1-q)x] (e1 = e ), and λqsen a q-gener- alised Lyapunov coefficient . The property we referred to is that
the entropy production per unit time Kqsen satisfies Kqsen = λqsen [8]. This generalises, for q ≠1, a relation totally analogous to the Pesin identity, which plays an important role in strongly chaot-
ic systems (i.e., qsen = 1). It is clear that Kq is a concept closely related to the so called Kolmogorov-Sinai entropy. They fre- quently, but not always, coincide. As we have shown, Sq can, for either q = 1 or q ≠1, be extensive under suitable conditions and lead to a finite entropy production per unit time. Other impor- tant properties are satisfied, such as concavity and Lesche-stability ᭡ Fig. 2: Schematic time-dependence of Sq for various degrees (or experimental robustness) [9]. Moreover, the celebrated of fine-graining ⑀.We are disregarding in this scenario the uniqueness theorems of Shannon and of Khinchine have also influence of possible averaging over initial conditions that been q-generalised [10,11], and the same has been done with might be necessary or convenient. central procedures such as the Darwin-Fowler steepest descent method [12]. In short, a consistent mathematical structure is in We choose one of those cells and in it we randomly pick place suggesting that the Boltzmann-Gibbs theory can be satis- M >> 1 initial conditions. As time t (assumed discrete, i.e., t = 0, factorily extended to deal with a variety of complex statistical 1, 2, …) evolves, these M points spread around into {Mi(t)} with mechanical systems. Since the first physical application [13] (to W(N) ∑i=1 Mi(t) = M. We can then define a set of probabilities {pi(t)} stellar polytropes), nonextensive statistical mechanics and its by pi ≡ Mi(t)/M. With these probabilities we can calculate Sq(N, related concepts have made possible applications to very many t; ⑀, M) for that particular initial cell. Then, depending on our natural and artificial systems, from turbulence to high energy focus, we may or may not average over all or part of the possi- and condensed matter physics, from astrophysics to geophysics, ble initial cells (both situations have been analyzed in the from economics to biology and computational sciences (e.g., literature). We consider now two different cases, namely strong signal and image processing). Recently, connections with scale- chaos (i.e., the maximal Lyapunov exponent is positive), and invariant networks, quantum information, and a possible weak chaos (i.e., the maximal Lyapunov exponent vanishes). q-generalisation of the central limit theorem [14,15] have been Both are illustrated in Figs. 1(c) and 1(d) for a very simple sys- advanced as well. In some of these problems, when the precise 2 tem, namely the logistic map xt +1 = 1 - ax t with 0 ≤ a ≤ 2, and dynamics is known, the indices q are in principle computable -1 ≤ xt ≤ 1. For infinitely many values of the control parameter from first principles. In others, when neither the microscopic a (e.g., a = 2), we have strong chaos (this is the case in Fig. 1(c) nor the mesoscopic dynamics is accessible, only a phenomeno- with a = 2). But for other (infinitely many) values of a, we have logical approach is possible, and then q is determined through weak chaos (this is the case in Fig. 1(d) with a = 1.401155…). As fitting. An interesting determination of this kind was recently we can see, it is quite remarkable how strongly similar all four carried out in the solar wind as observed by Voyager 1 in the dis- figures 1 are. This suggests the following conjecture: tant heliosphere [16]. Indeed, the q-triplet that had been conjectured was fully determined for the first time in a physical (2) system. The overall scenario which emerges is indicated in Fig. 3. as schematised in Fig. 2. (A is a positive constant.) We emphasize that this conjecture is built to some extent upon observations made on the time-dependence of low-dimensional systems, such as the logistic map and similar dissipative maps ( [5] and refer- ences therein) as well as two-dimensional conservative maps [6]. Whether similar behavior indeed holds for the time-dependence of high-dimensional dissipative or Hamiltonian systems with N >> 1 obviously remains to be checked. Conjecture (2) has two consequences. The first of them is that, since by definition of qsen it is lim ⑀ →0 Sqsen(t, ⑀) ~ Kqsent, we have that lim ⑀ → 0 lim M→ ∞ Sqsen(N, t; ⑀, M) ~ AKqsen Nt. This means, interest- ingly enough, that N and t play similar roles. The second consequence concerns the case when we have a fine but finite graining ⑀, for example that imposed by quantum considera- tions. Then we typically expect the expressions lim N →∞ lim t →∞
Sqsen (N, t,⑀, M) Sqsen (N, t,⑀, M) lim M→∞——— and lim t→∞ lim N→∞ lim M→∞ ——— ᭡ Fig. 3: Scenario within which nonextensive statistical Nt Nt mechanics is located (see [3] for more details) [18]. to coincide for typical q = 1 systems, and to differ for more complex
188 europhysics news NOVEMBER/DECEMBER 2005 8 .Bloi n .Robledo, and Baldovin A. [8] F. [18] T [11] S. dos Santos, [10] R.J.V. 5 .MyrladA Robledo, Mayoral and A. [5] E. 1]EGD Cohen, [17] E.G.D. Rajagopal, Abe and A.K. [12] S. important open problem mention important us let an of As anillustration does notmakestatistics thetaskeasy. questions are notyet understood fully even for Boltzmann-Gibbs thatsome basic fact The applications. specific [17]) andatthatof thedynamical origin thefoundations (e.g., atthelevelBoth of 4 .Ltr,M aagr .RpsraadC Tsallis, Rapisarda andC. A. Baranger, M. Latora, [4] V. AFRL. 1]LF ulg n ..Vinas, and Burlaga A.F. [16] L.F. cond-mat/0509229 Gell-Mann, Tsallis andM. C. Moyano, [15] L.G. [1] C. References Gell-Man M. C.Tsallis Sato Y. Acknowledgement applications. specific thathave already interesting shown and theKaniadakis statistics, theBeck-Cohen superstatistics Such isthecaseof literature. approacheseral thanthepresent one are already available inthe Letmention usfinally thatrelated or even more gen- welcome. problems such asthisone are very obviously Solutions of sion. thespace dimen- theforces andof of therange of as afunction isthevalue of andwhat how isapplicable, thepresent theory and if speaking, strictly unknown, itisstill able intheliterature, many Although favorable indicationsare avail- Hamiltonians. eur [14] C. Abe, [9] S. [7] [2] M. Insight VentureInsight Management. Fe Institute. 3 .Tals .Gl-anadY Sato, Gell-Mann and Y. M. Tsallis, [3] C. 1]AR lsioadA Plastino, and A. Plastino [13] A.R. [6] G. There is a plethora of open problems, as can be easily ascanbe guessed. open problems, There of isaplethora .Ltr,A aiad n .Tsallis, Rapisarda andC. A. Latora, V. oph The The 97 (2000). ap describe certain discontinuities certain describe thatoccur to however, we want, If as closepointwe to want. thecritical 15377 (2005). (2005). point Interdisciplinary Applications, Interdisciplinary and Heisenberg at ferromagnets htt For aregularly updated bibliography see 2004). he caseo T aai .TalsadF Baldovin, Tsallis andF. C. Casati, Ge A p T sal p://tsallis.cat.cbpf.br/biblio.htm. has benefited from aPostdoctoral Fellowship attheSanta b y sal r BG o l e, sics news lis, has benefited from by sponsoring SIInternational and l-M (e.g. some fractal dimensions connectedthe some fractal with (e.g. hs e.E Rev. Phys. ach. lis, P hoyepan,a swl nw,avreyo properties of avariety asiswellknown, explains, theory h J. ann andC. M y a upre yteCOUQ Foundation andby bywas supported theC.O.U.Q. f S s. tat. standar ilan J. et A Lett. Pramana Phys. NO .Mt.Phys. Math. J. 66 M d cr 271 VEMBER ath. 046134(2002). , T 52 als eds., sallis, it , , ical p 64 74 (2000). 479 (1988). 73 hs e.E Rev. Phys. hs e.E Rev. Phys. .Py.A Phys. J. 635(2005). , , Physica A 145 (2005). / hs et A Lett. Phys. he Ofr nvriyPes New York,(Oxford University Press, DECEMBER 38 nomena deserves acomment. T N 4104(1997). , c one E ,we adifferent need theoretical ), rc al cd c USA Sc. Acad. Natl. Proc. rpy.Lett. urophys. P 33 y.Rv E Rev. hys. 356 x 72 8733(2000). , 69 t e nsi 375(2005). , 026209(2005). , 045202(R)(2004). , 2005 174 long-range-interacting precisely atthecritical precisely ve Entropy - 384(1993). , 64 hs et A Lett. Phys. 72 , 056134 (2001). , 355 (2005). d s = 3Ising 102 273 , , q t v conceptThis isquite successfullyhas been and general applied to a [2]. by aconceptdescribed recently introduced as ‘superstatistics’ mo Nonequilibrium macroscopic phenomena with inho- scales. r odn ieEdRa,Lno 14S United Kingdom London E14NS, Mile End Road, London, I 3 2 Beck C. and superstatistics Atmospheric turbulence ok101 USA York 10021, ‘ A traveller frequently who travels between thethree a sees cities agiven mechanics with fixedtistical temperature valid. iscertainly sta- equilibrium locally though large scale, fluctuations on arather There are temperature spatio-temporal the sameattime. andFirenze New is York, It is unlikely thatthetemperature inLondon, theweather: of for example, Think, largeon scale. arather tems exhibit actually or spatial temporal temperature fluctuations othe intensive parameteroccur notonly for theweather but for many 1 e fl In turbulent temperature but any canbe relevant system parameter. In theinverse not be parameter generalneed thefluctuating region. which then differs completely from thatinanother spatial value, wheretain regions some system constant parameter hasarather mixt e y o ne reyo ytm,sc shdoyai ublne atmospheric such ashydrodynamic turbulence, systems, of ariety h okfle nvriy 20Yr vne e ok New New 1230 YorkYork, The RockefellerAvenue, University, NVSpA ....Frne Italy Firenze, U.A.A.V. ENAV S.p.A, University of Mary, Queen Sciences, Mathematical of School ows, for example, a very relevant system avery parameteristhelocal for example, ows, mp (r σ u ᭡ one week (green line) andthecorresponding parameter a constant system parameter–but many sys- nonequilibrium qiiru ttsia ehnc,theinverse temperature mechanics, statistical n equilibrium rg ( g dln) aswell asthecorresponding standard deviation ed line), ( 92[] exhibits kindsof fluctuationson spatio-temporal all 1962[1], f t F r dr t e ) ( bu otdln) bothfor a1hourwindow. ) (bluedotted line), ur e y dissipat ig nit aue.Sc yeo arsoi nooeiiso an macroscopic inhomogenities of of Such type ratures. δ e . 1 i ’ = 60min)r 1: ...Cohen E.G.D. , v e fanintensive parameter canoften effectively be ies of o nnnqiiru ytm swl.There are often cer- en nonequilibriumsystems aswell. f Time series ofatemporal series Time windvelocitydifference canonical ensembles corresponding to different local io n r ec at e or ⑀ ded by 5minfor anemometer Aevery , 2 w n .Rizzo and S. hic h, a c c o r ding to Kolmogorov’s theo- 3 FEATURES β ( t 189 ) β is features FEATURES
turbulence, pattern formation in Rayleigh-Benard flows, cosmic ray statistics, solar flares, networks, and models of share price evolution [3]. For a particular probability distribution of large- scale fluctuations of the relevant system parameter, namely the Gamma-distribution, the corresponding superstatistics reduces to Tsallis statistics [4], thus reproducing the generalized canonical distributions of nonextensive statistical mechanics by a plausible physical mechanism based on fluctuations. In this article we want to illustrate the general concepts of superstatistics by a recent example: atmospheric turbulence. Rizzo and Rapisarda [5, 6] analysed the statistical properties of turbu- lent wind velocity fluctuations at Florence Airport. The data were recorded by two head anemometers A and B on two poles 10 m high a distance 900 m apart at a sampling frequency of 5 min- utes. Components of spatial wind velocity differences at the two anemometers A and B as well as of temporal wind velocity differ- ences at A were investigated. ᭡ Fig. 2: Rescaled probability density of the fluctuating Analysing these data, two well separated time scales can be parameter β, as obtained for the Florence airport data. Also distinguished. On the one hand, the temporal velocity difference shown is a Gamma distribution (dashed blue line) and a u(t) = v(t + δ) - v(t) (as well as the spatial one) fluctuates on the lognormal distribution (solid red line) sharing the same rather short time scale τ (see Fig. 1). On the other hand, we may mean and variance as the data.The data are reasonably well fitted by the lognormal distribution. also look at a measure of the average activity of the wind bursts in a given longer time interval, say 1 hour, where the signal behaves approximately in a Gaussian way. The variance of the signal u(t) during that time interval is given by σ2 =
190 europhysics news NOVEMBER/DECEMBER 2005 and thelo versus amultiplicative definition of o We thatourresult see turbulence for atmospheric issimilarto lab- diff with with velocity corresponds Brownian the to longitudinal particle) the of ity l ferences theparameter fluctuations ones, andnotthermal since we are velocity analyzing turbulent Secondly, itself. velocity shows aGammaor (blue) thedashed line For comparison, 1hour. windowtime of dissipation rate energy thelocal it ismuch more related to asuitable power of Rather, perature asgiven by temperature theactual attheairport. aninverse tem- hence itdoes nothave thephysical meaningof themacroscopic fluctuationsand turbulent parameterof variance of probability The distribution between theanemometers A andB. fluctuations asgiven differences velocity by wind thelongitudinal stants (w sum general form x component velocity thewind (in the temporal fluctuationsof for interest: thisout for of twoand Rapisarda series time carried Rizzo different lengths. analysed windowscan be usingtime of thevariable all, First of the corresponding variables: there isafundamental difference intheinterpretation ofHowever, eur independent positive randomvariables log log so that For Taylor-Couette flow [8] shown rotating ithasbeen cylinders. such Taylor-Couette asaturbulent flowscales, asgenerated by two d casca Amultiplicative and between different scalesintheflow. spatial cells dissipation between neighboring theenergy fer mechanism of determined by thefluid thetrans- iscritically of motion turbulent the themechanism of but Kolmogorov following [1], turbulence, form thegeneral which is of solid (red) function linerepresentsThe distribution alognormal s environmental turbulence asmea- scaleof spatial The conditions. which are different under very performed turbulence, ry fluctuationsandlaborato- dissipation issimilarforwind turbulent It seems thattheabove mechanism for mentionedtransfer energy B w the Reynolds number measurements, fluctuates for the wind t p from andit exhibitshas strong aGaussian distribution deviations suitable rescaling become thelatter Gaussianfor large sum will y ive cascade process, i.e. if it can be represented itcanbe by a if i.e. ive cascade process, ur -direction) asrecorded attheanemometer A andalsothespatial r niyo theparameter of ensity r ut if log log ut if ing spat hereas in the laboratory experiments itiscontrolled. experiments hereas inthelaboratory o at nab ith W The probability The density T The lognormal distribution results distribution if lognormal The ngnrl for agiven nonequilibriumsystem theprobability In general, β e e iet(ft)tis(e i.4.In one models superstatistical 4). minent (‘fat’) Fig. tails(see oph β o d attheair he Gammadist r β e not sotie o h eprlcs ssoni i.2for a foras obtained thetemporal inFig. caseisshown de process isexpected distributed to to lead alognormally µ of e ry turbulence on experiments much space smaller andtime ry = i tcnlso a ece yRzoadRpsrai 5 6]) nt conclusion wasreached by Rizzo andRapisarda in[5, c l side onral itiue,seFg 3. Fig. see distributed, lognormally is indeed y w and = 1; y and n ∑ difference difference u sics news io-t β ic independent squared Gaussianrandomvariables g n (either temporal orasmeasured spatial) attheairport i e =1 is Gaussianthismeansthat e thatthedifference between theGammadistribution no s l …; b e l fitt prpit osat.Aprnl,thedataare rea- Apparently, appropriate constants. f log log mp above are related to r p ( mal dist β o n ⑀ ) ~ r o ) w ξ ed by a lognormal distribution (note thata distribution byed alognormal . The fluctuations of the variance parameter thevariance fluctuationsof The . smc agrta ntelbrtr,moreover t ismuch larger thaninthelaboratory, nteflw(ihrsailo eprl,notthe in theflow (either or temporal), spatial r i al d u oteCnrlLmtTerm under to LimitTheorem, Due theCentral . NO ribution resultsribution if ith meanz β χ c VEMBER nmc fthesystem under consideration. ynamics of -1 2 iuini setal hto anadditive thatof isessentially ribution dsrbto ucin hc so the which isof function, -distribution e β - β p is ult /b ( f with , u ( β ) of longitudinal wind velocity dif- velocity wind longitudinal ) of / (1/ ) ~ DECEMBER e imately determined by theunder- r ,i.e. o, n b . β and . So far there is no theory of farthere So isnotheory . β β β s β is lo ) exp[-(log( β can b c = 2005 ξ appropriate constants. is due to amultiplica- ∑ gnormally distributed. gnormally i i.e. , n i =1 e r X epresented by a β i 2 > 0. β = pr v / β µ Π (the veloc- oduct oduct )) T is alocal n i 2 =1 he c /(2 ξ s o i o 2 f )], X n- or β n β n . . i 2 me experi- wind asimilarkindastheatmospheric measurements of In oceanic thatconnection long range comparison with data. turbulence asare thelaboratory dissipationrate, energy ating K turbulencepheric dataseem roughly consistent with atmos- The also stretched exponentials tailsandmuch more. sible physical interpretation to theentropic index 7 .P ocadadM Potters, Bouchard andM. [7] J.-P. onie ihTals entropic index coincides Tsallis’ with ahrlresai-eprlsae In thelong-term one has run large spatio-temporalscale. rather [6] S. generalized canonical distributions ( generalized canonical distributions the and one ends upwith explicitely canbe integral evaluated, p iesaitclmcais i.e. sive mechanics, statistical 5 .RzoadA aiad,in Rapisarda, Rizzo and A. [5] S. it alsoal c fl where definegenerally often approximation agood provided is proportional to theaverage of tions p 1 ..Kolmogorov, [1] A.N. Ref < 3 .Bc,G eee,A aiad,adC Tsallis (eds.), andC. Rapisarda, A. Benedek, G. Beck, [3] C. sharply peaked distributions peakedsharply distributions thatfor shown itcanbe However, plicated behaviourarises. andmore com- evaluated cannotbe explicitly, theintegral case), that 8 .Bc,EGD oe,adHL wne,cond-mat/0507411 8 Swinney, andH.L. Cohen, E.G.D. Beck, [8] C. [4] C. Cohen, BeckandE.G.D. [2] C. and second moment of rection from arising thelocal superstatistics can lead to a variety of distributions distributions of to canlead superstatistics avariety no For other decay apower asymptotically with distributions law. These 4]. [2, Gaussian distributions whose inverse whose Gaussian distributions variance can understand thesetailssimply from of asuperposition t there are if Ontheother hand, mechanics arises. statistical nary then thoseare effectively by described e β ntns.But concept thesuperstatistics ismore generalinthat onstants). ( r uc olomogorov’s general ideas of a lognormally distributed fluctu- distributed alognormally olomogorov’s ideas general of mp u o 2 For casethat thespecial r > t icse eemgtb ntutv,testing yet larger scales. hereinstructive, nts discussed be might Quite generally, the superstatistics approach thesuperstatistics alsogives aplau- Quite generally, ) ~ mine t er a itiuinosre nFg n .General 2and3. inFig. observed mal distribution = uat WrdSinic igpr,20) 246 2005), Singapore, (World Scientific, tive Pricing, tive sivity, Metastability andNonextensivity,Metastability (cond-mat/0406684) Chaos C f e iz n .Rpsra in Rapisarda, Rizzo and A. p T ( < rature fluctuationssituations) (asinmostnonequilibrium ences ( < sal β β u io β lo saGmadsrbto,the ) isaGamma-distribution, t(ft)tis ..ntol oe asbt for example, notonly power laws but, i.e. nt (’fat’) tails, > ) exhibit broader tailsthanaGaussian. lis, ∫ > 2 ns in f 0 w ∞ = = es)C ek .Bndk .Rpsra n .Tsallis andC. Rapisarda, A. Benedek, G. Beck, C. (eds.) ( β s f onf f J. β ∫ ) (such as the lognormal distributions relevant inour distributions ) (such asthelognormal ( S β f 0 o 2 e tat. β ( )e , Cambridge University Press, Cambridge (2003) Cambridge UniversityCambridge Press, r othe rence, β he at al ) - - 2 1 P β nc β h d .FluidMech. J. u y l b 2 Flo β β e inthiscaseo r dist d s. β and u 52 r n eeial hs ye fdistribu- of thesetypes andgenerically , t e β , c,APCn.Proc. AIPConf. nce, β 479 (1988) r epciey lal,i there are no if Clearly, respectively. , ib is fi < Physica β f p C the8 of Proceedings utions ( 2 ( Theory of Financial Risk Financial andDeriva- of Theory mlxt,Mestasbility andNonexten- omplexity, > β x f u = e ( β 13 ) is a Gamma distribution the ) isaGammadistribution (1+ ) ~ d t β β WrdSinic igpr,2005) Singapore, (World Scientific, and ∫ 82(1962) , 322A ) a -dependent normalization ne just obtains f q o ac f ( epnnil)o nonexten- -exponentials) of ( β | β q q q u q ) 267(2003) , -exponential for ), β | β (up to some minor cor- o is anentropic parameter .For case thespecial > 1. ~ is nottoo large [2]. nstant v 2 ( as f d q q β -1) 742 bandb q (1) byobtained eq. o denote theaverage th r e – 2 1 , Experimental β 176 (2004) u xample thelo alue fluctuates on a q 2 ) = 1ando - q — Complexity, q 1 -1 FEATURES Onemay . β p where , 0 ( , u o p ) with ne has ( u 191 r ) is di g β - - ~ features FEATURES
Lattice Boltzmann and nonextensive diffusion Bruce M. Boghosian 1 and Jean Pierre Boon 2 * 1 Mathematics Department, Tufts University, Medford, MA 02155 2 Physics Department, Université Libre de Bruxelles, B-1050 Bruxelles, Belgium * [email protected], [email protected]
᭡ Fig. 1: Concentration fluctuations field in the onset of tatistical physics today is arguably in much the same situation fingering between two miscible fluids (flow direction South- Sthat Euclidean geometry found itself in the early nineteenth East) showing landscape of q-Gaussian ”hills and wells”(color century. Over the last decade, an increasing body of evidence has code indicating highest positive values (red) to largest indicated that denying a certain postulate of statistical physics – negative values (magenta) of c).These structures identify the existence of precursors to the fingering phenomenon as they the extensivity of the entropy – results not in a contradiction, but develop before any fingering pattern can be seen. rather in an entirely new family of mathematically consistent variants of the statistical physics developed by Boltzmann and − 1 equil Gibbs (see Editorial). - f j (r, t) = τ (f j (r, t) - f ), where f j is the single-particle distri- The mathematical formulation of these variants begins with a bution function for velocity vj, position r, and time t. generalization of the definition of the entropy in terms of the Lattice Boltzmann equations derived from underlying micro- microscopic state probabilities of the system under study (see Box scopic dynamical models with detailed balance possess a 1 in the Editorial). A family of such entropies has been posited, discrete-space-time analog of Boltzmann’s celebrated H-Theorem, parametrized by a single positive number q, such that the usual which establishes that when the system evolves towards equilibri- Boltzmann- Gibbs formulation is recovered when q = 1. More um it will reach a state of maximum entropy. That is – in precisely, whereas the Boltzmann-Gibbs entropy is expressed in mathematical terms – it is possible to identify a Lyapunov func- terms of the logarithm function, nonextensive variants are tion for the dynamics. From the point of view of the lattice expressed in terms of a q-deformed logarithm (defined in Box 1) Boltzmann equation as a physical model, this maintains an to which application of l’Hôpital’s rule confirms reduction to the important property possessed by the continuum Boltzmann ordinary logarithm as q → 1. Remarkably, many fundamental equation. However lattice BGK models are not based on any results of statistical physics, such as the Maxwell relations and underlying microscopic dynamical model, and one of the prop- Onsager reciprocity, are “q-invariant”; that is, they hold for any erties lost in the transition from lattice models with a microscopic statistical physics in the family. Other results, such as the Fluctua- basis to lattice models without one was the H-Theorem. The effort tion-Dissipation Theorem and the compressible Navier-Stokes to modify the lattice BGK model so that it has an H-Theorem led equations for viscous fluid dynamics, must be modified by the to the proposal of so-called entropic lattice Boltzmann models in addition of terms that vanish when q = 1. the late 1990’s. These models begin by positing an H function of trace form, depending on the single-particle distribution Entropic Lattice Boltzmann Models function, and by dynamically adjusting the relaxation time in the In the late 1980’s and early 1990’s, it was realized that lattice kinetic BGK operator to ensure that this H function does not increase. models could be used to great advantage in the construction of Researchers then focussed efforts on finding the most general new algorithms for computational fluid dynamics [1]. These are class of entropic lattice Boltzmann model that would produce models wherein particles hop about on a regular spatial lattice in correct macroscopic behavior, i.e. give rise to the Navier-Stokes discrete time steps according to deterministic rules, with velocities equations in the hydrodynamic limit [4]. These analyses restrict restricted to the lattice vectors, and with collisions conserving attention to collision operators of BGK form, but with variable mass and momentum. Unlike particles in a continuum fluid, relaxation time. They do not specify the precise form of the H whose velocities take on values in R3, the velocity space of a lat- function; rather, they assume that H is of trace form tice kinetic model consists of a finite set of points. Hydrodynamic H(t) = ∑r∑j h (f j (r, t)), where the outer sum is over the lattice and quantities – such as mass density and momentum density – may the inner sum is over the finite set of velocities, without specifying be obtained from such a discrete-velocity distribution by a finite the form of the function h. Boltzmann’s H function would corre- sum, rather than an integral. spond to the choice h(f ) = f lnf, but that assumption is not made It is possible to construct Boltzmann equations for the single- here. Of course the form of the equilibrium distribution function particle distribution function for lattice fluids. At first, researchers will depend on the choice of h. Remarkably, it is possible to carry restricted their attention to Boltzmann equations for microscop- out the entire Chapman-Enskog analysis for the above-described ic models of discrete-velocity particles, but it was then realized model without ever specifying h. This results in the Navier-Stokes that lattice Boltzmann equations for discrete-velocity fluids could equations with an extra factor multiplying the advective term, v · ∇ be constructed with idealized collision operators that did not v. This extra factor depends on h and its first two derivatives. The correspond to any underlying particulate model [2]; even a single Galilean invariance of the Navier-Stokes equations depends on relaxation time operator – à la Bhatnagar-Gross-Krook (BGK) – there being no such extra factor multiplying the advective term. replacing the full collision operator, could be used for this purpose The convective derivative operator is Galilean invariant. The [3] giving rise to so-called lattice BGK equation: f j (r + vj ∆t, t +∆t) above-described extra factor breaks that Galilean invariance. The
192 europhysics news NOVEMBER/DECEMBER 2005 sical response theory, and obtain a andobtain sical response theory, can p one to some raised power, inthedistribution byrather gradients but function, inthedistribution bysimply described gradients the advection-diffusion equation. obeys example below) –aswe know from linearresponse theory, as w in thenat hasresulted both, or perhaps points, spacevelocity to afinite of set the fr n idealized system model highly in (thathasfound importance but a physicallattice isnotanatural model Boltzmann system, It ishowever to acknowledge important thatanentropic model. diffusion coefficient rmfis rnils It emphasized shouldbe thatBoltzmann’s from first principles. demonstrated andcanbe naturally, arises form deformed functional 0. the diffuse andinthe entropy, second-order differential nonlinear ordinary equation for oydsrbto ucin whereas Boltzmann’s function, body distribution example of a statistical mechanical system astatistical from which the example of innonextensivetion [5]used physics! statistical func non-exponential distribution mechanics istheappearance of One o N the isnottheentropy!function entropy The of isafunctional factor depends on Since theextra demand factor equal to thatthisextra be one. we only need to restoreis straightforward Galileaninvariance; it however, But now, hydrodynamics. lattice of models particulate andthisproblem noted been had microscopic inearlier frame, constitutes apreferred Galileanreference thelattice itself after all, Galileaninvariance isnotcompletely unexpected– breaking of eur power What makes thisequation ageneralized diffusion equation isthe We then consider theprobability cle inafluid. parti- atracer Suppose we are interested inthediffusion of theory. mechanics classical linearresponse statistical of a generalization p For thediffusion from consideration arise first principle might [6]. how such non-exponential distributions interest inthequestion of Boltzmann’s entropy Box (see 1)andinBoltzmann’s function appears in logarithm thatjust asthenatural it isstriking Nevertheless, function. distribution thesingle-particle of functional and c entropy of certain complex certain systems. entropy of forminthe thissamefunctional ages –gives to theappearance rise of –wherein averagesergodicity time are notequal to phase space aver- possibly due to lossof phase space, of that similarfragmentation the solution is oess oe-a itiuin olw nagnrcmne,from manner, inageneric power-law follow, distributions rocesses, ume one T o fone makes thekey assumption thatthecurrent isnot Now if What thisanalysis shows isthatlattice hydrodynamics provides an oph his dist q e t -deformed logarithm function appears inthenonextensive appears function -deformed logarithm ag oedesnilyaogtesm ie freasoning asinclas- roceed along essentially thesamelinesof l o io r l sp x ical anal f r α me ns w responding function. distribution y t h hrceitcfaue fnonextensive statistical features of characteristic the ensiv ≠ sics news eak o ur r atpoint iuinfnto u,isedo rbblte,we could probabilities, instead of –but, function ribution ntat h ,andthegeneralized Einstein relation connecting the 1, ( al ap ith power-law tailsandthere considerable hasbeen f ) = y o fsaeit eua atc,or thereduction of space into aregular lattice, ion of f the concentration of a diffusing species (asinthe adiffusingspecies theconcentration of f e diffusionasnonlinearresponse sis). p f aac fthe earance of ln h T H r NO q he analysis described abovehe analysis described suggests thateither n t rttodrvtvs thisresults ina and itsfirst two derivatives, D at t f function forfunction anentropic lattice Boltzmann where ln , VEMBER to the underlying microscopic to theunderlying dynamics ime t g / iven atpointr thatitstarts DECEMBER q q is precisely the -d generalized diffusionequation e f o mdlgrtm It may be logarithm. rmed 2005 f q ( -logarithm func- -logarithm r , H t ; function isa function r H 0 0)to find , function, h 0 whose , at t ime N q H - - equation with aneffectiveequation with diffusion coefficient with with b the g no approachThis for wasused instance to investigate precursor phe- hydrodynamic equations) isnotpre-established. by of theset e.g. retical analysis where themacroscopic (such asgiven description theo- whichon isbased akinetic offers amesoscopic approach , thelattice method Boltzmann to other computational methods, In contrast to used can be simulate systems nonequilibrium [2]. thelattice above) BGKformulation with (described – particular thelattice equation –in Boltzmann From viewpoint, apragmatic Precursor statistics Gaussian: they have they thecanonical Gaussian: but thesolutions are not time, ment increases with linearly process isclassical inthesense thatthemean-squared displace- to leads This thatthediffusion thefact theequation. solution of but thediffusion coefficient equation”, media depends on the asthe diffusion thesamestructure equation hasformally “porous generalized The theprobabilitywhen (or concentration) vanishes. isunusualwhat isthattheeffective diffusion coefficient vanishes from thisdoes notdiffer radically classical phenomenology: tems, diffusiondensity-dependent coefficient complex to describe sys- it isquite common to introduce concentration-dependent and ilizat with p Concentration distribution fluctuations (b) Lower panel: ᭡ n el nFg1 thedashedlineis and wells inFig.1; plane cutthrough(solid line)obtainedby thehills asection me The generalized equation can be viewed asaclassical diffusion generalizedThe viewed equation canbe Fingering is a generic phenomenon isageneric Fingering thatresults from thedesta- F e q ig na intheo ne io + . r o 2: α n o alized diffusion equation [8]. w a pe ae:Concentration profile fluctuations (a) Upper panel: = 2. er la theinterface different betweentwo fluids with f w nse : P t o ( c ) f ∝ fi ngering andthedatawerengering analyzed using |c| - q with q q -exponential Box (see form 1) = 0.73(solidline). q -Gaussian; D α = α D FEATURES f α− 1 Since . 193
features FEATURES mobilities (because of their differences in viscosity or density) in systems such as a shallow layer or a porous medium, when the fluid with highest mobility is forced through the other fluid. As soon the Relaxation and aging in system responds non-linearly to the driving force, enhanced inter- nal fluctuations (such as concentration fluctuations) are produced characterizing the early stage of the fingering process. If the fluids a long-range interacting are miscible, the mixing zone at the interface between the two flu- ids grows as the fluid with high mobility displaces the other fluid, and there is a dynamical transition where the exponent of the system µ growth of the mixing length of the interfacial zone, Lmix ∝ t , changes from µ = 1/2 (the value typical of a diffusive process) to a Francisco A. Tamarit 1 and Celia Anteneodo 2 * larger value. In the diffusive regime (before any fingering pattern 1Facultad de Matemática, Astronomía y Física, Universidad becomes visible), the flow produces local concentration gradients Nacional de Córdoba, Ciudad Universitaria, 5000 Córdoba, which induce mobility fluctuations thereby triggering vorticity Argentina fluctuations. The concentration field in Fig.1 shows that a ”land- 2Departamento de Física, Pontificia Universidade Católica do Rio scape” of alternating hills and wells has developed. In each ”blob”, de Janeiro, CP 38017, 22452-970, Rio de Janeiro, Brazil the concentration field exhibits a two-dimensional q-Gaussian * [email protected]; [email protected] profile as illustrated in the upper panel of Fig.2 obtained by a sec- tion plane cut through the extrema in Fig.1. Such q-Gaussians are precisely solutions to the generalized diffusion equation. Now the omplexity refers to the quality that certain systems possess of remarkable fact is that the distribution that follows from a q- C being intricate and hardly predictable. Ranging from the exponential profile has a power law behavior. In two dimensions turbulent flows that form our atmosphere to the human languages, and for a q-Gaussian – as for the case of the concentration fluctu- our life has plenty of examples of natural complex behavior. ations c in the fingering pre-transitional regime – the distribution Statistical Mechanics, the area of physics that deals with the prob- is simply P(c) ∝ c -q, as illustrated in Fig.2. lem of explaining the macroscopic world from the dynamics of its The example presented here is representative of a generic class components, faces nowadays the challenge of applying the stan- of driven nonequilibrium systems where q-exponentials and dard reductionist program to all these fascinating systems. power law distributions are the signature of long-range interac- Even when the mathematical equations for describing its time tions and whose dynamical behavior is governed by non-linear evolution may be only a few, a complex system is composed of a equations, such as the generalized equation described above. huge number of interacting constituents. These constituents, usu- What has been shown is that during the onset of fingering, one ally very simple ones, interact giving rise to the emergence of an can identify precursors which exhibit statistical features typical of unexpected collective phenomenology, where cause and effect nonextensive statistics. Then the question arises as to whether there become subtle and where the long time behaviour is no longer is a possible physical interpretation of the origin of nonextensivity? obvious. For succeeding in the plan of explaining complex behav- The driving force produces a spatial sequence of alternating struc- ior from first principles, physicists have been looking for simplified tures, which, if they were independent, would exhibit an ordinary models, mathematically tractable and able to catch the essence of Gaussian profile originating from local diffusion centers (δ-func- complexity. tions), and would be described by a classical advection-diffusion Suppose you have such a complete knowledge on the micro- formulation. However, when growing, these structures develop into scopic details of certain system that you can write down its overlapping Gaussian blobs, and what the analysis shows is that by Hamiltonian. Now the natural question that arises is the following renormalizing the overlapping Gaussians, they are recast into a sum one: which are the mechanical conditions the system must fulfill in of scale invariant independent q-Gaussians. Similar statistical prop- order to guarantee that the statistical mechanics calculations would erties have been obtained in other nonequilibrium systems which predict, with an adequate degree of accuracy, the time averaged are discussed in companion articles in the present issue. quantities obtained from a laboratory experiment. And when trying to answer such an apparently simple question, one discovers that even the simplest systems can give place to very intricate behaviour. References Perhaps the simplest Hamiltonian model of interacting parti- [1] J.-P. Rivet and J.P. Boon, Lattice Gas Hydrodynamics (Cambridge cles one can image is the so called Hamiltonian Mean Field (HMF) University Press, Cambridge, 2001). model. Unlike most of the models we are used to deal with when [2] S. Succi, The Lattice Boltzmann Equation for Fluid Dynamics and modeling complexity, in this case, not only the dynamical vari- Beyond (Clarendon Press, Oxford, 2001). ables but also the interactions among them are extremely simple, lacking any trait of randomness or frustration. The system con- [3] Y. Qian, D. d’Humières and P. Lallemand, Europhysics Letters, 17, 479 (1992). sists of a set of N interacting particles or rotators of unitary mass, each one confined to move around its own unitary circle [1]. Each [4] B.M. Boghosian, P.J. Love, P.V. Coveney, S. Succi, I.V. Karlin and particle is then mechanically described by an angle θi and the J. Yepez, Phys. Rev. E, 68, R025103 (2003). corresponding conjugate momentum pi. The dynamics of the sys- [5] The precise value of q is the solution of a transcendental equation tem is ruled by the following Hamiltonian: with parameters depending on the spatial dimension, and the set of particle masses and velocities used. (1) [6] C. Tsallis Physica D, 193, 3 (2004). The first term is the kinetic energy associated with the motion of the [7] J.F. Lutsko and J.P. Boon, EuroPhys. Lett., 71, 906 (2005). particles, while the second one corresponds to the interaction poten- [8] P. Grosfils and J.P. Boon, Physica A, 358 (2005). tial (the summation running over all different pairs of particles).
194 europhysics news NOVEMBER/DECEMBER 2005 sit sen- theHMF isvery In particular, its longbehavior. time describe the canonical prescription seems to insuffi- cient be to short, in describe aswe will questionsthese fascinating mainly because, li or a awindow glass For instance, where thecanonical recipe fails. well established fact thatinanywell establishedfact tak cles [1-cos( parti- interacting apairof of in thesense thatthepotential energy U/N = e dimensionality. Second, the interaction is theinteraction Second, dimensionality. enough atleasthigh itative behavior, thermodynamical itsqual- keeping anyway track of many models, of thermostatistics the approach simplifiesthemathematicaltreatment of drastically thisunrealistic As itiswellknown, the distance betweenthem. independently of theothers, all with interacts which each particle in itrepresents connected afully system, In thefirst place, here. hsmdlcnb osdrdakntcvrino the considered canbe this model version akinetic of facontinuous at phase transition of predicts andthiscalculation theexistence canonical ensemble [1], sin ; vector atwo-dimensional magnetization particle local each we canassociate with In fact, systems. statistical studied themost any which iswithout doubt one of model, magnetic eur almost stat to occur faraway from equilibrium. But complexity unfortunately seems to systems equilibrium. intrue applied method thermostatistics the master the is assumption key of o phase ( andalow temperature ordered uniformly over distribute thecircle, with a thermal bathattemperature athermal with behaviour[2]. mo this despite itsapparent simplicity, one discovers that, In doing so, the t B sy inbyputting handanexternal dynamics thatwould force the instead of is, That proper deterministic microscopic dynamics. thisterm in(1)provides a theinclusion of However, ideal gas). asimple (that of isstraightforward thermodynamical quantities (micr ature disorderedphase would expect this last quantity towould coincide thislastquantity thetemperature expect with T mean kinetic energy per particle 2 particle per energy mean kinetic the twice wemeasure averagetime the of If given particle. per energy consider acompletely isolated system andprepare a with itinitially we may Alternatively, according to the Boltzmann–Gibbs measure. nsemble calculation, we can obtain the mean energy per particle per we themeanenergy canobtain nsemble calculation, igcl.OursimpleHMFis anexcellent for discussing prototype cell. ving f the original thermal bath (after suitable transients). This This bath(after thermal suitable transients). theoriginal f oltzmann–Gib he tmt ii hs pc codn,frisac,to theusual for instance, stem phase space to according, visit i There are a few features of the model thatare themodel mentioning worth There are afew features of The thermodynamics of this model can be easily canbe solved thismodel inthe thermodynamicsof The hr r lnyo physical systems which stay inmacroscopic There are of plenty M Let usassume thatthesystem equilibrium isinthermodynamical v e int e ipasasrrsnl ihvreyo complex dynamical of variety rich del displays asurprisingly oph r e to its initial preparation, specially just below the critical point. just below specially the critical e to preparation, itsinitial r θ mostat ost o ue dynamics Newtonian by simply integrating equations o i ) and correspondingly a global order) andcorrespondingly aglobal parameter: M cano → o a < y H/ sics news ≠ f c io N ) where rotators tend to synchronize their movements. 0), → θ on h iei nry hsi anybcueo the of ismainly This because count energy. thekinetic the mag ist nical, i nar p θ - > . c,thatthesystemmicroscopic visits configurations ics, . θ i i b j = = bs p ]tnst ycrnz hi oeet.Finally, )] tends to synchronize theirmovements. y simpl y stat canonical or grandcanonical) its contribution tocanonical or grandcanonical) p M i ne r NO y o c es (the bab t os VEMBER ic mo (characterized by y ass θ ilit i - uming, as we learn in any we as learn course on uming, y dist n, M dels in statistical mechanicsdels do instatistical not / p x DECEMBER resumably predictable ones) but sin measure based K r − T bto,we cannow investigate ibution, T Naogataetr,then one /N along atrajectory, c θ hn through thecanonical Then, . = 1/2between a i , f o r 1 ferromagnetic M 2005 → ≤ = i ) where rotators 0), → ≤ statistical theory statistical N XY high temper- high m → mean field in nature, i = (cos e rgodic (2) (3) θ T i relation C( function thetwo-time auto-cor- a complex system isthrough theanalysis of the crude two–timethe crude auto-correlation is function t Under thesecondi- and only differences time make physical sense. anal we decided to systems, states equilibrium inglassy observed out of already andthe described trajectories between thequasistationary Inspired by thestrong analogy for (see instance[4]). observed. amuch more complex dynamical behavior is exhibiting aging, w true usually referred to as features, diverge inthelimit sta thesystem may become conditions, on theinitial Depending isolat the motionof theequations of by obtained integrating perature), r the lo hap asit slowing down drastic indisordered observed systems, kind of itpresents the Actually, sanecontroversy [2]. of degree ing acertain includ- extensive research, of thesubject been had phenomenology thiscomplex analysis The of calculations. canonical thermostatistics ical q c thesystem inphase space is state The of [5]. the HMFmodel je cr attain t thesystem cannot function: thepotential energy of topology the that thisanomalous behavior understood canbe interms of Recently we have shown negative heatappear. specific regions with m closely curve resemblesalous caloric theonein observed thisanom- Interestingly, that expected inthecanonical ensemble. ions, one expects thatC( one expects ions, e ompletely thestate by characterized vector giving c he ainhpbtenteitra nryo thesystem anditstem- of betweenlationship theinternal energy it lirgetto fcutr fin raoi uli where ions or atomic nuclei, clusters of of ultifragmentation T ᭡ the state variable ( trajectories.The waiting timesare trajectories.The Data correspond to averages over 200ofsuch trajectory. that thesystem willgettrapped into aquasistationary energy andfor (subcritical) initialconditions that guarantee A v c t ical p he dep o pens, for instance, in spin glasses after asudden inspinglasses quenching into for instance, pens, k yz F r ries. Along these quasi–stationary solutions, whose lifetimes whose solutions, Along thesequasi–stationary ries. thermodynamical equilibrium, memory effects disappear effects memory equilibrium, thermodynamical e e e ig uant w t d int e e theb d sy r rue equilibrium because it gets trapped into asequence trapped itgets because of equilibrium rue < . … y simplewa oints of 1: e mp it endenc st > o no Normalized two-time auto-correlation of function y d m togydsges ntesbrtclrgo,with region, inthesubcritical strongly disagrees, em, stands f e e ha r oes notcoincideoes thevalue predicted with by the n-e C at v V/N o io ur ( e of quilibrium long-standing quasi-stationary tra- quilibrium long-standing quasi-stationary t + N fthetwo-time auto-correlation of function r of y o e p θ o → →∞ [3], ,t t, r a ; C p f characterizing therelaxation characterizing dynamics of f → w ae utemr,the Furthermore, hase. on b s ie for avalue ofthetotal time, ) vs. , ’ eaeoe eea elztoso the verage over several realizations of ,which canexhibit ), as alsov t h ieaeaeo any thermodynam- average thetime of , w t ) ,t oth timesise = aging. ’ ) < ≡ x → e C( ( rified inmany models. rified glassy t + For systems thathave attained t t w - t = 8x4 w ) ’ ). . viden x → However, for systems for However, ( n t with , w history-dependent ) t. calor > , x → ≡ n ic cur ( ;… 6. …; = 0; θ FEATURES → , → p ). v e Then, (the 195 (4)
features FEATURES
Fortunately, a large body of evidence suggests the existence of only a few dynamical universality classes associated with the out-of- equilibrium relaxation of a model, as occurs, for instance, in coars- ening dynamics or critical phenomena, from which one can extract, by analogy, valuable conclusions. Following these ideas we have looked for the functional dependence of C(t + tw,tw) on both times, tw and t, by trying different data collapses. In Fig. 2 we present the best data collapse obtained for the results of the three largest wait- ing times displayed in Fig. 1. The resulting scaling law shows that:
β C(t + tw ,tw) = f (t/tw) (5)
for the whole range of values of t/tw considered. Note that, for β β -λ t << tw it holds that f (t/tw) ~ (t/tw) . Surprisingly, this kind of scaling behavior is not usual in ordered systems, like the one here studied. Instead, this is the same kind of scaling observed in real spin glasses, which are characterized by the existence of high ᭡ Fig. 2: Auto-correlation function vs. scaled time.The data degrees of randomness and frustration [4]. The solid line corre- are the same shown in Fig. 1 for the three largest tw, but suitably scaling the time coordinate makes the data collapse sponds to the best fit of the data with the q-exponential function. into a single curve.The gray solid line corresponds to a q- This function naturally arises within the generalized thermosta- tistics introduced by Tsallis [6]. One sees that q = 1 + 1/λ, exponential fitting. Inset: lnq-linear representation of the same data, with q ≈ 2.35. Linearity indicates q-exponential yielding q ≈ 2.35. Notice that the q-exponential allows to fit the behavior. whole simulated time span, concluding that: β C(t + tw ,tw) ∝ expq (-t/t ) (6) dynamics. Afterwards, the auto-correlation function is suitable w centered and normalized to remain within the interval [-1, 1]. This affirmation is corroborated by the plot in the inset of Fig. 2, As we have previously mentioned, because we are dealing with where the same data of the main figure are represented as lnq[C(t β a well defined Hamiltonian system, it is possible to analyze its + tw ,tw)] vs. t/tw , yielding an almost perfect linear behavior. It is proper microscopic dynamics. But macroscopic systems always worth mentioning that similar fittings were obtained for other sys- involve such a huge number of interacting particles that the tem sizes. Later on, these simulations were remade by Pluchino, analytical integration of the equations of motion of all the con- Rapisarda and Latora [7] who verified that the sub-aging regime stituents is out of possibility. Then we must be able to integrate observed (β < 1) is mainly due to the contribution of the momen- their equations of motion with the help of computers (that is ta. Instead, the contribution of the angles to the correlation what we normally call a numerical simulation). Fortunately, for function displays the so called simple aging regime C(t, t’) ~ t/tw, the system we are interested in, the numerical integration of the which can be easily understood in terms of the angular coordinates. coupled equations of motion is a very simple task, which can be Finally, in Fig. 3, we present a further connection between carried out even on a modern personal computer, due to its dynamical anomalies and generalized Tsallis thermostatistics. This mean-field character. In the cases presented here, the system was plot, obtained from [8], presents the probability distribution always prepared in a “water bag” initial condition, that is, all the function (PDF) of the angular position, for a system of N = 1000 angles were set to zero while the momenta were randomly chosen rotators, initially prepared in the same initial conditions used for from a uniform distribution with zero mean and such that the analyzing aging, and at different times. It has been reported, some system has total energy U. Since for these initial configurations, years ago, that, in the quasi-stationary regime, the angles evolve the total energy is purely kinetic, we emulated the most drastic super-diffusively [9] (see also inset of Fig. 3, where the squared cooling down compatible with the chosen fixed energy. deviation is plotted as a function of time). Additional information In Fig. 1 we plot the normalized two-time autocorrelation is obtained looking not only at the squared deviation but at the function C(t + tw , tw) , for U/N = 0.69, N = 1000 and different whole distribution of angles. Curiously, after the meta-equilibrium waiting times (increasing from bottom to top). The value of the regime settles, the PDFs can be well described by q-Gaussian specific energy, together with the water bag initial condition, guar- shapes. The value of parameter q increases with time reaching a antees that, typically, the system will get trapped into a steady value q ≈ 1.5[8]. The ultimate relationship between this quasi-stationary trajectory as desired. In particular, for the values value of q and that obtained previously from the aging curves of U and N chosen, the discrepancy between canonical predic- remains an open question that deserves further analysis [10]. tion and microcanonical simulations is the most pronounced one. In conclusion, we have discussed two relaxational features We clearly note history dependence: for a given fixed tw, the (aging and spreading of angles) of the HMF, a paradigm of long- system remains in a quasi–equilibrium regime with temporal range couplings. Along the quasistationary solutions, both translational invariance up to a time of order tw. Thereafter, the features present traits of generalized exponential behavior. These auto-correlation function presents a slow algebraic decay and a traits may be a reflection of the complex structure of the phase strong dependence on both times. Furthermore, the longer the space regions where quasi-stationary states live. If that were the waiting time tw, the slower the decay of the correlation. case, then we would expect that the new generalized thermostatis- It is a well established fact that, despite its verified ubiquity in tics introduced by C. Tsallis in 1988 [6], inspired in multifractal nature, a careful analysis of the aging phenomenology can give valu- geometries, could offer a novel measure-based theory for predict- able information about the microscopic mechanisms involved in ing the mean values of a physical system when confined in the slowing down of the dynamics. In particular, since a general quasi–stationary long living states. Despite the proper complexity microscopic theory for aging is still lacking, scaling properties can of this enterprise, the manageability of the HMF model invites offer a qualitative description of the microscopic phenomenology. further and deeper investigations along these lines.
196 europhysics news NOVEMBER/DECEMBER 2005 1 .Atn n .Ruffo, Antoni andS. [1] M. References agenciesand Brazilian Faperj andCNPq. Agencia C´ordoba Ciencia (Argentina) SECYT-UNC (Argentina) from by supported CONICET(Argentina), grants partially works The here were described ful discussions andsuggestions. Pluchino for help- Rapisarda and A. We thank finally A. described. theworks here partially are who theco-authors of Moyano, L. Montemurro and We thankM. lus theseyears. along all We are indebted to Constantino Tsallis for hiscontinuous stimu- Acknowledgments eur 6 .Tsallis, [6] C. 1]C Tsallis, [10] C. [8] L. Anteneodo, Tamarit andC. F.A. Montemurro, [5] M.A. Anteneodo, andC. Stariolo A. D. Maglione, G. Tamarit, A. [3] F. 2 .Ltr,A aiad,adS Ruffo, S. and Rapisarda, A. Latora, [2] V. 7 .Puhn,V aoaadA Rapisarda, A. and Latora V. Pluchino, [7] A. 4 .Hman .Vnet .Dpi,M la .Ocioand M. Alba, M. Dupuis, V. Vincent, E. Hammann, [4] J. 9 .Ltr,A aiad,adS uf,P Ruffo, S. and Rapisarda, A. Latora, [9] V. de γ ᭡ the histogram isofthe the sameusedinpr Parameters andinitialconditions are of theHMFdynamics. oph ,signaling superdiffusion. > 1, F and Montemurro, Zanette andM.A. D.H. in Ph ..Cginoo .KrhnadM ´zr,in M´ezard, Kurchan andM. J. Cugliandolo, L.F. .TmrtadC Anteneodo, TamaritF. andC. .AtnooadC Tsallis, Anteneodo andC. C. Random Fields, Random Thermodynamics, .St,Po.Nt.Aa.S.USA Sc. Acad. Natl. Proc. Sato, Y. 69 Physica A E Rev. 031105 (2003). Tsallis, andC. Rapisarda, A. 1998). Entropy-Interdisciplinary Applications, C. ..Bouchaud, J.P. viation as a function of time. It follows It thelaw oftime. viation asafunction M ig , T o . y 056113 (2004). als Ofr nvriyPes xod 2004). Oxford, (Oxford University Press, sallis, A.F 3: y sics news sc o.50(pigr eln20) in 2001); Berlin 560(Springer, ysics Vol. n n .Atnoo inpreparation. Anteneodo, ano andC. 71 Hist . .Sa.Phys. Stat. J. V 036148(2005). , Physica A 338 in ~ ogram ofnormalized anglesat different times as, 60(2004); , .Py.Sc Jpn. Soc. Phys. J. P dtdb ..Yug(ol cetfi,Singapore,edited Young by A.P. (World Scientific, h NO y dtdb .AeadY kmt,Lecture Notes Okamoto, Abe and Y. edited by S. sica vosfiue.Notice that at longtimes, evious figures. 340 VEMBER 52 hs e.E Rev. Phys. q A , 1 (2004); C. Tsallis, M. Gell- Mann Gell- and M. Tsallis, C. 1(2004); , 479(1988); , Gusa om ne:squared Inset: form. -Gaussian 356 Physica D Phys. Rev. E E Rev. Phys. hs e.Lett. Rev. Phys. hs e.Lett. Rev. Phys. / , DECEMBER 375 (2005). 69 102 0 20) ..Bouchaud, J.P. 206(2000); , hs e.Lett. Rev. Phys. 52 193 hs e.E Rev. Phys. Nonextensive Mechanics and y.Rv Lett. Rev. hys. 57 20) ..Burlaga L.F. 15377(2005); , 2361(1995). , dtdb .Gell-Mann and edited by M. Physica A 315(2004); , 64 2005 056134(2001); , 84 80 0(00;V Latora, V. 208(2000); , N 5313(1998); , on-Extensive 67 Spin Glasses and Spin Glasses 340 80 hs e.E Rev. Phys. 031105(2003). , 83 σ 692(1998); , 187(2004); , hs e.E Rev. Phys. 2104(1999). , 2 ~ t γ Phys. , with 67 , that atleastinsome casesthenoise source could non be Gaussian. indicating systems, andbiological insensory particularly evidence, there some hasbeen experimental non Gaussiannoises, study of Ininterest to addition theintrinsic inthe Gaussiannoises. with t mainly due to non Gaussiannoises inhandling and thedifficulties w noise inthoseworks used wasone generated by a t those no no itwasassumedthat the sive [3], distribution thermostatistics w w c D * [email protected] Argentina Bariloche, S.C. and Centro Atómico Bariloche, Spain Santander, CSIC, Universidad deCantabria- Instituto deFísicaCantabria, Wio * Horacio S. nonextensivity phenomena and Noise induced C w adifferential equation stochastic differential equation(thatis, or aLangevin considering of form we thefollowing start used, thenon Gaussian noise to be In order to introduce of theform N q ha such noise source corresponds to variable, aGaussiandistributed ass such noise induced phenomena itwas thestudies of almost all in However, to namejust afew examples. noise sustainedpatterns, stochastic resonance [2], [1], noise induced transport transitions, like innoise induced phase order, phenomena or newof forms everyday new against intuition)noise or fluctuationscantrigger we have epne hti,i oto h ae,tevleo theparameter thevalue of thecases, inmostof is, That response. thesystems anenhancement and/or marked broadening of lines, sour the noise system’s response wasstrongly affected by adeparture of itwasfound thatthe thesystemsall andphenomena analyzed, ar where atclrfr fMroino eoyespoes,we have Markovian process), or memoryless of form particular o thep ime c repnigt asindsrbto) lal,thisresult Clearly, orresponding to aGaussiandistribution). ould b e hit ( ith random coefficients), with additive with noise ith randomcoefficients), on gaussiannoise optimizing thesystem’s response resulted q ising w v t ise T oee,motivated by work on previous based anonexten- However, l ume l asf ing ac - his article is a brief review on recent review studiessome about of isabrief his article e (me efo h asinbhvo,soigasito transition ashiftof showing ce from theGaussianbehavior, rn h atfwdcdso the20 thelastfew decades of uring community hasrecognized thatinmany situations (and t η ’ η orrelated) and non Gaussian noise source. The source The of orrelated) andnon Gaussiannoisesource. ) d thattheno ossib ( ( o ~ C e hig t ise ind t ithin ano ) wasano r some situations of biological interest. biological r some situationsof stesohsi rniesuc.Uuly itisassumedthat Usually, ) isthestochastic or noise source. o mo ( δ t r ( r - ilit t e l eeatfrmn ehooia plctos as hly relevant for many technologicalapplications, ryless) or colored (that is, with “memory”). This was This with “memory”). or colored (thatis, ryless) t lation lation - ’ y of obtaining some analytical results working when obtainingsome analytical y of exp[-( ) ~ uc t ’ ), e d p while for a typical Ornstein-Uhlenbeck process, forwhile atypical ne n Gaussianand no C ise sour tniesaitclpyisfaeok[] In xtensive physics statistical framework [3]. he ( t x - no . t t = f - ’ me ) = t c ’ )/ ( e ha ,t x, na w τ < η ], with with ], ) + ( d aGaussiandist t ) hen submitted to acolored (or η η ( t ( τ n M ’ t ) ), > the “correlation time”.the “correlation . th I f arkovian process (that century thescientific century the no ≠ iuin either ribution, ise is q 1 (with 1 (with -distribution FEATURES “w hit q ”(a e” 197 = 1 (1)
features FEATURES
The process η was analyzed in [4], and an effective Markovian approximation was obtained via a path integral procedure. Such an approximation allows different (quasi) analytical results to be obtained. Those results and their dependence on the different parameters in the case of a double well potential, were compared with extensive numerical simulations with excellent agreement. We will now briefly review some of the results obtained when studying a few of the noise induced phenomena indicate above. Stochastic resonance The phenomenon of stochastic resonance shows the counterintu- itive role played by noise in nonlinear systems as it enhances the response of a system subject to a weak external signal [2]. It was first introduced by Benzi and coworkers to explain the periodicity of Earth’s ice ages (see [2] and references therein). The study of stochastic resonance has attracted considerable interest due to its potential technological applications for optimizing the response in nonlinear dynamical systems, as well as to its connection with some biological mechanisms. A large number of the studies on stochas- tic resonance have been done analyzing a paradigmatic bistable one-dimensional double-well potential. In almost all descriptions the transition rates between the two wells were estimated as the inverse of the Kramers’ time (or the typical mean passage time ᭡ Fig. 1: The stationary probability distribution function for between the wells), which was evaluated using standard techniques. the non Gaussian distribution given by Eq. (3), for the value In almost all cases, noises were assumed to be Gaussian. τ/D = 1.The solid line indicates the Gaussian case (q = 1); the Consider the problem described by Eqs. (1) and (2), where dashed line corresponds to a bounded distribution (q = 0.5); , the external signal is while the dashed-dotted line corresponds to a wide distribution (q = 2). S(t) ~ A cos(ωt) and U0(x) is a double well potential. This prob- lem corresponds (for A = 0) to the case of diffusion inside the is with “memory”). It was shown that such non Gaussian and non potential U0(x), induced by the colored non Gaussian noise η. Markovian noise could be generated through the following We will not describe here the details of the effective Markovian Langevin equation Fokker-Planck equation (see [4, 5]); but it is worth indicating that . 1 d 1 such an approximation allowed us to obtain the probability η= - – —Vq(η) + –ξ(t); (2) distribution function of the process η, and to derive expressions τ dη τ for the Kramers time. Another useful approximation, the so-called where ξ(t) is a standard Gaussian white noise of zero mean and two-state approach [2], was also exploited in order to obtain ana- correlation <ξ(t)ξ(t’)> = Dδ(t-t’), while the potential lytical expressions for the power spectral density and the D τ η2 signal-to-noise ratio. V (η) = — ln[1 + –(q-1) —]. q τ (q-1) D 2 Figure 2 shows some of the main results. In the upper part is As this article is not the appropriate place to refer to all the depicted the theoretical results: on the left hand part for R – the properties of the process η, we make reference to [4] for details. signal-to-noise ratio – versus D, for a fixed value of the time However, it is instructive to show the stationary probability distri- correlation τ and various q. It is apparent that the general trend is bution function, which is given by that the maximum of the signal-to-noise curve increases when the
2 system departs from the Gaussian behavior (q < 1). The right st 1 τ η P (η) = — expq - — — (3) hand part again shows R vs D, but for a fixed value of q and q Z D 2 q [ ] several values of τ . The general trend agrees with previous results where expq(x) was defined in Box 1 (see the introduction by for colored Gaussian noises [2]: an increase of the correlation time C.Tsallis and J.P.Boon in this same issue), Zq being the normal- induces the maximum of the signal-to-noise ratio decrease as well ization constant. This distribution can be normalized only for q to shift towards larger values of the noise’s intensity. The latter fact < 3, its first moment is <η> = 0, while the second moment, is a consequence of the suppression of the switching rate for 2D increasing τ . Both qualitative trends were confirmed by Monte η 2 = ∫η2 P st (η)dη = — ≡ Dq, < > q τ(5-3q) Carlo simulations of Eqs. (1) and (2). The lower part of Fig. 2 is finite only for q < 5/3. Also, τq, the correlation time of the show the simulation results. The left hand side corresponds to process η, diverges near q = 5/3 and can be approximated over the same situation and parameters indicated in the upper left part. the whole range of values of q by τq ≈ 2τ /(5 - 3q). When q → 1 the In addition to the increase of the maximum of the signal-to- limit of η being a Gaussian, Ornstein-Uhlenbeck colored noise, is noise ratio curve for values of q < 1, it is also seen to be an aspect recovered, with noise intensity D and correlation time τ . that is not well reproduced or predicted by the effective Markov- Furthermore, for q < 1, the probability distribution function has a ian approximation: the maximum of the signal-to-noise ratio — curve flattens for lower values of q, indicating that the system, cut-off and it is only defined for |η| < —2D . In order to visu- √τ(1-q) when departing from the Gaussian behavior, does not require a alize the form of the probability distribution as function of η,in fine tuning of the noise intensity in order to maximize its response Fig. 1 it is shown for different values of q. to a weak external signal. On the right hand side, simulation
198 europhysics news NOVEMBER/DECEMBER 2005 eut ofimdms fthepredictions indicated above. results confirmed mostof Those non Gaussianwhite noise. caseof for theparticular above, noise introduced noise source built upto exploit of theform nance phenomenon wasanalyzed but usinganon Gaussian al it turns out that al itturns decreasing increases signal-to-noise ratio with signal-to-noise ratio) decreases with decreasessignal-to-noise ratio) with g cal Brownian motor are analyzing usuallystudied thefollowing atypi- of properties transport The interestand itsbiological [1]. researchers due to itspotential both technologicalapplications anincreasing number of theattention of non thatattracts –isanother noise equilibrium induced phenome- systems out of in induces directional transport and/or temporal symmetry, spatial –where thebreaking of Brownian motors or “ratchets” Brownian motors mately independent of (a)for afixed value of follows: are alsoshown. part upper right results for thesamesituationandparameters indicated inthe eur nest,theoptimal value of intensity, value of e ne shown in the upper part: on the left handsidefor ontheleft acorrelation time shown part: intheupper r q ᭡ a departure from the Gaussian from theGaussian a departure b sn ipeeprmna eu,in[6]thestochastic reso- Using setup, asimpleexperimental The numerical andtheoretical numerical The results as summarized canbe esults: ottom) oph .5addfeetvle ftecreaintm:(from top to bottom) = 0.75anddifferent values ofthecorrelation time: F r ig al st . y 2: q sics news o lef , the optimal noise theoptimal (thatone intensity maximizingthe , c τ Signal-to-noise ratio ( hastic differential equation hastic or Langevin t sidef = 0.25, m — d dt 2 x 2 or q = .5 1.5. 0.75, op NO τ - ≠ γ VEMBER = 0.1and(fr — 1. d dt x τ c o ie au fthenoise (c)for afixed value of ; - V q / DECEMBER ’ ( is independent of R x –q τ ) vs thenoiseintensity ( ) - om t h aiu au fthe themaximum value of , –bhvo) (from top to bottom) = 1–behavior): F q + op t and itsvalue isapproxi- ξ 2005 ( t o b ) + ott η q (b)for agiven ; ( om) τ t ), and ingener- q = 0.25, D ) for thedouble-well potential (4) 0.75, 1.0, τ higher higher d thestudies wasto analyze the mainobjective The of ly analyzed. or dispersion of the distribution increases with increases with thedistribution or dispersion of keeping Hence, Gaussian. thatisslower thana decays distribution asapowerability law, was analyzed in [7], with thedynamics of with was analyzed in[7], Brownian motor, atypical propertiesof before on thetransport correlatedtime forcing have considered been intheliterature [1]. ratchet several model different kindsof of For thistype rectified. themotionto allowing be equilibrium thermal the system out of func agvneuto 2.A icse eoe for 1< As before, discussed equation (2). Langevin obtained through the adiabatic through theadiabatic obtained Approximation -for e compared theGaussianOrnstein-Uhlenbeck process. with where done by the particle “against” the load force theload done by theparticle “against” thework unittime) (per of defined astheratio For theefficiency, the (sawtooth-like) ratchet potential, and ly, injected intoinjected thesystem through theexternal forcing ntedfeetprmtr,i atclr their dependence on inparticular, on thedifferent parameters, = 0.1anddifferent values oftheparameter xp 1.25; pnec fthemeancurrent ependence of q η Figur h feto h ls fthenon Gaussiannoise introduced theclass of of effect The By setting r .5 .5 .,12,whileontherighthandside for 1.25, 1.0, 0.75, = 0.25, ξ ( essio t τ t io ( ) is the time correlated) isthetime forcing zero (with mean)thatkeeps while onther t .5 .5 ..Lwrpr,Monte Carlo simulation Lower part, 1.5. 0.75, = 0.25, m q ) the thermal noise satisfying noise satisfying ) thethermal ns o , the stronger the “kicks” that the particle will receive will thattheparticle when thestronger the “kicks” , e 3, n usingana stems ftheparticle, is themassof o f m n thele q , = 0and t o g U ether with results of numerical simulations. numerical results of ether with igh 0 ft hand side, shows typical analytical results - analytical shows typical ft handside, ( diabat x ) = t sidef γ γ D ,theoverdamped wasinitial- regime = 1, — x 4 tenieitniy osat thewidth (the noise constant, intensity) ic ap 4 – or — x 2 proximation [7]. wasobtained 2 ( q .Theoretical results are .Theoretical J = = 0.75and(fr < ξ γ < ( F the friction constant,the friction t — d dx ) sacntn la”force, is aconstant “load” ξ t ( > t ) ’ η ) q F > and thee ( = 2 q (that indicates t to themeanpower ) described by the ) described enn ht the meaningthat, , om t γ q T δ ,theprob- < 3, op t ( fficiency ( t - FEATURES η J aclosed , o and ’ .Final- ). V 199 ε ( x as ε q ) ) . features FEATURES
᭡ Fig. 3: Left hand side: Current (a) and efficiency (b) for the Brownian motor described by Eq. (4), subject to the non Gaussian noise, both as functions of q.The solid line corresponds to an adiabatic approximation while the line with squares shows results from simulations. All calculations are for m = 0, γ = 1, T = 0.5, F = 0.1, D = 1 and τ = 100/(2π). On the right hand side: Separation of masses, results from Monte Carlo simulations for the current as a function of q for particles of masses m = 0.5 (hollow circles) and m = 1.5 (solid squares). Calculations are for three different values of the load force: F = 0.025 (a), F = 0.02 (b) and F = 0.03 (c).The values of the parameters were fixed at γ = 2, T = 0.1, τ = 0.75, and D = 0.1875.
A region of parameters similar to the ones used in previous occurs, that happens in the absence of a load force, and that it is studies was chosen, but considering a non-zero load force, leading enhanced when a non–Gaussian noise with q > 1 is considered. On to a non-vanishing efficiency. As can be seen, although there is not the right hand side of Fig. 3, part (a) the current J as function of q quantitative agreement between theory and simulations, the used for m1 = 0.5 and m2 = 1.5 is shown. It is apparent that there is an adiabatic approximation predicts qualitatively very well the optimum value of q that maximizes the difference of currents. This behaviour of J (and ε) as q is varied. As shown in the figure, the value, which is close to q = 1.25, is indicated with a vertical double current grows monotonously with q (at least for q < 5/3) while arrow. Another double arrow indicates the separation of masses there is an optimal value of q (> 1) giving the maximum efficien- occurring for q = 1 (Gaussian Ornstein-Uhlenbeck forcing). It cy. This fact could be interpreted as follows: when q is increased, was observed that, when the value of the load force is varied, the the width of the Pq(η) distribution grows and high values of the difference between the curves remain approximately constant but non Gaussian noise become more frequent, leading to an both are shifted together to positive or negative values (depending improvement of J. Although the mean value of J increases monot- on the sign of the variation of the loading). By controlling this onously with q, the width of Pq(η) also grows, leading to an parameter it is possible to achieve the situation shown in part (b), enhancement of the fluctuations around this mean value. This is where, for the value of q at which the difference of currents is the origin of the efficiency’s decay occurring for high values of q: maximal, the heavy “species” remains static on average (has J = 0), in this region, in spite of having a large (positive) mean value of while the light one has J > 0. In part (c) for the optimal q, the two the current for a given realization of the process, the transport of species move in opposite directions with equal absolute velocity. the particle towards the desired direction is far from being assured. Hence, the results indicated clearly show that the trans- Resonant gated trapping port mechanism becomes more efficient when the stochastic As indicated before, stochastic resonance has been found to play a forcing has a non Gaussian distribution with q > 1. relevant role in several biology problems. In particular, there are Regarding the situation when inertia effects are relevant (that is experiments on the measurement of the current through voltage- m ≠ 0), taking into account the results discussed above it is rea- sensitive ion channels in cell membranes. These channels switch sonable to expect that non Gaussian noises might improve the (randomly) between open and closed states, thus controlling the capability of mass separation in ratchets. Previous work has ion current. This and other related phenomena have stimulated analyzed ratchets with an Ornstein-Uhlenbeck noise as external several theoretical studies of the problem of ionic transport forcing (it is worth emphasising that it corresponds to q = 1 in the through biological cell membranes, using different approaches, as present case), and has studied the dynamics for different values of well as different ways of characterizing stochastic resonance in such the correlation time of the forcing, finding that there was a region systems. A toy model, sketching the behavior of an ion channel, was of parameters where mass separation occurs. This means that the studied in [8]. Among other factors, the ion transport depends on direction of the current is found to be mass–dependent: the the membrane electric potential (which plays the role of the barri- “heavy” species moves in one direction while the “light” one does er height) and can be stimulated by both dc and ac external fields. so in the opposite sense. We have analyzed the same system, but This included the simultaneous action of a deterministic and a considering the case of non Gaussian forcing, and focusing on the stochastic external field on the trapping rate of a gated imperfect region of parameters where (for q = 1) separation of masses was trap. The main result was that even such a simple (toy) model of a found. The main result was that the separation of masses indeed gated trapping process shows a stochastic resonance-like behavior.
200 europhysics news NOVEMBER/DECEMBER 2005 sib safnto fbt:thenoise and intensity both: of as afunction old and particles, thenotyet trapped of oehrwt togsiti h rniinln,as astrongline, together shiftinthetransition with theindicated re-entrance effect, result showed thepersistence of main The submitted to thenon Gaussiannoise indicated above. where bounded (as for noisecurrent intensities waslow small thetrapping as follows: contribution is modeled as ~ - contribution as~ ismodeled absorption The is(asusual)intensity thetuning parameter. andanother stochastic whose amplitude, asmall with periodic one accordingtraps to anexternal thathastwo field contributions, the dynamicalThe process consists intheopening or closing of generalized inorder to include theinternal trap’s dynamics. tions, eair thatis: behavior, irrelevant and irrelevant became thesignal of part (harmonic) thedeterministic ty r the st r maxim another andtheexistence of was alsoanalyzed, q for (2)wasused ian noise given by Eq. thecolored non Gauss- Finally, is the open trap “imperfect”). (i.e. is thesignal if is open or closed: eur Gaussian no non indicated previously resultsThe clearly show thattheuseof F the called A system, Noise induced transition from open the particles are trapped with aprobability unittime per with areopen trapped theparticles When is thetrap isalways noise thetrap without closed. that is: 0 to sy indicated by current, theabsorption of part theoscillating computing theamplitude of lar bounded character of the probability distribution function for function theprobability distribution characterbounded of t anincrease inthesystem response wasapparent white noise case, When compared the against thenoise. some intermediate value of opens, otherwise it is closed. The interesting The caseiswhen itisclosed. otherwise opens, diso (fr no the when In related previous works that, itwasshown studied. wasalso Gaussian white noise shows anoise induced transition, ag fvalues thanfor thecaseof of range with with occurs for values of system’s andthis response isenhanced or altered inarelevant way, t duce changes inthesystem’s significant response compared when h asincs.Mroe,i l ae,itwasfound thatthe cases, inall Moreover, o theGaussiancase. o etarded for ea inal c < 1c st g o l h eedneo themaximum of dependenceThe of The study The on wasbased thesocalled The stochasticThe resonance-like phenomenon by wasquantified s sa rsenUlnekoe are-entrance arose effect ise isanOrnstein-Uhlenbeck one, le t y t e ches optimalvalues ( η oph e m adiso η r the m r ∞ c d q c ud o w q > o fi e n uhart eando thesameorder alarger within and such arate remained of θ h aesse a tde n[] but itwas when samesystem The in[9], wasstudied . o was shown towas shown false. be um as a function of um asafunction r r w = 1. The transition wasanticipated for transition The = 1. ommen y ( nt B sos.Ta s a is, That esponse. y o ed state)ed as thenoise correlation time x sics news a a on o h nra”sohsi eoac:the stochastic resonance: hat wasfound for the “normal” probability distribution function), yielding thelargest yielding probability function), distribution nd ar ,hence ), ) –theHeaviside –determines function thetrap when r ith ar btdpstvl otert fovercoming thethresh- ibuted positively to therateof h aito ntersos fother related noise intheresponse thevariation of f q rdered state to an ordered one, and finally again to again andfinally a rdered state to anordered one, ises inman < 1. ∆ J e ts dcini h edfrtnn h os,simi- for intheneed eduction tuning thenoise, the optimum response the optimum g a gi ml.Hne there wasamaximum at Hence, small. was again ∆ A conjecture aboutre-entrance apossible effect io J NO n o was small too, while for while alarge noise intensi- too, was small ∆ genetic model, genetic q VEMBER J ( f indicating a departure from adeparture indicating Gaussian y no t ). The resulting The qualitative behavior was ). values of c q or do was o ise ind responding to anonGaussian and ub / γ DECEMBER le st ( t ) bserved, implying thatitispos- implying bserved, δ η q γ uc oc B ( ( x being awhitebeing noise [5]. w t hastic resonance effect resonance hastic that when submittedthat when to a sin( e ) ) = happens for he η ρ d p stochastic model stochastic ∆ . ( r ,t x, J γ 2005 ω e themaxim henomena could pro- ( ∗ t θ t ) on theparameter ); with with ); ) + q [ B . q τ sin( > 1, while itwas while > 1, was v η ≥ q resonant-like ρ ω ≠ η t the density q ar ) + c :Clearly, 1: um o departed ie the trap for reac- η d fr η c e - xists > f η o ∆ m γ c B ], q J ∗ , may said, be it noise from, how could to possible itbe such obtain of aform n,ascae ihi,to anadequate it, associated with and, 5 ..Fets .TrladHS Wio, Toral andH.S. R. Fuentes, [5] M.A. 3 e:M elMn n .Tals Eds., Tsallis, Gell-Mann andC. M. [3] See: 1]EP Borges, [10] E.P. ination oai .Wio Horacio S. About theauthor andassuming some ( variables, thefast through anadequateobtained of elimination adiabatic Gaussiannoises (white or colored) could be scales, time separated freedom two evolving well with severaltems of degrees with 7 .Bua n .S Wio, S. andH. Bouzat [7] S. [1] R. References CurieMarie Chair. C. Fuentes, M. M. Deza, R. Castro, F. Bouzat, authorThe thanksS. Acknowledgments 4 .A une,H .WoadR Toral, Wio andR. S. H. Fuentes, A. [4] M. H Europa andUSA. atseveralvisitor institutionsinLatin America, this conjecture requires work. some specific path integral integral path h edt eott q 2,andalsoto build upa (2), tothe need to resort Eq. without (1), such could a formalism to help directly study Eq. of It thattheuse isexpected mechanics studied. hasbeen statistical thenonextensivewhere andcalculus thealgebra associated with great interest. of be noise will non Gaussian induced to phenomenasubject such when akindof 6 ..Csr,MN uemn ..FetsadHS Wio, Fuentes andH.S. M.A. Kuperman, M.N. Castro, [6] F.J. [9] H.S. Wio; Revelli andH.S. J.A. Sánchez, [8] A.D. could result from the existence of a whole awhole could result from theexistence of aboveused noise Itconjectured canbe of thattheform properties. [2] wants t Stat the andof the Theoretical Physics 1992-1994, during Division CONICET andC tuto de Físicade Cantabria. e wasP To conclude, it is worth commenting itisworth on arelevant question: To conclude, An extremely relevant point isrelated to some recent work [10] T A. uemn .Rvli .Snhz .Tsoe .Toral and R. Tessone, C. Sánchez, A. Revelli, J. Kuperman, ist sal (2000); M Marchesoni, Jung andF. P. Häanggi, P. Gammaitoni, L. Phys., 351 64 169 I L ..Fets .Tsoe ..WoadR Toral, Wio andR. H.S. Tessone, C. Fuentes, M.A. and C. .Rian hs e.31 57(2002). 361, Rep. Phys. Reimann, P. nt .Atma n .Hänggi, Astumian andP. D. B e ical Ph e tters 051105(2001). , ulsar lis f e o thankt tastab W of the faster variables. However, the proof or rejection of or rejection theproof However, thefaster variables. of 165(2002). , 69(2005). , r dis io andR. r 70 o o 3 T c n .Gammaitoni, a andL. H.S. , iplinary Applications iplinary f r the als d.(ol cetfi,Snaoe 2005). Singapore, (World Scientific, Eds. sallis, 223(1998). , first principles? first y ilt esso 365 (2003). rmwr o hskn fstochastic process. framework for thiskindof isGopdrn 9620.He isnow attheInsti- sics Group 1986-2002. during y andNonextensivity, Physica A W rdae tteIsiuoBler,andhasbeen attheInstitutograduated Balseiro, h uoenCmiso o h wr fa European the o Commission for theaward of ir c r atI e io T nt o , r ol r J al, .A. tmc aioh,where he wasHead of o Atómico Bariloche, siuoBler,andSenior Scientistat nstituto Balseiro, lab P 340 eel n ..S´anchez, Revelli and A.D. h u.Py.J.B. Phys. Eur. rto n/rueu icsin.He and/or useful discussions. oration y sica D It thatindynamical known iswell sys- 5(04;H uai in Surayi, H. 95(2004); , (O P P y.Today hys. h 193 xf δ C. y s. o or exponential) correlation r B , Today Physica A d U 161 (2004) e Physica A Nonextensive Entropy - 41 hierarchical elim- adiabatic c hs et A Lett. Phys. ,G eee,A Rapisarda A. Benedek, G. k, .P 7(04,and 97(2004), , e ok 2004). New York, , 55 hierarchy 49 (11), (3), 295 303 P lc.andNoise Fluct. 39 (1996); 114(2001); , h 33 (2002); Complexity, 91(2002). , y 277 sica D of time scales time of nonextensive 304 , FEATURES e.Mod. Rev. hs e.E Rev. Phys. Physica A 1 68- 201
features FEATURES
by the modulus of the total magnetization, i.e. , goes Nonextensive from 1 to zero. Using standard procedures one can easily obtain the canonical equilibrium caloric curve, given by the relation U = T/2 + (1– M2)/2 . On the other hand, by numerically inte- thermodynamics and grating at fixed energy the equations of motion derived from the Hamiltonian, and starting the system close to equilibrium, the simulations reproduce well the theoretical prediction [3]. glassy behaviour However, the situation is quite different when the system is started with strong out-of-equilibrium initial conditions, as for Andrea Rapisarda and Alessandro Pluchino * example giving to the system all the available energy as kinetic Dipartimento di Fisica e Astronomia and INFN - Università di one. One way to do this is by considering all the angles = 0, thus Catania, 95123 Catania, Italy obtaining an initial magnetization M0=1 and V=0, and distribut- * [email protected], [email protected] ing all the velocities in an uniform interval compatible with the chosen energy density –water bag distribution. Adopting such ini- tial conditions, for an energy density interval below the critical hen studying phase transitions and critical phenomena one point, i.e. 0.5 ≤ U ≤ Uc , the microcanonical dynamics does have W usually adopts the canonical ensemble and exploits numeri- difficulties in reaching Boltzmann-Gibbs equilibrium: in fact, after cal Monte Carlo methods to predict the equilibrium behaviour. a sudden relaxation from an high temperature state, the system However it is also very interesting to follow the microcanonical remains trapped in metastable long-living Quasi-Stationary States ensemble and use molecular dynamics to investigate how the sys- (QSS) whose lifetime diverges with the system size N [4]. Along tem reaches equilibrium. This is particularly true for finite systems these metastable states, the so- called ‘QSS regime’, the system is and long-range interaction models since, in this case, extensivity characterized by a temperature lower then the equilibrium one, and ergodicity are not assured and deviations from standard ther- until, for finite sizes, it finally relaxes towards the canonical pre- modynamics are usually found. On the other hand recently many diction Teq. But, if the infinite size limit is taken before the data are available for phase transitions in finite systems, as for exam- infinite time limit, the system never relaxes to Boltzmann-Gibbs ple in the case of nuclear multifragmentation or atomic clusters, equilibrium and remains trapped forever in the QSS plateau at the and there is also much interest in studying plasma and self-gravi- limiting temperature TQSS. tating systems [1]. Moreover the generalized thermodynamics The QSS regime is characterized by many dynamical anomalies, introduced by Constantino Tsallis [2] to explain the complex such as superdiffusion and Lévy walks, negative specific heat, dynamics of nonextensive and non-ergodic systems provides fur- non-Gaussian velocity distributions, vanishing Lyapunov expo- ther stimuli and a challenging test in the same direction. In this nents, hierarchical fractal-like structures in Boltzmann µ-space, context an apparently simple but instructive model of fully-coupled slow-decaying correlations, aging and glassy features [3-11]. rotators, the so-called Hamiltonian Mean Field (HMF) model, has been intensively studied in the last decade [3-10] together with a generalized version with variable interaction range [11]. Such Hamiltonians have revealed a very complex out-of-equilibrium dynamics which can be considered paradigmatic for nonextensive systems [4,12]. We will illustrate in this short paper the interesting anomalous pre-equilibrium dynamics of the HMF model, focus- ing on the novel connections to the generalized nonextensive thermostatistics [5] and the recent links to glassy systems [10]. The HMF model: a paradigmatic example for long- range N-body classical systems The HMF model has an Hamiltonian H = K+V,with the kinetic energy
and the potential one .
→ In the latter is the orientation angle of the i-th spin si =(sin , cos ) and pi is the corresponding conjugate coordinate, i.e. the ᭡ Fig. 1: Time evolution of the HMF temperature for the angular momentum or the velocity, since the N spins have unitary energy density U = 0.69, N = 1000 and several initial mass. All the spins (rotators) interact with each other and in this conditions with different magnetization. After a very quick sense the system is a mean field model. The average kinetic term cooling, the system remains trapped into metastable long- — K provides information on the temperature of the system, which living Quasi-Stationary States (QSS) at a temperature smaller — can be calculated by the relation T=2 K/N. On the other hand, the than the equilibrium one.Then, after a lifetime that diverges potential part V is divided by the total number of spins in order with the size, the noise induced by the finite number of spins to consider the thermodynamic limit. At equilibrium this Hamil- drives the system towards a complete relaxation to the equilibrium value. Although from a macroscopic point of tonian has a second order phase transition: increasing the energy view the various metastable states seem similar, they density U=H/N beyond a critical point UC=0.75, characterized actually have different microscopic features and by a critical temperature TC=0.5, the system then passes from a correlations which depend in a sensitive way on the initial ferromagnetic (condensed) phase to a disordered (homogeneous) magnetization. one [3]. In correspondence, the order parameter given
202 europhysics news NOVEMBER/DECEMBER 2005 iltm vlto fthe evolution time tial of theini- U=0.69andN=10000, density for theenergy we report, where isnicely This inFig.2 illustrated differentin avery way. thesystem view behaves from amicroscopic pointof conditions, the macroscopic states metastable are present for theinitial all Although temperaturethe equilibrium reported line. asdashed netization M netization decreasesperature value thatrapidly according to mag- theinitial systemfrom The starts an initial tem- fortribution themomenta. theunitcircle dis- andusinganuniform of portions and wider over wider latterThe are realized by spreading angles theinitial M mohd by decreasing M smoothed, although thesefeatures seem to persist, Ontheother hand, [4]. T inated by thedifferent conditions initial anditsconnection with In order to explore microscopic thecharacteristic dynamics orig- C M eur T to atemperature according to curve, thecaloric sponding, this r For thesystem remains always homogeneous. configuration of different conditions0 out-of-equilibrium initial with N=1000andfor value for which theanomalies are more evident), in.Fractal-like characterizethe structures tions. but alsotheir dependence oncondi- theinitial the QSSregion, figure This how illustrates emerge structures andpersist in tions. b Tsallis appearsto statistics the be At variance, likely applicable. be p alstemsaitc,one canfocuson the thermostatistics, sallis est candidat r onnec Q e 0 0 These anomaliesThese stronglydepend onmagnetization theinitial SS . In Fig.1 we In show Fig.1 theQSStemperature plateaux for U=0.69(a . 0 thesystem already from plateau starts thelimiting corre- =0, tat oph 03.I l ae,atraln ieie thesystem relaxes to after along lifetime, In cases, all =0.38. easo o fmtsaiiyi em fVlasov equation [7]could interms of metastability ion of y tions to Tsallis thermostatistics sics news n thelatt e f 0 ni trahstemtsal S.Only for itreaches until QSS. themetastable , o r all theother cases. r all e NO r se VEMBER e ms t µ− 0 space for four different magnetiza- o be theonlyo be casewhere aninter- / ni,for M until, DECEMBER 2005 0 0 themicroscopic =0, µ− v elocity autocorre- elocity space for M ≤ M 0 ≤ 0 =1 1. almost exponential for M g Tsallis’ Ontheother hand, the Vlasov equation [7]. solution of is probably linked to thatthelatter thefact isastationary iin:i atw e au fq=1.5for M we avalue infact get of ditions: the dynamical correlations induced by thedifferent con- initial we how illustrate one cancharacterizeinaquantitative way fits, f M thedecay while isfaster for are similar, tion functions very tions M N=1000andseveral magnetiza- initial for U=0.69, in Fig.3(a) formed. One can see immediately Onecansee thatfor M formed. and anensemble average over 500different realizations wasper- states themetastable totruncated focusonly onof theproperties lation q are extremely well fitted by Tsallis’ Gsaitc srcvrd nFg3a,by meansof In Fig.3(a), isrecovered. BG statistics where z=–t/ , as fined the entropic index qisable to of quantify thedegree In thisnew context, phase space [4]. that live regionsof infractal systems theout-of-equilibrium dynamicsism indescribing of adequate to generalize theusual Boltzmann-Gibbs (BG)formal- sivity ifrn irsoi aueo theQSSinM different microscopic nature of while while thecurve, which indicates time of thebending characteristic l diminishprogressive- Ontheother handthey long decay. tailed statistics y f o eneralized isable formalism to characterizethedynamical = 1.2and r M 0 =0.2 andeven more for M o r init and 0 q function [5]. The latter The isplotted [5]. function ≥ steetoi ne 2.Actually, is theentropic index [2]. 0 . The initial fast relaxation illustrated in Fig.1 hasbeen fastrelaxation inFig.1 initial The illustrated . introduced by Tsallis to shown hasbeen particularly be 0.4 c ial magnetizations smaller thanM smaller ial magnetizations non-ergodicity q = 1.1f o r r elations exhibitelations nature along range andaslow o r M 0 ftednmc.For q=1thestandard thedynamics. of = 0.2andf 0 .Such aresult clearly indicates a = 0. 0 =0 [5,10]. These relaxation These decays =0 [5,10]. q-exponential o r M the QSSregime for M formed in andpersist are structures like Fractal- various cases. d ofthe properties illustr ar The initialfastparticles of spinsis 0.69 andthenumber energy isU= density The magnetizations. and f times (t=0,25,50,500) magnetiza when theinitial Ontheotherhand, > 0. ᭣ mo evolution oftheHMF c in ahomogeneous 0 thesy structure isevident.structure absent andno correlations are almost microscopiccase onfi ynamics inthe e plotted inred to F 0 ig del = 0r nonextensive thermo- nonextensive uain Inthis guration. or different initial . a 2: 0 0 t µ e themixing st ≥ = 0.4, -spac esp Initial time 0 em r 0.4 thecorrela- 10000. = N ae which = 0case, tion isM 0 q functions de-functions e τ -exponential ≥ c , e a emains t FEATURES t i τ .,while 0.4, o b v nonexten- l.Thus ely. t f being a being e 0 our come = 0 203
features FEATURES
anomalies observed not only for M0 = 1 but also for any finite initial magnetization. Actually there are several other results pointing in this direc- tion and in favour of Tsallis generalized statistics [4-6,12]. In the following, we want to discuss an interesting conjecture that gives further support to this interpretation. It concerns a link between the value of the entropic index q which characterize the velocity autocorrelation decay and the exponent of the anomalous diffu- sion γ [5]. In order to observe the diffusion process one can consider the mean square displacement of phases σ2(t) defined as σ2(t) = 2 <|θi(t)-θi(0)| > , where the brackets represent an average over all the N rotators. The mean square displacement typically scales as σ 2 ∝ t γ. In general the diffusion is normal when γ = 1, corre- sponding to the Einstein's law for Brownian motion, and ballistic (free particles) for γ = 2. Otherwise, the diffusion is anomalous and in particular one has superdiffusion if γ > 1. In Fig.3(b) we plot the mean square displacement versus time for U=0.69, N=2000 and different initial conditions. One can see that, after the ballistic regime of the initial fast relaxation, in the QSS regime and afterwards the system clearly shows superdiffu- sion for 0.4 ≤ M0 ≤ 1 and the exponent γ has values in the range 1.4-1.5. On the other hand, in the case M0 = 0 we get γ = 1.2. Actually by increasing the size of the system, diffusion tends to become normal (γ = 1 for N = 10000). Again this case seems to be quite peculiar and microscopically very different from the others studied, where anomalies are much more evident. Superdiffusion observed in the slow QSS regime seems to be linked with the q-exponential decay of the velocity correlations through the ‘γ-q conjecture’, based on a generalized nonlinear Fokker-Planck equation that generates q-exponential space- time distributions [5]. In this framework the entropic index q is related to the parameter γ by the relationship γ = 2/(3–q). Since in diffusive processes space-time distributions are linked to the respective velocity correlations by the well known Kubo formula, one could investigate the γ-q relation considering the entropic index q characterizing the velocity correlation decay in an anomalous diffusion scenario. In Fig.4 we illustrate the robustness of this conjecture by varying not only the initial conditions and the size of the system, but also the range of interaction. These calculations were done by considering the generalized α-XY Hamiltonian, with the parameter α, which modulates the range of the interaction, varying from 0 (HMF model) to 1 [11]. By plotting the ratio γ / [2/(3–q)] as a function of γ for various values of N, M0 and α at U=0.69, one can see that the γ-q conjecture is confirmed within an error of ± 0.1. This means that knowing the superdiffusion exponent one can predict the entropic index of the velocity correlation decay and viceversa. The simulations here discussed add an important piece of ᭡ Fig. 3: (a) Time evolution of the HMF velocity information to the puzzling scenario of the pre-equilibrium autocorrelation functions for U = 0.69, N = 1000 and dynamics of the HMF model and its generalizations, which different initial conditions are nicely reproduced by q- cannot be explained with the standard tools of the BG statistical exponential curves.The entropic index q used is also reported. (b) Time evolution of the variance of the angular mechanics. Although these results do not provide a rigorous displacement for U = 0.69, N = 2000 and different initial proof of the applicability of Tsallis generalized statistics, they conditions. After an initial ballistic motion, the slope strongly indicate that this formalism is at the moment the best indicates a superdiffusive behaviour with an exponent γ candidate for explaining the huge number of observed anom- greater than 1.This exponent is also reported and indicated alies for a wide class of out-of-equilibrium initial conditions. by dashed straight lines. Anomalous diffusion does not depend in a sensitive way on the size of the system. For both Links to glassy systems the plots shown, the numerical simulations are averaged The importance of the role of initial conditions in generating over many realizations. anomalous dynamics, together with the discovery of aging and dynamical frustration in the QSS regime [6,10], suggests also
204 europhysics news NOVEMBER/DECEMBER 2005 v w the t dynamical frustration. The The dynamical frustration. therotators due to the freezing of to of quantifythedegree rameter, systems extendedbeen has glassy by introducing anew order pa- and between theHMFmodel theanalogy Recently, function. slow relaxation autocorrelation inthevelocity observed time tion enda .I nlg ihtebhvoro the thebehaviourof with In analogy . pat Ed defined as cage theso-called i.e. one, to theTsallis’ mentary theHMFdynamics comple- of another non-ergodic description eur breaking 1] Such aphenomenon produces of asort [10]. compete more among each other intrapping andmore particles clusters which ahierarchical of distribution by theexistence of inoe h pncngrtos to the obtain tion over theNspinconfigurations, from M thesystem when starts theHMFmodel, In of theQSSregime thesystem time. acertain during whichholdof get astraps act one thus thatthesestates expects barriers: energy by high rather minima corresponding to configurations metastable surrounded many with local system aglassy isextremely rough, of landscape Itaccepted iswidely thattheenergy complex landscape. energy the longinsome forof times regions remain very trapped thesystem but nevertheless can breaking ergodicity case, true asinthe may equilibrium regions inwhich local achieved, be notbrokenthe phase space isa-priori into mutually inaccessible lib intheout-of-equi- Ontheother hand, M. to themagnetization e phase in the SK model, the emerging dynamical theemerging frustration phase intheSKmodel, q al ith while themagnetization to M goes zero as ᭡ metastable QSS regime. In this case, increasing the size of the system, the polarization remains the polarization constant around avalue increasing thesize ofthesystem, Inthis case, metastable QSS regime. but inthe (b) The samequantities plotted vs in (a)are thesize here ofthesystem, reported order parameters are identical. uilibrium regime of the HMF model the polarization thepolarization theHMFmodel of regime uilibrium r Then one can average the module of the elementary polariza- theelementary Then one canaverage themoduleof wards Anderson order parameter intheSherrington-Kirk- r oph τ i F um QSSr ic , e that recalls the pictorial explanation of aging made by aging the that recalls explanation thepictorial of model for structural glasses [13]andthus could glasses justifythe formodel structural i ig mp o k (SK)mo = 1, f . y 0 5: the s 1iiilcniin,such amechanism isreproduced conditions, =1 initial sics news scenario of glassy systems [13]. The latter The occurs when systems glassy [13]. scenario of o r 2, a h antzto n h oaiainpaepotd steeeg est o =00 teulbim thetwo (a) The magnetization pare Mandthepolarization plotted vs theenergyfor density N=10000at equilibrium: al a … uc gm,wihpashr h oeo theSpin-Glass which plays here therole of egime, v c N rg,integrated over inter- time anopportune erage, essi d , e and t l o NO v oiin feach spin of e positions f VEMBER an infi 0 b e elementary polarization elementary ing aninit / iernesi-ls 1] inthe [13], nite spin-glass range DECEMBER ial t 2005 ransient time [10]. time ransient N dynamical frustra- dynamical -1/6 weak ergodicity- weak . was defined as polarization p polarization p is e q ual M lib unhsterltv oino h pnvcos Thus thespinvectors. quenches therelative of motion introduces aneffective and randomnessintheinteractions ne no the for several values oftheparameter the errors ofthecalculations. This ratio isalways onewithin (seeprevious figure). function from therelaxationindex qisextracted ofthecorrelation diffusion e [11], of theinteraction ofageneralized version oftheHMFmodel ᭡ 0 t r n null value as in the equilibrium condensed phase, while mag- while n null condensed value asintheequilibrium phase, In Fig.5 weIn of plotthebehaviour Fig.5 1iiilcniin.At Mand equilibrium conditions. =1 initial izat i F um (a)andv ig the figure illustrates theratio of the anomalous io r . modynamic limit. 4: n, F v or diff anishes w xponent nteQSrgm b,for U=0.69andthe (b), s NintheQSSregime er en γ ytmszsadiiilcniin,and t system sizes andinitialconditions, ith thesiz divided b e No y 2/(3-q)vs α p thesystem andiszero in f and Mversus Uatequi- which fixes therange γ . The entropic p p assume the FEATURES 0.24 ~ p has a 205
features FEATURES same values for both the ferromagnetic and paramagnetic phase. Instead, in the QSS regime magnetization correctly vanishes with N-1/6 while polarization remains constant at a value 0.24 ±0.05. Complexity of seismicity This does strongly indicate that we can consider the QSS regime as a sort of glassy phase for the HMF model. Again, it is impor- tant to stress the role of the initial conditions in order to have and nonextensive dynamical frustration and glassy behaviour. Actually, glassy features are very sensitive to the initial kinetic explosion that produces the sudden quenching and dynamical frustration. In statistics particular, reducing M0 from 1 to 0.95 the polarization effect and the hierarchical clusters size distributions become much less Sumiyoshi Abe 1, Ugur Tirnakli 2 and P.A. Varotsos 3 1 evident, until they completely disappear decreasing further M0 Institute of Physics, University of Tsukuba, Ibaraki 305-8571, [10]. In this sense, the M0=1 initial conditions seems to select a Japan special region of phase space where the system of rotators de- 2 Department of Physics, Faculty of Science, Ege University, 35100 scribed by the HMF Hamiltonian behaves as a glassy system. We Izmir, Turkey note in closing that this result gives also a further support to the 3 Solid State Section and Solid Earth Physics Institute, Physics broken ergodicity interpretation of the QSS regime of Tsallis Department, University of Athens, Panepistimiopolis, Zografos themostatistics. 157 84, Athens, Greece Conclusions Summarizing the HMF model and its generalization, the α-XY eismic time series is basically composed of the sequence of model, provide a perfect benchmark for studying complex Soccurrence time, spatial location and value of magnitude of each dynamics in Hamiltonian long-range systems. It is true that sev- earthquake. In other words, seismic moments (their logarithms eral questions remain glassy dynamics. being values of magnitude) as amplitudes are defined at discrete spacetime points. Seismicity is therefore a field-theoretic phenom- enon. However, unlike a familiar one such as an electromagnetic References field, it is inherently probabilistic in both magnitude and locus in [1] T. Dauxois, S. Ruffo, E. Arimondo, M. Wilkens eds., Dynamics and spacetime. It is characterized by diverse phenomenology, accord- Thermodynamics of Systems with Long Range Interactions, Lecture ingly. Known classical examples are the Gutenberg-Richter law and Notes in Physics Vol. 602, Springer (2002). the Omori law. The former states that the frequency of earthquakes [2] Tsallis J. Stat. Phys. 52 (1988) 479. with magnitude larger than M, N (> M), is related to M itself as log N (> M) = A - bM, where A and b are constants and, in particular, [3] T. Dauxois, V. Latora, A. Rapisarda, S. Ruffo and A. Torcini, in ref.[1] p. 458 and refs. therein. b ≈ 1 empirically. Magnitude is related to the emitted energy, E, as M ~ log E. The Omori law describes the temporal rate of after- [4] V. Latora, A. Rapisarda and C. Tsallis, Phys. Rev. E 64 (2001) 056134; shocks following a main shock. The number of aftershocks between V. Latora, A. Rapisarda and A. Robledo, Science 300 (2003) 249. t and t + dt, dN (t), after a main shock at t = 0, empirically decays -p [5] A. Pluchino, V. Latora, A. Rapisarda, Continuum Mechanics and in time as (t + t0) , where t0 and p are constants with p varying Thermodynamics 16 (2004) 245 and cond-mat/0507005. between 0.6 and 1.5 according to real seismic data. It is noticed that [6] M.A. Montemurro, F.A. Tamarit and C. Anteneodo, Phys. Rev. E 67, both of them are power laws, i.e., they have no characteristic scales, (2003) 031106 . pronouncing complexity and criticality of the phenomenon. Nonextensive statistics (Tsallis statistics or q-statistics) [1] has [7] Y.Y.Yamaguchi, J. Barrè, F. Bouchet, T. Dauxois, S. Ruffo, Physica A 337 (2004) 36. been attracting continuous attention as a possible candidate the- ory of describing statistical properties of a wide class of complex [8] H. Morita and K. Kaneko, Europhys. Lett. 66 (2004) 198. systems. It is a generalization of Boltzmann-Gibbs equilibrium [9] P.H. Chavanis, J. Vatterville, F. Bouchet, Eur. Phys. Jour. B 46 (2005) statistical mechanics, and is concerned with nonequilibrium sta- 61. tionary states of complex systems. Since complex systems reside at [10] A. Pluchino, V. Latora, A. Rapisarda, Phys. Rev. E 69 (2004) 056113, the edge of chaos in the language of dynamical systems, cond-mat/0506665 and A. Pluchino and A. Rapisarda, Progr. Theor. ergodicity, which is a fundamental premise in Boltzmann-Gibbs Phys. (2005) in press, cond-mat/0509031 . statistics, may be broken. That is, at the level of statistical mechan- ics, long-time average and ensemble average of a physical quantity [11] For a generalization of the HMF model see: A. Campa, A. Giansan- ti, D. Moroni, J. Phys. A: Math. Gen. 36 (2003) 6897 and refs. do not coincide. Clearly, the concept of ergodicity does not apply therein. to seismicity because of the absence of the ensemble notion in its nature. [12] F.D. Nobre and C. Tsallis, Phys. Rev. E 68 (2003) 036115. Though the complete dynamical description of seismicity is [13] L.F. Cugliandolo in Slow relaxation and non-equilibrium dynamics still out of reach, some models have been proposed in the litera- in condensed matter, Les Houches Session 77, J-L Barrat et al.eds., ture to explain some features. Among others, the spring-block Springer-Verlag (2003), condmat/0210312 and refs. therein. model, the self-organized criticality model and mean-field models such as the coherent noise model are frequently discussed (the last one will be visited here later). Criticality and nonergodicity of real seismicity may lead to the question if some of its physical aspects can phenomenologically be described by nonextensive statistics and related theoretical treat- ments. The answer to this question turns out to be affirmative.
206 europhysics news NOVEMBER/DECEMBER 2005 eaino aftershocks exhibits the relation of “natural time”. waiting main shock occurs. The averages The are andthevariance defined by main shock occurs. ranges. The The ranges. n .Talsfrtedfiiin fthe Tsallis for thedefinitions of and C. Boon P. inthepresent Box case[see intheEditorial by J. quakes, thedistancebetween two orinterval successive thetime earth- i.e., fornia. In nonextensive statistics, the theories with with thetheories In nonextensive statistics, fornia. inJapan both andCali- real isfound to close seismicity be to 2, o sos andhisc ter iscalled whichintroduced“natural hasbeen time”, by Varot- Such parame- atime thediscrete series. time parameter of time well by the thedataare fitted extremely of both over ranges, thewhole that, have They found thedatataken inJapan andCalifornia. of sets distancespatial between two successive earthquakes by usingtwo tributions both inJapan both tributions andCalifornia with sees that the basic random variable is thatthebasicrandomvariable sees par Com- events taken under regime consideration. inside theOmori c The a definite it also obeys scalingproperty. In addition, below). explained be (this phenomenon time to waiting thenatural will exp c duality”.exhibits “spatio-temporal where the alsoobeys thecalmtime distance, spatial asthe andhave [2]that, ascertained interoccurrence time), (or, termed “calm time” between twointerval successive earthquakes, thetime have they analyzed of long-timestatistics further Then, q eur σ < itiuin 1,where [1], distributions 2 - the interest isthatthesum of Another pointof principle. c ad the hand, Ontheother including themaximum Tsallis entropy principle. toknown derivable be from several independent approaches which is Zipf-Mandelbrotso-called power-law distribution, S under consideration. variable therandom at afinite value of ieo the of time In thisequation, event-event correlation proposed function inRef.[3] isgiven by The aftershocks law apeculiar obeys for event-event correlation. ha isreferred to as Abe andSuzuki “Omori regime”. Recently, shocks, law inwhich theevents the Omori obey for after- series, time t Such features are mainly due to af- ingeneral. structured, highly sho e t o o o im -exponential distributions are-exponential distributions themaximum Tsallis entropy f m m r 2 r nsid mple ve discovered [3] that nonstationarity of the time series of series thetime ve discovered of [3]thatnonstationarity A st The The b n uui[]hv tde ogtm ttsiso the Abe andSuzuki [2]haveof long-timestatistics studied T sho the dist > eainfnto nE.(1)is found to thefunctional satisfy relation inEq. function q n q 1 ihteodnr uoorlto ucin one autocorrelation theordinary function, (1)with ing Eq. oph q r = ultane u,nonextensive wellexplains statistics spatio-temporal hus, ( (> 0)are saidto be t-time scale, however, the seismic data is nonstationary and theseismic dataisnonstationary however, scale, t-time = (1/ x < k olwn ansok.Atm nevlo theseismic of interval Atime cks mainshocks. following n t nes,the ) anditsinverse, β riking property reported inRef.[3] property riking isthatevent-event cor- e t q xit m red to realized be only by themaximum Tsallis entropy y 2 is apositive constant and -exponential distribution with with -exponential distribution > sics news - q y o r < ous d M t ibutions of the spatial distance thespatial in andthecalmtime ibutions of -exponential distribution with 0< with -exponential distribution q q n m ol -exponential distributions with 0< with -exponential distributions -exponential distribution hastheform: -exponential distribution heet In other words, th event. > f ) 2 lab respectively. , ∑ r eal se e M o r t k n =0 i - r 1 vation of both both of vation at is the time when the when is thetime NO t o m ismicit VEMBER s[] Accordingly, rs [4]. + dual k q , steidxo theTsallis entropy [1]. is theindex of < q t lgrtmcfnto,ln function, -logarithmic m nisln-iebhvo.In the y initslong-timebehavior. M oec te.Teeoe seismicity Therefore, to each other. t / m DECEMBER ’ > stenme fthesuccessive is thenumber of Q aging phenomenon aging = (1/ q is apositive randomvariable, > 1and0< t n n which istheoccurrence , q M q ly oeo acertain plays arole of n > 1isequivalent to the -exponential function, 2005 n th shock after agiven ) w ∑ q in E q > 1over thewhole M k q -exponential dis- =0 - 1 < 1hasacutoff q q. q .Hr,the Here, < 1. t (1) ist ~ m < 1casesis with respect with + q q exp ( k (> 0)and x t q )]. m -indices q ’ +k e (- r β me and Q (1) ), d called the scaling relation, and the associated function andtheassociated function thescalingrelation, called h s ftentrltm,nottheconventional time. time, thenatural the useof aft and scalingr theabove-mentioneding thesefeatures phenomenon with aging Combin- law. theOmori process due to thepower-law nature of It isaslow relaxation process. nonstationary nonequilibrium q regarded Amainshock canbe asakindof atfaults. distribution hasacomplex crust The regarding landscape thestress tershocks. relation: sho g e the anal In theGutenberg-Richter both describe law law. and theOmori isalready model This to known andscalingproperties. the aging theaftershocks by itcanexhibit described order to examine if simple mean-fi p these properties are universal for complex catastro- systems with isto ismore what examine important how physical viewpoint, From the mentioned above for modeling. give criteria stringent phenomena andrelations novel The laws, phenomenology. richer e between nonextensive dynamics. mechanics andglassy statistical adeep connection whichsome suggests theexistence evidence of thereis Rapisarda cond-mat/0507005), andLatora (e-print no, Pluchi- Baldovin and Robledocond-mat/0504033) (e-print of and par xp ferhuks Real ishowever seismicity by characterized much earthquakes. l of eagmn stre h cln ucin For earthquake after- le argument istermed thescalingfunction. ecigpoes Aftershocks amainshock give following to rise uenching process. hes. r The ensembleaverage over 120000numerical isperformed ᭡ q scaling function isofthe function scaling cur atn ie ne:thesemi- Inset: waiting time. event for correlation different functions values ofthenatural T eety Tirnakli and Abe [5]have reanalyzed numerically a Recently, The Gutenberg-RichterThe law atouchstone hasbeen for any mod- e uns t -logarithmic function).The straight lineimpliesthat the function).The -logarithmic c onent r icular int he ab F ks, shocks may of be v ig M . e (seeB A . f o y 1: ( ll quantities are dimensionless. C sis, d n o γ e ( v w Data collapse of the aging curves oftheevent-Data collapse oftheagingcurves . n l sim ) ise eutsesnwlgto h hsclntr faf- e result on thephysical new light sheds nature of I lto,one recognizes thatthemechanism governing elation, a mainshock isidentifiedandtheassociated Omori t isemphasized thattheseproperties are revealed by e + o r s ic,according to therecent investigations of est since, x intheedit n eld model called thecoherent called model eld noise in model ulation may useful for be thispurpose. mpir w , n w ) = ical g lassy d l C y g ~ q or ( -exponential form with n/f i ial pap v q e ( n b -log plotofthecollapsed-log y n namics. w ) h eaino thiskindis relation The of )). y er f f ( n or thedefinitionof w ) T ~ his o ( n w ) bse γ w q rvation isof rvation ith ap ~ – FEATURES C ~ 2.98. fasin- of osit 207 i v e features FEATURES regime is fixed. As expected, the model is nonergodic and, ac- cordingly, the time average and the ensemble average of a phys- ical quantity are different from each other. We have ascertained Sq entropy and self- that, for the event-event correlation function (defined by the natural-time average) in Eq. (1), the model well reproduces the aging and scaling properties (together with the form of the scal- ~ gravitating systems ing function, C) discovered in Ref.[3] for real seismicity. On the other hand, an intriguing feature was found for the event-event A.R. Plastino 1, 2 * correlation function defined by the ensemble average with 1 Departament de Física, Universitat de les Illes Balears, 07122 respect to numerical runs. To distinguish such a correlation func- Palma de Mallorca, Spain. tion from the one with the natural-time averages in Eq. (1), it is 2 Physics Department, University of Pretoria, Pretoria 0002, denoted here by D(n + nw, nw). This quantity also turned out to South Africa. exhibit the aging phenomenon with respect to the natural wait- * [email protected], [email protected] ing time, that is, the smaller the value of the natural waiting time is, the faster correlation decays. So, the system has an internal clock. Fig. 1 shows that the aging curves become collapsed to a ore than fifteen yeas ago, a generalized thermostatistical single curve by the rescaling of the natural time: D(n + nw, nw) = Mformalism (usually referred to as nonextensive statistical ~ 1.05 D(n/(nw) ), establishing the scaling property. The inset presents mechanics) based on a power-law entropic measure Sq was its semi-q-log plot. The straight line there implies that the scal- advanced by Constantino Tsallis [1]. Increasing attention has been ~ ing function, D, is given by the q-exponential function. paid to the Tsallis proposal in recent years, because it has been Now, according to Tsallis [6], there may be “q-triplet” {qstat, qsen, hailed by many researchers as a useful tool for the description of qrel} for a system described by nonextensive statistics, where qstat is certain aspects of physical scenarios exhibiting atypical thermo- the entropic index appearing in the maximum Tsallis entropy dynamical features due, for instance, to the presence of long range distribution as well as the Tsallis entropy itself, qsen is the index interactions. When the Tsallis formalism appeared in 1988, it was characterizing sensitivity of a nonlinear dynamical system to not at all clear what possible physical applications it might have. the initial condition and qrel controls the rate of relaxation and The first hint pointing towards a relationship between the Tsallis' decay of correlation. In the case of a simple system described by ideas and the thermodynamics of systems with long range inter- Boltzmann-Gibbs-type statistics, the q-triplet may be given by {qs- actions came in 1993 [2], when it was realized that the Tsallis tat, qsen, qrel} = {1, 1, 1}. In the case of a complex system, physical entropic functional is closely related to a family of distribution quantities are often expressed empirically in terms of the q- functions well known by astronomers studying the dynamics of exponential function: e.g., probability distributions (qstat), the dis- stellar systems: the polytropic stellar distributions. In point of fact, tance between two trajectories of a dynamical system (qsen) and the discovery of the connection between Tsallis entropy and the relaxation or correlation (qrel), with the values all different from stellar polytropic distribution constituted the first application of unity. An example is provided by the recent work done by the the Tsallis' formalism to a concrete physical problem. Stellar people from NASA [7], who have discovered a non-Boltzmann- systems, such as stellar clusters or galaxies, are important examples Gibbs case in a single physical setup. Analyzing the fluctuating of astrophysical self-gravitating systems, where the gravitational magnetic field strength observed by Voyager 1 in the solar wind, interaction between the constituents of the system play a funda- they have found that {qstat, qsen, qrel} = {1.75 ± 0.06, -0.6 ± 0.2, 3.8 mental role in determining the system's properties. The ± 0.3}. exploration of the connections between the Tsallis thermostatis- Therefore, the q-exponential scaling function obtained for af- tical formalism and the physics of self-gravitating systems has ~ tershocks in the coherent noise model, D, with qrel ~– 2.98, which been the focus of a considerable research activity in recent years is notably different from unity, could be seen as a fingerprint of [3-7]. further relevance of nonextensive statistics. Nonextensive statistical mechanics is built up on the basis of Science of complexity certainly enables one to reveal novel the nonextensive, power-law entropy Sq [1]. The entropic index aspects of real seismicity. Nonextensive statistics is expected to q (also called the Tsallis' entropic parameter) characterizes the offer a guiding principle for a deeper understanding of complex statistics we are dealing with. In the limit q → 1 the usual Boltz- dynamical systems with catastrophes, in general and complexity mann Gibbs (BG) expression is recovered: S1 = SBG (for the of seismicity, in particular. definitions of the basic quantities associated with the q-entropy see the Box in the editorial introduction by Tsallis and Boon to the present Issue). Optimizing the entropic measure Sq under the References constraints imposed by normalization and the mean value of the [1] Nonextensive Statistical Mechanics and Its Applications. edited by energy, one obtains the probability distribution associated (in S. Abe and Y. Okamoto (Springer-Verlag, Heidelberg, 2001). the context of Tsallis' formalism) with the thermal equilibrium or [2] S. Abe and N. Suzuki, J. Geophys. Res., 108 (B2):2113, 2003; S. Abe metaequilibrium of the system under consideration. The main and N. Suzuki, Physica A, 350: 588-596, 2005. property of this q-generalized maximum entropy distribution is that it exhibits a power-law like dependence on the microstate [3] S. Abe and N. Suzuki, Physica A, 332: 533-538, 2004. energy ⑀i, instead of the exponential dependence associated with [4] P.A.Varotsos, N.V. Sarlis, and E.S. Skordas, Phys. Rev. E, 66: 011902, the standard Boltzmann-Gibbs thermostatistics. 2002. Galaxies can be regarded as self gravitating N-body systems [5] U. Tirnakli and S. Abe, Phys. Rev. E, 70: 056120, 2004. that are trapped for a long time in a non-collisional regime (un- [6] C. Tsallis, Physica A, 340: 1-10, 2004. til collisional effects finally become important) where the stars move under the influence of the mean potential Φ generated by [7] L. F. Burlaga and A. F. Viñas, Physica A, 356: 375-384, 2005. the whole set of stars, Φ being a function of the spatial position x.
208 europhysics news NOVEMBER/DECEMBER 2005 n ban h stemlshr itiuin which has sphereone theisothermal obtains distribution, total massandenergy, by imposed of theconservation straints standar to maximize one the tries if However, thesystem. of and energy constraints for stellar systemsnatural spherical are thetotal mass The sonable to usetheconserved asconstraints. quantities itisrea- maximum entropy prescription thefinalrelaxed state, to infer one by tries recourse to a if Consequently, quantities. ab the sol ten as( The Vlasov writ- equation canalsobe each star. individual mass of namical p systems exhibiting non extensive thermody- for thestudy of e (1-2). preserved under theevolution governed by equations course, of is, given by thesystem, of energy where coupled to function anequation relating thedistribution concerned) the Vlasov equation stellar dynamics is(asfarascollisionless systems are equation of generalized thermostatistical formalism based on based formalism generalized thermostatistical constituted This thefirst clue suggestingthatthe expected. cally moving in the gravitational potential inthegravitational moving star asingle evaluated of along theorbit function, distribution potential potential instanc N function a distribution any collisionless system isgiven by astate of of description The eur q t [8]. mass andenergy ftebhvoro an thebehaviourof of Vlasov-Poisson system provides only anapproximate description stressed thatthenon-collisional dynamics governed by the where a self-gravitating a self-gravitating thedynamics of Acomplete (Newtonian) account of systems [8]. stellar various this approximation useful for isvery thestudy of sy function tribution with theNewtonian potential with gravitational oyrpcshr itiuin hc,fracranrneo the of for range acertain spherepolytropic which, distributions tems is investigated numerical via I e n g q nsi par tmi ht ie ucinlo theform given of afunctional stem isthat, uilib ou c W The Vlasov equation (1) for self gravitating systems, with the with systems, gravitating The Vlasov equation (1)for self n[] twssonta h xrmlzto fthenon ex- thattheextremalization itwasshown of In [2], ouple oph e v ne hen stellar systems relax to (or to anequilibrium ameta- t the mtr are endowed with ameter, e u v G rium) state, one expects them to “forget” all information all them one to expects “forget” state, rium) e, dF/dt ral, this approach to the dynamics of self-gravitating sys- self-gravitating thisapproach to thedynamics of ral, otmn-ib nrp fthesystem under the con- d Boltzmann-Gibbs entropy of q y t io -e is the velocity vectoris thevelocity and sics news d (N [7]). is Newton's constant and gravitational ir init Φ r Φ so the Vlasov equation (1)verify ns of nt o sgvn nasl-ossetwy ntrso thedis- interms of inaself-consistent way, is given, perties due to interactions. long range perties ) r ie y() snnier since thegravitational isnonlinear, given by (2), orb. o e w ial c p y und ,maigta h oa iedrvtv fthe meaningthatthetotal derivative time of = 0, t o N in qain fmto fthe motion of nian) equations of F onditions with the exception of the conserved the theexceptionwith onditions of -particle system isprovided-particle by of set thefull NO nipratfaueo the Vlasov-Poisson feature An of important . N e VEMBER r thesameconstraintsto leads thestellar F pril efgaiaigsse.However, system. self-gravitating -particle ( x , v , / t DECEMBER ) ina6-dimensional phase space fi nite N t the time. The fundamental The the time. bd iuain se for -body simulations (see, mass ande Φ aihs It hasto be vanishes. , 2005 Φ , dC/dt eg,asphysi- nergy, m S = 0. q denotes the N is relevant -par T F he t ( infinite x t ic , v otal les. (3) (4) (2) (1) , t ) ilrneo values of of cial range the the value of the radial coordinate theradial value of for each corresponds, cut-off The has aclear physical meaning. the connection between the that is, they constitute distribution functions inthe functions distribution constitute they that is, q-MaxEnt distributions, of theform to exhibit happen distributions nt tla ytm,theestablishedconnection between the finite stellar systems, case; T the Boltzmann-Gibbs-Shannon measure. generalization of The in order to represent aphysically sensible entropic measure [1]. basic p measuredard inanappropriate logarithmic limitand(ii)the st (i)entropic power incorporating forms law consideration of proposed by recourse to the general argumentsvery dealing with and o motivated by any other nor wasit form wasnotmotivated systems, by self-gravitating ees the meters, therelevant para- after anappropriate identification of exhibits, (5) distribution polytropic The spaces. andvelocity in position quantity [3-7], dist ast their propertieshasconstantly interested theoretical of tigation inves- The systems. self-gravitating from thetheoretical study of way Polytropic natural inavery arise distributions other reasons. e h tnadslau fatohsc tdns Now, astrophysics students. the standard of syllabus lated to index aiieteetoi ucinlS functional theentropic maximize e (5)isan distribution inthepolytropic cut-off The emerges. and velocity vectorsand velocity respectively theelements within st [5] obtaining these mo plausib [8]. velocity tional escape o imposed by the conservation of mass and energy [2]. massandenergy of by theconservation imposed stettleeg prui as fa niiulsa,and star, anindividual unitmass)of (per is thetotal energy where t (hence o remember that the original path leading to pathleading o remember the that theoriginal xample o ntropy isinteresting distributions andstellarfor polytropic aking thisint f bse e r uc The limit The tla oyrpcshr itiuin r ftheform sphereStellar polytropic are distributions of oyrpcdsrbtoscnttt h ipet physically Polytropic constitute distributions thesimplest, l c ohscssdrn h atoehnrdyas[] Polytropic rophysicists thelastone hundred during years [8]. r lar p q r o mostat vtoa aaascae ihra aais nsieo this, In spite of real galaxies. dataassociated with rvational n ib q -maximum entropy when distributions t sdrbeitrs.Bsdstentbefc ht for aspe- that, Besidesthenotable fact nsiderable interest. n ur etoyi ut aua n,t oeetn,unique to some extent, -entropy isaquite and, natural and thest oete htafntoa ftheprobabilities should have of thatafunctional roperties tosaesiltefcso anintense research activity utions are thefocusof still (usually called the (usually called orsod otes aldShse pee for = 5corresponds to Schuster thesocalled sphere; emdl o efgaiaigselrsses[] Alas, stellar systems forle models [8]. self-gravitating ltoi itiuin hsctofaie aual,and naturally, arises thiscut-off distributions olytropic es (inspir q f d ( /) finiteness naturally for massandenergy < 7/9), f x e ist q ls d , w n -MaxEnt form. The entropic The parameter -MaxEnt form. v c,a Talsctofpecito” which affects prescription”, as “Tsallis cut-off ics, o a ) hat iskno n →∞ d ud o notp (see Figure(see 1)by identifying 3 cut itisremarkable thattheextremalization ccount, xd e d on multifractals) andreducingd on multifractals) to thestan- y o 3 (hence v f denotes the number of stars with position stars with denotes thenumber of q their basic properties constitutestheir basicproperties of apart specific non-extensive to leads thermostatistics , r w o v n, polytropic index polytropic q id S = 1)recovers sphere theisothermal ihntefil fnon extensive of thefield within nacrt ecito fthe of e anaccurate description q system. The entropic The form system. entropy is andstellar polytropes r to thecorresponding gravita- , q under thenaturalunder constraints r osat.The ) are constants. q n -3/2 with -3/2 with .In thecaseof < 1. ( It isimportant x , v q S d FEATURES ) q 3 can be re-can be polytropic space that space x e q ntropic and /(1- S A n q , 209 was < 5 Φ d (7) (6) (5) q S 3 ), 0 v q , features FEATURES of the q-measure leads to a family of distribution functions of polytropic distributions (that is, Tsallis' q-maxent distributions) considerable importance in theoretical astrophysics. In a sense, with an evolving polytropic index (i.e., evolving q-parameter). astrophysicists, when studying Newtonian self-gravitating Taruya and Sakagami found that the alluded sequence of systems, have been dealing with q-MaxEnt distribution functions polytropic distributions provides a good description both of the for over a hundred years without being aware of it. The link be- density profile, and of the single-particle energy distribution of tween the Tsallis non extensive q-entropy and stellar polytropic the transient states of the system. Even more interesting, they distributions consitutes a clue (among several others) indicating also found numerical evidence that the same kind of behaviour that the entropic measure Sq is not just an ad hoc mathematical occurs if the outer boundary is removed. This suggest that the construction. polytropic distributions may play an important role as As already mentioned, many interesting contributions have quasi-attractors, or quasi-equilibrium states of an evolving self- been made in recent years in connection with the application of gravitating system [7]. Tsallis' thermostatistics to self gravitating systems in general, and Summing up, we have seen that the connection between to the stellar polytropes in particular [3-7]. Sau Fa and Lenzi have Tsallis entropy and self-gravitating systems has been an active obtained the exact equation of state for a two-dimensional self research field in recent years, generating a considerable amount gravitating N-particule system within Tsallis thermostatistics [3]. of new results. For sure, there are still several open questions to An interesting analysis of the Jean's gravitational instability of a be addressed. For instance, is there any role to be played here by self-gravitational system characterized by a q-gaussian velocity the super-statistics formalism (see the article by Beck, Cohen, distribution was done by Lima, Silva, and Santos [4]. A detailed and Rizzo in this Issue)? Another interesting question, in our study of the gravothermal catastrophe of self-gravitating systems opinion, concerns the possible relationship between the physics in conection with Tsallis' q-entropy was performed by Taruya of systems with long range interaction, and the recently advanced and Sakagami (see [5] and references therein). It has been proposal by Tsallis, Gell-Mann and Sato [9], that special occu- recently pointed out by Chavanis and Sire [6], that the criterion pancies of phase space may make the Sq entropy additive. for nonlinear dynamical stability for spherical stellar systems gov- erned by the Vlasov-Poisson equations resembles a criterion of thermodynamical stability. On the basis of this, it is possible to References develop a thermodynamical analogy to study the nonlinear dy- [1] C. Tsallis, J. Stat. Phys. 52, 479 (1988). namical stability of spherical galaxies. Within this approach, the [2] A.R. Plastino and A. Plastino, Physics Letters A 174, 384 (1993). concepts of entropy and temperature would be essentially effec- tive. In [6], the Tsallis functional is interpreted as a useful H- [3] K. Sau Fa and E.K. Lenzi, J. of Math. Phys. 42, 1148 (2001). function connecting continuously stellar polytropes and isother- [4] J.A.S. Lima, R. Silva, and J. Santos, Astron. and Astrophys., 396 309 mal stellar systems. (2002). On the basis of long term, N-body simulations of self gravi- [5] A. Taruya and M. Sakagami, Cont. Mech. and Therm. 16 279 (2004). tating systems, Taruya and Sakagami have obtained important [6] P.H. Chavanis and C. Sire, Physica A 356 (2005) 419. numerical evidence for a connection between the physics of self- gravitating systems and the Tsallis formalism [7]. Taruya and [7] A. Taruya and M. Sakagami, Phys. Rev. Lett. 90, 181101 (2003). Sakagami have shown that the evolution of a stellar system con- [8] J. Binney and S. Tremaine, Galactic dynamics (Princeton University fined within an adiabatic wall (starting with initial conditions Press, Princeton, 1987). away from the Boltzmann-Gibbs distribution, and before the sys- [9] C. Tsallis, M. Gell-Mann, and Y. Sato, Proc. Natl. Acad. Sc. USA 102, tem enters the gravothermally unstable regime and undergoes 15377 (2005). core collapse) can be fitted remarkably well by a sequence of
᭣ Fig. 1: Stellar polytropes are simple models for stellar systems exhibiting a particle density in position-velocity space that is a power-like function of the energy. The conco-mitant exponent is n-3/2, where n is called the polytropic index. This polytropic distribution can be obtained by optimizing the system's q-entropy under the constraints imposed by the system's total mass and energy. The polytropic index n is here plotted as a function of the Tsallis' entropic paramenter q.The red star corresponds to the so called Schuster sphere, with n = 5 and q = 7/9. For n < 5 (q < 7/9) the polytropic distributions have finite mass and energy. All the depicted quantities are dimensionless.
210 europhysics news NOVEMBER/DECEMBER 2005 ttmeaue ffew keV (1keV at temperatures of addressed such ashow nuclear reactions occur instellar plasmas the b Since and playroles inmostastrophysical important contexts. solar c produceddetected neutrinos by pp andCNOreactions inthe directly confirmed by thathave experiments several terrestrial hasbeen thisdescription dates production: for thestellar energy s thequantum andBethe effect tunnel by meansof barriers and relative velocity discusses se on effects fusion reaction ratesand dramatic canyield quantities, where Italy I-10129 Torino, Abruzzi 24, Italy I-09042 Monserrato (CA), 1 tions. such devia- consequences of theorigin canbe andthen what at we first discusswhy even tiny devi- In thefollowing, Maxwellian. is where theiondistribution velocity nuclear plasma, anelectron i.e. main-sequence stars like theSun have acore, itiscommonly accepted that In fact, the Maxwellian form. ution andproposed could thatsuch from adistribution deviate and the theions of andmomentum distributions theenergy to describe r meas that d It was soonrealized networks dominate production. theenergy several MeV reactions andwhat or reaction of Coulomb barriers C T 3 2 Marcello Lissia rates andfusion functions iondistribution plasmas: Nuclear astrophysical eur I Thermonuclear inplasmasanddistribution tails reaction p r n ag e ea uccessfully proposed theCNOandthen thepp cycle ascandi- Politecnico di Torino, Dipartimento di Fisica, C.so Duca degli Duca C.so diFisica, Dipartimento Politecnico di Torino, Km1, Sestu S.P. diFisica, Dipartimento Università diCagliari, Istituto Nazionale diFisica Nucleare (I.N.F.N.) Cagliari er unit volume isgiven andunittime) by la levant fusion cross sections, but also the use of statistical physics statistical but alsotheuseof fusion crosslevant sections, ions from theMaxwellian canhave distribution important F I Gamow understood thatreacting nuclei Coulomb penetrate c y n thepasto oph usio tion rate per particle pair is defined as the thermal average pairisdefined as thethermal particle tion rateper Maxwellian exponential tail, while leaving unchanged while bulk Maxwellian exponential tail, from howthe deviations illustrates small very his article t o ur e as w etailed answersetailed to such questions involved notonly good ore [2]. ,Haubold) have distrib- theenergy examined critically n, g σ ir scr y e inning o n r me sics news = ith ea σ v nts o e e ( e tosaetefnaetleeg oreo stars source are energy fundamental ctions the of r n v al me ning [1]. 1 stencercosscino h ecin The thereaction. ) isthenuclear cross of section ,2 1, nl ( unu ehnc,basicquestions were quantum mechanics, f n r q and Piero Quaratiand Piero e uhr eg,dE tisn Kacharov, Atkinson, d'E. y afew authors (e.g., 2 ) par NO c v r uant hanisms thatcancauses thereaction rate , = (1+ VEMBER t ic um me les o δ 12 / DECEMBER f ) -1 c t y aia nesadn fthe hanical understanding of n pe 1 (2) per cubic 1(2)per centimeterpe 1 ,3 1, n 2 r < ≈ σ tenme freactions (the number of v 11.6 2005 > , uch deviations. x 10 6 K) against (1) the t bar state. theequation of e.g., cles: thatreceiveproperties comparable contributions from parti- all where the particle distribution function function distribution where theparticle o st cross do sections notdepend present and farfrom resonances, is When barrier noenergy thereacting particles. of function by cross andby thespecific section distribution thevelocity rmprilswt nryo theorder of of energy from with particles n the specific form of form n thespecific r 11.83 MeV and 11.83 MeV going from (energy peak Gamow window) moves to higherenergies different energies.The select reactions windows ofparticle lower panel(b) shows how different particles.The reacting horizontal red bandindicates theenergy range ofthe cur rapidly increasing dashblue distribution.The Maxwellian isthe panel(a)theexponentialupper thinblackcurve ( 3 √ ᭡ multiplied times10 note have that thethree curves been (much) lower; them visibleonthesamesc product of the two curves (Gamow peak) times10 peak) (Gamow ofthetwo curves product o Therefore, the reaction rate per particle pair thereaction rate particle per Therefore, h iuto svr ifrn ntepeec faCoulomb different situation isvery The inthe presence of H E r — E ng G F ie e emperature [3]. G v = 45.09 MeV, blue). Correspondingly, the peaks become thepeaks Correspondingly, blue). = 45.09MeV, ig — ,we h ecigprilsaecagd asinthe fusion are thereacting charged, when particles r, → = l e sho y o . E 1: 4 ) f H n thee T e +2 or thesolarr w he G s thebehavior ofthepenetration factor p + p amo eg.Ms fthecontribution to Most of nergy. N p ( E → = 10 G w p = 11.83M 5 < ,10 σ d 9 v eac + ).The red thick curve shows red the thickcurve ).The f a n nryslcin Inthe eak andenergy selection. > 17 ( and10 , = v tion ) isw + ∫ 0 ale. e ∞ eV + f 3 ( H ( E eak. The same is true for bulk sameistrue The eak. e) and red), , v 26 G e + ) epciey to make respectively, , 9 e,gen,to green), = 493keV, σ 3 vdv H f e ( kT v → , ) is a local function of function ) isalocal p andthedependence , 4 + H < σ e +2 14 N v > → is determined 8 p .The < N FEATURES 15 ( σ E O + x e G v 3 He + > = xp (- c γ o mes 211 (2)
features FEATURES
which is called the most effective energy, since most of the react- ing particles have energies close to E0. Figure 1 gives a pictorial demonstration of how the Gamow peak originates and how different reactions select different parts of the distribution tail and can be used to probe it. In the upper panel (a) the exponentially decreasing function (thin black curve) is the Maxwellian factor; the rapidly growing function (dash blue curve) is the penetration factor (for graphi- cal reason multiplied by 109) of one of the most important 3 3 4 reactions in the Sun, He + He → He + 2p (EG = 11.83 MeV), 2 1/3 which corresponds to a most effective energy E0 = (EG(kT) /4) = 17.036 kT for kT = 1.293 keV = 11.6 x 106 K; the product of the two functions (Gamow peak) is the thick red curve. Note that the Gamow peak, and therefore the rate, is very small (it has been multiplied by an additional 108 factor to make it visible on the same scale of the other curves), since at the most effective energy E0 both the cross section and the number of particles are expo- nentially small. At this point it is important to remark that the area under the Maxwellian curve for energies within the Gamow peak (the energy window indicated by the red band) is of the order of 0.1% of the total area: only a few particles in the tail of the distributions contribute to the fusion rate. The fact that the penetration factor effectively selects particles in the tail of the distribution is the more dramatic the larger the charge of the reacting ions: for the p + 14N → 15O + γ (the leading reaction of the CNO cycle, which dominates the energy produc- tion in main-sequence stars larger or older than the Sun) the contributing particles are few in a million. The effect on the Gamow peak when increasing the charges of the reacting nuclei is shown in the lower panel (b) of Figure 1. The ᭡ Fig. 2: Effects of tiny changes in the tail of the distribution green, red, and blue curves show the Gamow peak multiplied by 5 17 26 for the reaction p + 14N. Using the parametrization of Eq. (5), 10 ,10 , and 10 , respectively, for three fundamental reactions in + 3 the upper panel (a) shows the effect of taking q = 1.002 main-sequence stars: p + p → d + + e (EG = 493 keV), He + 3 4 14 15 (green) and q = 0.998 (red), while the lower panel (b) shows He → He + 2p (EG = 11.83 MeV), and p + N → O + γ (EG = the effect of q = 1.01 (green) and q = 0.99 (red); black curves 45.09 MeV)). It is immediately evident that the larger the charges correspond to the usual Maxwell distribution (q = 1). In the of the ions the higher is the energy of the particles that contribute lower panel (b) the green (red) Gamow peak (thick lines) is to the rate, the (much) lower is the peak and, therefore, the (much) divided (multiplied) by an additional factor of five. Note that smaller is the rate. In fact the maximum of the Gamow peak E0 ∝ 1/3 2/3 thin curves (exponentials and q-exponentials) have been EG ∝ (Z1Z2) and its height is proportional to exp (-3E0 /kT). multiplied by 109 to enounce their tiny differences. A convenient parametrization of deviations from the Maxwell distribution is the deformed q-exponential: reactions that power stars [7]. The penetration of large Coulomb E E 1/(1 - q) barriers (Zα/r is of the order of thousands in units of kT when r expq (- —) = (1-(1 - q) — ) (5) is a typical nuclear radius) is a classically forbidden quantum kT kT effect. The penetration probability is proportional to the which naturally appears in Tsallis' formulation of statistical —— Gamow factor exp [-√EG/E], where the Gamow energy mechanics [4]. This particular deformation of the exponential has 2 2 EG = 2µc (Z1Z2απ) , α is the fine structure constant, µ is the the advantage of describing both longer tails (for q > 1) and cut- reduced mass, and Z1,2 are the charges of the ions. The cross sec- off tails (for q < 1), while reproducing the exponential in the tion is exponentially small for E << EG and grows extremely fast limit q → 1. with the energy; therefore, one usually defines the astrophysical S Figure 2 shows the effect of substituting into the Maxwellian factor, whose energy dependence is weaker distribution exp(-E / kT) the distribution Nq expq(-E/kT), where N is the normalization factor that conserves the total number of S(E) —— q σ(E) = — e-√EG/E (3) particles. We show the effect for the p + 14N reaction and the values E (a) q = 1 ± 0.002 and (b) q = 1 ± 0.01. This reaction determines The two factors in the integrand in Eq. (2) that carry most of the the rate of energy production from the CNO cycle, dominant at energy dependence are the Maxwellian distribution ∝ e-(E/kT), older stages and, therefore, also determines the time of the exit >> which is exp——onentially suppressed for E kT, and the penetration from the main sequence. Black curves refer to the exponential √ E /E factor e- G , which is exponentially suppressed for E << EG. (q = 1), red curves refer to the cut-off (q < 1) exponential, and Contributions to the rate come only from an intermediate region green curves refer to the longer-tail (q > 1) exponential. (Gamow peak) around the temperature-dependent energy Note that all exponentials have been multiplied times a huge 6 2 1/3 10 factor to emphasise their tiny differences: these values of q E G (kT) E 0 = (— ) (4) produce functions that are almost indistinguishable from the 4 exponential unless one looks very far in the tail.
212 europhysics news NOVEMBER/DECEMBER 2005 ing o I inthisdilute andthestellar studied interior canbe limit. neglected environment) atzero order themany-body correlations canbe correction due to theastrophysical plasma examplewell-known of a theresidual inter- of thattheeffects thefact In spite of energies. w systems thatare dilute in the appropriate and variables derived: tofirst good approximation. avery neglected canbe equilibrium from such andtemporal deviations scalethatspatial asmall the scaleo sufficiently slow on but notalways, nuclear reactions are often, ever, How- since thetemperature decreases from core to surface. local, thequasiequilibriumisonly In addition, nuclear out. isburned fuel anditendsthe when thestarlifetime, theorder of of life-time, state metastable This hasalong equals theheatproduction rate. ar can p therates even by few percent some of changes of robust, rather theoverall while is stellar structure In fact, stars. evolution of and structure the mental forquantitative a understanding of nuclear reaction ratesinstellar interiors isfunda- calculations of particles [7]. particles n se stellar core andare thehelio- inagreement with properties of from Maxwellian do the momentum notmodify distribution their excess [5]. energy o In quasi-stellar andmechanical eigenfrequencies [2]. luminosity, • Gamow peaks are shifted towards higher energies, when the when • Gamow are peaks shifted towards energies, higher • from islargerthe effect thelarger theexponential thedeviation star B isalmostuniversally distribution velocity taken to a Maxwell- be theparticle Onthisground, equilibrium. thermodynamical good andisin isnon-relativistic, for ions), and completely negligible i.e., isnon-degenerate, such asthatintheSun, Normal stellar matter, Deviations distribution from Maxwellian eur • onthe effect rate isalready large for (andtherates) becomethe peaks for correspondingly q> higher • n thislimitthev ction cannot be neglected (theelectron screening factor isa neglected cannotbe ction One canmake several remarks: the samescale! (multiplied)by divided 0.01) hasbeen five to make them fiton distribution hasataillonger thantheexponentialdistribution (green curves, nentials (red curves, (the larger q eoe ue(oeta atro 10)for becomes huge (more thanafactor of 1 and smaller for1 andsmaller q<1; httegen(e)pa nlwrpnl()o figure 2( that thegreen (red) inlower peak (b)of panel b uc oltzmann (MB)distribution. hose r sooycntans[] u a fetteeauto fthe but may of theevaluation affect constraints [6], ismology e p je At rigorously canbe leastinone limit theMBdistribution C hs eitossol ecrflyetmtd since reliable carefully shouldbe estimated, deviations These As already shown in the recent past, very small deviations small very As already intherecent shown past, ) hyaesitdtwrslwreege o u-f expo- are they shifted towards lower for energies cut-off > 1); oph s ar lear fusio quantum effects are small (in fact, they are for they small electrons quantum are effects (infact, small o c ossib r ts like Jupiter, deviations could even deviations be larger andexplain ts like Jupiter, n thes nc o e mo d y erning the thermodynamical equilibrium, main sequence main equilibrium, thermodynamical the erning esid uc sics news f thermal andmechanical exchanges thermal andtake place on f le, e d r | ual int q -1 prifso rsbifso rpriso the uperdiffusion or subdiffusion propertiesof e p n r .. ntecs fthesolarphoton andneutrino inthecaseof e.g., e tectable discrepancies, when precise when measurements tectable discrepancies, recisely in a stationary state whererecisely theluminosity inastationary at e | es thatma ); locity distribution istheMaxwellian distribution one. locity e NO r a q ction is small compared issmall ction to the one-body VEMBER < 1); y b / DECEMBER e nhanc do eltd depend- ored depleted, 2005 | | 1 -q 1 -q | | .1 note = 0.01; .0,it = 0.002, | 1 -q | = stat thatare functions to different distribution from theMBone ina isalsoMaxwellian. for rate calculations, quantity relevant the is which deduce relative-velocity the that distribution, to are necessary assumptions correlationsabout between particles additional isMaxwellian, distribution energy one-particle even the when Finally, theparticles. of momentum andenergy tive two-body anddepends isnon-local interaction on the (1)and(2)are theresulting notvalid effec- but if momentum, of andatermindependent variables, other the and independent of theparticle inthemomentum aterm of quadratic as thesum of thesystem could expressed be of andthatthetotal energy to (3), corresponding different are independent, particles probabilities of one usestheassumptionthatvelocity mechanics approach, In statistical theequilibrium lective andfields). variables to istransferred col- energy amountof (nosignificant particles thecolliding freedom of usingonly of when thedegrees served con- islocally and(4)thatenergy Stosszahlansatz), (Boltzmann's atthesamepointare two notcorrelated particles velocities of (3)thatthe issufficiently (2)thattheinteraction local, sions, collision ismuch time between colli- thanthemeantime smaller one assumes (1)thatthe In approach, akinetic assumptions [7]. ubiquitous Maxwell-Boltzmann are on distribution based several the dr p independent thesystem by theproduct isnotdescribed of tion of tributions. system andthere exist many approaches to derive therelevant dis- anisms thatgenerate thesecorrelations depend on theparticular microscopic specific The mech- such more generaldistributions. tak lis lik Druyvenstein or Tsal- different from theMaxwell (e.g. distribution amo cr collision elastic the ions andthedependence on momentum of collisions between of counts to isthetype decide thedistribution (stable or andwhat metastable) are stationary these distributions sig er powerser of the e Even when relation [10]. holds between momentum andenergy anuncertainty due to plasmamany-body effects, tant because, is impor- distinction This on momentum thanon energy. rather the fa powers of se the of depend on strongly effects theenergy These non-local. makes effective and effects) dependent interactions time (memory In theplasma addition theplasma. thatdescribe quasi-particles the different canbe from of thedistributions fusion reaction, e therel- of thedistributions makes thecollisions not independent, [7]. function thedistribution deplete tailof thehigh-momentum equationsisto factors effect in thekinetic whose enhance or to forces) random introduces of ingeneral, micro-fields or, electric atdgeso reo bevd .. theonesby selected a e.g., freedom observed, of degrees vant r s etos(olm,sree olm,enforced Coulomb, screened Coulomb, (Coulomb, oss sections le o a fnnlclt nteFke-lnkeuto.We intheFokker-Planck stress non-locality that equation. nal of C In we thefollowing give arguments andmechanisms that lead the one shouldkeep inmindthatderivations of However, I T n motn n la xml fthislastpointisgiven by andclear example of One important eas ftemn-oyntr fteefciefre,which theeffective forces, themany-body nature of of Because es int io napoc htue h okrPac qain which n anapproach thatusestheFokker-Planck equation, c bab he p o ng othe t ne e dist e nar tta aypoess uha ula uinisl,depend such asnuclear fusion itself, thatmanyct processes, r ift d par r rg lte ftecmoet,are ingeneralresponsible for thecomponents, of ilities ltosbtenprils sothattheprobability distribu- betweenparticles, elations o a r J y stat itiuinmitisisMxela xrsin the maintainsitsMaxwelliany distribution expression, ese ( p r p ib c t ) anddiffusio r ihrta h oetodr[] h rsneo high- presence The of thanthelowesthigher order [8]. c icles andon thecollisional frequency. nc ) rtepeec fion-ion correlations [9]. or thepresence of s), p utions) can be obtained, when when obtained, utions) canbe ount thea , i.e., higher derivative terms, can be interpreted canbe asa derivative higher terms, i.e., , e. frno ed eg,dsrbtoso random of distributions random (e.g., fields e of v e n r g fet ftheenvironment through age of effects D ( p ) c o efficients, stationary solutions stationary efficients, J ( p ) and D FEATURES ( p ) inc l 213 ud e features FEATURES momentum distribution can be different in the high energy tail. In fact, this quantum uncertainty effect (not Heisenberg uncer- tainty) between energy and momentum p, caused by the Critical attractors and many-body collisions and described by the Kadanoff-Baym equation, implies an energy-momentum distribution of the form q-statistics (6) A. Robledo with Instituto de Física, Universidad Nacional Autónoma de México, Apartado Postal 20-364, México 01000 D.F. México (7) where is the mass operator of the one-particle Green he singular dynamics at critical attractors of even the simplest function. After integrating in d the product of fQ ( , p) and the T one dimensional nonlinear iterated maps is of current interest Maxwellian energy distribution, we obtain a momentum distrib- to statistical physicists because it provides insights into the limits of ution with an enhanced high-momentum tail. Although this validity of the Boltzmann-Gibbs (BG) statistical mechanics. This approach produces a deviation from the MB distribution, the state dynamics also helps inspect the form of the possible generaliza- represented by fQ(p) is an equilibrium state [11, 12]. The tions of the canonical formalism when its crucial supports, phase Maxwellian distribution is recovered in the limit when δγ ( , p) space mixing and ergodicity, break down. becomes a δ function with a sharp correspondence between The fame of the critical attractors present at the onset of chaos momentum and energy. in the logistic and circle maps stems from their universal proper- Distributions different from the Maxwellian one can also be ties, comparable to those of critical phenomena in systems with obtained axiomatically from non-standard, but mathematically many degrees of freedom. At these attractors the indicators of consistent, versions of statistical mechanics that use entropies chaos withdraw, such as the fast rate of separation of initially close different from the Boltzmann-Gibbs one [4, 13]. by trajectories. As it is generally understood, the standard expo- We have argued that it is not sufficient to know that the nential divergence of trajectories in chaotic attractors suggests a Maxwellian distribution is a very good approximation to the par- mechanism to justify the property of irreversibility in the BG ticle distribution. We must be sure that there are no corrections statistical mechanics [1]. In contrast, the onset of chaos imprints to a very high accuracy, when studying reactions that are highly memory preserving properties to its trajectories. sensitive to the tail of the distribution, such as fusion reactions The dynamical nature of trajectories is appraised on a regular between charged ions. Several mechanisms have been outlined basis through the sensitivity to initial conditions ξt, defined as (others need to be studied) that can produce small, but important ξt(x0) ≡ lim (∆xt /∆x0), t large, (1) deviations in the tail of the distribution. ∆x0 → 0 where ∆x0 is the initial separation of two trajectories and ∆xt that at time t. For a one-dimensional map it has the form ξt(x0) = exp(λ1t), References with λ1 > 0 for chaotic attractors and λ1 < 0 for periodic ones. The [1] R. d.Escourt Atkinson and F. G. Houtermans, Z. Physik 54 (1929) number λ1 is called the Lyapunov coefficient. At critical attractors λ1 656. = 0 and ξt does not settle onto a single-valued function but [2] V. Castellani et al, Phys. Rept. 281 (1997) 309 [arXiv:astro-ph/9606180]. exhibits instead fluctuations that grow indefinitely. For initial posi- tions on the attractor ξ develops a universal self-similar temporal [3] D. Clayton, Principles of Stellar Evolution and Nucleosynthesis (New t structure and its envelope grows with t as a power law. York: McGraw-Hill Book Company, 1968). It has been recently corroborated [2]-[5] that the dynamics at [4] C. Tsallis, J. Statist. Phys. 52 (1988) 479. the critical attractors associated with the three familiar routes to [5] M. Coraddu et al, Physica A 305 (2002) 282 [arXiv:physics/0112018]. chaos, intermittency, period doubling and quasiperiodicity [6], [6] S. Degl'Innocenti et al, Phys. Lett. B 441 (1998) 291 [arXiv:astro- obey the features of the q-statistics, the generalization of BG sta- ph/9807078]. tistics based on the q-entropy Sq [7]. The focal point of the q-statistical description for the dynamics at such attractors is a ξt [7] M. Coraddu et al, Braz. J. Phys. 29 (1999) 153 [arXiv:nucl- associated with one or several expressions of the form th/9811081]. ξ (x ) = exp [λ (x ) t], (2) [8] G. Kaniadakis and P. Quarati, Physica A 192 (1993) 677; ibid. 237 t 0 q q 0 (1997) 229. where q is the entropic index and λq is the q-generalized Lyapunov [9] F. Ferro and P. Quarati, Phys. Rev. E 71 (2005) 026408 [arXiv:cond- coefficient. Also the identity K1 = λ1 (where the rate of entropy mat/0407665]. production K1 is given by K1t = SBG (t) - SBG (0) with SBG the Boltz- [10] V. M. Galitskii and V.V.Yakimets, JEPT 24 (1967) 637. mann-Gibbs entropy) generalizes to [11] A. N. Starostin, V. I. Savchenko, and N. J. Fisch, Phys. Lett. A 274 Kq = λq, (3) (2000) 64. where the rate of q-entropy production Kq is defined via Kqt = [12] M. Coraddu et al, Physica A 340 (2004) 490 [arXiv:nucl- Sq(t) - Sq(0) [3], [4], [7]. th/0401043]. [13] G. Kaniadakis et al, Phys. Rev. E 71 (2005) 046128 [arXiv:cond- Tsallis q index & Mori’s q-phase transitions mat/0409683] The central issue of research in q-statistics is perhaps to confirm the occurrence of special values for the entropic index q for any given system and to establish their origin. In the case of critical
214 europhysics news NOVEMBER/DECEMBER 2005 than exponential [2]. The two The different behaviors couched canbe than exponential [2]. u frpdydcesn tegh 4,[5]. decreasing rapidly strengths [4], but of the spectrum ‘order parameter’ Lyapunov coefficients generalized of thefunction Legendreother via transformation, different regions of the multifractal attractor. themultifractal different regionsof entropic index values for thespecial the and theseare identifiedasthesource of right-hand side ( right-hand the attractor [6]. As with ordinary thermal 1 thermal ordinary As with [6]. the attractor se t indexes of to amultiplicity be thesituation ismore complicated andthere appear chaos, onset of doubling accumulationthe period pointandinthequasiperiodic asin attractors, For multifractal critical develops intermittency. via the chaos threshold whereas atthetangent bifurcation chaos doublings thatculminate in period thesequencecations form of c e gen- (one hump) unimodal mapsof tangentThe bifurcations of Tangent &pitchfork bifurcations v m 80’s functions [8])are thespectral q theallowedattractors values for eur eae oteocrec fdynamical ‘ related to theoccurrence of come They out inpairs and are scalingfunctions. the attractor dynamic counterparts of the multifractal dimensions themultifractal dynamic of counterparts variance a single well-defined valuea single for theindex there is thepitchfork andthetangent bifurcations, simpler cases, For the class parameters to belongs. sality which theattractor hand sid io o ( -phase transitions (developed by-phase transitions Mori andcolleagues inthelate ral nonlinearity nonlinearity ral nsit c q ᭡ ndit The main quantities in the thermodynamic formalism of mainquantitiesinthethermodynamicformalism The ns, = 1- ). oph F ig A i a v io c ity to initial conditions, i.e. a sensitivity thatgrows asensitivity faster i.e. to conditions, initial ity . q y t ns, 1: q -p sics news e ( ually an infinite family of such transitions takes such transitions place aninfiniteually family of v in the ( (a) ofcontrolThe mapattractor classic logistic asafunction parameter hase t q i.e. x ) < ≡ f p ( q x α d o ‘ x r . The The . fr c λ λ w ansit ) that characterize the geometric structure of structure ) thatcharacterizethegeometric ) o > e ( ( er-law convergence of orbits when attheleft- when orbits er-law convergence of z e energy’ NO q q x f ,andadivergence at ), > 1displa )/d c fthebifurcation there isa ‘super-strong’ ) of h on ftangency the pointof io VEMBER q λ -phase transitions connect qualitatively-phase transitions n isindicat q ( q . The functions functions The . ,given by ), ψ / q ( DECEMBER λ y weak insensitivity to initial y weak insensitivity but with precisebut values with given by q ), are from obtained theuniver- φ a disc e ( yascino linearslope d by of asection q ) and λ q ( q o [2]. The pitchfork The bifur- [2]. q -phase’ transitions [4], transitions -phase’ 2005 nt φ ) q ( ≡ inuity at inuity ψ st in the ‘susceptibility’ q x order phase transi- ) and ( d λ c φ . ,related to each ), H ( q owever at the )/d ψ q ( q λ = D andthe , ) are the ( q (a) q in the ) and as a la ye eety[] 4.By condition takingasinitial [4]. lyzed recently [3], dynamics The attheFeigenbaum ana- hasbeen attractor 1. Fig. t 2 period is found thattheresulting of orbit in generic states with statesin generic with incontrast to thethorough coverage alous phase-space sampling, causinganom- confining trajectories or expelling of has theeffect This thatproducesmap atthetransitions monotonic trajectories. e power laws intertwined reproduce thatasymptotically the made of chaos attheaccumulation isobtained point onset of aroundhood incomplete neighbour- mixinginphase space interval (asmall or impeded Therefore, iterates from anadjacent chaotic region. of intermittency inthatthere transitions isnofeedback mechanism thetangent bifurcation differs from other studies of treatment of The only two universality classes for maps[2]. unimodal tence of actually alsothatfor itsconjugateactually value values for thepitchfork bifurcations nonlinearity For mapof aunimodal accumulationPeriod-doubling point je b period-doubling theprecursorples together with cascade of often theFeigenbaum called which reappears attractor inmulti- to conditions. initial sitivity pitchfork The bifurcations display weak insen- the tangency point. associated I the o each of window isequal to cascades within theperiod number of has the pitchfork bifurcation one has the index t wasestab io ntire period-doubling cascadentire thatoccurs period-doubling for ifur c c For thetangent bifurcations one hasalways ne t ed o z q r cat ries that interpose the chaotic attractors beyond thechaotic thatinterpose attractors ries -phase transition with indexes with -phase transition d, = 2) with extremum= 2)with at b q it thate -e io the appearance of a specific value for aspecific the the appearance of xp ns intheinfi q q µ lishe -Lyapunov coefficients are for valid all (b)Enlargement ofthebox in(a). . nnil,o which the of onentials, x egsa h agn iucto tisoeig See merges atthetangent bifurcation atitsopening. = d that x c rssfo h pca tnec’saeo the of shape from ) arises thespecial ‘tangency’ λ ξ 1 nit t > 0. has p e n z umber of windows of periodic tra- periodic windows of umber of x > 1andtherefore define theexis- r = 0andcontrol parameter e cise q /.Ntby theseresults for Notably, = 5/3. λ q l q y thef and 2- µ z q eedtrie.Asmen- were determined. n -ind > 1 (e.g. the logistic map thelogistic > 1(e.g. Q , n ∞ = 2- = 1, 2, …, [6]. This is This [6]. …, 2, = 1, ossso trajectories consists of o e q r xes of andthesets m o µ for thetwo sides of q q < ) turns out to) turns be = 3/2, while for while = 3/2, e finter- of aset f µ ∞ x q 0 (se ind µ FEATURES = 0at ∞ i.2b) e Fig. (b) µ fthe of e x (and ∞ µ . µ T 215 the ∞ he µ it features FEATURES
᭡ Fig. 2: (a) Absolute values of positions in logarithmic scales of the first 10000 iterations for the trajectory of the logistic map at the onset of chaos µ∞ with initial condition x0 = 0.The numbers correspond to iteration times.The power-law decay of time subsequences mentioned in the text can be clearly appreciated. (b) Positions θ(t) vs t in logarithmic scales for the orbit with initial condition θ(0) = 0 of the critical circle map for the golden-mean winding number.The labels indicate iteration time t where Fn is the Fibonacci number of order n. due to the occurrence of q-phase transitions between ‘local attrac- 2 - q as they are tied down to the expressions for λq in ξt. Again, tor structures’ at µ∞. The values of the q-indexes are obtained from the dominant dynamical transition is associated to the most the discontinuities of the universal trajectory scaling function σ. crowded and sparse regions of the multifractal, and the value of its — This function characterizes the multifractal by measuring how q-index is [5] q = 1- lnwgm/2 lnαgm where wgm = (√5 - 1)/2 is the n adjacent positions of orbits of period 2 approach each other as n golden mean and αgm is the universal constant that plays the → ∞ [9]. The main discontinuity in σ is related to the most same scaling role as α F . crowded and most sparse regions of the attractor and in this case q = 1 - ln 2/(z - 1) ln αF, where αF is the universal scaling constant Structure in dynamics associated with these two regions. Furthermore, it has also been The dynamical organization within critical multifractal attractors is shown [3], [4] that the dynamical and entropic properties at µ∞ difficult to resolve from the consideration of a straightforward are naturally linked through the q-exponential and q-logarithmic time evolution, i.e. the record of positions at every time t for a tra- expressions, respectively, for the sensitivity ξt and for the entropy jectory started at an arbitrary position x0 within the attractor. In this Sq in the rate of entropy production Kq. case what is observed are strongly fluctuating quantities that persist in time with a scrambled pattern structure that exhibits memory Quasiperiodicity & golden mean route to chaos retention. Unsystematic averages over x0 would rub out the details A recent study [5] of the dynamics at the quasiperiodic onset of of the multiscale dynamical properties we uncovered. On the other chaos in maps with zero slope inflection points of cubic nonlin- hand, if specific initial positions with known location within the earity (e.g. the critical circle map) shows strong parallelisms with multifractal are chosen, and subsequent positions are observed only the dynamics at the Feigenbaum attractor described above. at pre-selected times, when the trajectories visit another selected Progress on detailed knowledge about the structure of the orbits region, a distinct q-exponential expression for ξt is obtained. within the golden-mean quasiperiodic attractor, see Fig. 2b, helped determine the sensitivity to initial conditions for sets of Manifestations of q-statistics in condensed matter starting positions within this attractor [5]. It was found that ξt is problems made up of a hierarchy of families of infinitely many intercon- There are connections between the properties of critical attractors nected q-exponentials. Here again, each pair of regions in the referred to here and those of systems with many degrees of multifractal attractor, that contain the starting and finishing freedom at extremal or transitional states. Three specific examples positions of a set of trajectories, leads to a family of q-exponentials have been recently developed (see Table). In one case the dynam- with a fixed value of the index q and an associated spectrum of q- ics at the chaos threshold via intermittency has been shown to be Lyapunov coefficients λq. related to that of intermittent clusters at thermal critical states As in the period doubling route to chaos, the quasiperiodic [10]. In the second case the dynamics at the noise-perturbed dynamics consists of an infinite family of q-phase transitions, each period-doubling onset of chaos has been found to show paral- associated to trajectories that have common starting and finishing lelisms with the glassy dynamics observed in supercooled positions located at specific regions of the multifractal. The spe- molecular liquids [11]. In the third case the known connection cific values of the variable q (in the thermodynamic formalism) at between the quasiperiodic route to chaos and the localization which the q-phase transitions take place are the same values for transition for transport in incommensurate systems is analyzed the entropic index q in ξt. The transitions come in pairs at q and from the perspective of the q-statistics [12].
216 europhysics news NOVEMBER/DECEMBER 2005 to conditions initial compatible with att cality manifest through acrossovercality from canonical to rtclcutro order parameter clustercritical of r amplitude scale thecluster isexpected to evolve by increasing itsaverage again be represented be again by acluster divergence intheexpression for ap faster thanexponential cluster growth with size the order parameter = size of ty excess or densi- magnetization system unstable cluster asingle of theentire critical connection examined canbe instead when of The for toed attractor intermittency. thedynamics atacritical or fluid system amagnetic isseen to relat- be standard of models state in critical anequilibrium fluctuations of dynamicsThe of Critical clusters &intermittency eur r freedom by many of degrees round consideration thedetailed of findsaway thismethod As itiswellknown, theory. the Langevin by aBrownian isusuallydescribed particle motionof erratic The Feigenbaum attractorGlassy dynamics&noise-perturbed Increments in more spinsupthandown). freedom (e.g. microscopic of degrees the sents anenvironment unevenness inthestates with of integrated integrated is This two lengths inspace. expressed canbe between instability The asaninequality cluster. the dominate are of evaluated asthesedetermine aninstability in connection with reconsidered hasbeen recently Contoyiannis andcolleagues [13], codnl n ail o When thedivergence isreached at accordingly so. andrapidly function most probable configurations from partition acoarse-grained e t the s ty closer toty ime e e epresenting the effect of collisions with molecules inthefluid molecules in collisions with epresenting of theeffect tniiyo a of xtensivity ne R p Amongst anddynamical thestatic properties for such single An important element intheanalysis isthedeterminationAn of important r oph actor for intermittency which implies an atypical sensitivity actor for intermittency which impliesanatypical r wal p the p o ubsy ach to criticality andtheinfinite-sizeach cluster to criticality limitatcriti- v ol y sics news u r oesw banapcueo intermittency [10]. rocess we apicture obtain of Z st R o t φ . The conditions The under which theseconfigurations . – R e io fi φ is considered. The analysis, initially developed initially by analysis, The is considered. slctd But asubsequent would fluctuation m islocated. h oiac fthisconfiguration in thedominance of , le and/or size n o ugssnnxesvt ftheBGentropy but suggests nonextensivity of φ – φ for fixed q ( f r -e φ ) nolo q NO ntropy expression. 2) The findingthatthe 2)The ntropy expression. φ is d -statistics [10]. -statistics ( VEMBER r falarge cluster only by its withholding ) of escr R ng R because thesubsystembecause studied repre- takes theposition e ib r r d 0 e / DECEMBER > d b escr > φ φ ( R y thed φ r ( where , ftesm ye From this thesametype. ) of ib r ( ). In a coarse-grained time In acoarse-grained ). r es thespat emnin[0:1)The ) we mention [10]: q saitc.3 oh the 3)Both, -statistics. ynamics o 2005 r 0 r stelcto fa is thelocation of 0 for thesingulari- ial regionwhere R fthespace- of f Z q the cr -statistics. decreases itical r 0 t 2) A mapequivalent to arelationship betweenthe relaxation amplitude cooled liquids. In this analogy thenoise amplitude In thisanalogy cooled liquids. p themost threshold to shown hasbeen with exhibit parallels f discr mapformula general seen The to canbe represent a tices [9]. like coupled intheso-called maplat- other systems coupled to it, basic f and a diffusion, sufficient to determine dynamical correlations, andtheseare isproduced equilibrium thermal by forces, random approach The to moveswhich theparticle by anoise source. time correlationstime when 1) Two-step andintheir two- intrajectories relaxation occurring p similar to thetemperature difference tem- from transition aglass one control parameter man external to noiseused canbe states model insystems with of effect spir o ime andtheconfigurational entropy. erature. Specifically ourresults Specifically are [11]: erature. r ᭡ applications. r o T c mine T e it, he d y d able: ’ ete form for a Langevin equation with nonlinear equation with ete for form aLangevin ‘friction t o att e e r r m [11]. m f ynamics of noise-perturbed logistic mapsatthechaos logistic noise-perturbed ynamics of g tfaue fgas yaisi,freape super- for example, dynamics in, glassy nt features of r eso reo.In only aone-dimensional mapwith freedom. rees of coso nonlinear low-dimensional mapsunder the actors of Summary.The three routes to chaos, properties and properties three routes to chaos, Summary.The σ o r thefl ol etogtt ersn h feto many tocould represent thought be of theeffect uc uto-ispto hoe.In thesame tuation-dissipation theorem. σ µ → h osdrto fexternal noise of the consideration of 0. in thenoise-p < ᭣ square displacement evolution ofthemean (b) Time (red). trajectory c (a)Repeated- map. logistic r initial conditions with 1000 trajectories the noiseamplitude. bylabeled thevalue of are Curves 1]. inside [-1; andomly distr x ell map(blue)and t F > ig 2 for anensembleof . 3: G lassy diffusion er ibut σ turb plays arole FEATURES ed ed < x t 2 > 217 -
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3) Both, trajectories and their two-time correlations obey an ‘aging’ scaling property typical of glassy dynamics when σ → 0. 4) A progression from normal diffusiveness to subdiffusive behav- Nonextensive statistical ior and finally to a stop in the growth of the mean square displacement as demonstrated by the use of a repeated-cell map. (see Fig. 3) The existence of this analogy is perhaps not accidental mechanics and complex since the limit of vanishing noise amplitude σ → 0 involves loss of ergodicity. Localization & quasiperiodic onset of chaos scale-free networks One interesting problem in condensed matter physics that Stefan Thurner exhibits connections with the quasiperiodic route to chaos is the Complex Systems Research Group HNO, Medical University of localization transition for transport in incommensurate systems, Vienna,Währinger Gürtel 18-20, A-1090 Vienna, Austria where the discrete Schrödinger equation with a quasiperiodic potential translates into a nonlinear map known as the Harper map [14]. In this equivalence the divergence of the localization ne explanation for the impressive recent boom in network length corresponds to the vanishing of the ordinary Lyapunov Otheory might be that it provides a promising tool for an coefficient. It is interesting to note that the basic features of understanding of complex systems. Network theory is mainly q-statistics in the dynamics at critical attractors mentioned here focusing on discrete large-scale topological structures rather than turn up in the context of localization phenomena. on microscopic details of interactions of its elements. This view- point allows to naturally treat collective phenomena which are Acknowledgements often an integral part of complex systems, such as biological or Partially supported by CONACyT and DGAPA- UNAM, Mexican socio-economical phenomena. Much of the attraction of network agencies. theory arises from the discovery that many networks, natural or man-made, seem to exhibit some sort of universality, meaning that most of them belong to one of three classes: random, scale-free References and small-world networks. Maybe most important however for [1] There are some parallels between the evolution towards an attractor the physics community is, that due to its conceptually intuitive in dissipative maps and the irreversible approach to equilibrium nature, network theory seems to be within reach of a full and in thermal systems. It should be recalled that for the chaotic coherent understanding from first principles. dynamics of a conservative system (such as the hard sphere gas) Networks are discrete objects made up of a set of nodes which there is no dissipation nor strange attractor. It is of course this are joint by a set of links. If a given set of N nodes is linked by a dynamics that is relevant to Boltzmann’s assumption of molecular fixed number of links in a completely random manner, the result chaos. is a so-called random network, whose characteristics can be rather [2] F. Baldovin and A. Robledo, Europhys. Lett. 60, 518 (2002); easily understood. One of the simplest measures describing a net- A. Robledo, Physica D 193, 153 (2004). work in statistical terms is its degree distribution, p(k), (see box 1). [3] F. Baldovin and A. Robledo, Phys. Rev. E 66, 045104 (2002); E 69, In the case of random networks the degree distribution is a Pois- 045202 (2004). sonian, i.e., the probability (density) that a randomly chosen node λke-λ _ [4] E. Mayoral and A. Robledo, Phys. Rev. E 72, 026209 (2005). has degree k is given by p(k) = —, where λ = k is the average k! [5] H. Hernández-Saldaña and A Robledo, Phys. Rev. E (submitted) and degree of all nodes in the network. However, as soon as more com- cond- mat/0507624. plicated rules for wiring or growing of a network are considered, [6] See, for example, R.C. Hilborn, Chaos and nonlinear dynamics, 2nd Revised Edition (Oxford University Press, New York, 2000). Some network measures [7] C. Tsallis, A.R. Plastino and W.-M. Zheng, Chaos, Solitons and The degree ki of a particular node i of the network is the Fractals 8, 885 (1997). number of links associated with it. If links are directed they [8] H. Mori, H. Hata, T. Horita and T. Kobayashi, Prog. Theor. Phys. either emerge or end at a node, yielding the diction of out- or Suppl. 99, 1 (1989). in-degree, respectively. The degree distribution p(k) is the probability for finding a node with degree k in the network. In [9] See, for example, H.G. Schuster and W. Just, Deterministic Chaos. An th (unweighted) networks the degree distribution is discrete and Introduction, 4 Revised and enlarged Edition (John Wiley & Sons often reads, p(k) = p(1)e k/ q with > 0 being some charac- Publishers, 2005). q teristic number of links. Apart from the degree distribution, [10] A. Robledo, Physica A 344, 631 (2004); Molec. Phys. 103, 3025 important measures to characterize network topology are the (2005). clustering coefficient ci, and the neighbor connectivity knn. [11] A. Robledo, Phys. Lett. A 328, 467 (2004); F. Baldovin and The clustering coefficient measures the probability that two A. Robledo, Phys. Rev. E (in press) and cond-mat/0504033. neighboring nodes of a node i are also neighbors of each other, [12] H. Hernández-Saldaña and A Robledo, in preparation. and is thus a measure of cliquishness within networks. The neighbor connectivity is the average degree of all the nearest [13] Y.F. Contoyiannis and F.K. Diakonos, Phys. Lett. A 268, 286 (2000); neighbors of node i. When plotted as a function of k, a non- Y.F. Contoyiannis, F.K. Diakonos, and A. Malakis, Phys. Rev. Lett. trivial distribution of the average of c allows statements about 89, 035701 (2002). 7 hierarchic structures within the network, while knn serves as a [14] J.A. Ketoja and I.I. Satija. Physica D 109, 70 (1997). measure of assortativity.
218 europhysics news NOVEMBER/DECEMBER 2005 new nodeto anexisting node a isrealized model The by thelinkingprobability setting of nodes. ing no exist- nodeisnotonly added a newly proportional to of thedegrees (se for systemsby mechanics. classical described statistical akey andthus canviolate assumption homogeneously populated, states possible not all (configurations) are equallyprobable or inthesense that systems, toy-model for ‘nonergodic’ of some sort dynamical complex networkscanpotentially isthatthey provide sp mig c epcneto fcomplex the networks with a deep connection of there for issubstantialevidence there exists yet, nocomplete theory t networks which poten- of thevariety of portion asmall review will In theremainder we seem to exhibit auniversality inthisrespect. kinds various of models real-world networks aswellof tions of distribu- Degree growing andpreferentially networks. organizing cal mechanics are often called networks of isthereason This why thesetypes complex systems. phenomena toes associations andscalingphenomena critical in which rais- any scale, characteristic bare of becomes apower-law, inmany distribution casesthedegree In particular, involved. anetwork canbecome rather simpleconceptthe seemingly of eur thedistancebeing between the new andnode node par asy complex networks noticed ithasbeen distributions thatdegree t o itseems almost show power-law asymptotic distributions, degree real-world networks andmodel number thatavast of and thefact networks Given theabove of characteristics recently proved in[3]. mo solution to the analytical The also allow for preferential rewiring. node no meaningthatany new preferentialso-called attachment rule, a growing networks with in[4]describes model The [5]. tials h ereo node of the degree the maximum entropy –one extremizes principle where where entropy isdefined as mechanics, mann-Gibbs statistical Boltz- Nonextensive mechanics of isageneralization statistical 2]. complex systems andnonergodic systems [1, phenomena, e nonextensive mechanics. statistical of forandubiquity theimportance sign for canonical ensembles tributions compatible with compatibletributions with e al aeantrlln onnetniesaitc.Even though haveially linkto anatural non-extensive statistics. mb o bvious tobvious for look aconnection betweennetworks andnonex- nsi nst o ebigaddt h ytmlnsisl to analready existing tode added being thesystem linksitself Over thepasttwo decades theconcept of H S Recently in[6]preferential attachment networks have been e b mp ame del hasa oph nds t inc ht b e o v dd r e stat i w o aints, t e,but alsodepends on theEuclidean distance between des, BG in the network with aprobability thatisproportionalin thenetwork to with ot x ine h eybgnigo therecent of efforts modeling of beginning e thevery t e y e e d er er v h oe n[]wihhsn erc.In acareful in[4]whicho themodel hasnometrics. d inE sics news clyflo oe-as[] or even exactly followically power-laws [4], er, it has been found ithasbeen thatnetworks exhibiting dis- degree er, stands for q ist ue t the c b ioilpprb .TalsadJP on.Another Tsallis Boon). andJ.P. byditorial paper C. ical p q e has been extremelyhas been successful inaddressing critical -e uc ing fi o the fact that the most general Boltzman factor thatthemostgeneralBoltzman o thefact complex networks. complex xponential as a result, with thenonextensivity with xponential asaresult, iensae where theattachment probability for lidean space, o h i rresponding distribution isthe distribution rresponding i.e. , y xed uniquely by parameters. themodel NO sics. Boltzmann-Gibbs. VEMBER p link (extensive) q ∝ -exponentials are limited notatall to k / i i DECEMBER In wasextended [5]thismodel to . to be A further intriguing aspect of aspect intriguing A further is the p link f–inthephilosophy of If q -exponentials innature 2005 ∝ q epnnil aswas -exponential, nonextensive statisti- nonextensive k i / r α q i S ( q ≠ α q i under certain ; -e 1 instance in α ≥ q xp -e = 0corre- 0), with with 0), xp o ne o ne nt (1) ial n- r i nonextensivity parameter ward in[8]alsounambiguously exhibit nlglgsae lal xiiigapwrlw Figure 3shows clearly exhibiting apower-law. scale, in log-log g discrete states insome further hasbeen model The ‘phase space’. asjumps between of conserved thought which canbe – strictly links isnot thenumber thismodel of to Due thenature of [9]. theexisting nodes inthenetwork and islinked randomly to any of At anew nodeisintroduced thesametime to thesystem removed. scerfo h orlto ofceto the is clear from thecorrelation coefficient of int wasproposed in[7]amodel thatbuilds [5]but upon networks, networks random are recovered intheBoltzmann-Gibbs limit. i.e., d c no sing (undir nodesinan In afixed thismodel number of inrealtransfer gases. to the energy-momentum transferedinanalogy linksget nodes, two where collision uponan (inelastic) of ‘gas’ of type as acertain dist q have distributions degree theresulting Again, Euclideannot being but internal distance. ne b theresulting networks have of analysis distributions thedegree ne els exhibitingels network of thistype snapshot of makes the straight lines (inset) thatthere lines(inset) exists anoptimalvalue of straight h havetions subjectedto been analysis astatistical where the o nrlzdt xii itnedpnec si 6,however eneralized to exhibit adistance dependence asin[6], -lo y e e linking rate of linking ᭡ q is smallt r e g e p tw A model forA model nongrowing networks which wasrecently put for- na fott nesadteeouino socio-economic In to aneffort understand theevolution of d roduces scheme which depends arewiring on the r n sho -exponential degree for distribution ded to connect the two nodes. The emerging degree distribu- degree emerging The ded to connect thetwo nodes. ree distribution isa ree distribution ributions. This model wasmotivated model This by interpreting networks ributions. othesis o garithms of the degree distribution for several values distribution thedegree of of garithms esp es; le no F o ig r ce)ntokcn‘eg’ .. two fuseinto nodes one networkected) can i.e., ‘merge’, o k distanc the linkc . 1: nding (cum wn to be wn d o mak ,wihkesteuino ik fthetwo original linksof which keeps theunion of e, Snapshot ofanon-growing dynamic networkwith q lgrtmalna ucini ,showing that the ink, alinear function -logarithm f q -e e b e c _ r xp o q q nnecting thetwonnecting nodesbefore themerger is = 1, for details see [8, 9].The shown network 9].The for detailssee[8, = 1, epnnildge itiuin.The -exponential distributions. degree -e o onnection patterns visible.onnection e ltv)dge itiuini hw nFg 2 ulative) inFig. isshown distribution degree xp ne tw o ntials couldntials rejected. notbe q e ne ntonds .. h ubro steps thenumber of i.e., en two nodes, -e q q xp epnnilfr.I i.1we show a In Fig. -exponential form. In thelarge . ntials with aclear with ntials o nential. pars propars toto α N limit, = 256 nodes anda = 256nodes q -exponential degree α for themany mod- -d q q e -logarithm with -logarithm approaches one, p e dneo the ndence of FEATURES q int which , ernal 219 q . It r i features FEATURES
onto nodes and links. In this view the basis of nonextensive sys- tems could be connected to a network like structure of their ‘phase space’. It would be fantastic if further understanding of network theory could propel a deep understanding of nonextensive statis- tical physics, and vice versa, making them co-evolving theories.
References [1] C. Tsallis, J. Stat. Phys., 52, 479-487, 1988. [2] M. Gell-Mann and C. Tsallis, eds., Nonextensive Entropy - Interdisciplinary Applications (Oxford University press, N.Y., 2004); C. Tsallis, M. Gell-Mann, and Y. Sato, Proc. Natl. Acad. Sc. USA 102, 15377 (2005). [3] R. Hanel and S. Thurner, Physica A, 351, 260- 268, 2005. [4] A.-L. Barabasi and R. Albert, Science, 286, 509- 512, 1999. ᭡ Fig. 2: Log-log representation of (cumulative) degree [5] R. Albert and A.-L. Barabasi, Phys. Rev. Lett., 85, 5234-5237, 2000. distributions for the _same type of network with various [6] D. J. B. Soares, C. Tsallis, A. M. Mariz, and L. R. da Silva, Europhys. system sizes N and r = 8.The distribution functions are from Lett., 70, 70-76, 2005. individual network configurations without averaging over identical realizations. On the right side the typical [7] D. R. White, N. Kejzar, C. Tsallis, J. D. Farmer, and S. White. exponential finite size effect is seen. A generative model for feedback networks, Phys. Rev. E (2005), in press cond-mat/0508028. A quite different approach was taken in [10] where an ensem- [8] S. Thurner and C. Tsallis, Europhys. Lett., 72, 197-203, 2005. ble interpretation of random networks has been adopted, [9] B. J. Kim, A. Trusina, P. Minnhagen, and K. Sneppen, Eur. Phy. J. B, motivated by superstatistics [11]. Here it was assumed that the 43, 369-372, 2005. average connectivity λ in random networks is fluctuating [10] S. Abe and S. Thurner, Phys. Rev. E, 72, 036102, 2005. according to a distribution Π(λ), which is sometimes associated with a ‘hidden-variable’ distribution. In this sense a network [11] C. Beck and E. G. D. Cohen, Physica A, 322, 267-275, 2003. with any degree distribution can be seen as a ‘superposition’ of [12] H. Hasegawa, Physica A, in press, 2005; cond-mat/0506301. random networks with the degree distribution given by [13] G. Palla, I. Derenyi, I. Farkas, and T. Vicsek, Phys. Rev. E, 69, 046117, ∞ λke-λ p(k) = dλΠ(λ) —. In [10] it was shown as an exact example, 2004. ∫0 k! that a power-law functional form of Π(λ) leads to degree distrib- [14] C. Biely and S. Thurner. Statistical mechanics of scale-free networks at a critical point: complexity without irreversibility. cond-mat/ 0507670. 1 utions of Zipf-Mandelbrot form, p(k)∝ —γ , which is equivalent (k0+k) to a q-exponential with an argument of k / · and given the sub- stitutions, = (1 - q)k0 and q = 1 + 1/γ. Very recently a possible connection between small-world networks and the maximum Tsallis-entropy principle, as well as to the hidden variable method [10], has been noticed in [12]. In yet another view of networks from a physicist’s perspective, networks have recently been treated as statistical systems on a Hamiltonian basis. It has been shown that these systems show a phase transition like behavior [13], along which network structure changes. In the low temperature phase one finds networks of ‘star’ type, meaning that a few nodes are extremely well connected resulting even in a discontinuous p(k); in the high temperature phase one finds random networks. Surprisingly, for a special type of Hamiltonian, networks with q-exponential degree distributions emerge right in the vicinity of the transition point [14]. While a full theory of how complex networks are connected to q ≠ 1 statistical mechanics is still missing, it is almost clear that ᭡ Fig.3: q-logarithm of the (cumulative) distribution such a relation should exist. It would not be surprising if an function from the previous figure as a function of degree k. Clearly, there exists an optimum q which allows for an understanding of this relation would arise from the very nature of optimal linear fit. Inset: Linear correlation coefficient of ln networks, being discrete objects. More specifically, it has been q P(≥ k) and straight lines for various values of q.The optimum conjectured for nonextensive systems that the microscopic value of q is obtained when ln P(≥ k) is optimally linear, i.e., dynamics does not fill or cover the space of states (e.g. Γ-space q where the correlation coefficient has a maximum. A linear lnq (6N dimensional phase space) for Hamiltonian systems) in a means that the distribution function is a q-exponential; the homogeneous and equi-probable manner [2]. This possibly slope of the linear function determines . In this example makes phase space for nonextensive systems look like a network we get for the optimum q = 1.84, which corresponds to the itself, in the sense that in a network not all possible positions in slope γ = 1.19 in the previous plot. space can be taken, but that microscopic dynamics is restricted
220 europhysics news NOVEMBER/DECEMBER 2005 ics [1]. maximizat founded on underpinning an avariational with tion “entropy” ferent ineach. cases except different are are aspects utilizedthegoals because dif- standing aninterplaybeen between thetwo research andtheir under- efforts w t features These are notutilized inthesamemanner inquan- space. state andthetensor principle productsuperposition of structure the tions incomposite systems from features arising thetwin of quantum correla- Fundamentally of thisisanother aspect rapid. slowinitially but over thepasttwo decades progress very hasbeen development Aparallel inquantum was information below). have been isstudied given inthelastcentury intheBox (see some that Alistof etc. superconductivity, such asferromagnetism, body systems are physical manifested incharacteristic properties many- andcorrelations interactions The among theconstituents of DC Washington, Naval Research Laboratory, Rendell, andR.W. Rajagopal K. A. to quantum information implications mechanics : Nonextensive statistical eur um man ith p ohaeso investigation are on based aprobabilistic founda- areasBoth of oph r o y T . ete fmany-body systems isgiven there because has of perties sics news y-b he sp T o,which may Mechan- Quantum Statistical called be ion, he basicq o d e y p cifi h c f sc.I h o,acorresponding list parallel In theBox, ysics. NO uantum mechanical principles applyuantum mechanical to principles both o VEMBER r m o h nrp safntoa fthe theentropy of asafunctional f / DECEMBER 2005 ab the system - “energy”. to leads This thefamiliarexponential prob- theHamiltonian of constraintscertain such astheaverage value of von Neumann entropy to subject then given by amaximization of andassociated quantities. matrix density initions of Table See made explicit be presently. will Ifor def- matrix density e r Non-equilibrium properties such astheanomalous equation. p not o Tsallis andBoon). Box intheIntroduction of the (See power- law [2]. probabilities incontrast to theBGtype to which leads q, theTsallis entropy anentropic with index, of e q r laxat ᭡ ese uilib h qiiru rpriso themany-body systems are properties equilibrium of The nqatmifrain themaximum entropy scheme is In quantum information, ilit F ntl f use because its origins are elsewhere as will be made clear arebe elsewhere itsorigins aswill usebecause f ig ies o io r .1: ium are properties by studied aquantum evolution time y n in time aren intime often thequantum version analyzed with . T Properties andFeaturesProperties ofaSingle Quantum Bit he e f the B v lto srpae ypoesn finformation olution isreplaced by processing of oltzmann-Gib bs (BG)f Schrieffer; oosy Rosen; Podolsky, Inf Transparency Inducedtically Condensation; EIT BCS BEC QI EPR ᭣ the terms B or – Quantum o – Electromagne- – Einstein, – B – B ma x: Explanation of ose –E ar tion; r.Any non- orm. en Cooper, deen, inst FEATURES ein 221
features FEATURES accomplished by quantum operators, such as unitary or mea- coding, error correction, etc. associated with transmission and surement operators, enabling the passage of given initial reception of information, and a unit of information, the classical information to a known destination. The energy level (band-) bit, log2; and (4) an additive measure of information, which structure of the states plays the central role in determining the quantifies average information per symbol, the von Neumann global physical characteristics of the many body system which led N entropy, S = -∑ p(xi) log p(xi). When more than one source is to the Silicon-based classical computer. In this development, the i=1 classical information structure was sufficient in its construction considered, generalization of these concepts lead to the notions and operation. In contrast, the quantum information structure of (a) marginal probabilities, (b) conditional probabilities, and exploits the state function of the many-body system and the related entropies, and (c) relative entropy which enables compar- corresponding development of a quantum computer is presently ison of two sources. The above description is for digital sources. at an infant stage and it will perhaps be based on quantum optics There are also continuous sources (e.g. light) which describe or state-space aspects of condensed matter. The Box gives a flavor analog systems. All this changes dramatically when quantum of this mutual relation between many-body systems and informa- physics is the underpinning structure. To appreciate this change, tion theory, highlighting the twin aspects of the quantum energy we first display the foundations of quantum theory that subsumes level structure and the quantum state. classical theory (see Table 1 for the relevant definitions). Classical information theory is a description of communicating Quantum theory involves (a) superposition principle, (b) information (signals, alphabets, etc. denoted by real numbers xi, uncertainty principle, and (c) the system density matrix govern- i=1, 2,…N) from one place to another using systems governed by ing the probabilistic description of the system. Physical quantities classical physics. It rests on four intuitive ideas: (1) probability associated with the system are represented by Hermitian operators structure of a message source, p(xi); (2) a notion of additive infor- whose average values defined in terms of the system density mation content, I(xi)= -log p(xi), where it is the sum of each if matrix give their measured values. The density matrix replaces the there are independent sources; (3) use of a binary arithmetic (see probability of occurrence of events in classical theory. The classi- Fig. 1 - yes, no - classical bit), which is sufficient to develop cal bit now takes on a more general representation, because in the quantum description the superposi- tion principle comes into play. See Fig. 1, for a pictorial representation of a qubit. Conceptually these three fea- tures give a more general description of the system than the classical theory. Thus the classical information theory based on probabilities associated with signals and the consequent the- oretical structure defining entropy as the information measure, algorithms, coding of information, etc. are all generalized in the quantum version with important consequences. The superposition principle precludes cloning and deletion of information and significantly improves the classi- cal search algorithm. The uncertainty principle places conditions on mea- surements of a certain class of physical variables of the system and when there is more than one signal or source, gives conditions for “inde- pendence” or “separability” of the systems. In fact, classically correlated signals become generalized to include entanglement and other nonlocal features in the quantum context. See Fig. 2 for a description of these con- cepts in the simple case of two qubits. More precisely, quantum entangle- ment implies that the parts do not determine the whole. This feature gave rise to dense coding, “tele- portation” and novel quantum cryptography that have no classical counterparts. Grover’s quantum search algorithm exploits the super- ᭡ Table 1: Quantum Density Matrix Description position principle and makes the search faster than the classical version.
222 europhysics news NOVEMBER/DECEMBER 2005 d T p the Table asasum of written 1)canbe and ρ ρ lo states asfol- of defined theseparability Werner tonor isitguaranteed succeed. itdoes notquantify it entanglement, While detects thisviolation tum theory. quan- have established thisfeature of inequality theBell testsExperimental of to indicate quantum non- locality. wastaken violation This realistic model. that could explained notbe by any local correlations asequencemeasuring of due toby inequality Bell acertain of aviolation ly thiswasstated interms of Original- states andtheirmeasurement. entangled ence of isthecharacterization sci- problem inquantum information Afundamental ly correlated behavior. systems canexhibit randomyet perfect- whereby distant involves entanglement, which oftenquantum non- locality sult [3]. one may con- For exposition, adetailed isanother successfulrithm application. algo- quantum Shor factorization The eur w not clear thatadditive measures such asvon Neumann entropy itis thesystem isentangled, If Table 1). itscomponents (see ing measures of thecorrespond- defined by thesum of A discussion to thisday [4]. of the subject pointis This measure for thispurpose. that one could employ anadditive itisnotobvious making upasystem, correlations amongintrinsic theparts mann e f originally formation”, “entropy of of measureglement isoften stated interms entan- The then thesystem isentangled. t semi-definiteness under of theaction positive does of notretain itsproperty d the if wasfirst statedproperty by Peres: ,7 ,9.Tbe2gvsasmayo hs eut.In thisTable, w theseresults. Table of 2gives asummary 9]. 8, 7, 6, state where the von Neumann measure gives thewronganswer [5, known acertain of criterion in correctly theseparability obtaining e theTsallis entropy was entanglement, non-additive feature of b components, T m eeslo n fthesubsystems, one of ime reversal of ruae ntrso thevon Neu- ormulated interms of mployed to characterize it. This was shown to wasshown This more be successful mployed to it. characterize e e ab r ( ( e hose composite systems for which this ould b e c diiemaueo asystem is n additive measure of w o A,B A,B c nsit h alako quantum physics is of hallmark The o oph d le 1. ntangled. A criterion for testing Acriterion this ntangled. s: ∑ o mp cso h est arcso its of matrices thedensity of ucts i sdrasml pca opst ie tt ftwo qubits compositensider asimplespecial mixed state of ), ) = y mat if w y o nt osit e ap i S sics news f the sy ,then thesystem isseparable = 1, ∑ inc r a c i opy for pure states given in io p w i fthecomposite state of rix omposite system (AB)(see e entanglement isdue toe entanglement n d ρ r i o ρ i ( p st i A ( o it ngnrl niiaigtepsiiiyo a of Anticipating thepossibility in general. riate A e ) & es notholdare said to m density matrix, m density ) NO ⊗ ρ ρ VEMBER i i ( ( B B ) intheform ), 0 ≤ / DECEMBER w i ≤ 1, 2005 ᭡ e ᭡ xample T F able 2: ig .2: Illustration of quantumIllustration entanglement basedonasingletwo quantum-bit . S Comparison of Separability Criteria ofthe Comparison Criteria ofSeparability Werner State alien t f ea tures ofpure andmixed state matrices are density alsoillustrated. tain values of the P andthesearemethods compared result theexact by obtained with entangled pure stateentangled for ab thissystem equal prob- occur with thefour statesmeant thatall of and ad pure state matrix density which isasum of the called Werner state, lt,14 hssaehsteitrsigpoet fbeing an state This hastheinteresting property of 1/4. ility, e r es criterion inTablees criterion 2. e nsit y mat F . The separability condition is deduced separability The by various . r ix r e rsnignie ( presenting noise: F =1, but a separable mixed but aseparable state forcer- =1, I 2 ⊗ I 2 )/4. FEATURES B y no ise is 223
features FEATURES
Another fundamental issue is the proper discriminating measure when two systems are under consideration. In classical information theory, one employs the Kullback – Leibler relative entropy for this Entropy and statistical purpose which also has its quantum version. These are also additive measures and the Tsallis counterparts of these have been put forward and employed in the quantum context as well [10, 11]. complexity in brain There is promise in future work using the Tsallis approach to problems arising in quantum information theory, especially in the areas of quantum algorithms and quantum computing. activity There has been some discussion of the thermodynamics of information, in particular quantum information. Since there are A. Plastino 1, 2 and O.A. Rosso 3, 4 hints that quantum entanglement may not be additive, and since 1 Departament de Fisica. Universitat de les Illes Balears. E-070701 the concept of entropy has been introduced into the discussion, an Palma de Mallorca, Spain; examination of maximum Tsallis entropy subject to constraints 2 Facultad de Ciencias Exactas and Instituto de Física La Plata such as the Bell-Clauser- Horne-Shimony-Holt observable was Universidad Nacional de La Plata and CONICET, Argentina. studied for purposes of inferring quantum entanglement [5, 6]. C.C. 727, 1900 La Plata, Argentina * 3 Institut für Mathematik, Universität zu Lübeck, Wallstrasse 40, Acknowledgements D-23538 Lübeck, Germany AKR and RWR are supported in part by the US Office of Naval 4 Instituto de Cálculo, Facultad de Ciencias Exactas y Naturales, Research and AKR by the Air Force Research Laboratory, Rome, Universidad de Buenos Aires. Pabellón II, Ciudad Universitaria. New York. 1428 Buenos Aires, Argentina. * * Permanent address About the authors A.K. Rajagopal obtained his Master’s degree from the Indian Institute of Science, Bangalore, India and a Doctor’s degree from lectroencephalograms (EEG) are brain-signals that provide us Harvard University, Massachusetts, USA. In the past twenty years E with information about the mean brain electrical activity, as he has been a Research Physicist at the Naval Research Laborato- measured at different sites of the head. EEGs not only provide ry, Washington D. C. Recently the focus of his work has been in insight concerning important characteristics of the brain activity Statistical Mechanics and Quantum Information. but also yield clues regarding the underlying associated neural R. W. Rendell obtained a PhD in Physics from the University of dynamics. The processing of information by the brain is reflected in California at Santa Barbara. For the past twenty years he has been dynamical changes in this electrical activity. The ensuing activity- a Research Physicist at the Naval Research Laboratory, Washing- variations are found in (i) time, (ii) frequency, and (iii) space. It is ton D.C. Recently the focus of his work has been in Nanoscience then very important to have theoretical tools able to describe qual- and Quantum Information. itative and quantitative variations of these brain-signals in both time and frequency. Epileptic signals (ES) are specially important sources of brain information. We concentrate on ES in this article. References The EEG-signal is what mathematicians call a non stationary time- [1] A. K. Rajagopal in Lecture Notes in Physics, (eds. Sumiyoshi Abe and series (ST). Powerful analytical methods have been developed over Yuko Okamoto) Springer-Verlag, New York (2001) gives an the years to extract information from ST. The brain ST is contami- account of Tsallis entropy as a unified basis of both quantum sta- nated by another body-signals (called artifacts) due mainly to eye tistical mechanics of many-particle systems as well as quantum movements and muscle activity. Artifacts related to muscle con- information. (Lecture series were held during February 15-19, tractions are specially troublesome in the case of epileptic seizures 1999) that exhibit rigidity and convulsions (called tonic-clonic seizures). [2] M. Gell-Mann and C. Tsallis (eds.) Nonextensive Entropy – Interdisci- The troublesome artifacts acquire here very high amplitudes that plinary Applications, Oxford Univ. Press, New York (2004); contaminate the whole recording. A drastic way of preventing this Physica D 193, (2004), Special Issue on Anomalous Distributions, contamination is by injecting curare to the patient. The classic work Nonlinear Dynamics and Nonextensivity, edited by H. Swinney of this type is that of Gastaut and Boughton (GAB) [1], who and C. Tsallis. described the characteristic frequency pattern of a tonic-clonic [3] M. A. Nielsen and L.L. Chuang, Quantum Computation and Quan- epileptic seizure (ACES) in patients subjected to muscle relaxation tum Information, Cambridge Univ. Press, Cambridge (2000) from curarization and artificial respiration. They found that, after a [4] P.W. Shor, Commun. Math. Phys. 246, 453 (2004) short period characterized by phase desynchronization of the brain’ signals, a typical feature appears in the records, baptized by them as [5] A. K. Rajagopal, Phys. Rev. A 60, 4338 (1999) an “epileptic recruiting rhythm” (ERR, at about 10 Hz). Later, as [6] S. Abe and A. K. Rajagopal, Phys. Rev. A 60, 3461 (1999) the seizure ends, they detected a progressive increase of lower [7] A. K. Rajagopal and S. Abe, Phys. Rev. Lett. 83, 1711 (1999) frequencies associated with the convulsive phase. The TCES pro- ceeds as follows: about 10 s after seizure onset, lower frequencies [8] S. Abe and A. K. Rajagopal, Physica. A 289, 157 (2001); C. Lloyd, and M. Baranger, Phys. Rev. A 63, 042103 (2001); S. Abe, Phys. Rev. A (0.5-3.5 Hz) are observed that gradually diminish their activity. The 65, 052323 (2002) convulsive activity is associated to generalized polyspike bursts from muscle-jerks. Very slow irregular activity dominates then the EEG, [9] A. K. Rajagopal and R. W. Rendell, Phys. Rev. A 66, 022104 (2002) accompanied with a gradual frequency increase of up to (3.5 - 12.5 [10] S. Abe, Phys. Rev. A 68, 032302 (2003) Hz), indicative of the end of the seizure. [11] S. Abe and A.K. Rajagopal, Phys. Rev. Lett. 91, 120601 (2001) In Fig. 1.a we depict a typical EEG signal (sample frequency ωs = 102.4 Hz, for signal acquisition details see Ref. [2]) corre-
224 europhysics news NOVEMBER/DECEMBER 2005 mltd f100 amplitude of basis o a is called what interms of which expresses signal theoriginal use another Math tool called “wavelet isamethod This analysis”. For computing thesequantifiers we quantifiers for EEGanalysis. q 1to Fig. Thus we go from the “bare”-signal of information. an ab c Recordings were under video performed sponding to aTCES. eur at80 seizure The starts theseizure. ent stages of at ar theconvulsive phase of to itispossible establish the beginning er, Howev- theEEGmore due difficult to muscle artifacts. analysis of amplit slow waves lower by fastones with superposed of “discharge” q mathe- allows one to Information evaluate specific called Theory matics then thebranch of aPD, one isinpossession of If series. (PD)to aprobability thistime associating distribution idea thatof It wasourmain coefficients thatwe derive from theEEGsignal. basic e The 1. Fig. like formsignals information able to thatof extract a “Wavelet approach thatis Transform –Information Theory” we follow iscalled what thatone regards asatime-series, signals, clonic) transition. imp o atfir htcntl saltaotteEG We three devised uantifiers thatcantell usalotabout theEEG. uant corresponding abrupt activity decreases bands inthelow frequency corresponding abrupt activity the 10 “epileptic recruiting rhythm”- discernible. frequency bands frequency E inl.From thisfigure itisclearthat theseizure isdominated bands by themiddlefrequency EEG signal). epileptic seizure. 155 ᭡ nt In the mathematical characterization of these brain electrical thesebrain In of themathematicalcharacterization oph ossib F ound 125 r li re ohv nacrt eemnto fthediffer- ol inorder to have anaccurate determination of r ig it up s ude. This discharge This lastsapproximately 8 ude. eet ftheMath we are used wavelet called transform lements of f oieta,vsal,thedramatic (tonic transition stage)to from convulsions rigidity (clonicstage)around 125 visually, Notice that, . ies, . y an spa 1: sics news ea ftesga’ mltd.Notice thatitisalmost thesignal’s amplitude. t decay of le t a) cal o v EEG signal for a tonic-clonic epileptic seizure.The seizure starts at 80 seizure starts EEG signalfor atonic-clonic epilepticseizure.The b) le s eo ucin.Wavelets are just anappropriate functions. ce of , is d q Time-evolution oftherelative wavelet energy corresponding to EEGnoise-free signal(withoutcontribution of n h n ftheseizure at 155 and theend of ually detectually therigidity-to-convulsions (tonic- µ V uantifiers, that contain otherwise inaccessible thatcontain otherwise uantifiers, B NO 1 fewrs h ezr ped,makingthe theseizure spreads, Afterwards, . and VEMBER B 2 representing contributions12.8 > frequency , / DECEMBER Hz (shadowed area in the figure).The vertical lines represent the start andtheendingof linesrepresent vertical thestart (shadowed area inthefigure).The 2005 s s and hasamean , where there is s , with a with , wavelet coefficients represent theelements each resolution level ( the signal p ={ under analysissignal isgiven by sampled-values est tion andefficiently provide and adirect information full both approximations atscales P We decided to regard these coefficients are given by not islinkedtheusual energy with theconcept of functions, of 2 N M W cy planes[2, phenomena andfrequen- specific thetime inboth characterizing thatconstitutessity aconvenient tool for and detecting dist we define thenormalized ly, elements here called of basis, signal-energy can be obtained inthe fashion obtained canbe signal-energy the sig b out over frequency-resolutioncarried pertinent all levels (denoted B y anindex j r 6 ω = { a } collected using a uniform time grid. The wavelet-expansion The is } collected grid. time usingauniform e = lo Since thefamily { imation of signal-energies at different frequencies. The brain- The atdifferent signal-energies frequencies. imation of and v ions derived from Fourier’s theory for such spaces. The wavelet The ions derived from Fourier’s for such spaces. theory t Hz iuinfrteeeg.Clearly, forribution theenergy. e e . ρ le d as the local residuald asthelocal errors between successive signal- j nal b corresponding mainly to muscular activity that blurthe corresponding mainlyto muscularactivity g t E } can be considered} canbe probability den- energy asatime-scale B 2 6 ( ne M (3.2 -0.8 s rgy y theamplit ). and theclonicphaseat 125 j 4, ) andw t T ) corresponding to thefrequencies 2 (R 5]. he wa WE), Hz ψ j ).This behavior associated be can with ).This ie s ( riten as velet coefficient { series 1 …,- = -1, j,k ρ ( ud C t j )} is an orthonormal basisfor thespace)} isanorthonormal = j j ( -itiuini uhabss[] The insuch abasis[3]. e-distribution and k ε ρ ρ <, ) =< j j / j vle,which represent the -values, ψ tdfeetsae,asaprobability atdifferent scales, , ε B N tot j,k j 3 t , will be be will , .It contains on information + 1. and ( . ) = t T ,andallow for characterizing ), ψ ∑ his R ∑ j,k B j ρ 4 n h inleeg,at > andthesignal-energy, -1 j (12.8 -3.2 j = WE isour s ε = 1andthedist -N .The seizure endsat .The j = C ∑ ε ∑ j C ( k t k ot j k C ( fthis distribu- ) of s k = | j C is notclearly ( )} canb Hz k ∑ j ( fi ) k r j ,witha ), j -1 ψ )| st quant s < 0 FEATURES ω n 2 , j . The total The . ,k n ε ( ribution Relative t ≤ e int j ,…, = 1, Final- . ), | ω with ifi 225 e | e r- ≤ r . features FEATURES
Information theory introduces tools, called entropic information ordered process can be represented by a periodic mono-frequency measures, that provide useful criteria for analyzing and comparing signal (signal with a narrow band spectrum). A wavelet representa- different probability distributions. In looking for degrees of “disor- tion of such a signal will be resolved at just one unique wavelet der” in our brain-signal, we devised a math-tool, our second resolution level j, i.e., all relative wavelet energies will be (almost) zero quantifier, that we call a Generalized Escort-Tsallis Entropy (GWS) except at the wavelet resolution level j which includes the represen- [4]. It is written as tative signal’s frequency. For this special level, the relative wavelet energy will be (in our chosen energy units) almost equal to unity. As (1) a consequence, the GWS will acquire a very small, vanishing value. A signal generated by a totally random or chaotic process can be taken as the representative of a very disordered behavior. This kind is a normalization constant that enforces the convenient of signal will have a wavelet representation with significant contri- inequalities 0 ≤≤1, that simplify the analysis to be performed. butions coming from all frequency bands. Moreover, one could The GWS is a measure of the degree of order/disorder of the expect that all contributions will be of the same order. Consequent- signal and thus yields useful information concerning the underlying ly, the relative wavelet energy will be almost equal at all resolutions dynamical brain-process associated with the signal. Indeed, a very levels, and the GWS will acquire its maximum possible value.
᭡ Fig. 2: Temporal evolution of two quantifiers a) the normalized escort-Tsallis wavelet entropy (GWS) and b) Jensen escort-Tsallis wavelet statistical complexity measure (JGWC), corresponding to an EEG noise-free signal (see caption Fig. 1).The behaviour of the GWS clearly varies with q (see Figs. 2.a) in the temporal domain. During the pre- and post-ictal stages, these normalized GWS-values acquire a rather regular, constant behaviour, with a dispersion that diminishes as q grows. For all q ≥ 1, the normalized GWS (JGWC) values during the ictal stage are much smaller (greater) than those pertaining to the pre-ictal stage.This difference is better appreciated in the time range corresponding to the “epileptic recruiting rhythm”(represented by a shadowed area in the figure). These features suggest that the escort-Tsallis entropy measure constitutes the appropriate tool for characterizing the tonic and clonic stages.The minimum absolute value of the normalized entropy is to be found in the vicinity of ~ 125 s, in agreement with the medical diagnosis: in that neighbourhood one encounters the tonic-clonic “phase transition”. c) Ratio between the temporal mean value corresponding to ictal and pre-ictal epochs as function of the parameter q for the normalized escort-Tsallis wavelet entropy (GWS). d) Same for Jensen-escort-Tsallis wavelet statistical complexity measure (JGWC).This behaviour clearly illustrates the superiority of the q > 1 techniques that magnify differences between ictal and pre-ictal stages, critical for clinical purposes.
226 europhysics news NOVEMBER/DECEMBER 2005 signal (Fig. 1.a) without contaminant artifact-contributions ( contaminant 1.a) without artifact-contributions (Fig. signal tion probability distribu- theunderlying reflectedbe inthefeatures of physical exists structure thatshould of degrees possible of range In between thesetwo instances special awide (zero complexity). to of structure order speak no possess andmaximalrandomness perfect the opposite extremes of Certainly, physical description. theaccompanying are nottotally of independent aspects one, correlations on theother andstructural on theone hand, ness, Random- 7]. 6, [5, thebrain) physicalthe pertinent system (here, toposition capture therelationship between thecomponents of to ina be i.e., thecorrelationalall thatmay structures present, be a system isnotautomatically tantamountto adequately grasping le We denote these 6band-resolution priate wavelet analysis [3]. and o “ t fr Once thehigh related thatblur the EEG[2]. to muscular activity pcso lengths epochs of “ theabove listed three wavelet quantifiers for the tion of quantifier. our far” w where [ w (4)) that distribution (4)) thatdistribution (2)to we consider for thecomplexity (Eqs. evaluation if obtained P constantis anormalization (0 distribution Q eur t and above) represent the amount of “disorder”. The quantity The “disorder”. above) represent theamountof and Calbet [7]forand Calbet agiven probability distribution Mancine (SCM)introduced by López-Ruiz, complexity measure” relativelyassociated with large entropic values. order one hasinmindhere, “new” thekindof example of typical life isa Biological for which theentropy small. isvery structures, crystal with for instance, isnottheone associated, “order” of type This situations. spontaneouslyincertain thatarise structures off-equilibrium “order”. It refers to non-equilibrium measure of called Complexity isa “complexity”. introduced inthisrespect, W,GSadJW,we ignore the contributions from the GWS andJGWC, RWE, thethree quantifiers thatwe have introduced above: tion of o r” en ifrn rmzr fthere exist or “privileged”, different being fromure”, zero if mo remaining” signal. For this purpose the signal is divided into isdivided For thesignal thispurpose signal. remaining” f e e a vels by he [ . The disequilibrium disequilibrium The . [ q B P Figure 1.bdisplays thequantifier RWE corresponding to theEEG Ascertaining the degree of unpredictability and randomness of andrandomness unpredictability Ascertaining of thedegree For EEG-work sixfrequency for bandsare anappro- important We formfor adopt functional the thefollowing “statistical P v P oph ue e 1 r 6 ] = r B A , P ). le e e lik w P 2 [ ncy ar t statistical complexity measure complexity t statistical Q hr,ta o h E) A new notion recently hasbeen that for theEEG). (here, ( e P ith respectto bands (>12.8 ] astheJ y T Q P ] = sics news stands f Bj efloig ealddsrpino thesignal-analysis of description detailed he following, e 0 is located (inthisspace) from distribution theuniform l y” · ( P ∑ t P | ifa j 1 e stat | | =1, N j P [ =1 mn h cesbesae ftesse,and thesystem, among theaccessible states of t r lmntd we cananalyze evolu- thetime arects eliminated, P e 2 o ] represent the nsen-escort-Tsallis divergence [8]given by , es amo ( r theso-cal L p P = 2.5 NO j e ,6,andproceed asfollows intheevalua- 6), …, ) P ] isd 1/ P C 2 q Hz VEMBER Q . given by the RWE. The JGWC The isour given by theRWE. (b [ T P gteacsil ns Here we choose ng theaccessible ones. would reflect on thesystems’s “architec- ) thatcontain frequency high artifacts s oth discrete given distributions) by he c ] = e each ( fi ne le H orresponding / dsqiiru”and d “disequilibrium” d asadistancefrom the DECEMBER ≤ [ q P M -Kullback escort-Tsallis entropy Q ] · = 256 ≤ Q ( 1). [ JGW P ] , Q 2005 data [ C), P J e ] tells usjust “how ns ). C en-escort-Tsallis ( J,q G P ) is inthiswa : H (d unif e fi third or ne (2) (4) (3) Q B B m d y 3 0 1 ca tgs(nraeo ttsia infiac) ( significance); statistical stages (increaseictal of difference between themeanvalues corresponding to pre-ictal and decrease, specially in pre-ictal period. This fact ( fact This inpre-ictal specially period. decrease, the s nemtet the intermittent, p the ic behavior, constant these normalized GWS values acquire regular, arather stages, thepre-(ictal) During andpost-crisis the temporal domain. seizure, when the when seizure, ucino theparameter of function p where between thetem- theratio 2.cand2.d, appreciated inFigs. clearly behaviourcanbe This for thepre- periods). andpost-ictal the convulsive in (clonic) phase iscorrelated increased activity with the of 1.bthatthestart We inFig. alsoobserve more important. theother frequency while bandsbecome values lower than10%, (12.8-3.2 B [ (pre-ictal: lowrhythms pre-ictal) phasecharacterizedis by a dominanceof omittedcan be inafirst reading.) We (called thattheinitial see f B around 125 the e any filteringmethod. curare or thatourresultsthe fact were theuseof without obtained We emphasize rhythm (ERR)[1](shadowed area inthefigure). theabove recruiting epileptic ior associated canbe with described values aho n ohfeunybnsbcm lal oiat The fashion frequency andboth bandsbecome clearly dominant. low frequency bands seizure isdominated by frequency themiddle bands We conclude from this example thatthe phase. out thepost-ictal agl epnil o rgntn h n ftheseizure). largely theend responsibleof for originating is inhibition factors, preponderanceneuronal ratewith of firing Tsallis’ these t For q all 2.aand2.c). Figs. ty. This discharge This lastsapproximately 8 ty. slow waves on low superimposed voltage fastactivi- discharge of maxima ar Two relative 1. Fig. thebare record of but notby of inspection thatone“phase candetectby atthepatient looking transition”, one encounters in thatneighbourhood thetonic-clonic diagnosis: resented inour analysis by rep- thelow frequency activity, at90s, Starting theRWE. 80% of which reaches inthefrequency bandsB5andB6, “activity-rise” ations at145and155 ations amechanism entirely different from theone that produces vari- of b crisis. theepileptic thetonicfor characterizing andclonic stages of e Onemay therefore suggest thattheescort-Tsallis in thefigure). r contributions due to contaminant frequency high bands( ar or or ang nt on ntevcnt f~125 ~ of e found inthevicinity o r 2 5 e d ). e-ic r T Our JGW h iiu bouevleo thenormalized entropy isto minimumThe absolute value of and r q al meanv B o T he othe e corresponding to theERR(represented by ashadowed area up < 1thefl 4 e pile ymaue htw s ee constitutes theappropriate tool thatwe usehere, py measure, he b
tal stage are much smaller than those pertaining total stage are the much thanthosepertaining smaller tal stag rqec ad After 140 frequency band. pic imes ar q with a dispersion thatdiminishes adispersion with B e nfc,w e rmFg.2.band2.dthattheJGWS yields we from see Figs. In fact, . irt fthe of riority 6 p with a dispersion thatdiminishes adispersion with Hz e nFg.2aad2b In we their evaluation ignore 2.aand2.b. ted inFigs. e frequency bandsmaintainthispredominance through- t havior of the GWS clearly varies with theGWS with clearly varies havior of bevda 15sad~155 at~145sand~ e observed ic se r q s ), with a corresponding abrupt activity decrease inthe acorresponding activity with abrupt ), C-quantifier results numerical alsodepend uponthe B al e. (transition from(transition tonic to clonic stage) are theresult uctuations inthesequantifiersuctuations increase and for 5 e asso uant ues c + T zr,rsetvl.Changes intheEEGseries respectively. izure, his diff B B B 6 ifi 6 o ] 5 itdwt h nso ( theends of ciated with frequency activity alsoincreases rapid frequency invery activity e rresponding to andpre-ictal ictal epochs as B ciiyrssu gi ilteedo the theend of till upagain rises activity ~ q r 5 ,GSadJW,a ucino time, of asafunction GWS andJGWC, s, s > 1T 50%). The seizure starts at80 seizure The starts 50%). e and nuoa ftge,adces fthe (neuronal adecrease of “fatigue”, r ≥ q B ence isbetter appreciated inthetime for GWS and JGWC is shown. That is That for GWS andJGWC isshown. 5 ,thenormalizedGWS values during 1, B and sal 6 (3.2-0.8 lis-t s s , when clonic when discharges become , B , in agreement with themedical in agreement with 6 decreases to abruptly relative , echniques that magnify suchechniques thatmagnify s as as theTsallis’ Hz and produces amarked q .Cery thisbehav- Clearly, ). grows (inparticular i s ii ) theERRand( . ) clearly illustrates q i As stated above, ) emphasizes the (se e Fig q FEATURES grows (see B 3 s s. and B with a with 2.a) in 1 q 227 and > 1 ii B ) 4 features FEATURES differences, that are critical for clinical purposes. However, the mean JGWS values are significatively larger in the ictal than in the pre- and post-ictal epochs for all q ≥ 1. Long-range memory The present article described informational tools derived from the orthogonal discrete wavelet transform and their application to the analysis of brain electrical signals. The quantifier (relative and nonextensivity in wavelet energy) RWE provides information concerning the relative energy associated with different frequency bands that are to be found in the EEG and enables one to ascertain their corresponding financial markets degree of importance. Our second quantifier, normalized wavelet entropy (GWS), carries information about the degree of order/dis- Lisa Borland order associated with a multi-frequency signal response. Finally, Evnine-Vaughan Associates, Inc., 456 Montgomery Street, Suite our third quantifier, the statistical wavelet complexity (JGWC), 800, San Francisco, CA 94104, USA provides us with a measure that reflects the intricate structures hid- [email protected] den in the brain-dynamics. In particular, it becomes clear that the ERR behavior reported by Gastaut and Broughton [1] for generalized TCES is accurately erhaps one of the most vivid and richest examples of the described by the RWE quantifier. Moreover, the reported study P dynamics of a complex system at work is the behavior of finan- does not require the use of curare or of digital filtering. In addition, cial markets. The price formation process of a publicly traded a significant decrease in the entropy was observed in the recruit- asset is clearly the product of a multitude of evasive interactions. ment epoch, indicating a more rhythmic and ordered behavior of Individuals around the globe post orders to buy or sell a particular the EEG signal, compatible with a dynamical process of synchro- stock at a particular price. Transactions are cleared at a certain price nization in the brain activity. In addition the recruiting phase also at a given time, either by passing through the hands of a specialist exhibits larger values of statistical complexity. on the trading floor, or automatically on the many electronic It is well established that an EEG is directly proportional to the markets which have flourished along with technological advances local field potential recorded by electrodes on the brain’s surface. over the past few years (Fig. 1). Apart from fundamental properties Furthermore, one single EEG electrode placed on the scalp records of the company whose stock is being traded, factors such as sup- the aggregate electrical activity from up to 6 cm2 of the brain surface, ply and demand clearly must affect the price of stocks, as well as and hence from many millions of neurons. With such large num- general trends in the particular industry in question. Stock specif- bers, is seems quite natural to model the neocortex as a continuous ic events, such as mergers and acquisitions, have a big impact, as do sheet of neurons (neuronal matter) whose activity varies with time. world events, such as wars, terrorist attacks and natural disasters. Taking into account the available results for (i) the chaoticity index Time series of financial data exhibit highly nontrivial statisti- (the largest Lyapunov exponent with stationary constraints cal properties. What is quite fascinating is that many of these removed) as a function of time and (ii) the largest Lyapunov expo- anomalous properties appear to be universal, in the sense that nent for selected portions of the EEG signal, one can confidently they are present in a variety of different asset classes, ranging for assert that a chaotic behavior can be associated with the whole EEG example from commodities such as wheat or oil, to currencies and signal. This chaoticity becomes smaller during the recruiting phase individual stocks. Furthermore they are present across the geo- [2]. As pointed out by many authors (see for instance [9]), the graphical borders, and can be observed among others in US, coexistence of chaos with ordering and increasing complexity for European and Japanese markets. extended system is a manifestation of self-organization. We can thus Finding a somewhat realistic model of price variations that can suggest, on the basis of experimental EEG data and using appropri- capture the spectrum of interesting statistical features inherent in ate statistical tools, that in the case of tonic-clonic epileptic seizures, real data is a challenging task, important for many real-world the epileptic focus triggers a self-organized brain state characterized reasons, such as risk control, the development of trading strategies, by both order and maximal complexity. option pricing and the pricing of credit risk to name a few. Bache- lier’s random walk model in 1900 was the first attempt of a mathematical model of price variations. While a century ago this References Gaussian stochastic process was state-of-the-art, and indeed lies at [1] H. Gastaut and R. Broughton, Epileptic Seizures. Charles C. Thomas, the bottom of the celebrated Black-Scholes option pricing formal- Springfield, 1972. ism, we now know Figure 2: The empirical distribution of daily [2] O. A. Rosso, M. L. Mairal, Physica A 312 (2002) 469. returns from the stocks comprising the SP 100 (red) is fit very well by a q-Gaussian with q = 1.4 (blue). that it is way too simple to [3] S. Mallat, A Wavelet Tour of Signal Processing, Academic Press, San describe the properties of real data. In fact, during the past decade, Diego, 1999. there has been an increasing and widespread access to data extract- [4] O. A. Rosso, M. T. Martin, A. Plastino. Physica A 313 (2002) 587. ed from financial markets. This includes for example every single [5] O. A. Rosso, M. T. Martin, A. Plastino. Physica A 320 (2003) 497. trade and quote of all stocks traded on the New York Stock [6] P.W. Lamberti, M. T. Martin, A. Plastino, O. A. Rosso, Physica A 334 Exchange, records from various electronic markets, the entire order (2004) 119. book data from the London stock exchange, to name just a few sources. These vast amounts of historical stock price data have [7] R. López-Ruiz, H. L. Mancini, X. Calbet. Phys Lett A 209 (1995) 321. helped establish a variety of so-called stylized facts [1, 2], which can [8] P.W. Lamberti, A. P. Majtey. Physica A 329 (2003) 81. be seen as statistical signatures, of financial data. [9] H. Haken, Information and Self-Organization, Springer–Verlag, The best known stylized fact is perhaps the distributions of Berlin, 2000. returns (defined as logarithmic relative price changes). On time scales ranging from minutes to weeks these have fat tails, exhibiting
228 europhysics news NOVEMBER/DECEMBER 2005 ffreedom. of t n)=(1 +n), p incontrast to evidence. reversaltime empirical symmetric man le st Multifractal pastreturns. ly anautoregressive of function GARCH winning prize inwhich models thevolatilityisessential- Engle’s Nobel for thesimplestof sameholdstrue The the data. in observed but notthelong memory reproduce fattails, process, v such astheHeston where model the chastic volatilitymodels, scales, which induce time- jumps andthus fattailson short processes, Popular approaches include Levy the GaussianBachelier model. to capture fattailsandvolatilityclustering which don’t exist in differentiateseries thepastfrom thefuture. price thatfinancialtime fluctuations!) present of inmostmodels (which however fact obvious therather implying isnot sal, under rever- time asymmetry aswell asthestatistical power law), followed by after-shocks according decaying inmagnitude to a large volatility shocks tend to be earthquakes (inother words, law theOmori for of afinancialanalogue are scaling, multifractal Examples subtle ones which have elucidated been inrecent years. are alsomore In addition to facts, thesestylized leverage. called aneffect large pastprice changes imply large future volatilities, correlations there inthat are asymmetric certain Also, log-normal. volatilityisclose theinstantaneous to more of thedistribution andfurther- phenomenon alsoreferred to asvolatility clustering, a returns, or lower higher of series volatilityinthetime bursts of to leads This thevolatility decays only slowly asapower law. of tion theautocorrela- because evident involatility fluctuations, memory there isalong range In addition, does becomecloser to aGaussian. thedistribution the price changes increases to months or years, Student form power-law decay andare quite modeled by well theTsallis or eur 1 t w themost isstill This Nobel-prize Black-Scholes formula. winning the calculatedbe thepublication reliably somewhat of with options could But itwasn’t 1974thatthefair price until of bubble. were quite abit traded by speculators to prior thefamoustulip w make ahuge profit reason hehad when to believe thattheharvest options on call olives used to theGreek mathematician, Thales, options were exploited already by theRomans hasitthat andstory similarto Contracts inthefuture. theexpiration) (called time tion) to buy astock thestrike)atacertain price (called atacertain und inst options are financial In short, purely speculative opportunities. aswelloffer control, andrisk respectto hedging with functions financial important fill They volumes. inhigh globally are traded o fordesirable reasons such asefficiently thefairprice calculating is tractability Analytic analytically. to notimpossible deal with if snei sbsdo asinmdlfrsokrtrswih as (since on for itisbased aGaussianmodel stock returns which, o theSt ime-scale incr olat The TsallisThe (alsoreferred distribution to asaq-Gaussian)isequivalent options or other financialderivatives which right intheirown f e catcvltlt oes(iia ocsaemdl fturbu- (similartoochastic volatilitymodels of cascademodels nt fl udb atclrygo.I oln nte10s tulip options In Holland inthe1600s, good. particularly ould be id r I Several different have models proposed been [3]inanattempt haps theE n a oph e ruments which dependruments insome contingent fashion on the e y o ilit l r y use o lying stock or other asset class. The simplestexample The is stocklying or otherclass. asset b ddit w) ar f the stylized facts, lacking mainly in that they are lacking mainly inthatthey strictly facts, thestylized f y isass y u ud sics news t c e io w d o o nt dist he e anothe ,ms ftheabove are mentionedmodels difficult mostof n, n 1 v ur p r umed toumed follow mean-reverting itsown stochastic eases andd Fg 2).As over thetime-scale which one calculates (Fig. olve too quickly asthe to theGaussiandistribution e nisap inpiigmdl o eas fitsaccuracy of not because model, pricing tion opean call option. This is the right (notobliga- istheright This option. opean call r ib NO u r p inweee sartoa ftheform(3+ tion whenever of qisarational VEMBER stv nee eoigtenme fdegrees ositive integer denoting thenumber of r o mising candidat o notp / DECEMBER eetvltlt lseig Sto- resent volatilityclustering. 2005 c 2) reproducing [2]), e (cf t thisquan- thatprices used arewidely often quoted just interms of theBlack-Scholes volatilityisso senting option prices interms of shap price isthus aconvex notconstant but instead mosttypically fi mathematical animpressive school of In fact, assumptions). (which exists due to thesameGaussian tractability matical isunrealistic) butdue to rather itsmathe- we above, discussed the st o for theBlack-Scholes to model match empirical this intuitively; Traders seem to correct for underestimated. be current price will the probabilityexpire thatthestock priceatstrikesfarfrom will its for returns, formula on isbased aGaussiandistribution pricing Scholes paradigm. base these B t,mostoften referred to simply as ity, ption prices, higher volatilities must be used the farther away volatilities higher mustthefarther used be prices, ption nanc Gaussian with Gaussian ᭡ ᭡ st execute electronically. on optionsare fed andthe trader infromcan themarket B o e F F e, d larg cks c cause r ig ig r e hasb ik o la .1: .2: ft e p kShlsipidvltlte safnto fthestrike ck-Scholes volatilities implied of asafunction nrfre oa h oaiiysie hswyo repre- way This of en referred to asthevolatilitysmile. ompr The empirical distribution ofdailyreturns fromThe distribution the empirical yia lcrnctaigsre.Hr,live quotes Here, trading screen. electronic A typical e r eal st l ic y on notions stemming from the famousBlack- e e isfr e ising the SP 100 (red) is fit very wellising theSP100 (red) by isfitvery a n d q c eun xii a al,yet theBlack-Scholes ock returns exhibit fattails, = 1.4(blue). e o eoe vrteps he eae,andis veloped over thepastthree decades, m thecur etsokpievle Aplotof rent stock price value. the v ol . FEATURES q - 229
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sentiment. Mathematically, it leads to a non-linear diffusion equa- tion for the price. Exact time-dependent solutions to this equation can be found resulting in a Tsallis distribution for price changes at all times, and volatility clustering is also present. The entropic index q which characterizes the resulting distribution depends on the power of the statistical feedback term. If q = 1, the power vanishes so there is no statistical feedback and the standard Gauss- ian model is recovered. If q > 1, the power is negative and fat tails are present. This model has been quite successful for the purpose of option pricing, again largely due to the fact that one can actu- ally calculate a lot of things analytically, and in particular one obtains closed-form solutions for European options. Since a value of q = 1.4 nicely fits real returns over short to intermediate time horizons (corresponding to 4 degrees of freedom with the Student formulation), this model is clearly more realistic than the standard Gaussian model. Using that particular value of q as calibrated from the historical returns distribution, fair prices of options can be calculated easily and compared with empirical trad- ed option prices, exhibiting a very good agreement. In particular, while the Black-Scholes equation must use a different value the ᭡ Fig.3: Theoretical implied Black-Scholes volatilities from volatility for each value if the option strike price in order to repro- the q = 1.4 model (triangles) match empirical ones (circles) duce theoretical values which match empirical ones, the q = 1.4 very well, across all strikes and for different times to expiration. model uses just one value of the volatility parameter across all strikes. One can calculate the Black-Scholes implied volatilities cor- From all that has been said up to now, it is really quite clear that responding to the theoretical values based on the q = 1.4 model, the true model of stock price fluctuations has many challenging and a comparison of with the volatility smile observed in the mar- statistics to reproduce, in addition to correctly pricing derivative ket will reflect how closely the q = 1.4 model fits real prices (Fig. 3). instruments. Furthermore, it would be desirable that the mecha- Although quite successful, this model is not entirely realistic. nisms of such a model are somewhat intuitive. With these goals The main reason is that there is one single characteristic time in in mind, the field of nonextensive statistical mechanics has made that model, and in particular the effective volatility at each time is some progress in recent years although of course there is still a related to the conditional probability of observing an outcome of long way to go both within and beyond this framework. the process at time t given what was observed at time t = 0. For As already mentioned in passing, returns (once demeaned and option pricing this is perfectly reasonable; one is interested in the normalized by their standard deviation) have a distribution that is probability of the price reaching a certain value at some time in very well fit by q-Gaussians with q ≈ 1.4 [4], only slowly becoming the future, based entirely on one’s knowledge now. But this is a Gaussian (q → 1) as the time scale approaches months or years. shortcoming as a model of real stock returns; in real markets, Another interesting statistic which can be modeled within the traders drive the price of the stock based on their own trading nonextensive framework, is the distribution of volumes, defined as horizon. There are traders who react to each tick the stock makes, the number of shares traded. A q-exponential multiplied by a ranging to those reacting to what they believe is relevant on the simple power of the volume presents power laws at both high and horizon of a year or more, and of course, there is the entire spec- low volumes and fits very well to the data [4]. These results are trum in-between. Therefore, an optimal model of real price encouraging, albeit they are macroscopic descriptions of the data movements should attempt to capture this existence of multiple and a dynamical description of the underlying processes is of course time-scales and long-range memory. desirable. For the volumes, such a model was recently proposed. For Indeed, by including a kind of statistical feedback over multiple stock prices, a class of models that has had some success in option timescales a model is obtained which seems to account for most pricing was introduced a few years ago [5], based upon a statistical stylized facts of financial time series (Fig. 4). The distribution of feedback process. Recently, that model was extended to incorpo- returns are fit well by Tsallis-Student distributions. As the time rate memory over multiple time-scales [6] (recovering a class of horizon of returns increase, the distribution approaches Gauss- long-ranged GARCH models [7]) and seems to reproduce most of ian in the same way as empirical data. Long range volatility the stylized facts of financial time series. Other interesting models clustering is present, with a decay that matches real data. The dis- related to the nonextensive thermostatistics include an ARCH tribution of instantaneous volatility is close to log normal. Subtle process with random noise distributed according to a q-Gaussian as effects like the multifractal spectrum and Omori analogue are well as some state-dependent additive-multiplicative processes [8]. reproduced. In particular, the time-reversal asymmetry is inher- These models do capture the distribution of returns, but not nec- ent. Although some of these statistics can be calculated essarily the empirical temporal dynamics and correlations. analytically, most are obtained through numerical simulation. In In the statistical feedback model, price fluctuations are assumed principle, the model can be used for option pricing via Monte- to evolve such that the Tsallis entropy is maximized. This leads to Carlo simulations, but analytic option pricing formulae would be an instantaneous volatility which is proportional to a power of the very welcome. Obtaining these is still an open problem. probability of the most recent price: It is large when price moves A final interesting remark on the implications of this model is are exceptionally large (or rare); conversely, the volatility is small- that the parameters which calibrate to empirical data put the er if the price moves are more moderate (or common). This model close to an instability. This suggests that the dynamics of mechanism is an attempt to model the collective behavior of mar- financial markets are operating on the brink of non-stationarity. ket players. The statistical feedback tries to reflect market In fact, this mathematically perhaps undesirable property is
230 europhysics news NOVEMBER/DECEMBER 2005 7 .Zmah .Lynch, P. Zumbach, [7] ]G. 6 .BradadJP Bouchaud, andJ.P. Borland [6] L. Borland, [5] L. 1 .Peo,P oirsnn ..Aaa,M ee,HE Stanley, H.E. Meyer, M. Amaral, L.A. Gopikrishnan, P. Plerou, [1] ] V. References well-established ideas inanew light. andwe rethink mustdard perhaps some very paradigm markets infinancial forcesthe long-memory usbeyond thestan- In asimilarfashion, must go beyond thestandard framework. one ispresent, interactionsor long-range long-memory If ry. memo- andshort-term interactions for short-range systems with works well mechanics the notionthatBoltzmann-Gibbsstatistical however.states otherwise volatilitymodel long-memory The influence on investor behavior. reflected canhave intheprice andthatthepastprice history no relevantinformation to thestock and isimmediately absorbed questionsThis theefficient market which hypothesis states thatall which wasverified empirically. volatility andpastprice changes, isthatitpredictsmodel alarge correlation between thepresent the Another implicationof can completely break down andcrash. they andaswe haveappear only quasi-stationary seen historically, markets financial indeed inreality, philosophically quite desirable; eur 2 ..Buhu n .Potters, Bouchaud andM. [2] J.P. [8] S.M. [3] F 4 f .Gl-anadC Tsallis, Gell-Mann andC. M. [4] Cf. reproduces ofthereal data. facts most ofthestylized simulated series daily returns since 1965oftheSP500.The feedback multiple time-scale (top) model andthe memory ᭡ To bring an analogue to physics, this fact isinasense akinto thisfact To to ananalogue physics, bring oph F o 151 (SP J t nar P b D Fluctuations in Econophysics andFinance, inEconophysics Fluctuations t .-P i ions inec ig eeec e ..J Hull, J. r reference e.g. see ac h ve Pricing, e y r .4: k model for volatility fluctuations, for volatility k model ocad .Gbi n ..MCue,Po.O SPIE 5848, Of Proc. McCauley, GabaixandJ.L. X. Bouchaud, . y y D i s. v sics news A at uar R p A time series ofreturns simulatedA timeseries withthelong- e i E elna,W 05.[physics/0503024]. WA 2005). Bellingam, IE, p v .E v. t lications, es, e Quie nmc:annxesv ttsia prah inNoise and approach, statistical anonextensive onomics: Quantitative Finance Quantitative P C 60 r amb e 6519 (1999); nt r os, NO ic O ig nvriyPes 2004. Universityridge Press, e H xf VEMBER .Atnoo .Tsallis, C. Anteneodo, C. o al Quantitative Finance Quantitative r d U l, 2004. F iest rs,N,2004. NY, niversity Press, ut Theory of Financial Risks Financial andDeriva- of Theory / On a multi-timescale statistical feed- statistical amulti-timescale On DECEMBER ures, Options and other Financial andother Options ures, Nonextensive Entropy -Interdiscipli- 2 1-3,(2002). 415-431, , p h ysics/0507073 (2005). d.D Abbot, D. eds. 2005 , 3 P 2,(2003). 320, , o w e r -la w dist ribu- CZ-182 21Praha 6,Czech Republic CZ Academy ofSciencestheCzech78 Republic, Strizkovska I ENS NL-1098 SMAmsterdam, The Netherlands 404 Kruislaan Amstel Institut, •IT-70126University ofBari Italy Bari, Via CP64•IT-00044 Galilei, Galileo Frascati, Italy I oftheNCM10Conference,Chairman F-91191 Gif-sur Labor Physics •ESCAMPIG-18 inIonized Gases Contact: August 2006•Amsterdam, The Netherlands 20 -26 GIREP 2006ModelinginPhysics andPhysics Education Website: Email: Tel: Contact: 1 1 Website: Email: T Contact: May 2006•Frascati, Italy 8 -12 Atmospheric Science Conference 2006 Europhysics Conferences Website: Email: Tel: Contact: 18-22 September, 2006•Praha, Czech Republic MaterialsNon-Crystalline •NCM10 10th In Website: Email: Tel: C 26 Technology of Thin Films •IWTF2 2nd International Workshop onPhysics and W Email: Tel: or F Contact: June2006•Paris, France 19 -23 ConferenceEuropean XRay Spectrometry •EXRS2006 Website: Email: Tel: 2006 S nstitut nstitut el: ax: 2 -16 July2006•Lecce, Italy 2 -16 on 8 ebsite: -180 00Praha 8-Liben,Czech Republic Dr. Alexandre Simionovici t -30 June2006•Prague, Czech Republic -30 h -L + +420 220318530• +33 472728697• +31 205255963• +39 0805442101• +420-28484 2448• tac +33 169082619• E 39 0694180544• atoire NationalBecquerel, Henri CEASaclay y uropean Conference onAtomic andMolecular [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] on, 46alléed'I e ofP e ofP t: ternational Conference of ontheStructure Prof. TonEllermeijer, M.Cacciatore, ofChemistry CNR-IMIPc/oDepartment Prof. ClausZehner, ESA/ESRIN, Prof. Ladislav Cervinka, EXRS 2006Secretariat Dr http://earth.esa.int/atmos2006 www.icaris.info/NCM10 w www.girep2006.nl http://escampig18.ba.imip.cnr.it www.fzu.cz/activities/workshops/iwtf2 p ww.nucleide.org/exrs2006 . Z onsor h h denek Ch y y sics AV CR,NaSlovance 2 sics - Y v ett , ed C e talie •69007Lyon, France, , F Fax: Fax: Fax: Fax: voj, F Fax: Email: r ax: anc onf +31 205255866 +33 472728677 +39 0805442024 +420-28484 2446 + +420 233343184 e 39 0694180552 exrs2006@c er enc es ea.fr FEATURES 231
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1 J.P. Ansermet, M.C.E. Huber, C. Rossel and Ingrid Kissling-Näf 4 The editors: G. Morrisson and Claude Sébenne at work 2 M. Gell-Mann receiving the Einstein medal at the opening ceremony 5 The magnetic cannon of Europhysics Fun 3 World Year of Physicss 2005 flag floats over Bern 6 The main building at the University of Bern 2
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C C J US-MD 20771, R506Teslnk,Greece GR-54006 Thessaloniki, A D DE-07734 Jena, Germany Germany DE-07734 Jena, M F Theoretisch-Physikalisches Institut D U NASA /GSFC Department Research &Scientific Support ESA, K80 ahsC Denmark C, DK-8000 Aarhus . . riedrich-Schiller-Universität . ecretary & ecretary Treasurer . . .J. Borde, hair hair . r o niv .A. Tammann, epartment ofPhysicsepartment epar t .C.E. Huber, ax-Wein-Platz 1 Bicak, Bicak, L. Palle, istotle University of Thessaloniki M T F. Schutz, S omic & Molecular Physicsomic &Molecular V ar Germany S I eeie Spain Tenerife, D I United Kingdom P S G Denmark Denmark hsc,Gl e osa,Germany Potsdam, Bei Golm Physics, R singanos taly taly chutz, eneziano, olett witzerland t. ulanier oland epublic ak vik er arlsson, arlick ermany do,United Kingdom idcot, ersit d A A P G ..Andersen L.H. P tment ofPhysics andAstronomy donio X X k s lha@ph blf [email protected] nvriyo xod United Kingdom University ofOxford, arov, M et . o [email protected] Schaefer . [email protected] Palle +30 31998185/995384 +49 3641947114/102 +45 89423605/ 86120740 +1 3012864098/860264 +34 922605384/210 C v, v, o, o, y ersbur e [email protected] European (Virgo), Gravitational Observatory alsUiest,Czech Republic University, harles y La Laguna- University ofRome “La Sapienza” mb I ..Lue,Belgium Leuven, K.U. U of A bevtied ai,Lsa France Lesia, Observatoire deParis, O .Kokkotas K. siuoAtosc eCnra,LaLaguna nstituto Astrofisica deCanarias, uhrodApeo aoaoy Chilton Appleton Laboratory, Rutherford Osser , U , University Of Glasgow, United Kingdom United Kingdom Glasgow, University Of niv bser A O P Friedrich-Schiller-Universitaet Jena, nvriéPer tMreCre France Université Curie, Pierre etMarie , M Imperial College,UnitedImperial Kingdom iest ai od France niversité Paris Nord, , Polish Academy ofSciences,Wroclaw, U oihAaeyo cecs Poland Polish Academy ofSciences, • A P U ulk c ers U bser y , CERN, ax Planck Institute For Gravitational niv G ar Mailcode 682.3 Mailcode aul Scherrer Institute,Villigen,Switzerland adem str Universität Hannover, Germany Germany Universität Hannover, ersit niv s U g niv r U .au v hus Russia , o v eenb niv oph atory oftheCopenhagen University,atory ersit ieste ae,Binningen, niversitaet Basel, troAtosc iActi Firenze, atorio Astrofisico diArcetri, vo Astronomical Observatory, ersit v ersit .dk y tied ai,Muo,France Meudon, atoire deParis, ersitaet G y S fWls College ofCardiff,of Wales, y y witzerland l,USA elt, eeie Spain Tenerife, of y sik om.nasa.go P DIRECTORY EPS y B O of P of A Fleck . cecs nrjv Czech Ondrejov, Sciences, f alisches InstitutP Oslo adova, Italy Italy adova, thens a,Austria raz, , N orway , v Greece , Italy Italy , otsdam, 05/06 235
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Vice-Chair R. Grimm O. Dulieu, University of Paris-sud, France M. Scheffler, Max-Planck-Gesellschaft, Berlin-Dahlem, Institute of Experimental Physics D. Gerlich, Technische Universitaet Chemnitz, Germany Germany University of Innsbruck K. Hansen, University of Chalmers, Sweden W.M. Temmerman, Daresbury Laboratory,Warrington, and IQUQI Austrian Academy of Sciences T. Softley, University of Oxford, United Kingdom United Kingdom AT-6020 Innsbruck, Austria E. Tiemann, University of Hannover, Germany TEL/FAX +43 5 125 076 300 / +43 5 125 072 921 Liquids Section EMAIL [email protected] Electronic & Atomic Collisions Section Chair D. Frenkel Chair T.D. Märk FOM, Institute for Atomic and Molecular Physics Treasurer U. Becker Institut für Ionenphysik Kruislaan 407 Fritz Haber Institute der Max-Planck-Gesellschaft Universität Innsbruck NL-1098 SJ, Amsterdam,The Netherlands Faradayweg 4-6 Technikerstr. 25 TEL/FAX +31 206 081 234 / +31 206 684 106 DE-14195 Berlin, Germany A-6020 Innsbruck, Austria EMAIL [email protected] TEL/FAX +49 3 084 135 694 / +49 3 084 135 695 TEL/FAX +43 5 125 076 240 / +43 5 125 072 932 EMAIL [email protected] EMAIL [email protected] Secretary & Co-Opted Member G.J.Vroege University of Utrecht Board Members Secretary & Co-opted Member C. Guet Faculteit Scheikunde J.M. Hutson, University of Durham, United Kingdom Département de Physique Theorique et Appliquee Padualaan 8 J.M. Launay, University of Rennes, France CEA Ile de France • BP 12 NL-3584 CH Utrecht,The Netherlands F. Martin, Universidad Autonoma de Madrid, Spain FR-91680 Bruyeres le Chatel, France TEL/FAX +31 302 533 406 / +31 302 533 870 G. 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Szmytkowski, Technical University of Gdansk, Poland S. Dietrich, Max-Planck-Institut für Metallforschung, H. Hotop, Technical University of Kaiserslautern, Stuttgart, Germany Germany Vice Chair G. Pichler H.N.W. Lekkerker, University of Utrecht,The J. Tennyson, University College London,United Institute of Physics Netherlands Kingdom Bijenicka cesta 46 G.J. Vroege, University of Utrecht,The Netherlands HR-10000 Zagreb, Croatia G. Wennerstrom, Lund University, Sweden Atomic Spectroscopy Section (EGAS) TEL/FAX +385 14 698 888 / +385 14 698 889 Chair H. Hotop EMAIL [email protected] Low Temperature Section Technische Universitaet Kaiserslautern Chair H. Adrian FB Physik • Postfach 3049 Condensed Matter Division Institut für Physik DE-67653 Kaiserslautern, Germany Johannes Gutenberg-Universität Mainz TEL/FAX +49 6 312 052 328 / +49 6 312 053 906 Chair E.P. O’Reilly Staudingerweg 7 EMAIL [email protected] Tyndall National Institute (NMRC) DE-55099 Mainz, Germany Lee maltings, Prospect Row TEL/FAX +49 61 313 923 637 / +49 61 313 924 076 Secretary C. Blondel IE-Cork, Ireland EMAIL [email protected] CNRS • Laboratoire Aimé Cotton TEL/FAX +353 214 904 413 / +353 214 270 271 Campus d’Orsay - Bât. 505 EMAIL [email protected] Board Members FR-91405 Orsay Cedex A. Barone, University of Naples “Federico II”, Italy TEL/FAX +33 169 352 056 / +33 169 352 100 Board Members T. Dietl, Polish Academy of Sciences, Poland EMAIL [email protected] M. Ausloos, Universite De Liege, Belgium A. Gilabert, Université de Nice, France D. Bideau, Universite De Rennes I, France M. Krusius, Helsinki University of Technology, Finland Board Members O. Fischer, Université De Genève, Switzerland C.J. Lambert, Lancester University, United Kingdom C. Bordas, University of Lyon 1,Villeurbanne, France S. Hess, Technische Universität Berlin, Germany M. Charlton, University of Wales, Swansea, United I. Ipatova, A.F. Ioffe Physico-Technical Institute, St- Macromolecular Physics Section Kingdom Petersburg, Russia Chair G. Strobl W. Ernst, Graz University of Technology, Austria E.P. O’reilly, National Microelectronics Research Physikalisches Institut J. Eschner, Universitat Autonoma de Barcelona, Spain Centre Lee Maltings, Cork, Ireland Albert-Ludwig-Universität D. Hanstorp, Göteborg University, Sweden S. Wright, Imperial College, London, United Kingdom Hermann-Herder-Strasse 3 E. Lindroth, University of Stockholm, Sweden DE-79104 Freiburg, Germany F. Merkt, Eth Hönggerberg, Zürich, Switzerland Ex-Officio Members TEL/FAX +49 7 612 035 857 / +49 7 612 035 855 K. Pachucki, University of Warsaw, Poland H. Adrian, J. Gutenberg Universität Mainz, Germany EMAIL [email protected] A. Sasso, Universita di Napoli Federico II, Italy J.L. Beeby, University Of Leicester, United Kingdom V. Shabaev, St-Petersburg State University, Russia A. Fasolino, University Of Nijmegen,The Netherlands Board Members R. Shuker, Ben-Gurion University, Beer Sheva, Israel D. Frenkel, Fom, Amsterdam,The Netherlands A. Cunha, Universidade do Minho, Portugal K. Taylor, the Queen’s University Belfast, United M.R. Ibarra, Universidad De Zaragoza, Spain A.M. Donald, University of Cambridge, United Kingdom Kingdom T. Janssen, University of Nijmegen,The Netherlands J. A. Manson, EPFL, Switzerland G.R. Strobl, Albert-Ludwigs-Universität Freiburg, R. Schirrer, Institut Charles Sadron, Strasbourg, France Webmaster H.P.Garnir Germany G. J. Vancso, University of Twente,The Netherlands University of Liège, Belgium K. Wandelt, Universität Bonn, Germany Place du 20 Août Co-opted Members BE-4000 Liège Co-Opted Members A.R. Khokhlov, Moscow State University, Russia TEL/FAX +32 43 663 764 / +32 43 662 884 A. Zawadowski, Technical University of Budapest, G.H. Michler, Martin-Luther-Universität, Germany EMAIL [email protected] Hungary A. Pavan, Politecnico di Milano, Italy
Chemical & Molecular Physics Section Invited Magnetism Section Chair J. Tennyson V.Sechovsky, Charles University, Prague, Czech Republic Chair M.R. Ibarra Department of Physics and Astronomy Institut of Nanoscience of Aragon (INA) University College London Gower Street Electronic & Optical Properties Of Solids Condensed Matter Physics department & Institute of UK-WC1E 6BT, London, United Kingdom Chair J.L. Beeby Material Science (ICMA) TEL/FAX +44 2 076 797 155 / +44 2 076 797 145 University of Leicester Pedro Cerbuna 12 EMAIL [email protected] University Road ES-50009 Zaragoza, Spain UK-LE1 7RH, Leicester, United Kingdom TEL/FAX +34 976 762 777 Secretary D. Gerlich TEL/FAX +44 1 462 522 323 / +44 1 462 558 291 EMAIL [email protected] Institut für Physic EMAIL [email protected] TU Chemnitz Board Members Semiconductors & Insulators Section DE-09107 Chemnits, Germany A.J. Craven, University of Glasgow, United Kingdom Chair L. Sorba TEL/FAX +49 3 715 313 135 / +49 3 713 513 103 R. Del Sole, Universita degli Studi di Roma Tor Laboratorio TASC-INFM EMAIL [email protected] Vergata, Italy S.S. 14 km 163,5 A. Georges, Ecole Normale Supérieure, Paris, France I-34012 Basovizza (Trieste), Italy Board Members E.I. Lindau, Lund University, Sweden TEL/FAX +39 040 375 6439 / +39 040 226 767 P. Casavecchia, University of Perugia, Italy A. Muramatsu, Universitaet Stuttgart, Germany EMAIL [email protected]
236 europhysics news NOVEMBER/DECEMBER 2005 NL-1009 DB, Amsterdam,The Netherlands Amsterdam,The NL-1009 DB, DE-53115 B Wegelerstrasse 12 Universität Bonn P E C/o NIKHEF D H InstitutfürFestkörperforschungMax-Planck A.W. Kleyn, Kleyn, A.W. Kirschner, J. B Co-opted Members E P. Varga, U. Valbusa, N.V J. Jupille, Board Members EMAIL TEL/FAX Chair Section & Interfaces Surfaces P R A. S Co-opted Members Y R. Pick, S H. Board Members EMAIL TEL/FAX S EMAIL TEL/FAX C S C M F. Peeters, J. Kotthaus, A G Board Members eur EMAIL TEL/F W E. de Wolf, U G. R. J. Lambourne, P H. F Board Members EMAIL TEL/FAX C Education M. Sancrotti, M. J J E N I University ofNijmegen SE-106 91S A ofPhysics Department University, Stockholm Institut fürP nstitute for Theoretical Physics W Niemantsverdriet, .W. .J. Pireaux, .O . . . ur . . .W. Lovesey, ecretary . tructural & Dynamical Properties of Solids ofSolids Properties &Dynamical tructural . . hair hair lbaN Fasolino, . . eisenbergstrasse 1 L-6525 ED Nijmegen,The Netherlands L-6525 EDNijmegen,The . -759Sutat Germany E- 70569Stuttgart, M. Wachter, . ebmast Ho Vlieg, Michel, Torre, Tejedor, Gumhalt T M. Bastard, Kuzmany, U K U N Paix, Belgium Belgium Paix, Sweden . Skolnick, Skolnick, Sz ophysics Foundation . ibell t er B i nit niv ether oph R V ernber ngdom y o A ymonsk Titulaer dinande ichardson, y T o E K.Wandelt er x X ed@nik k [email protected] [email protected] [email protected] Janssen . ed Kingdom sochansk . va University Centre ersit nvriéP tM ui,France Curie, Université P. etM. ej@ph 41 882 , J +49 7116891673/091 +31 243652854/2120 +31 205922117/ 205925155 +46 855378670/601 +49 228732253/515 U U (pr ehiceUiesttWe,Austria Technische Universität Wien, ohansson U aoaor NS France Laboratoire CNRS, lands y er niv U niv. Firenze, Italy Italy Firenze, niv. A niv Universiteit Antwerpen, Belgium Belgium Universiteit Antwerpen, NIKHEF, Amsterdam,The Netherlands Amsterdam,The NIKHEF, U sics news er, t E onn, e-univ E g Università di Genova, Italy Italy Università diGenova, iesddAtnm eMdi,Spain niversidad Autonoma deMadrid, y nvriyo edn theNetherlands University ofLeiden, Bussmann-Holder . h o F U Ludwig Maximilians Universität, Germany Germany Ludwig Universität, Maximilians E HZrc,Switzerland TH-Zurich, iesddAtnm eMdi,Spain niversidad Autonoma deMadrid, a-lnkIsiu,Hle Germany Halle, Max-Planck-Institut, oeNraeSprer,France cole Supérieure, Normale , U , ersity of Nijmegen,The Netherlands Netherlands ofNijmegen,The ersity nvriatWe,Austria Universitaet Wien, acultés Universitaires Notre-dame dela y ck ACIF,ret,Italy TASC-INFM,Trieste, Linz, i, ersit .W niversity of Nijmegen, the Netherlands theNetherlands niversity ofNijmegen, y U hef (univ R iest fSefil,UK niversity ofSheffield, U , sik sto.se U tefr pltnLbrtr,United Appleton Laboratory, utherford n U holm, niv .A. G iv i niv niv .nl orbur U alische Chemie The Open University, Milton Keynes, Milton The University, Open i, ersit ermany y Austria ersit niv Lingeman ersit riyo arb Croatia ofZagreb, ersity U of H ersit ersit z hgor riyo t nrw,UK Andrews, ofSt. ersity S y y w y U sec elsink y i et Belgium eit Gent, of La eden section), Johannes Kepler Johannes section), iest fEnhvn the niversity ofEindhoven, NO of C o tion), eau@nn d VEMBER tvia, i, U aa,Poland racaw, Finland niv U L psala University, e atvia v rsity, Ukraine Ukraine rsity, .nl / DECEMBER 2005 A Altenbergerstrasse 69 KeplerUniversityJohannes Linz I NL-3730 AED Sciamach KNMI U M. Gil B. Diffey, T J G. Br D. Balis, Board Members EMAIL TEL/FAX Chair Environmental Physics G. S A. Oelme, J. Dunin-Borkowski, B EMAIL TEL/F Chair P EMAIL TEL/F Webmaster E M. Vollmer, I E. Sassi, G. Planincic, R H.J. Jodl, B EMAIL TEL/FAX C P R. L Burkhardt, H. R. G. Barr Board Members EMAIL Secretary EMAIL TEL/FAX C High E P J I. Isaksen, KLno W A,United Kingdom UK-London SW72AZ, S Imp IFIC - D U Walton Hall ofPhysics andAstronomyDepartment T Wilhelminalaan 10 Wilhelminalaan KS/AS Department ES- 46100B A deFisicaDepartamento teorica S U nstitute for Theoretical Physics . . .P . . . he Open Universityhe Open -52,Upaa Sweden Uppsala, E-75121, outh Kensingt . -00Ln,Austria T-4040 Linz, . v hair hair re-University Section re-University Section oar oard Members ppsala University -itnKye K A,United Kingdom Keynes MK76AA, K-Milton e Strzalkowski, niversity Section niversity Section Burrows, T L Da de Wolf, Lambourne, . Barlo enida D par S Finland G P Germany B U aalas ev P eitner auer er pain oland adnug Germany randenburg, nited Kingdom nited Kingdom reece ommereau, asseur, vies d A AX V ai, , ial College .Goede A. J. G.Tibell U tmen X [email protected] [email protected] [email protected] [email protected] d jose alencia INT M eir ner .M.Titulaer .M.Titulaer B w .l.w +34 963544553/543381 +31 302206425/210407 +46 184713849/513 + + nvriyo als Italy University ofNaples, , AUTH -University Aristote of Thessaloniki, , RMKI, emb , er Universität Kaiserslautern, Germany Germany Universität Kaiserslautern, y U uhmHsias United Kingdom Hospitals, Durham , , a, FMI -F 44 1908653229/+44654192 43 A, U r. Moliner 50 Moliner r. nvriyo so Norway University ofOslo, abr,Sweden Varberg, .b U IPNP .Wark D. N g niv C nabeu [email protected] t U urjassot-V niv LIP University ofApplied Sciences niv Institut MPI -M [email protected] o-Pl y ke,Asedm theNetherlands Amsterdam, ikhef, .Lambourne R. 73 of R nvriyo jbjn,Slovenia University ofLjubljana, iest fBee,Germany niversity ofBremen, e ers ersität G EN eea Switzerland Geneva, CERN, & B , ersit , ersit on c ibn Portugal Lisbon, B T W 224 688551/+4373585 udap rge Czech Republic Prague, innish M P adia he Open University, Milton Keynes, Milton University, he Open ilt,The Netherlands ilt,The CNRS, arsaw University of Technology, ar y • • o ampus y xPac nttt Hamburg, ax Planck Institut, PO Box 201 PO Box 535 ticle P of E t aswUiest,Poland Warsaw University, est of M N alencia, ion Sciences iessen, Germany Germany iessen, acional TecnicaAeroespacial, F , s nla United Kingdom ast Anglia, r Hungary et ance anchest eor h ysics S olo pain er gic United Kingdom , al Institut e, B RISOE National Laboratory DK-4000 R U D andPhysical ofElectronics School Sciences U F B GANIL N Z Kernfysisch Versneller Instituut-KVI Treasurer EMAIL TEL/FAX Secretary EMAIL TEL/FAX C Nuclear Physics F I. Wormser, J N P. Vilain, J P E. Rabinovici, Pokorski, S. F P Linde, F. M. Cieplak Board Members EMAIL TEL/FAX S EMAIL TEL/F C P G. Rosner, O G. Viesti, B Y. N.Novikov, R. G.L M. Leino, A. M. A. Co-opted Members R. Wyss, N. VanGiai, A. Z. Sujkowski, J C. Leclercq-Willain, B EMAIL C EMAIL TEL/F DK-2800 L 309 B. Institute ofPhysics, the D DE-01062 D Technische Universität Dresden und Institut fürKern- Teilchenphysik ...... -47 anCdx France R-14076 Caen Cedex, iosy ernikelaan 25 ernikelaan oulevard Becquerel Henri ecr . hair hair hair oar h niversity ofSurrey -ulfr,Sre,G27H United Kingdom GU27XH, Surrey, K-Guildford, L-9747 AAG epartment ofPhysicsepartment . bser Tuominiemi, Wess, S Pauss, Zwirner, Osland, Sphicas, Rubio T Wermes, Kugler, Gridnev, y Belgium Israel R S R P Poland Studies, D Hass, tamenov, ürler, t oland sics inLif ussia epublic etar ebr chl,Sweden ockholm, d A A stems Department anish man-elec R P o v X X scholt [email protected] henr [email protected] [email protected] [email protected] M .A. .Johnson C. . v er (N ec U +31 503633552/634003 +44 1483689375/686781 +45 46774104/109 +45 45253285/931669 +33 231454598/665 , KTH -Royal Institute of Technology, as IE rxle,Belgium Bruxelles, IIHE, y IHF mtra,theNetherlands Amsterdam, NIKHEF, E W embers embers nvriyo aoa Italy University ofPadova, T U iest fMnc,Germany niversity ofMunich, nvriyo yäkl,Finland University ofJyväskylä, Lindgar yngb H uih Switzerland Zurich, TH, en, osk N U echnic .Lewitowicz M. .Scholten O. H. nvriyo lso,United Kingdom University ofGlasgow, , imn nttt fSine Rehovot, eizmann Institute ofScience, , U U niv T ik.fl r P A ra,France Orsay, LAL S iest fBre,Norway niversity ofBergen, cerPyisIsiueAC,Rz Czech Rez, uclear Physics Institute ASCR, U Hungar nvriéd ai-u,Osy France Orsay, Université deParis-Sud, echnic nvriyo asw PolandUniversity of Warsaw, esden, iest fAhn,Greece niversity ofAthens, iest fRm,IF,Italy INFN, niversity ofRome, e uPEC olish A itPtrbr nvriy Russia aint-Petersburg University, F B Hungar iest fBn,Germany niversity ofBonn, ilde [email protected] r The Institute Andrzej Soltan for Nuclear ersit erwUiest,Jrslm Israel Jerusalem, Hebrew University, e ly t Petersburg Nuclear Physics Institute, St. oningen,The Netherlands oningen,The [email protected] AS, ,Denmark y, U t vbjer Scienc H. iest fHlik,Finland niversity ofHelsinki, lUiest fMnc,Germany al University ofMunich, , d C) oa Bulgaria Sofia, D y al U F Germany ian A cademy ofSciences,Warsaw, enmar Université Libre deBruxelles, of r y g eiesleb V niv P DIRECTORY EPS es alencia, c ersity adem k • BP 5027 en y S pain of Sciences, 05/06 237
directory EPS DIRECTORY 05/06
V. Croquette, Laboratoire de Physique Statistique, Quantum Electronics & Optics D.L. Weaire, Trinity College, Dublin, Ireland Paris, France H.S. Wio, Centro Atomico Bariloche, Argentina T. Duke, Cavendish Laboratory, Cambridge, United Chair P. French Kingdom Imperial College Co-opted Members H. Flyvbjerg, RISOE National Laboratory, Roskilde, Physics Department G. Ahlers,University of California, Santa Barbara, USA Denmark Femtosecond Optics Group P. A. Lindgard, RISOE National Laboratory, Denmark H. Gaub, University of Munich, Germany Prince Consort Road T. Viczek, ELTE, Budapest, Hungary A. Maritan, SISSA,Trieste, Italy UK-SW7 2BW London, United Kingdom P. Ormos, Institute of Biophysics, Szeged, Hungary TEL/FAX +44 2 075 947 706 / +44 2 075 947 714 F. Parak, Technische Universität München, Germany EMAIL [email protected] GROUPS J. Zaccai, CEA-CNRS, Institut de Biologie Structurale, Grenoble, France Secretary G. Björk Accelerators Group QEO IMIMT Plasma Physics School of Communication and Information Chair C.R. Prior Technology ASTeC Intense Beams Group Chair J.B Lister Royal Institute of Technology Rutherford Appleton laboratory CRPP / EPFL KTH Electrum 229 UK-Oxon OX11 0QX, Chilton, Didcot, United Kingdom CH-1015 Lausanne, Switzerland SE-164 40 Kista, Sweden TEL/FAX +44 1 235 445 262 / +44 1 235 445 607 TEL/FAX +41 216 933 405 / +41 216 935 176 TEL/FAX +46 87 904 060 / +46 87 904 090 EMAIL [email protected] EMAIL jo.lister@epfl.ch EMAIL [email protected] Chairman-elect C. Biscari Vice-Chair R. Bingham Board Members INFN–LNF Rutherford Appleton Laboratory C. Björk, Royal Institute of Technology, KTH, Sweden Via E. Fermi,40 Chitton Didcot R. Blatt, Universität Innsbruck, Austria IT-00044 Frascati (Roma), Italy UK- OX11 0QX, United Kingdom A.D. Boardman, University of Salford, United Kingdom TEL/FAX +44 1 235 445 800 / +44 1 235 445 848 A. Brignon, Thales Research and Technologies, Orsay, Executive Secretary & Treasurer C. Petit-Jean-Genaz EMAIL [email protected] France CERN • AC Division V. Buzek, Slovak Academy of Sciences, Bratislava, CH-1211 Geneva 23, Switzerland Board Members Slovakia TEL/FAX +41 227 673 275 / +41 227 679 460 C. Hidalgo, EURATOM-CIEMAT, Madrid, Spain P. French, Imperial College London, United Kingdom EMAIL [email protected] S. Jacquemot, CEA/DAM – Ile de France, Bruyères-le- D. Lenstra, Vrije Universiteit, Amsterdam, the Chatel, France Netherlands Board Members J. Ongena, EURATOM – Belgium State, NFSR, Belgium P. Mataloni, Universita di Roma la Sapienza, Italy C. J. Bocchetta, Sincrotrone Trieste, Italy J. Stockel, Academy of Sciences of the Czech R. Menzel, Universität Potsdam, Germany R. Brinkmann, DESY,Germany Republic, Prague G. Roosen, Université Paris Sud, IOTA, Orsay, France H. Danared, Stockholm University – MSI, Sweden M. Tendler, Royal Institute of Technology, Stockholm, L. Torner, Universitat Politecnica de Catalunya, J.P. Delahaye, CERN - AB Division, Switzerland Sweden Terrassa, Spain D.J.S. Findlay, Rutherford Appleton Laboratory,UK A. Tuennermann, Friedrich-schiller Universität, Jena, J.M. Lagniel, CEA Saclay, France Co-Opted Members Germany M. Minty, DESY,Hamburg, Germany T. Märk, Science Faculty Innsbruck, Austria I. Veretenicoff, Vrije Universiteit Brussels, Belgium S.P. Moller, Arhus Universitet – ISA, Denmark M.E. Mauel, Columbia University, New York, USA A. Mosnier, CEN Saclay, France Co-opted Members C. Pagani, INFN - IASA, Italy Ex-Officio Members A. Arimondo, Universita degli Studi di Pisa, Italy F. Perez, CELLS,Spanish Synchrotron Light Source,Spain G.M.W. Kroesen, Eindhoven University of Technology, J. Dalibard, ENS – UPMC, Paris, France U. Ratzinger, Johann Wolfgang Goethe Universität, The Netherlands M. Dawson, Institute of Photonics, University of Germany J. Meyer-Ter-Vehn, MPI – Max-Planck-Institute for Strathclyde, Glasgow, United Kingdom L. Rivkin, Paul Scherrer Institut, Switzerland Quantum Optics, Garching, Germany S. de Silvestri, Politecnico di Milano, Italy F. Ruggiero, CERN– AB Division, Switzerland M. Ducloy, Institute Galilee, Université Paris nord, Y. Shatunov, Budker Institute of Nuclear Physics, Russia Beam Plasma and Inertial Fusion Section Villetaneuse, France S.L. Smith, ASTEC, Daresbury Laboratory, United Chair J. Meyer-ter-Vehn C. Fabre, Université Pierre et MarieCurie, Paris, France Kingdom Max-Planck-Institut für Quantenoptik J. Faist, Université de Neuchâtel, Switzerland U. van Rienen, Universität Rostock, Germany Hans-Kopfermann-Str. 1 D.R. Hall, Heriot-Watt University, Edinburgh, United A.Wrulich, Paul Scherrer Institute, Switzerland DE-85748 Garching, Germany Kingdom TEL/FAX +49 8 932 905 137 / +49 8 932 905 200 G. Huber, Universität Hamburg, Germany Computational Physics Group EMAIL [email protected] U. Keller, Swiss Federal Institute of Technology – ETH Zurich, Switzerland Chair P.H. Borcherds Board Members P. Laporta, Politecnico di Milano, Italy The University of Birmingham • Physics Department S. Atzeni, Universita "La Sapienza" di Roma, Italy G. Leuchs, Universität Erlangen-Nürnberg, Germany Edgbaston J. Honrubia, Polytechnic University of Madrid, Spain P. Loosen, Fraunhofer Institut für Lasertechnik, UK-Birmingham B15 2TT, United Kingdom P. Mora, Centre de Physique Théorique (UPR 14 du Aachen, Germany TEL/FAX +44 1 214 753 029 CNRS), France P. Mandel, Université Libre de Bruxelles, Belgium EMAIL [email protected] P. Norreys, Central Laser Facility, Rutherford Appleton E. Riedle, Ludwig-Maximilians-Universität München, Laboratory, United Kingdom Germany Secretary A. Hansen J. Wolowski, Institute of Plasma Physics and Laser P. Torma, Helsinki University of Technology, Finland Instituut for Fysikk Microfusion, Poland V.N. 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Tel, Eötvös University, Budapest, Hungary CH-1211 Geneva 23, Switzerland A. Vulpiani, Universita degli Studi La Sapienza, Rome, TEL/FAX +41 227 672 581 / +41 227 674 400 Italy EMAIL [email protected]
238 europhysics news NOVEMBER/DECEMBER 2005 T302Tise Italy IT-34012 Trieste, S S W.D. Klotz, T S. Kim, T. Katoh, N. K J.W. Humphrey, K.T. Hsu, W D.P. Gurd, J.P. Froideveaux, B J. Farthing, S. Dasgupta, P.N. Clout, M. Clausen, R. Chevalley, N. Charrue, W R A G. Baribaud Contacts N. Yamamoto, R D Co-opted Member R G. Raffi, R S I L Board Members EMAIL TEL/FAX France FR-91191 Gif-sur-Yvette Cedex, D C Treasurer EMAIL TEL/FAX Secretary eur J M. R. Tanaka, C. A. S R. Steiner, J M. S T. Schmidt, B I. R G. R E M. Rabany, M. P R. P M. Panighini, O Y R. Mueller, M. M A. M W.P. Mcdowell, R. M B R.P J A. Luchetta, S. L D R. Lackey, E H. . .O . .R. L . . nrtoeTise Elettra incrotrone Trieste, S 4–K.135 Basovizza 163.5, 14–Km. .S. . . .A. Kuper, . . .F . .M. . . . EA-Saclay . . . APNIA/SIS . S. Ko, .P . Sk K Na Hoff, R Lackey, Tanaka, P T S Müller, Buenger, Blecha, Lecorche, Bulfone, N K . Germany Germany Kingdom Or United Kingdom C Japan Korea Germany T Busse r iquez, . ewis ak .J. Heubers, imura, o chirmer affi, ose ytin, ammery, M aupp ohler, ana Uhomoibhi, huot, inin, en erio ucillo elly c oph ezger se, M vr utz, oua g ada, annix, Cla j d aniza tr ar atil, [email protected] , SAMSUNG, South Korea South SAMSUNG, B P ya, [email protected] , E , J , J ,India e, + +39 0403758579/565 Institut ohang University ofScience & Technology, European Southern Observatory (ESO), European Observatory Southern , INR, t echal, okae ainlLbrtr,USA rookhaven National Laboratory, B La SRCC,Taiwan SRCC,Taiwan oin ur t CERN, INFN, E ceeao aoaoy Japan KEK -Accelerator Laboratory, , CNRS -IN2P3, , 33 169087032/+33088138 T y , B Hahn-M chey L , r F N plainNurnSuc,USA Spallation Neutron Source, M em ainlLbrtr,USA Fermi National Laboratory, O .F Gournay J.-F. Japan A .Bulfone D. GSI, Germany Germany GSI, J AR,Japan JASRI, NIRS, TRIUMPF it oto ytmIc,USA ControlVista System Inc., A he Or o BESSY, Berlin, Germany Germany Berlin, BESSY, os A P SY eln Germany Berlin, ESSY, pa otenOsraoy Germany Observatory, Southern opean C U wr sics news SF FranceESRF, riNtoa ceeao aoaoy USA Nationalermi Accelerator Laboratory, R , SI Japan ASRI, S EN Switzerland CERN, Consorzio RFX, Italy Italy Consorzio RFX, Culham Science Centre, Jet Facilities, Jet Culham Science Centre, tion, ational Accelerator Centre, South Africa South ational Accelerator Centre, EN Switzerland CERN, , P sraor eGnv,Switzerland bservatoire deGenève, DESY ax-P t ok str aul-Scher U z Consorzio RFX, Italy Italy Consorzio RFX, sinAaeyo cecs Russia ussian Academy ofSciences, R nrtoeTise ltr,Italy Elettra, incrotrone Trieste, niv GANIL, J CERN, obs Switzerland rofibus, Bhabha,Variable Energy Cyclotron Bhabha,Variable cii,Switzerland Acqiris, CERN-P Institut ech Societa Italiana Avionica, Italy Italy Avionica, Italiana Societa enc itIsiuefrNcerRsac,Russia oint Institute for Nuclear Research, uther niv High Energy Accelerator Research , ha onomic I ao ainlLbrtr,USA lamos National Laboratory, Institut ST D ron ainlLbrtr,USA Argonne National Laboratory, taly NIKHEF U riyo brk,Japan ofIbaraki, ersity e Japan acle Center, United Kingdom United Kingdom acle Center, LC USA SLAC, lack-Institut FüP J T ersit , v olfi Q France Worldfip HQ, niv e apan f ehia nvriy Czech Republic Technical University, eitner Institut he Q G rHg nryPyis Russia or HighEnergy Physics, nNtoa aoaoy USA en National Laboratory, t B omic Energy Research Institute, S f , er France ivision, or C ersit ree aoaoy USA Laboratory, erkeley witz e s y many anada r d D o ula eerh Russia for Nuclear Research, al O rIsiu,Switzerland er-Institut, o ueens U , of D F the Netherlands the Netherlands ivision, A r de F e y anc NO pplet r of bser land or S witz isic VEMBER e W tmund v est England niv on Lab Switzerland , ,Brazil a, atory of Trieste, Italy Italy of atory Trieste, G er lasmaph ersit ermany land , Germany Germany , rtr,United oratory, y / DECEMBER of Belfast, y , sik, UK 2005 I ainlHlei eerhFudto,Greece National Research Hellenic Foundation, A. Suz F I. Nadr F. Brouillard, M.C. A B EMAIL TEL/F Chair Physics for Development Group A.K C.Symeonidis, J. Sebesta, A.Moreno Gonzalez, P. Lazarova, J L. Kovacs, R. Karazija, E D K N L. J A.Angelopoulos, B EMAIL S EMAIL TEL/FAX C ofPhysicsHistory Group Z J. Zhao, C L. Yamazaki, N F White,K. E D K H. G. Orlandi, C. Rossel J.P. Huignard, G. Demor B EMAIL TEL/F Chair T E L. Hasselg G. Furlan, V G. DelgadoBarrio, C L. Woeste, A U Switzerland CH-1015 Lausanne, Université deLausanne F Institut dePhysique delaMatière Condensée T O50 egn Norway NO-5007 Bergen, - uln Ireland E-2 Dublin, . .H. Worm, . aculté desScienc rinity Collegerinity echnolo . . . .L. F ecretay . o-opt .Y. Yao, hair . .J llega oard Members o oar . . niv . . Lajz P Vesztergombi, Lillethun, Holmber Zhiyuan, Gr Van Houtte, Yamamoto, W Hoffmann, Ef ard Members . L B M USA Facility, United Kingdom Greece Physics S eulc Czech Republic Republic, Mario Boella, Italy Italy Boella, Mario F . iuzzi, . W undamen brtr,China aboratory, weden B elgium W ersity of Bergen, Department ofPhysics Department ofBergen, ersity enninger, thymios, adr andin, d A A ar o ittamor er .Vaagen J. Chergui M. D or- t roblewski, breu ESilva, X X majed.chergui@epfl.ch [email protected] [email protected] jans chal Members Members y en 55 ed M a, .L.Weaire a, o id nttt fHg nryPyis China Institute ofHighEnergy Physics, +47 55582724/589440 +41 216923664or678/635 + H , CEA–S W wicz, P Thomas Jefferson National Accelerator G 353 16081055/+353711759 tier Spain , IBM Z g nvriaTise Italy Universita Trieste, r efei National Synchrotron Radiation U ezey olg,Sobtey Hungary Szombathely, Berzsenyi College, .v • olyt ri nvriä eln Germany Freie Universität Berlin, , en, einer oeisUiest,Bailv,Slovakia Bratislava, Comenius University, .Vlahakis C g T inu nvriy Lithuania University, Vilnius y College Green T n I A aagen@fi Bulgaria or emb U R China NR, ainlIsiuefrFso cec,Japan National Institute for Fusion Science, etv lcrncSsesS,Switzerland reative Systems Electronic SA, , h , Université Catholique deLouvain, iv c Group e, e, tal, U niv N M U HLSTosnCF FranceTHALES-Thomson CSF, e ino In echnic adem LPPS, K • CERN, ersit nvriyo tes Greece University ofAthens, U C niv acla ur ational Research Hellenic Foundation, niv T Department ofPure andApplied Department K-AclrtrLbrtr,Japan EK -Accelerator Laboratory, xPac-ntttBri,Germany ax-Planck-Institut Berlin, R , ACdrce France EA Cadarache, niv r SC Spain CSIC, ersit ers U o C W nvriyo tes Greece University ofAthens, isk c eerh Switzerland ich Research, s BSP es, ersit y ersit niv R,Switzerland ERN, Instituto deMatematica yFisica y y U arsa ersité J al S t y el, U , er of Zar al Institut iesdd oAgre Portugal niversidade doAlgarve, S France Universidad Complutense de y .uib of ersité deP psala U witzerland y nazionale U of B y w w of H Scienc nited Kingdom of N edish Academy ofSciences, .no U oseph F ago er iest,Poland niversity, lik,Finland elsinki, gen, amur iest,Sweden niversity, e z es oftheCzech ,Spain a, of P ar Istituto Superiore , N , is-Sud –LPP uir France ourier, Belgium Belgium orway orway h y is Romania sics, , F rance M3506Yrvn Armenia AM-375 036 Yerevan, Institut A A A K 70, A Universitätsplatz 5 A Hima15 D. 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A P Atominstitut derOsterreichischen Universitäten S A aculty ofNaturalaculty Sciences ee addr raterallee 2 T-1090 T-8010 G T-8010 G -00we,Austria T-1020 wien, ulevardi Zogu 1 ecretary ecretary ecretary General ecretary ar r lik rmenian Physicalrmenian Society -iaa Albania L-Tirana, L L elar niversiteti I Tiranes a a epartment ofPhysicsepartment ustr r lbanian Physical Society esiden menian P tional A tional Academy Of Sciences Of Belarus tional Academy Sciences Of Of - - l-F F hanian B . iaa Albania Tirana, iaa Albania Tirana, S A A ranzens Universität Graz usian P ian Physical Society k X X d p a [email protected] [email protected] v r offi[email protected] [email protected] er e e [email protected] aryna Avearyna o [email protected] [email protected] spahiu@y [email protected] +355 4227914 +355 427669 +375 172840954/030 +43 3163805192/809816 +43 1427751700/779517 +43 15880114100/199 +43 6764849053/3163809816 +374 2341347/350030 W [email protected] ess oftheS für P of M gl@ap t r a,Austria raz, c ien, ..Lippitsch M.E. A ..Ershov-Pavlov E.A. P. Berberi P Rauch H. Avakian R. z Austria az, adem N .A. Deda . .C.Tscheliessnigg E. r a h tesS.2 others St. insk, olecular andAtomic Physics h ATIONAL SOCIETIES h t A A y y er .univie.ac.at y er sik panase sical Society Society sical .Spahiu D. G.Vogl ustr sic ialphysik y phi.am a Belarus Belarus hoo.com of E al Society al Society ia ecretary General General ecretary . Babakhanyan Babakhanyan . Scienc A Albania Belarus Austria P DIRECTORY EPS vich rmenia es of Belarus es ofBelarus 05/06 239
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BY-220072 Minsk, Belarus HR-10000 Zagreb, Croatia Treasurer E.H. Pedersen TEL/FAX +375 172 840 654 / +375 172 393 131 TEL/FAX +385 14 698 861 / +385 14 698 889 University of Aarhus EMAIL [email protected] EMAIL [email protected] Institute of Physics and Astronomy Nordre Ringgade 1 Vice-Presidents President A. Hamzic DK-8000 Aarhus C, Denmark V. M. Anischik, Belarussian State University, Belarus University of Zagreb EMAIL [email protected] TEL +375 172 066 001 or 206 690 Faculty of Science, Department of Physics N. M. Olehnovich, National Academy of Sciences, Bijenicka 32 Belarus HR-10000 Zagreb, Croatia Estonia TEL +375 172 841 203 TEL/FAX +385 14 605 544 / +385 14 680 336 EMAIL [email protected] Estonian Physical Society Secretariat Belgium Vice-President Z. Roller-Lutz Estonian Physical Society University of Rijeka Tähe 4 Belgian Physical Society Medical School EE-51010 Tartu, Estonia General Secretary J.Ingels Brace Branchetta TEL/FAX +372 7 428 182 / +372 7 383 033 Belgisch Intituut voor Ruimte-Aëronomie HR-51000 Rijeka, Croatia EMAIL efs@fi.tartu.ee Institut d'Aéronomie Spatiale de Belgique TEL/FAX +385 51 651 210 / +385 51 675 806 Editor Physicalia Info EMAIL [email protected] President A. Kikas Ringlaan 3 University of Tartu BE-1180 Ukkel, Belgium Treasurer D. Veza Institute of Physics TEL/FAX +32 23 730 378 / +32 23 748 423 University of Zagreb Riia 142 EMAIL [email protected] Faculty of Science, Department of Physics EE-51014 Tartu, Estonia Bijenicka 32 TEL/FAX +372 7 428 182 / +372 7 383 033 President V .Pierrard HR-10000 Zagreb, Croatia EMAIL [email protected] Institut d'Aéronomie Spatiale de Belgique TEL/FAX +385 14 605 535 / +385 14 680 336 3, Avenue Circulaire EMAIL [email protected] Vice-Presidents BE-1180, Uccle, Belgium I.Kink, University of Tartu, Estonia TEL/FAX +32 23 730 418 / +32 23 748 423 TEL/FAX +372 7 428 886 / +372 7 383 033 EMAIL [email protected] Czech Republic EMAIL ilmar.kink@fi.tartu.ee P. Suurvarik, Tallinn Universityof Technology, Estonia Vice-President P.Wagner Czech Physical Society TEL/FAX +372 6 203 000 / +372 6 202 020 Limburg Universitair Centrum Administrative Secretary A. Bernathova EMAIL [email protected] Instituut voor Materiaallonderzoek Czech Physical Society Wetenschapspark 1 Na Slovance 2 Treasurer P.Tenjes BE-3590 Diepenbeek, Belgium CZ-182 21, Prague 8, Czech Republic University of Tartu TEL/FAX +32 11 268 895 / +32 11 268 899 EMAIL [email protected] Tähe 4 EMAIL [email protected] EE-51010 Tartu, Estonia General Secretary A. Cieply TEL/FAX +372 7 375 576 / +372 7 375 570 Treasurer J. Hellemans Nuclear Physics Institute ASCR Katholieke Universiteit Leuven CZ-250 68, Rez near Prague, Czech Republic Academische Lelarenopleiding Natuurkunde TEL/FAX +420 266 173 284 / +420 220 940 165 Finland Naamsestraat 61 EMAIL [email protected] BE-3000, Leuven, Belgium Finnish Physical Society TEL/FAX +32 16 324 250 / +32 16 324 254 President J. Dittrich General Secretary E. Hänninen EMAIL [email protected] Institute of Physics University of Helsinki Academy of Sciences of the Czech Republic Finnish Physical Society Narodni 3 FI-00014 University of Helsinki , Finland Bulgaria CZ-117 20 Prague 1 TEL/FAX +358 919 150 523 / +358 919 150 553 TEL/FAX +420 220 941 147 / +420 220 941 130 EMAIL finphys@helsinki.fi Union of Physicists in Bulgaria EMAIL [email protected] Secretary S. Jordanova President K. Hämäläinen Union of Physicists In Bulgaria • Administrative Council Vice-Presidents University of Helsinki, Department of Physical Sciences 5, James Bourchier Blvd D. Slavinska, Charles University, Prague, Czech Republic Division of X-ray Physics • PO Box 64 BG-1164 Sofia, Bulgaria EMAIL [email protected] FI-00014 University of Helsinki, Finland TEL/FAX +359 2 627 660 / +359 29 625 276 D. Novotny, University of J.E. Purkyne, Usti nad Labem, TEL/FAX +358 919 150 640 / +358 919 150 610 EMAIL [email protected]fia.bg Czech Republic EMAIL keijo.hamalainen@helsinki.fi EMAIL [email protected] Executive Secretary T. Popov Vice-President K.A. Suominen Sofia University Treasurer P. Bydzovsky University of Turku, Department of Physics 5 James Bourchier Blvd Nuclear Physics Institute ASCR FI-20014 University of Turku, Finland BG-1164, Sofia, Bulgaria CZ-250 68, Rez near Prague, Czech Republic TEL/FAX +358 23 335 782 / +358 23 335 070 TEL/FAX +359 28 161 880 TEL/FAX +420 266 173 283 / +420 220 940 165 EMAIL kale-antti.suominen@utu.fi EMAIL [email protected]fia.bg EMAIL [email protected]
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B G eerbr .Jonson B. B J.P. Ansermet t eas .g u . hysique Expérimentale . . .Albietz S. details oftheS h ener G asel, nd Braunecker Akkus y y yalog alo .zuett sik r S , ibourg, Département de Physique Département ibourg, .Züttel A. g,Switzerland ugg, S witzerland es al g@unibas Institut fürP w Switzerland , e.ansermet@epfl.ch • e Switzerland L. el@unifr Department ofPhysics Department den A Sweden T P DIRECTORY EPS mon ur ecr k .ch .ch e h etary General etary y .tr y sik 05/06 243
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