1 Physica D 233 (2007) Bandt Pompe approach to the classical-quantum transition A.M. Kowalski a, M.T. Martín b, A. Plastino b, O.A. Rosso c, a ...

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Bandt–Pompe approach to the classical-quantum transition A.M. Kowalski a , M.T. Mart´ın b , A. Plastino b , O.A. Rosso c,∗ a Instituto de F´ısica, Facultad de Ciencias Exactas, Universidad Nacional de La Plata and Buenos Aires’ Comisi´on de Investigaciones Cient´ıficas (CICPBA),

C.C. 67, 1900 La Plata, Argentina b Instituto de F´ısica, Facultad de Ciencias Exactas, Universidad Nacional de La Plata and Argentina’s National Research Council (CONICET),

C.C. 727, 1900 La Plata, Argentina c Instituto de C´alculo, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Pabell´on II, Ciudad Universitaria,

1428 Ciudad de Buenos Aires, Argentina Received 30 January 2006; received in revised form 7 January 2007; accepted 14 June 2007 Available online 29 June 2007 Communicated by A. Pikovsky

Abstract By regarding the celebrated classical-quantum transition problem as one pertaining to the domain of dynamic systems’ theory, we present a discussion that exhibits the superiority of the Bandt–Pompe approach to the extraction of a probability distribution from time series’ values. c 2007 Elsevier B.V. All rights reserved.

Keywords: Semiclassical theories; Quantum chaos; Entropy; Statistical complexity

1. Introduction Thequestion of the relation of quantum and classical mechanics is a large and important one. The transition between the corresponding regimes of applicability is not abrupt but gradual (a transition region exists), as was realized from the very beginning of quantum mechanics. Its earlier manifestation is that of the WKB approximation. No sooner was wave mechanics abroad, than a semiquantum method of applying it to the most important problems of the day was devised. These problems were the new phenomenon of tunnelling through a potential barrier, and the energy eigenstates of a potential well, either of an oscillator or the radial problem in atomic spectra. Solving problems in wave mechanics generally meant the solution of differential equations, for which even in one dimension there were no analytical solutions, except in a few special cases. By approximating the wave function as a oscillatory wave depending on a phase integral, many useful problems could be solved by a mere quadrature. Almost ∗ Corresponding author. Tel.: +54 11 4786 8114; fax: +54 11 4786 8114.

E-mail addresses: [email protected] (A.M. Kowalski), [email protected] (M.T. Mart´ın), [email protected] (A. Plastino), [email protected] (O.A. Rosso). c 2007 Elsevier B.V. All rights reserved. 0167-2789/$ - see front matter doi:10.1016/j.physd.2007.06.015

simultaneously, Wentzel [1], Kramers [2] and Brillouin [3] published applications of this theory to the Schr¨odinger equation. Their initials give the term WKB approximation. In this paper we wish to describe original details of the classicalquantum transition region that we have obtained by recourse to information theory. Kolmogorov and Sinai [4,5] converted information theory into a powerful tool for the study of dynamic systems. Consequently, Shannon’s information measure [6] became a “quantifier” of the dynamics. In more recent times, other quantifiers, also based on information theory concepts, like different entropic forms and statistical complexities [6–13], have been proved to be useful for the characterization of the dynamics associated to time series. Indeed, information theory measures and probability spaces Ω are inextricably linked. The central point that will here occupy our attention is the fact that, in the evaluation of the above mentioned quantifiers, the determination of the probability distribution P associated to a given dynamic system or time series is the basic “starting” stage. Many schemes have been proposed for a proper selection of P ∈ Ω . We can mention, among others, procedures based on amplitude-statistics [14], symbolic dynamics [15], Fourier analysis [16], and wavelet transform [17]. Their applicability

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depends on data characteristics like stationarity, length of the series, parameters’ variation, level of noise contamination, etc. In all these cases the global aspects of the dynamics can be somehow “captured”, but the different approaches are not equivalent in their ability to discern all the relevant physical details. One must also acknowledge the fact that the above techniques are introduced in a rather “ad-hoc fashion”, and not derived directly from the dynamical properties themselves of the system under study. Bandt and Pompe [18] proposed, instead, a method for evaluating the probability distribution P associated to a given time series based, precisely, in the peculiar facets of the attractor-reconstruction problem. In such a way, causal information is duly incorporated in the construction-process that yields P ∈ Ω . A notable Bandt and Pompe result is a clear improvement in the performance of the information quantifiers obtained using their P-generating algorithm. Of course, one must assume with them that the system is stationary and that enough data are available for a correct attractor reconstruction. The goal of the present communication is to delve further into the methodological questions associated to the “P ∈ Ω determination-problem”. Our consideration will be illustrated by a really tough problem, whose difficulties go well beyond those posed by the simple dynamic examples commonly employed to discuss new methodological aspects of the information theory approach to dynamical systems, namely, the so-called classical-quantum border [19]. This problem can be tackled without recourse to statistical quantifiers, and thus offers a unique possibility of getting an unbiased assessment. The classical limit of quantum mechanics (CLQM) continues attracting the attention of many theorists and is the source of much exciting discussion (see, for instance, Refs. [20, 21] and references therein). In particular, the investigation of “quantum” chaotic motion is considered important in this limit. Recent literature provides us with many examples, although the adequate definition of the underlying phenomena is understood in diverse fashion according to the different authors (see Ref. [22] and references therein). Here, by recourse to the employment of information based quantifiers, we address the subject using a “semiquantal” model advanced by Cooper et al. [23–25]. We use two quantifiers, namely, • the normalized Shannon entropy, HS [6], and • the intensive MPR-statistical complexity CJS [12], • both evaluated (herein lies our claim to originality) with the permutation probability distribution methodology proposed by Bandt and Pompe [18]. It is important to remark that the normalized entropy is not taken as a measure of the system complexity (a common identification founded in the bibliography). In the present work an specific quantifier – the MPR-Statistical Complexity – is consider. Why are we interested in using these quantifiers? Because we will show that there are details of the classical-quantum transition that one can not appreciate even having at hand the exact solutions for the present problem. These solutions have

been studied and reported by us in previous efforts ([53] and references therein). Semiclassical methodologies, that date back to the pioneer years of quantum mechanics, have been of immense help in allowing for an intuitive grasp of many quantum features. Among semi-classical descriptions we single-out here the socalled semiquantal ones, characterized by a mixture of dynamic regimes. We face two interacting systems, one of which is amenable to a classical treatment (for instance, because it is macroscopic) while the other requires a strict quantum one. More generally, quantum effects are considered to be negligible for one of the two systems. Many important systems have been satisfactorily addressed in this fashion during the last 60 years. One can mention the Bloch equation [26], for instance. In quantum optics we encounter two-level systems interacting with an electromagnetic field within a cavity and also the Jaynes–Cummings model [27–32]. Reference can be made to the micro-maser [33], the collective nuclear model [34], the molecular motion of perturbed nonlinear dimers, [35], some cosmological models (quantum field theory in curved space–times) [36], etc. As for the Cooper model we can cite the work of Bonilla and Guinea for the description of wave functions collapse as observed with a classical instrument [37]. Also the treatment of quantum chaos (i) using the Born–Oppenheimer approximation [38] and (ii) by recourse to the effective Pattanayak–Schieve Hamiltonian [39,40]. We revisit the Cooper mentioned model in order to illustrate, in what we hope is a convincing way (given the abstruse nature of the classical-quantum limit), the usefulness of the Bandt and Pompe approach for the evaluation of probability distribution associated to the time series generated by the system. A comparison with results obtained with an alternative technique, using probability distributions based on wavelet analysis, will also be made [41,42]. 2. Methodology 2.1. Statistical complexity measures In a recent contribution, L´opez-Ruiz, Mancini and Calbet (LMC) have proposed a statistical complexity measure, based on the notion of “disequilibrium”, as a quantifier of the degree of physical structure in a time series [10]. Given a probability distribution P associated to a system’s state, the LMC measure CLMC is the product of a normalized entropy H (normalized Shannon-entropy) times an Euclidean “distance” from P to the uniform distribution Q. CLMC vanishes both for a totally random process and for a purely periodic one. Mart´ın, Plastino and Rosso (MPR) [11] improved on this measure by suitably modifying the distance-component (in the concomitant probability space). In Ref. [11], Q is built-up using Wootters’ statistical distance [43]. Regrettably enough, the two statistical complexity measures above mentioned are neither intensive nor extensive quantities, although they yield useful results. Also, a reasonable complexity measure should be able to distinguish among

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different degrees of periodicity and it should vanish only for the simplest degree of periodicity. In order to attain such goals any natural improvement should give this statistical measure an intensive character. In Ref. [12] Lamberti et al. obtained a MPR-statistical complexity measure that is (i) able to grasp essential details of the dynamics, (ii) an intensive quantity, and (iii) capable of discerning among different degrees of periodicity and chaos. This statistical complexity measure is the one to be employed here. 2.2. Selecting the probability distribution The essence of symbolic dynamics is to associate a symbolsequence with each trajectory of a continuous or discrete dynamic system. This is done by means of a suitable partition of the state–space. Such a process is described within the context of a delay embedding of the time series into a D-dimensional space (see Ref. [44]). Special “generating partitions” yield, in the limit of an appropriately fine resolution, the Kolmogorov–Sinai entropy. However, these partitions are very difficult to ascertain even in the case of two-dimensional systems. Bandt and Pompe [18] advanced a method that “naturally” determines the adequate symbol-sequence from the time series’ values, without additional model assumptions. They determine suitable partitions of the state–space by recourse to an adequate comparison between neighbouring series’ values. For any given series they look for certain ordinal patterns of order D. From the symbol-occurrence frequency they deduce a permutation probability distribution. The advantages of Bandt and Pompe’ method reside in (a) its simplicity, (b) the associated extremely fast calculation process, (c) its robustness and (d) its invariance with respect to nonlinear monotonous transformations. Although the Kolmogorov–Chaitin algorithmic complexity is another excellent tool for the purpose at hand, that one could profitably use in order to face the problem we are focusing our attention on, it poses a much more difficult task from a practical viewpoint. 2.3. The MPR-statistical complexity plus Bandt–Pompe approach The intensive MPR-statistical complexity measure [12] can be viewed as a functional CJS [P] that characterizes the probability distribution P associated to the time series generated by the dynamic system under study. It quantifies not only randomness but also the presence of correlational structures [10–12]. The intensive MPR-statistical complexity is of the form CJS [P] = Q J [P, Pe ] · HS [P],

(1)

where, to the probability distribution P = { p j ; j = 1, . . . , N }, we associate the entropic measure !, N X HS [P] = S[P]/Smax = − p j ln( p j ) Smax , (2) j=1

with Smax = S[Pe ] = ln N , (0 ≤ HS ≤ 1). Pe = {1/N , . . . , 1/N } is the uniform distribution and S is Shannon’s

entropy. The disequilibrium Q J is defined in terms of the extensive Jensen–Shannon divergence [12] and is given by Q J [P, Pe ] = Q 0 {S[(P + Pe )/2] − S[P]/2 − S[Pe ]/2}

(3)

with Q 0 a normalization constant (0 ≤ Q J ≤ 1) given by Q 0 = −2

−1 N +1 ln(N + 1) − 2 ln(2N ) + ln N . N

(4)

Thus, the disequilibrium Q J is an intensive quantity. For evaluating the probability distribution P associated to the time series (dynamic system) under study we follow the methodology proposed by Bandt and Pompe [18] and consider partitions of the D-dimensional space that will hopefully “reveal” relevant details of the ordinal structure of a given one-dimensional time series. Given the time series {xt : t = 1, . . . , M} and an embedding dimension D > 1, we are interested in “ordinal patterns” of order D [18,45] generated by (s) 7→ xs−(D−1) , xs−(D−2) , . . . , xs−1 , xs , (5) which assign to each time s the D-dimensional vector of values at times s, s − 1, . . . , s − (D − 1). Clearly, the greater the D-value, the more information on the past is incorporated into our vectors. By the “ordinal pattern” related to the time (s) we mean the permutation π = (r0 , r1 , . . . , r D−1 ) of (0, 1, . . . , D− 1) defined by xs−r D−1 ≤ xs−r D−2 ≤ · · · ≤ xs−r1 ≤ xs−r0 .

(6)

In order to get a unique result we set ri < ri−1 if xs−ri = xs−ri−1 . Thus, for all the D! possible permutations π of order D, the probability distribution P = { p(π )} is defined by p(π ) =

]{s|s ≤ M − D + 1; (s), has type π} . M − D+1

(7)

In this expression, the symbol ] stands for “number”. The normalized entropy HS and the intensive MPR-statistical complexity CJS are then evaluated for this “permutation” probability distribution. 2.4. Embedding considerations and length of the time series The Bandt and Pompe’s methodology can be apply to any type of time series (regular, chaotic, noisy or reality based), with a weak stationary assumption [18]. It is important to remark that for the applicability of Bandt and Pompe’s methodology we need not assume that the time series under analysis is representative of low dimensional dynamic systems. In this methodology the embedding dimension D plays an important role in the evaluation of the appropriate probability distribution. This is so because D determines the number of accessible states D!. Also, it conditions the necessary length M of the time series that one needs in order to work with reliable statistics. In relation to this last point, we propose that the condition M D! be satisfied. In particular, Bandt and Pompe propose work with 3 ≤ D ≤ 7.

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In the present work we are dealing with a low dimensional system (see Section 3). Then we can establish the optimal embedding dimension as well as the length of the time series to be considered by recourse of the nonlinear dynamic tools. The purpose of the time delay embedding is to “unfold” our projection process back to a multivariate state space that is representative of the original system. The general topological result of Ma˜ne´ and Takens [46,47] states that, when the attractor has dimension D A , all orbits’ self-crossings will be eliminated if one chooses D > 2D A . One may view the method of delays as a bridge connecting (i) the temporal fluctuations of a single observable O to (ii) the spatial characteristics of the dynamic system that O represents [44]. The result of Ma˜ne´ and Takens yields only a sufficient condition and provides no assistance when selecting a reconstruction delay for experimentally obtained data. ¿From a mathematical point of view the attractor will be unfolded if we use either the minimum embedding dimension D (min) (see below), or any D ≥ D (min) . In summary, the embedding dimension must obey D ≥ (2D A + 1) ≥ D (min) . In practice, working in any dimension larger than the minimum required by the data leads to excessive computation when we evaluate any metric parameter (R´enyi dimensions, Lyapunov exponents, etc.) that we may be interested in. Larger dimensions also increase contamination by round-off or instrumental errors. This “noise” will intensively populate and “dominate” things in the additional D − D (min) dimensions of the embedding space, where no dynamics is operating [44]. For the determination of D (min) one can follow the method of False Nearest Neighbors proposed by Abarbanel and coworkers [48]. In practice, the correlation dimension, DC , has become a widely used measure in the pertinent literature, and can be profitably employed as an estimator of the attractor’s dimension D A and, a posteriori, of the embedding dimension D. The correlation dimension DC can be computed by applying the useful method described by Grassberger and Procaccia [49]. The estimation of DC provides a lower bound to the actual number of variables required to model the system. Moreover, Eckman and Ruelle [50] have established the relation between the dataset “size” and fundamental limitations for estimating dimensions and Lyapunov exponents in dynamical systems. In particular, they have shown that the Grassberger and Procaccia algorithm will not produce reliable dimensions larger than DCmax

= 2 ln(M)/ ln(1/ρ),

(8)

with ρ = r/(2R) 1. Here r is the radius of the ball containing the nearest neighbours, for which a “plateau” manifests itself (see Refs. [49,50]), R the radius of the reconstructed attractor, and M the dataset’s size. 3. A semi-classical model and the CLQM: A review 3.1. Hamiltonian and dynamic equations We will illustrate the preceding ideas with reference to a trivial generalization of the semi-classical hamiltonian that represents the zero-th mode contribution of a strong external

field to the production of charged meson pairs [23–25]. The Hamiltonian reads ! 2 2 1 p ˆ P A 2 2 Hˆ = + m q ω xˆ , (9) + 2 mq m cl where (i) xˆ and pˆ are quantum operators, (ii) A and PA classical canonical conjugate variables and (iii) ω2 = ωq 2 + e2 A2 an interaction term that introduces nonlinearity, ωq being a frequency. The quantities m q and m cl are masses, corresponding to the quantum and classical systems, respectively. As shown in Refs. [51,52], in dealing with (9) one faces an autonomous system of nonlinear coupled equations (that here plays the role of a dynamic system) ˆ dhxˆ 2 i h Li dh pˆ 2 i ˆ = , = −m q ω2 h Li, dt mq dt PA dPA dA = , = −e2 m q Ahxˆ 2 i, dt m cl dt ˆ h pˆ 2 i dh Li 2 2 =2 − m q ω hxˆ i , Lˆ = xˆ pˆ + pˆ x. ˆ dt mq

(10)

The system of equations (10) follows immediately from Ehrenfest’s relations [51,52]. By suitably varying the parameters entering the Hamiltonian one can study the transition from a purely quantal regime to the classical one [22,41,42]. Of course, we are mostly interested in the intermediate region. To study the classical limit we need to also consider the classical counterpart of Eq. (9) # " PA 2 1 p2 (11) + m q (ωq2 + e2 A2 )x 2 , H= + 2 mq m cl where all the variables are classical. Recourse to Hamilton’s equations allows one to find the classical version of Eq. (10) (see Refs. [51,52] for details). The classical limit is obtained by letting the “relative energy” Er =

|E| → ∞, I 1/2 ωq

(12)

where E is the total energy of the system and I an invariant of motion related to the Uncertainty Principle I = hxˆ 2 ih pˆ 2 i −

ˆ 2 h Li . 4

(13)

A classical computation of I yields I = x 2 p 2 − L 2 /4 ≡ 0. A measure of the convergence between classical and quantum results in the limit described by Eq. (12) is given by the norm N of the vector 1u = u − u cl [51,52] N1u = |u − u cl |,

(14)

ˆ is the “quantum” part of the where u = (hxˆ 2 i, h pˆ 2 i, h Li) solution of the system equation (10) and u cl = (x 2 , p 2 , L) is its classical partner. 3.2. Reaching the classical limit: Morphological details A detailed study of the classical quantum transition was performed in Refs. [51–53]. We summarize their main results,

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as needed for our present methodological discussion. In plotting diverse dynamical quantities versus (growing) Er , one finds an abrupt change in the system’s behaviour for a special value of Er , to be denoted by Er cl . From this value onwards, the quantum dynamics starts converging to the classical one. One can thus assert that Er cl provides us with an indicator of the presence of a quantum classical “border”. Thus, Er > Er cl

(15)

will be considered as an “almost-classic zone”. The zone Er < Er cl ,

(16)

corresponds to the regime investigated in Ref. [53]. This regime, in turn, is characterized by two different subzones [51, 52]: (1) one of them is an almost purely quantal one, in which the microscopic quantum oscillator is just slightly perturbed by its interaction with the classical one, and (2) the other exhibits a transitional nature (semiquantal). The border between these two subzones can be characterized by a “signal” value Er P . For Er > Er P chaos is always found. The relative number of chaotic orbits (with respect to the total number of orbits) grows with Er and tends to unity for Er → ∞ [51–53]. The region Er P < Er < Er cl ,

(17)

is not of a quantal character, due to the presence of chaos. Neither can it be regarded as classical, since convergence to the known classical results has not even started yet. We call it the semiquantum zone. As Er grows from Er = 1 (the “pure quantum instance”) to Er → ∞ (the classical situation), and specially from Er P onwards, a significant series of morphology changes is detected. The concomitant orbits exhibit features that are not easily describable in terms of Eq. (14), which is a global measure of convergence in amplitude (of the signal). Can these morphological features be adequately captured by a dynamic system’s approach based on the analysis of a representative time series? We answer the question below. 4. Bandt–Pompe methodology’s answer The method proposed by Bandt and Pompe [18] for evaluating the probability distribution P is based on the details of the attractor-reconstruction procedure. Bandt and Pompe consider a partition of the D-dimensional state space determined by the intersections of D! hyper-planes of R D : x1 = x2 , . . . , x1 = x D ; x2 = x3 , . . . , x2 = x D ; . . . ; x D−1 = x D . Each permutation π of order D can be associated with one of the connected pieces determined by the partition. In other words an “ordinal pattern” represents one connected piece of R D , and the union of all pieces is the total state space R D . The probability distribution P of “ordinal patterns” is given by the frequency, in the attractor structure, of each

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piece (pattern). P is assigned by “counting” the times that the attractor visits each piece (see Eq. (7)). In particular, if the attractor is symmetrical with respect to the hyper-planes, all the connected pieces have the same frequency and thus the distribution of ordinal patterns is uniform: the attractor “visits” all the partition pieces with the same frequency. Consequently, the information provided by the time series so as to predict geometrical locations of successive D-strings vanishes and the entropy is maximal (Smax = ln D! and HS = 1). On the other hand, if the situation is such that the attractor remains always within just one of the connected pieces, one can “predict” with certainty (HS = 0). The statistical complexity given by Eq. (1) vanishes both for Q J = 0 and for HS = 0. Complexity and entropy are functionals of the probability distribution P. How to best select P is the issue we are discussing in this communication. A particular selecting criterion is provided by the relative wavelet energy (see Refs. [41,42]). With it, the ensuing complexity measure refers to the behavior of the frequency-bands present in the series under scrutiny. This behaviour might not be the best representative of the overall dynamic structure that underlies the series. The features of a dynamical system manifest themselves primarily in the geometry of its attractor, as illustrated by Fig. 1, a graph that depicts the time-series and their attractors for different Er -values. Recourse to the Bandt and Pompe methodology requires first of all to ascertain the embedding dimension D, a critical factor in order to assign our probabilities P from which, we repeat, our quantifiers HS and CJS will be evaluated. The time series of Fig. 1 refers to the temporal evolution of the expectation value hxˆ 2 i (left). The attractors (right) correspond to an embedding dimension D = 3 and time lag τ = 1, for several Er -values. Using distinct delays τ provides one with different details of the time series. The results of Fig. 1 suggest that these attractors “live” in a space of dimension D A ≡ DC ∼ 2. Thus, for an embedding dimension D = 5 our attractors will have “unfolded” to a sufficient degree, an assumption compatible with the Eckman–Ruelle estimation [50] for the maximum correlation dimension DCmax ∼ 7 of a time series of length M = 5000 (one further needs to assume that the density for the attractor is ρ ∼ 0.1). The CLQM model we are dealing with here is represented by five non-linear coupled equations (see Eq. (10)). A correct reconstruction of the attractor associated with the integration of this system is obviously achieved using an embedding dimension D = 5. Since the model possesses two invariants of motion [53], only three independent variables remain. Thus, the “real” attractor dimension will be D A ≤ 3. The choice DC ≈ 2 does imply D = 5. We will consequently adopt an embedding dimension D = 5 for evaluating first P and then HS and CJS . Notice that the condition M = 5000 D! = 120 is satisfied. Fig. 2 depicts the relative probability frequencies of N = D! = 120 partition-connected pieces, Pj =

j X k=1

pk

with j = 1, . . . , 120,

(18)

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(a) Er = 1.01.

(b) Er = 3.32.

(c) Er = 6.00. Fig. 1. Raw signal (time series corresponding to hxˆ 2 i with M = 5000 data) and the corresponding reconstructed attractor in embedding dimension D = 3 and time lag τ = 1 for different Er -values. The subfigures (g)–(i) correspond to the classical zone. Subfigure (i) depicts results that to all practical effect can be regarded as classical.

as a function of Er [54]. In this figure the spaces between successive curves represent the relative frequencies of the concomitant ordinal patterns, where the first pattern j = 1 is associated to what one may call the “bottom space”, the next pattern ( j = 2) to the space between the first and the second curve (counting from below), an so on up to the top “space” j = 120. Differences between distributions at distinct Er values closely resemble the qualitative differences between the original signals at these precise Er values (compare the six signals and attractors of Fig. 1). In particular, the high degree of visibility of both the bottom and top patterns ( j = 1 and j = 120), respectively, at Er < 21.52-values, is a sign of the predominance of monotonous behaviour in the original signals.

Fig. 3 refers to the quantity P j , It nitidly exhibits three regions: quantal, transitional, and classical. In each of them the P j -behaviour is of a quasi constant nature. The Er -specific values that delimit these regions will be determined below, in discussing entropic and complexity results. In the quantum zone only two “ordinal patterns” predominate, corresponding to p1 and p120 . The remaining ordinal patterns have an almost null participation. The series associated with Er ' 1 is very regular and its attractor has a small and regular elliptical shape, with a sharp boundary (see Fig. 1). It consists of only two symmetrical pieces ( p1 ' p120 ). Starting at Er ' 3.32 the attractor maintains its overall shape but becomes elongated. This is reflected in a probability increase for the occupancy of other regions and a corresponding

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(d) Er = 6.81.

(e) Er = 8.09.

(f) Er = 10.52. Fig. 1. (continued)

diminution of p1 and p120 . A transitional zone insinuates itself (Figs. 1 and 2). The Er ' 3.32 value was referred to previously as the “signal point” Er P , at which chaos begins to rise its head, as duly reflected by the time series’ values for specific types of initial conditions. For Er ' 6 some intermediate { pi : 1 < i < 120} acquire higher relevance. For Er ' 10 curves belonging to the attractor’s interior show deformation signs and generate “holes” (see Fig. 1). The attractor dimension decreases and one begins to appreciate the emergence of a more symmetrical ordinal patterns distribution, although the bottom and top patterns ( p1 and p120 ) still distinguish themselves from the rest. The D-strings are of a more and more homogeneous character in the distribution of the partition pieces, as illustrated by Fig. 2. Eventually, for about half (about 50) of the 120 patterns we are here considering, an almost uniform distribution

prevails. Also, the attractor becomes symmetric with respect to some of the planes we are dealing with. The concomitant pieces are visited with similar frequency, although for p1 and p120 this happens more frequently. For relative energies Er > 21.52 the associated time series exhibits convergence towards known classical results (Cf. Fig. 1) and the pertinent (relative) probability frequencies P j (Fig. 2) correspond to that of the classical case, characterized by the (also relative) dominance of approximately 50 of our regions. The value Er = 21.52 corresponds to the quantity we called Er cl above. Fig. 3 exhibits results obtained by replacing the original time series by its ordinal transforms. We follow here the Keller and Sinn work [54], who concoct a clever and at the same time simple algorithm in order to conveniently order all the patterns involved in tackling the task we are here facing. Accordingly,

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(g) Er = 21.55.

(h) Er = 33.28.

(i) Er = 60.00. Fig. 1. (continued)

the ordinal pattern at a given time t is now coded by a number n D (t) that adopts the values 1, 2, . . . , D!. The “ordinal transform” is a map that assigns the time series {(ν D (t))t : t = 1, . . . , M} to the original one {xt : t = 1, . . . , M}, where n D (t) ∈ [0, 1]. (19) D! This Keller and Sinn transform “extracts” from the original series its ordinal information content, which, in turn, allows one to distinguish between either (1) a stochastic origin or (2) a deterministic one for our series. Fig. 3 depicts, for each Er , the corresponding transformed series. Pay attention to the vertical lines of dots. For Er −values corresponding to the quantum zone, the vertical line contain just a few dots (the ordinal transform can adopt just a few ν D (t) =

values), while, as one approaches the classical zone the dots become almost uniformly distributed in the interval [0, 1]. A transition zone is easily distinguishable. Fig. 3(b) focuses on the details of the interval 1 < Er < 11. For 1 < Er < 4 a dot-concentration is appreciated, indicative of ordinal patterns containing an important amount of information. As Er grows, a dot-dispersion process can be detected, as new ordinal patterns make their appearance. Fig. 4(a) and (b) depict, respectively, the permutation entropy HS and the intensive MPR-statistical complexity CJS versus Er . Our three regions are clearly appreciated in Fig. 4: • Region I: quasi-quantal, for small enough Er ’s, • Region II: transitional starting at Er ' 3.32, in which a slope-change is detected, and • Region III: the region of convergence towards classicality, beginning at, approximately, Er cl = 21.52.

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Fig. 2. Relative frequency probabilities P j , for D = 5 and τ = 1, as a function of Er -values. The distribution corresponding to Er = 60 is the classical one, to all practical effects.

Fig. 4. (a) Normalized entropy HS and (b) intensive MPR-statistical complexity CJS as a function of Er -values (D = 5 and τ = 1). Notice that to the right of the second vertical dash line we encounter the classical zone.

Fig. 3. Ordinal Pattern, for D = 5 and τ = 1, as a function of Er -values.

It is time now to compare the present results (Bandt and Pompe approach) to those previously obtained by recourse to a wavelet analysis in [41,42], in order to be in a position to pass judgment on the “Bandt–Pompe improvements”. With reference to those old results (wavelet approach) the present results are clearly better in the first region, where the results using wavelets exhibit an inconsistent behavior not amenable to a reasonable interpretation. As different from the analysis presented in [41,42] here both region I and region III exhibit small and similar degrees of complexity, as one should expect, as the pertinent dynamics

are, respectively, (i) regular in the quantum instance and (ii) chaotic in the classic one. The complexity cannot vanish neither in (i), because of frequency superposition, nor in (ii), due to the chaotic character of the motion (obviously, chaotic and random are not equivalent concepts in this respect). These observations confirm the ones one reaches by simply solving the equations of motion for the model of Cooper et al. [53]. A new contribution to our understanding is here given, though: the higher complexity degree of the transitional region II vis-a-vis that of I and III, where CJS grows with Er till the value Er ' 10 is reached, peaks up there, and decreases thereafter. This higher complexity can be interpreted as due to the prevailing “mixing” of the quantum and classical regimes. As the quantal “influence” diminishes, so does the complexity. Also, the present results (Bandt and Pompe approach) display a more rapid convergence rate to classicality vis-a-vis the old results using wavelet approach. The present behaviour agrees, as stated, with that obtained by recourse to purely dynamic considerations (i.e. not using statistical tools) [53]. Our stronger argument in favour of the present methodology is to be found in its ability to detect without ambiguity just where convergence to classicality starts, again in agreement with purely dynamic reasoning [53]. This is best illustrated by reference to Fig. 5, that depict the complexity-entropy plane. The continuous lines refer to strict upper and lower bounds for the complexity quantifier [13]. Classicality is clearly located at

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References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] Fig. 5. The Entropy-Complexity plane HS × CJS . Continuous lines represent, respectively, the maximum and minimum values of complexity for fixed values of entropy. Note the piling-up effect that takes place beginning at Er , corresponding to Ercl . This evidence of convergence to the classical instance, HS = 0.80402955 and CJS = 0.29651476.

the right of the graph, the quantum regime to the left. Maximum complexity “signals” the transition stage. The quantum regime is obviously “more ordered” than the classical one, as expected. 5. Conclusions “Just” having the exact solutions for the Cooper model without the proper information-theoretical tools, important details of the classical-quantum transition cannot be clearly observed [53]. Thus the importance we are attributing to the statistical complexity concept as a helpful tool to unravel complicated dynamic features. We have shown that a statistical analysis based on the Bandt–Pompe methodology neatly reproduces the most important features of the classical-quantum transition in the model of Cooper et al., that had been previously described by simply solving the pertinent equations derived from Ehrenfest theorem [53], without recourse to statistical concepts. But much more is learned. In particular, a slope-change of the statistical complexity is detected at the special point Er P . From that Er -point onwards, as one ventures into the transition region, the complexity reaches a maximum value first, and diminishes afterwards, on its way to its classical value. Beyond the Ercl -point the complexity initially exhibits some oscillations and afterwards a smooth convergence to its classical value. Most importantly, the Bandt–Pompe approach to dealing with our model is seen in this work to provide revealing insights into the classical-quantum transition process, details that cannot be obtained even if one exactly solves the pertinent equations of motion. Acknowledgment This work was partially supported by CONICET (PIP 5687/05 and 6036/05), Argentina.

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