TThe Wigner distribution and 2D classical maps
Jamal Sakhr
Department of Physics and Astronomy,University of Western Ontario, London, Ontario N6A 3K7 Canada (Dated: November 7, 2018)
Abstract
The Wigner spacing distribution has a long and illustrious history in nuclear physics and inthe quantum mechanics of classically chaotic systems. In this paper, a novel connection betweenthe Wigner distribution and 2D classical mechanics is introduced. Based on a well-known cor-respondence between the Wigner distribution and the 2D Poisson point process, the hypothesisthat typical pseudo-trajectories of a 2D ergodic map have a Wignerian nearest-neighbor spacingdistribution (NNSD) is put forward and numerically tested. The standard Euclidean metric is usedto compute the interpoint spacings. In all test cases, the hypothesis is upheld, and the range ofvalidity of the hypothesis appears to be robust in the sense that it is not affected by the presenceor absence of: (i) mixing; (ii) time-reversal symmetry; and/or (iii) dissipation.
PACS numbers: 05.45.Mt, 02.50.Ey, 05.40.-a, 05.45.Ac a r X i v : . [ n li n . C D ] M a y . INTRODUCTION Random matrix theory (RMT) [1–3], which was originally formulated in the 1960s tobetter understand nuclear spectra, has more recently emerged as a fundamental theoreticaltool for understanding the spectra of quantum systems whose classical analogs are chaotic(see Refs. [4–9] for examples and discussions). The most basic mathematical object of thetheory is the probability density P ( S ) of the spacing S between adjacent energy levels. Theaforementioned probability distribution P ( S ), commonly referred to as the nearest-neighborspacing distribution (NNSD), has been computed for a vast number of experimental andtheoretical systems (mostly bounded conservative systems), and its overall shape variesdepending on: (i) the presence or absence of time-reversal symmetry; and (ii) the degree ofchaoticity in the classical dynamics. When there is time-reversal symmetry and the classicaldynamics are strongly-chaotic, the sample NNSD is typically well modeled by the Mehta-Gaudin distribution, which is the theoretical NNSD for the eigenvalues of a random matrixchosen from the so-called Gaussian orthogonal ensemble (GOE). The latter distribution,denoted here by P GOE ( S ), cannot be expressed in closed-form (i.e., in terms of a finitenumber of elementary functions). An excellent analytical approximation to P GOE ( S ) ishowever given by the Wigner distribution [3, 8] P W ( S ) = π S exp (cid:16) − π S (cid:17) , (1)which happens to be the theoretical NNSD for eigenvalues chosen from a Gaussian ensembleof real symmetric 2 × like ” (or “GOE- like ”) NNSDs. Inview of this fact, a “Wigner- like ” NNSD is now widely regarded to be a “generic” propertyof time-reversal-invariant (TRI) quantum systems having strongly-chaotic classical limits[49]. The underlying reasons for this observed statistical behavior nevertheless remain elu-sive. It is presumed that, in “generic” TRI systems, the Wignerian shape of the NNSD (aproperty of the quantum eigenvalues) derives solely from the chaoticity of the classical dy-namics (a property of the classical trajectories). It is however not fundamentally understood how classical chaos is itself responsible for producing the observed Wignerian shape of theenergy-level spacing distribution (a point that is often glossed over in the “quantum chaos”literature). 2hile there have been significant advancements in understanding the observed agreementwith other predictions of the random matrix model, such as the spectral rigidity [12] and thetwo-point correlation function [13–19], there has been comparatively little material progressin understanding the observed agreement (most often semi-quantitative) with the eigenvaluespacing statistics of the classical random matrix ensembles. The nearest-neighbor spacingdistribution is particularly challenging mathematically. An analytical means of understand-ing the spacing statistics of quantum chaotic systems, based on semiclassical periodic orbittheory, has been attempted in Ref. [20]. However, due to the asymptotic techniques em-ployed therein, the semiclassical formulas for the k th-nearest-neighbor spacing distributions P ( S ; k ) are good approximations to the corresponding distributions obtained from RMTonly for large values of S and k ; for small k and in particular for k = 1 the obtainedsemiclassical formulas do not reproduce the well-known RMT results. In short, a clear-cuttheoretical justification for why a Wigner- like NNSD is a common “quantum signature ofchaos” is still lacking.Curiously, the Wigner distribution also appears in the seemingly unrelated subject ofspatial point processes, and more specifically, in the context of the homogeneous Poissonpoint process in R (henceforth denoted by P ): the Wigner distribution is the nearest-neighbor spacing distribution (NNSD) for P [8, 21]. To the author’s knowledge, this resultwas first mentioned in a paper by Grobe, Haake, and Sommers [22] published in 1988 andwas subsequently useful in understanding certain spectral fluctuation properties of quantumdissipative systems (see, for instance, Chapter 9 of Ref. [8]). The intent of this paper is tointroduce a connection between P and two-dimensional (2D) ergodic maps and to therebyexplicate the fundamental significance of P W ( S ) to 2D classical mechanics. II. ERGODIC TRAJECTORIES OF 2D CLASSICAL MAPS: THE P MODEL
The link between P and 2D classical mechanics to be introduced below is based on asimple observation: Trajectories of discrete maps are discrete point sets. The most well-known example is the Poincar´e return map. In numerical investigations of discrete maps, itis common to compute numerical trajectories (i.e., ‘pseudotrajectories’) for a large numberof initial conditions and to then generate a so-called phase portrait (a plot consisting ofall computed pseudotrajectories overlaid in a single window whose boundary defines the3 D ( S ; ) FIG. 1: (Top) A numerical realization of a two-dimensional homogeneous Poisson point process(of intensity 10000) consisting of 10000 random points uniformly distributed in the unit square.(Bottom) Interpoint nearest-neighbor spacing density histogram for the point pattern shown above.The smooth curve is the Wigner distribution [Eq. (1)]. phase space of the map). For both symplectic and dissipative maps, the signature phaseportrait of an ergodic map is an apparently random scatter of points (often generated from one initial condition). This signature point pattern has become ubiquitous in numericalinvestigations, especially in numerical Poincar´e surface-of-section (SOS) computations forchaotic Hamiltonian systems, and the understanding in that context is that the apparentlyrandom scatter of points is a numerical rendering of how an ergodic (or nearly ergodic) orbitexplores the interior of the energy shell in the full phase space. In the specific context of2D maps, such point patterns are visually indistinguishable from any numerical realizationof P (see, for example, the top panel of Fig. 1). This begs the question: For a 2D ergodic ap, do the interpoint nearest-neighbor spacings of any typical pseudotrajectory of that maphave a Wignerian distribution? Intuitively, the affirmative answer is correct. Any typical pseudotrajectory of a 2D ergodicmap will densely and uniformly cover the entire phase space of that map (excluding perhapsa zero-measure subset of the phase space) [50]. The interpoint spacing statistics of sucha pseudotrajectory should therefore be consistent with the spacing statistics theoreticallypredicted for P . In particular, the NNSD of such pseudotrajectories should be consistentwith the Wigner distribution. These arguments should hold regardless of the presence orabsence of: (i) mixing; (ii) time-reversal symmetry; and (iii) dissipation. Furthermore,pseudotrajectories need not necessarily cover densely all of the available phase space inorder for their NNSDs to be Wignerian. Pseudotrajectories evolving ergodically in anypositive-measure subset of the full phase space should also possess NNSDs consistent withthe Wigner distribution. The preceding claims naturally require verification and indeed theintent in the following sections of the paper is to validate these claims numerically. III. GENERIC EXAMPLES OF 2D ERGODIC MAPSA. Example 1: Reversible Symplectic Anosov Map
The cat map q n +1 p n +1 = q n p n , mod 1 (2)is area-preserving, ergodic and mixing (in fact hyperbolic), and time-reversal invariant [23].In studying ‘typical’ (non-periodic) pseudotrajectories of map (2), the set of initial conditionsconsisting of all pairs of rational numbers chosen from the unit interval is excluded. This isa set of measure zero in R , and thus almost all initial conditions chosen from [0 , × [0 , ξ i = ( q i , p i ) and ξ j = ( q j , p j ) in phase space is de-fined here using the two-dimensional Euclidean metric: ∆( ξ i , ξ j ) = (cid:112) ( q i − q j ) + ( p i − p j ) .The distance between a given point ξ i and its nearest neighbor is then defined by5 (1) i = min { ∆ ( ξ i , ξ j ) : i, j = 1 , . . . , N ( j (cid:54) = i ) } , and similarly the distance between ξ i andits furthest neighbor is defined by d ( N ) i = max { ∆ ( ξ i , ξ j ) : i, j = 1 , . . . , N ( j (cid:54) = i ) } . If, foreach given point ξ i , the spacings { ∆ ( ξ i , ξ j ) : i, j = 1 , . . . , N ( j (cid:54) = i ) } are sorted by size(in ascending order), then the k th-nearest-neighbor spacing is the k th element of the set { d (1) i , d (2) i , . . . , d ( k ) i , . . . , d ( N ) i } . For numerical comparisons, the sample k th-NNSD is definedin terms of the scaled spacings S ( k ) i = d ( k ) i / ¯ d ( k ) , where ¯ d ( k ) = (1 /N ) (cid:80) Ni =1 d ( k ) i , is the mean k th-nearest-neighbor spacing. The k th-nearest-neighbor spacing density histogram is thenconstructed by binning all N values of S ( k ) i and normalizing the area under the histogramto unity.For the pseudotrajectory shown in the top left panel of Figure 2, the density histogramof the (scaled) nearest-neighbor spacings is shown in the top right panel of Figure 2. Thelatter is clearly in accord with the Wigner distribution [Eq. (1)].Small nonlinear perturbations of map (2) that preserve the time-reversal symmetry donot alter the above result. Consider, for example, the perturbed cat map [24] q n +1 p n +1 = q n p n + K π cos(2 πq n ) , mod 1 (3)where K is a free perturbation parameter. Map (3) is symplectic and time-reversal-invariant[25]. It is also hyperbolic when K ≤ (cid:0) √ − (cid:1) / √ K = 0 . B. Example 2: Reversible Symplectic Map that is Ergodic but not Mixing
Consider next the following 2D map that describes skew translations (or rotations) onthe unit two-torus: q n +1 p n +1 = q n p n + α , mod 1 . (4)When the real parameter α is an irrational number, map (4) is (uniquely) ergodic and not mixing [26, 27]. Map (4) is also time-reversal invariant [28]. The left panel of Figure 3shows a typical pseudotrajectory of map (4) when α = ln(2), and the density histogram of6 S D ( S ; ) S D ( S ; ) FIG. 2: (Top Left) A typical pseudotrajectory of the unperturbed cat map defined in Eq. (2).The pseudotrajectory was evolved from the initial point ( q = ( √ − / , p = ( √ − / K = 0 .
3. The pseudotrajectorywas evolved from the initial point ( q = ( √ − / , p = ( √ − / the (scaled) nearest-neighbor spacings is shown in the right panel of Figure 3. The latteris again consistent with the Wigner distribution. Incidentally, the nearest-neighbor spacingdistribution of the eigenphases that come from quantizing map (4) is not well-defined in thesemiclassical limit [29]. 7 Q P S D ( S ; ) FIG. 3: (Left) A typical pseudotrajectory of the non-mixing ergodic map defined in Eq. (4)when α = ln(2). In this instance, the pseudotrajectory was evolved from the initial point ( p =1 / √ , q = 1 / √ C. Example 3: Reversible Symplectic Map with an Ergodic Component
2D area-preserving maps of the form y n +1 = y n + Kf ( x n ) , mod 1 , (5a) x n +1 = x n + y n +1 , mod 1 , (5b)where K is a free parameter, and f ( x n ) is some prescribed function, generally possess amixed phase space consisting of commingled regular and chaotic regions [30]. (The “standardmap” obtained by prescribing f ( x n ) = π sin(2 πx n ) is the most well-known and well-studiedmember of this family of maps.) The form and distribution of the regions generally dependon both the function f and the parameter K . When the function f ( x n ) is a piece-wise linearfunction of the interval x n ∈ [0 , f ( x n ) = − x n if x n ∈ (cid:2) , (cid:1) − + x n if x n ∈ (cid:2) , (cid:1) − x n if x n ∈ (cid:2) , (cid:3) , (5c)8 S D ( S ; ) FIG. 4: (Left) A typical pseudotrajectory of map (5) with parameter K = 2. The pseudotrajectorywas evolved from the initial point ( x = 1 / √ , y = 1 / √ the phase space consists of one regular region and one ergodic region separated only by theborder of the regular region, which in this case is non-hierarchical [32]. In the present context,the most important feature of map (5) is that its phase space has an ergodic region (i.e.,there exists a positive-measure subset of the full phase space wherein the dynamics of map(5) is ergodic). As argued in Sec. II, the interpoint NNSD of any typical pseudotrajectoryinitiated in the ergodic region should be consistent with the Wigner distribution. One suchpseudotrajectory of map (5) with K = 2 is shown in the left panel of Figure 4, and the densityhistogram of the (scaled) nearest-neighbor spacings is shown in the right panel of Figure4. Although the goodness-of-fit of the Wigner distribution appears to be lower here thanin the previous examples, the Kolmogorov-Smirnov (KS) test does not reject the Wignerdistribution at the 99% significance level (KS test statistic=0.0070, p -value=0.1776) nor doesthe more sensitive Anderson-Darling (AD) test (AD test statistic=2.3712, p -value=0.0579)at the same significance level. 9 . Example 4: Non-Reversible Symplectic Anosov Map The area-preserving hyperbolic cat map q n +1 p n +1 = q n p n , mod 1 (6)is not time-reversal invariant [33]. (In fact, the only symmetry possessed by map (6) isparity.) The top left panel of Figure 5 shows a typical pseudotrajectory of map (6) and thedensity histogram of the (scaled) nearest -neighbor spacings, which is again consistent withthe Wigner distribution, is shown in the top right panel of Figure 5.From an RMT perspective, it is also interesting to consider small nonlinear perturba-tions of the above cat map that break so-called pseudosymmetries [34]. A simple exam-ple (following Ref. [34]) is constructed by composing the unperturbed cat map given byEq. (6) with two shears: a momentum shear P p ( q, p ) = ( q, p + k p F ( q )), and a position shear P q ( q, p ) = ( q + k q G ( p ) , p ), where k p , k q are free parameters that determine the strength ofthe perturbations, and F ( q ) and G ( p ) are periodic functions. If A denotes the unperturbedcat map given by Eq. (6), then the perturbed cat map is defined as follows: q n +1 p n +1 = ( A ◦ P p ◦ P q ) q n p n , mod 1 . (7a)More explicitly, the above map can be written as q n +1 p n +1 = q n p n + k q G ( p n ) + k p F ( q n + k q G ( p n )) , mod 1 . (7b)To break the parity symmetry of A using the momentum shear P p (which is always invariantunder time reversal), F ( q ) should be an even function. To ensure that the perturbed maphas no classical symmetries (and no pseudosymmetries when quantized), the functions F ( q )and G ( p ) were chosen as follows: F ( q ) = 12 π (cid:16) cos(2 πq ) − cos(4 πq ) (cid:17) , (7c) G ( p ) = 12 π (cid:16) cos(4 πp ) − cos(2 πp ) (cid:17) . (7d)It is well-known that cat maps are structurally stable [35], which means that, if the pertur-bation is sufficiently small, the perturbed cat map possesses the same dynamical properties10 Q P S D ( S ; ) Q P S D ( S ; ) FIG. 5: (Top Left) A typical pseudotrajectory of the unperturbed cat map defined in Eq. (6).The pseudotrajectory was evolved from the initial point ( q = 1 / √ , p = 1 / √ k p = k q = 0 . q = ( √ − / , p =( √ − / as the unperturbed cat map; in other words, the hyperbolicity of the unperturbed map ispreserved under sufficiently small perturbations. Map (7) is thus symplectic and possessesno symmetries (classical or otherwise), and (for small values of k p and k q ) the map is hyper-bolic (and thus also ergodic). A typical pseudotrajectory of map (7) with k p = k q = 0 . S D ( S ; ) FIG. 6: (Left) A typical pseudotrajectory of the dissipative cat map (8) with perturbation pa-rameter ε = 0 . q = ( √ − / , p =( √ − / times. (Right) Nearest-neighbor spacing density his-togram for the pseudotrajectory shown to the left. The smooth curve is the Wigner distribution[Eq. (1)]. nearest-neighbor spacings is shown in the bottom right panel of Figure 5. The latter isagain consistent with the Wigner distribution. The same result is obtained even when theparity symmetry of A is not broken (i.e., when F ( q ) is an odd function). E. Example 5: Non-Reversible Dissipative Anosov Map
Nonlinear perturbations of cat maps are generally not area-preserving. For example, thefamily of perturbed cat maps given by q n +1 p n +1 = q n p n + ε f ( q n , p n ) f ( q n , p n ) , mod 1 (8a)where ε is a perturbation parameter, and f ( q n , p n ) and f ( q n , p n ) are periodic functions, isgenerally dissipative [36]. (Note that the unperturbed map is not reversible.) A simple butconcrete example is furnished by taking (see Ref. [37]) f ( q n , p n ) = 12 π (cid:16) sin [2 π ( q n + p n )] + sin (2 πq n ) (cid:17) , (8b) f ( q n , p n ) = 0 . (8c)12ap (8) with ε (cid:54) = 0 is dissipative with local volume contraction rate given byΛ( q n , p n ) ≡ ln (cid:12)(cid:12) det J ( q n , p n ) (cid:12)(cid:12) = ln (cid:12)(cid:12)(cid:12) ε (cid:110) cos [2 π ( q n + p n )] + 2 cos (2 πq n ) (cid:111)(cid:12)(cid:12)(cid:12) , (9)where J ( q n , p n ) is the Jacobian of map (8). For sufficiently small values of ε (see Ref. [35]for a precise criterion), map (8) is furthermore hyperbolic (and thus ergodic). A typicalpseudotrajectory of map (8) with ε = 0 .
033 is shown in the left panel of Figure 6 and thedensity histogram of the (scaled) nearest-neighbor spacings is shown in the right panel ofFigure 6. The latter is again consistent with the Wigner distribution.Note that longer pseudotrajectories ( N = 10 compared to the usual N = 25000 used inall prior examples) were used in the analysis of this dissipative case. The reason is that theorbits of this dissipative map have a non-trivial spatial structure, and this structure takessome time to materialize. (The spatial structure of the orbits in the prior symplectic cases iscomparatively trivial.) Shorter pseudotrajectories of this map (e.g., N = 25000) also possessWignerian NNSDs, but their statistical significance (as measured by goodness-of-fit tests)is lower. Longer pseudotrajectories (e.g., N = 10 ) produce results very similar to thoseshown in Figure 6, and their corresponding statistical significances are also similar. F. Higher-Order Spacing Distributions
Although not directly relevant to the main premise of the paper, it is worthwhile tobriefly comment on and discuss higher-order spacing distributions. For all the examplesconsidered in this paper, the higher-order spacing statistics are found to be consistent withthose theoretically predicted for P . Note however that the higher-order spacing statisticsof P do not generally coincide with higher-order GOE spacing statistics. For example,the second -NNSD for GOE eigenvalues is the same as the NNSD for eigenvalues from theGaussian symplectic ensemble (GSE) [38]: P GOE ( S ; k = 2) = P GSE ( S ; k = 1) . (10)The second -NNSD for P , on the other hand, coincides with the eigenvalue NNSD for theGinibre ensemble of 2 × P G ( S ) = 3 π S exp (cid:18) − π S (cid:19) . (11)13 S D ( S ; ) Data P (k = 2) GOE (k = 2) S D ( S ; ) Data P (k = 3) GOE (k = 3)
FIG. 7: (Top)
Second -nearest-neighbor spacing density histogram for the pseudotrajectory shownin the left panel of Fig. 3. The unbroken (blue) curve is the Ginibre distribution P G ( S ) [Eq. (11)].The dashed (red) curve is the Wigner surmise P W ( S ; k = 2 , β = 1) [Eq. (12)]. (Bottom) Third -nearest-neighbor spacing density histogram for the pseudotrajectory shown in the left panel ofFig. 3. The unbroken (blue) curve is the Wigner surmise P W ( S ; k = 1 , β = 5) [Eq. (13)]. Thedashed (red) curve is the Wigner surmise P W ( S ; k = 3 , β = 1) [Eq. (14)]. The ‘Wigner surmise’ approximation for Eq. (10) is given by (using the notation of Ref. [21]) P W ( S ; k = 2 , β = 1) = P W ( S ; k = 1 , β = 4) = 2 π S exp (cid:18) − π S (cid:19) . (12)14s an example, consider again the skew map of Sec. III B. For the pseudotrajectory shownin the left panel of Fig. 3, the density histogram of the (scaled) second -nearest-neighborspacings is shown in the top panel of Fig. 7. The agreement with the Ginibre distribution[Eq. (11)] and discordance with the Wigner surmise P W ( S ; k = 2 , β = 1) [Eq. (12)] is evident.The result for the third -nearest-neighbor spacings is shown in the bottom panel of Fig. 7.Although not widely known, the third -NNSD for P coincides with the Wigner surmise forthe β = 5 Hermite ensemble and is given by [21] P W ( S ; k = 1 , β = 5) = 15 π S exp (cid:18) − π S (cid:19) . (13)(The spacing distributions of the β -Hermite ensembles are discussed in Ref. [39].) The GOEWigner surmise of order k = 3 is given by [21] P W ( S ; k = 3 , β = 1) = 2 π S exp (cid:18) − π S (cid:19) . (14)The agreement with the third -NNSD for P [Eq. (13)] and discordance with the Wignersurmise P W ( S ; k = 3 , β = 1) [Eq. (14)] is again evident.In general, as k (i.e., the order of the spacing) increases, the discrepancy between the k th-NNSD for P and the GOE Wigner surmise of order k [ P W ( S ; k, β = 1)] increases. Notethat the higher-order spacing distributions for P and the higher-order Wigner surmises forthe GOE do not coincide at any common order k >
1, but there are in fact a countablyinfinite number of cases in which a coincidence occurs between different order members fromthese two families of distributions (see Eq. (13) of Ref. [21]) [51].
IV. A NON-GENERIC 2D ERGODIC MAP
The ergodic maps considered in Sec. III can be regarded as “generic” 2D ergodic maps.It is expected that there will be a zero-measure set of “non-generic” 2D ergodic maps whosepeculiarities (for example, the possession of special symmetries) render them incompatiblewith the current hypothesis. Irrational (incommensurate) translations or rotations on atwo-torus, which are symplectic and time-reversal-invariant, are a classic example: q n +1 = q n + a, mod 1 , (15a) p n +1 = p n + b, mod 1 , (15b)15 Q P Q P S D ( S ; ) FIG. 8: (Top) A typical pseudotrajectory of map (15) with a = √ b = √
3. The pseudotra-jectory was evolved from the initial point ( q = sin(2 / , p = cos(5 / . , . × [0 . , . a and b are such that, for any two given integers k (cid:54) = 0 and k (cid:54) = 0, thenumber k a + k b is not an integer. Map (15) is known to be ergodic (and not mixing), butits number-theoretical properties give rise to ergodic trajectories that are highly ordered.The peculiarity of these so-called quasi-periodic trajectories in the present context is thatthey possess a countable number of distinct interpoint spacings, and therefore cannot havea Wignerian NNSD. Figure 8 illustrates this point using map (15) with a = √ b = √ three distinguishable nearest-neighbor spacings resulting in thespiked distribution shown in the bottom panel of Fig. 8. The results shown in Fig. 8 areexemplary. More generally, the number and position of the spikes will depend on the specificvalues of a and b . V. DISCUSSION AND CONCLUSION
To summarize, the hypothesis that any typical pseudotrajectory of a 2D ergodic map willhave a Wignerian nearest neighbor spacing distribution, was put forward and numericallytested. In all test cases, the hypothesis is upheld, and the range of validity of the hypothesisappears to be robust in the sense that it is not affected by the presence or absence of: (i)mixing; (ii) time-reversal symmetry; and/or (iii) dissipation. Furthermore, pseudotrajec-tories need not necessarily cover densely all of the available phase space in order for theirNNSDs to be Wignerian in nature. As the example in Sec. III C demonstrates, pseudo-trajectories evolving ergodically in any positive-measure subset of the full phase space alsopossess NNSDs consistent with the Wigner distribution.The ergodic maps considered in Sec. III can be regarded as “generic” 2D ergodic maps.It is expected that there will be a zero-measure set of “non-generic” 2D ergodic maps whosepeculiarities render them incompatible with the current hypothesis. Irrational translations orrotations on the two-torus are a classic example. The generic examples of Sec. III representin some sense the ideal 2D simply ergodic map. The phase space of other generic mapsmay possess certain features (e.g., sticky structures [40]) not present in the ideal examplesconsidered here that can result in small but not insignificant deviations from a Wignerian17NSD. Identifying and understanding the effect(s) of such features on the NNSD is a problemthat warrants its own study.It is important to stress that an ergodic map need not be chaotic. Skew maps (like the oneof Sec. III B), for example, are ergodic, and while they do exhibit zero autocorrelation anda particular kind of sensitivity to initial conditions [41], they have zero Lyapunov exponentand zero entropy [42]. By virtue of the last two properties as well as the fact that theyhave no mixing properties, skew maps are regarded as being non-chaotic. Likewise, chaoticmaps are not necessarily ergodic. In fact, most of the well-known chaotic maps [43] arenot ergodic. Most chaotic maps do not generate trajectories that densely and uniformlycover their phase spaces (or positive-measure subsets thereof). Thus, in general, boundedaperiodic pseudotrajectories of 2D chaotic maps will not have Wignerian NNSDs. There arehowever a number of important special cases where such pseudotrajectories are expected tohave Wignerian NNSDs. These shall be described in a follow-up paper.The Wigner distribution has played a fundamental role in characterizing the quantum spectral fluctuations of classically chaotic systems. While it is interesting that the samedistribution characterizes the ergodic trajectories of 2D classical maps, there appears tobe no deeper correspondence with the random matrix model of quantum chaos. To clarifythis point, it is useful to refer to maps (3) and (7). When the time-reversal invariantmap (3) is quantized, the NNSD of the eigenphases is expected to be consistent with theNNSD for eigenvalues drawn from the circular orthogonal ensemble (COE), denoted here by P COE ( S ). In the limit where the matrix size goes to infinity, the COE reverts to the GOE,and hence P W ( S ) ≈ P COE ( S ). (Note that the sample NNSD of the eigenphases that comefrom quantizing map (3) is indeed consistent with P W ( S ) [24].) On the other hand, whenthe non-reversible map (7) is quantized, the NNSD of the eigenphases is expected to beconsistent with the NNSD for eigenvalues drawn from the circular unitary ensemble (CUE),denoted here by P CUE ( S ) [34, 44]. In the limit where the matrix size goes to infinity, theCUE reverts to the Gaussian unitary ensemble (GUE), and in this case the usual analyticalapproximation to P CUE ( S ) (the so-called Wigner surmise for the GUE) differs significantlyfrom P W ( S ). The presence or absence of time-reversal symmetry thus has a significant effecton the NNSD of the quantum eigenphases. Time-reversal symmetry produces no comparableeffect in the present context. Regardless of whether or not time-reversal symmetry is present,the pseudotrajectories of these 2D Anosov maps are Wignerian. It should be emphasized18hat the ideas and results of the present paper are not aimed at addressing the open questionof why a Wigner- like NNSD is a common “quantum signature of chaos”, nor should they(at present) be interpreted as being any kind of theoretical justification for this commonlyobserved quantum phenomenon. The goal here is merely to introduce the novel way in whichthe Wigner distribution enters the domain of classical mechanics.The numerical results of Sec. III describe the interpoint nearest-neighbor spacings of pseudo trajectories. This begs the question: Do the same results hold for the exact tra-jectories of a 2D ergodic map? This is a question that is difficult to answer definitivelywithout detailed analysis. Chaotic pseudotrajectories emulate the true dynamics of a givensystem only when ‘shadowed’ by exact trajectories of the system, and are otherwise onlymeaningful in a statistical sense. Shadowing of numerical trajectories is a fundamental is-sue, in particular, for strictly non-hyperbolic chaotic systems. In hyperbolic systems, whichis an extreme and rather exceptional case, the existence of shadowing trajectories for allpseudotrajectories is guaranteed (see, for example, Ref. [45]). For such systems, conclusionsabout the statistical properties of numerical trajectories will also generally apply to exacttrajectories. Long shadowing trajectories are not precluded for all non-hyperbolic chaoticsystems (for example, the shadowing property has been established for the standard map[46]), but their consideration introduces technical questions about how accurate and forhow long numerical trajectories are valid. The maps used in Secs. III A, III D, and III Eare all 2D Anosov maps, which are uniformly hyperbolic. Together with the appropriateshadowing theorems for hyperbolic systems, the numerical results of Secs. III A, III D, andIII E thus motivate the following proposition:
Typical trajectories of 2D Anosov maps havea Wignerian nearest-neighbor spacing distribution [52].The idea that the ergodic trajectories of 2D maps can be modeled by a homogeneous 2DPoisson point process can be generalized to higher (or lower) dimensions. For a d -dimensionalergodic map, the pertinent model is then the homogeneous Poisson point process in R d (henceforth denoted by P d ). When d (cid:54) = 2, the Euclidean
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47] J. Sakhr and J. M. Nieminen, Phys. Rev. E , 036201 (2006).[48] J. Sakhr (unpublished).[49] There are cases for which the NNSD is qualitatively different from P W ( S ) (for classic examples,see Refs. [10, 11]). Such cases are deemed to be “special” or “non-generic”.[50] In this paper, the term “ergodic” should be understood as a shorthand for simply ergodic,which means that typical pseudotrajectories will (after sufficient time) cover the phase spaceuniformly. Non-simple ergodicity arising from dense but non-uniform coverage of the phasespace occurs in many well-known 2D maps (e.g., the Sinai map), but the treatment of thistype of ergodicity requires certain refinements that lie beyond the scope of the present paper.[51] For example, the seventh -NNSD for P is equal to the fourth -order Wigner surmise for theGOE.[52] It is important to emphasize that the interpoint spacings implicitly being referred to in thisstatement are measured using the standard Euclidean metric.[53] See the Introduction of Ref. [47] for a derivation of this fact.metric.[53] See the Introduction of Ref. [47] for a derivation of this fact.