Dynamical localization of chaotic eigenstates in the mixed-type systems: spectral statistics in a billiard system after separation of regular and chaotic eigenstates
aa r X i v : . [ n li n . C D ] J u l Dynamical localization of chaotic eigenstates in themixed-type systems: spectral statistics in a billiardsystem after separation of regular and chaoticeigenstates
Benjamin Batisti´c and Marko Robnik
CAMTP - Center for Applied Mathematics and Theoretical Physics, University ofMaribor, Krekova 2, SI-2000 Maribor, Slovenia, European UnionE-mail: [email protected], [email protected]
Abstract.
We study the quantum mechanics of a billiard (Robnik 1983) in the regimeof mixed-type classical phase space (the shape parameter λ = 0 .
15) at very high-lyingeigenstates, starting at about 1.000.000th eigenstate and including the consecutive587654 eigenstates. By calculating the normalized Poincar´e Husimi functions of theeigenstates and comparing them with the classical phase space structure, we introducethe overlap criterion which enables us to separate with great accuracy and reliabilitythe regular and chaotic eigenstates, and the corresponding energies. The chaoticeigenstates appear all to be dynamically localized, meaning that they do not occupyuniformly the entire available chaotic classical phase space component, but are localizedon a proper subset. We find with unprecedented precision and statistical significancethat the level spacing distribution of the regular levels obeys the Poisson statistics,and the chaotic ones obey the Brody statistics, as anticipated in a recent paper byBatisti´c and Robnik (2010), where the entire spectrum was found to obey the BRBstatistics. There are no effects of dynamical tunneling in this regime, due to the highenergies, as they decay exponentially with the inverse effective Planck constant whichis proportional to the square root of the energy.PACS numbers: 01.55.+b,02.50.Cw,02.60.Cb,05.45.Pq, 05.45.Mt, 47.52.+j
Submitted to:
J. Phys. A: Math. Gen.
1. Introduction
In quantum chaos [1, 2, 3] of general (generic) time-independent (autonomous) Hamiltonsystems in the strict semiclassical limit we can conceptually separate regular and chaoticeigenstates. This picture goes back to the work by Percival in 1973 [4] and Berryand Robnik in 1984 [5]. One of the main results in quantum chaos is the fact thatin classically fully chaotic (ergodic, autonomous Hamilton) systems with the purelydiscrete spectrum the fluctuations of the energy spectrum around its mean behaviour ynamical localization in chaotic systems t H = 2 π ~ / ∆ E , where h = 2 π ~ is the Planck constant and ∆ E is the mean energylevel spacing, such that the mean energy level density is ρ ( E ) = 1 / ∆ E . This statementis known as the Bohigas - Giannoni - Schmit (BGS) conjecture and goes back to theirpioneering paper in 1984 [8], although some preliminary ideas were published in [9].Since ∆ E ∝ ~ d , where d is the number of degrees of freedom (= the dimension of theconfiguration space), we see that for sufficiently small ~ the stated condition will alwaysbe satisfied. Alternatively, fixing the ~ , we can go to high energies such that the classicaltransport times become smaller than t H . The role of the antiunitary symmetries thatclassify the statistics in terms of GOE, GUE or GSE (ensembles of RMT) has beenexplained in [10], see also [11] and [1, 2, 3, 6]. The theoretical foundation for theBGS conjecture has been initiated first by Berry [12], using the Gutzwiller periodicorbit theory (trace formula) [13] (for an excellent exposition see [1]) and later furtherdeveloped by Richter and Sieber [14], arriving finally in the almost-final proof proposedby the group of F. Haake [15, 16, 17, 18].On the other hand, if the system is classically integrable, Poisson statistics applies,as is well known and goes back to the work by Berry and Tabor in 1977 (see [1, 2, 3]and the references therein, and for the recent advances [19]).In the mixed-type regime, where classical regular regions coexist in the classicalphase space with the chaotic regions, being a typical KAM-scenario which is the genericsituation, the so-called Principle of Uniform Semiclassical Condensation (of the Wignerfunctions of the eigenstates; PUSC) applies, based on the ideas by Berry [20], and furtherextended by Robnik [3]. If the stated semiclassical condition is satisfied, the chaoticeigenstates are uniformly extended, and consequently the Berry-Robnik statistics [5, 21]is observed - see also [3]. If the semiclassical condition stated above requiring that t H is larger than all classical transport times is not satisfied, the chaotic eigenstates willnot be extended but localized and the Berry-Robnik statistics must be generalized asexplained in [22, 23, 3, 24, 25] and in this paper.The relevant papers dealing with the mixed-type regime after the work [5] are [21]- [31] and the most recent advance was published in [24]. If the couplings between theregular eigenstates and chaotic eigenstates become important, due to the dynamicaltunneling, we can use the ensembles of random matrices that capture these effects[32, 24]. As the tunneling strengths typically decrease exponentially with the inverseeffective Planck constant, they rapidly disappear with increasing energy, or by decreasingthe value of the Planck constant. In this work we shall deal only with high-lyingeigenstates, and therefore we can neglect the effects of tunneling.However, quite generally, if the semiclassical condition is not satisfied, such that t H is no longer larger than the relevant classical transport time, like e.g. the diffusion time infully chaotic but slowly ergodic systems, we find the so-called dynamical localization , ynamical localization in chaotic systems Chirikov localization . Dynamical localization was discovered in time dependentsystems [33]. It was intensely studied since then in particular by Chirikov, Casati,Izrailev, Shepelyanski and Guarneri, in the case of the kicked rotator as reviewed in[34]. See also the references [35]-[38], and the most recent work [39]. For a generaloverview of the time dependent Floquet systems see also [1, 2]. It has been observedthat in parallel with the localization of the eigenstates one observes the fractionalpower law level repulsion (of the quasienergies) even in fully chaotic regime (of thefinite dimensional kicked rotator), and it is believed that this picture applies also totime independent (autonomous) Hamilton systems and their eigenstates [39]. (See theexcellent review of localization in time independent billiards by Prosen in [40].) Indeed,this has been analyzed with unprecedented precision and statistical significance recentlyby Batisti´c and Robnik [24] in case of mixed-type systems, and the present work is beingextended in the analysis of separated regular and chaotic eigenstates. An early attemptof separation of eigenstates in the billiard system has been published in [41], using adifferent approach at much lower energies and with much smaller statistical significance.In [42] mushroom billiards were studied at much lower energies and with much smallerstatistical significance, where the aspects of tunneling were investigated in the first place,but not the dynamical localization, although a clear deviation from the GOE statisticswas found in the chaotic eigenstates.In this paper we introduce a criterion for classifying eigenstates as regular andchaotic, and moreover, we show that the regular levels obey the Poisson statistics,whilst the chaotic dynamically localized eigenenergies obey exceedingly well the Brodydistribution [43], with the Brody parameter values β within the interval [0 , β = 0 yields the Poisson distribution in case of the strongest localization, and β = 1 givesthe Wigner surmise (2D GOE, as an excellent approximation of the infinite dimensionalGOE), which describes the extended chaotic eigenstates. It turns out that the Brodydistribution introduced in [43], see also [44], fits the empirical data much better thane.g. the distribution function proposed by F. Izrailev (see [35, 34] and the referencestherein).It is well known that Brody distribution so far has no theoretical foundation, butour empirical results show that we have to consider it seriously in dynamically localizedchaotic eigenstates, thereby being motivated for seeking its physical foundation, and ananalogous result was obtained in the recent work of Manos and Robnik (2013) [39] in theanalysis of the quantum kicked rotator, where the object of study are the eigenstatesof the Floquet operator and the statistical properties of the spectrum of eigenphases(quasienergies) in classically fully chaotic regime.In the Hamilton systems with classically mixed-type dynamics, which is the genericcase, we have classically regular quasi-periodic motion on d -dim invariant tori ( d is thenumber of freedoms) for some initial conditions (with the fractional Liouville volume ρ ) and chaotic motion for the complementary initial conditions (with the fractionalLiouville volume ρ = 1 − ρ ). The chaotic set might be further decomposed into severalchaotic regions (invariant components) in case d = 2, whilst for d > ynamical localization in chaotic systems ~ eff by the following formula for the gap probability E ( S ), E ( S ) = E r ( ρ S ) E c ( ρ S ) (1)and the level spacing distribution P ( S ) (see e.g. [3]) is of course always given as thesecond derivative of the gap probability, namely P ( S ) = d E ( S ) /dS , so that we have P ( S ) = d dS E r ( ρ S ) E c ( ρ S ) = d E r dS E c + 2 dE r dS dE c dS + E r d E c dS . (2)This factorization formula (1) is a direct consequence of the statistical independence,justified by PUSC. Here by E r ( S ) = exp( − S ) we denote the gap probability for thePoissonian sequence with the mean level density one. By E c ( S ) we denote the gapprobability for the chaotic level sequence with the mean level density (and spacing) one.Note that the classical parameter ρ and its complement ρ = 1 − ρ enter the expressionas weights in the arguments of the gap probabilities.Using the Bohigas-Giannoni-Schmit conjecture we conclude that in the sufficientlydeep semiclassical limit E c ( S ) is given by the RMT, and can be well approximated bythe Wigner surmise P W ( S ) = πS (cid:18) − πS (cid:19) , F W ( S ) = 1 − W W ( S ) = exp (cid:18) − πS (cid:19) , (3)such that E c ( S ) is equal to E W ( S ) = 1 − erf (cid:18) √ πS (cid:19) = erfc (cid:18) √ πS (cid:19) , (4)where erf( x ) = √ π R x e − u du is the error integral and erfc( x ) its complement, i.e.erfc( x ) = 1 − erf( x ). In the equation (3) W W ( S ) denotes the cumulative Wigner levelspacing distribution W W ( S ) = R S P W ( x ) dx and F W its complement. The explicitBerry-Robnik level spacing distribution (in the special case of one regular and onechaotic component) follows immediately, P BR ( S ) = e − ρ S (cid:26) e − πρ S (cid:18) ρ ρ + πρ S (cid:19) + ρ erfc (cid:18) √ πρ S (cid:19)(cid:27) . (5) ynamical localization in chaotic systems ~ eff )is by now very well established in highly accurate numerical calculations for all E ( k, L )probabilities, not only the gap probability [21].In the present work the above basic BR formula (1) is generalized as in [24] tocapture the dynamical localization effects, responsible for the deviation from the BRregime.At not sufficiently small ~ eff (e.g. in billiards this means at low energies) the chaoticeigenstates (their Wigner functions in the phase space) are not uniformly extended overthe entire classically allowed chaotic component, but are dynamically localized. Thus wesee the transition from GOE in case of extended chaotic states to the Poissonian statisticsin case of strong localization. The level spacing distribution in such a transition regimeof localized chaotic eigenstates can be described by the Brody distribution with the onlyone family parameter β , P B ( S ) = C S β exp (cid:0) − C S β +1 (cid:1) , W B ( S ) = 1 − exp (cid:0) − C S β +1 (cid:1) , (6)where the two parameters C and C are determined by the two normalizations < > = < S > = 1, and are given by C = ( β + 1) C , C = (cid:18) Γ (cid:18) β + 2 β + 1 (cid:19)(cid:19) β +1 (7)with Γ( x ) being the Gamma function. As mentioned before, if we have extended chaoticstates β = 1 and RMT (3) applies, whilst in the strongly localized regime β = 0 andwe have Poissonian statistics. Again, by W B ( S ) we denote the cumulative Brody levelspacing distribution, W B ( S ) = R S P B ( x ) dx . The corresponding gap probability is E B ( S ) = 1( β + 1)Γ (cid:16) β +2 β +1 (cid:17) Q β + 1 , (cid:18) Γ (cid:18) β + 2 β + 1 (cid:19) S (cid:19) β +1 ! (8)where Q ( α, x ) is the incomplete Gamma function Q ( α, x ) = Z ∞ x t α − e − t dt. (9)By choosing E c ( S ) in equation (1) as given in (8) we are able to describe the localizationeffects on the chaotic component. Such approach has been already proposed in the paperby Prosen and Robnik in 1994 [22, 23] and the resulting level spacing distribution,emerging from this assumption, was called Berry-Robnik-Brody (BRB). It has twoparameters, the classical parameter ρ and the quantum parameter β . It will turnout that this description is indeed excellent, and has been verified to high accuracy byBatisti´c and Robnik [24].The applicability of the Brody distribution in this context is theoretically still notwell understood, but we shall see that the theory describes very well the empiricaldata from the highly accurate energy spectra of billiards at energies around and belowthe deep semiclassical (Berry-Robnik) regime. Therefore, the resulting theory is of ynamical localization in chaotic systems P I ( S ) = A (cid:18) πS (cid:19) β exp (cid:20) − βπ S − (cid:18) B − πβ (cid:19) S (cid:21) , (10)where the constants A and B are determined by the normalizations < > = < S > = 1.As we shall show the dynamical localization effects can persist up to very high-lyingeigenstates, even up to one million, whilst - as mentioned before - the tunneling effectsoccur usually only at very low-lying eigenstates, due to the exponential dependence onthe reciprocal effective Planck constant, ∝ exp( − const./ ~ eff ), and thus can be neglectedin our case.The paper is structured as follows: In section 2 the billiard system is definedas introduced by Robnik [46, 47] with shape parameter λ = 0 .
15, and we describethe Poincar´e Husimi functions, in section 3 we introduce the method of classifyingand separating regular and chaotic eigenstates in terms of normalized Poincar´e Husimifunctions (which are Gaussian-smoothed Wigner functions), in section 4 we present theresults, and in section 5 we conclude and discuss the results.
2. Introducing the model system and the definition of the problem
As an interesting and frequently studied model system we have chosen the billiardintroduced by Robnik in references [46]-[47], whose boundary is defined by the quadraticconformal map of the unit circle | z | = 1 of the z -complex plane onto the w -complex plane(which is the physical plane) as follows w = z + λz . (11)The choice of this quadratic map has two reasons: (i) it allows for an elegant method [47]to solve the Helmholtz equation in the w -plane by transforming back to the z -plane, and(ii) it is the simplest one with nontrivial classical dynamics [46]. The shape parameter λ goes from 0 (circle; integrability) to 1/2 (cardioid billiard; ergodicity: full chaos [48]).For 0 ≤ λ ≤ / λ from 0 we have at λ = 1 / z = −
1, and thus w = − λ . By a theorem due to J. Mather this is ynamical localization in chaotic systems
7a sufficient condition for the destruction of Lazutkin’s caustics and of the underlyinginvariant tori near the boundary, which in turn is a necessary condition for the ergodicityof the classical billiard dynamics. However, at λ ≥ / λ = 1 /
2, the ergodicity was proven rigorously byMarkarian [48].We are interested in the mixed-type case, within the interval 0 < λ < / λ = 0 .
15, which is one of the most frequently studied casesin [26, 27, 22, 23, 25, 52, 53, 41]. The fractional phase space volume ρ of the classicallyregular part of the phase space (not to be confused with the area on the Poincar´e surfaceof section!) is equal to 0 . ρ was estimated numerically as ρ = 0 .
36, which is due to the technicaldifficulties in distinguishing the regular regions and slowly diffusing chaotic regions dueto the sticky objects in the classical phase space. These difficulties were overcome in [52]and [24], using the new methods based on the ideas and approach in [54]. The estimate ρ = 0 .
175 is now believed to be accurate within at least one percent relative error.The classical mechanics of 2D billiards is studied in the Poincar´e-Birkhoffcoordinates ( s, p ), where s is the arclength parameter going from 0 to L , the perimeterof the billiard domain, in our case counted anticlockwise from the point w = 1 + λ ,and p is simply the sine of the reflection angle, which goes from -1 to +1. The bouncemap is defined by the free motion between the collision points on the boundary, obeyingthe specular reflection law upon each collision. Thus, the complete information onthe classical dynamics is contained in the structure of the bounce map on the cylinder( s, p ). When analyzing the quantum mechanics of this system, we would like to find ananalogous two dimensional space which also contains the complete information aboutthe quantum mechanics, namely about the eigenfunctions. This is the space of the so-called Poincar´e Husimi functions (see [55] and the references therein) that we introducebelow.The quantum mechanics of the billiard system comprises the study of the solutionof the Helmholtz equation for the billiard domain B ,∆ ψ + k ψ = 0 , (12)with the Dirichlet boundary condition ψ = 0 on the boundary ∂ B . We have used anumber of methods, like in [24], to calculate the eigenenergies E j = k j , where j is thecounting index j = 1 , , , . . . , and the associated eigenfunctions are denoted by ψ j ( r ).Introducing the important quantity u ( s ), the normal derivative of the eigenfunction ψ on the boundary, from here onwards called boundary function , u ( s ) = n · ∇ r ψ ( r ( s )) , (13) ynamical localization in chaotic systems n is the unit outward vector normal to the boundary at position s , we can show[56] that the eigenvalue problem (12) is equivalent to the following integral equation u ( s ) = − I dt u ( t ) n · ∇ r G ( r , r ( t )) . (14)Here r ( t ) is the position vector at the point s = t on the boundary, whilst r is the positionvector inside the billiard B . G ( r , r ( t )) is the free particle Green function, namely G ( r , r ′ ) = − i H (1)0 ( k | r − r ′ | ) , (15)where H (1)0 ( x ) is the zero order Hankel function of the first kind. It is important toknow that knowing u j ( s ) for a certain eigenfunction j , we can immediately calculatethe wave function ψ j ( r ) within the interior of the billiard domain by ψ j ( r ) = − I dt u j ( t ) G ( r , r ( t )) . (16)Thus, in certain analogy to the classical mechanics, the quantum mechanics is completelydescribed by the boundary functions u j ( s ). Finally, there is the important identity [56]12 I dt n ( t ) · r ( t ) u j ( t ) = k j . (17)The quantum analogy of the classical phase space is the space of Wigner functions[57] or of other phase space representations of quantum states in general. The Wignerfunctions are real but not positive definite functions, and exhibit lots of oscillationsaround the zero level also in the regime where the quantum probability density is lowand thus such structures often obscure the main physical aspects of the phenomena.Nevertheless, they do uniformly condense on the classical invariant objects, accordingto the mentioned PUSC [3] in the introduction. There are different ways of definingpositive definite phase space functions, but Husimi functions [58] are perhaps the bestway to do it. They are in fact Gaussian smoothed Wigner functions. In general, wedefine them by the projection of the wave function onto a coherent state. One of thepossible formulations can be found in [55], whose definitions and the notation we shalluse in what follows. The idea and the approach (in slightly different form) goes back tothe works [59], [60] and [61]. The most important idea is to define the one-dimensionalcoherent states onto which we project the boundary functions u j ( s ). For this reason, anddue to the analogy with the classical dynamics and its Poincar´e surface of section on the cylinder ( s, p ), the underlying Husimi functions are called Poincar´e Husimi functions [55]. The key idea is to introduce one-dimensional coherent state as a function of thecoordinate s on the boundary ∂ B , localized at ( q, p ) ∈ [0 , L ] × R , which is properly periodized , as introduced by Tualle and Voros [60], but here following the notation from[55] we define c ( q,p ) ,k ( s ) = X m ∈ Z exp { i k p ( s − q + m L ) } exp (cid:18) − k s − q + m L ) (cid:19) . (18) ynamical localization in chaotic systems s with the period L is now obvious. Here we have dropped allnormalization factors, because in the end we shall normalize the Poincar´e Husimifunctions anyway. Then, using this, the Poincar´e Husimi function associated with the j -th eigenstate represented by the boundary function u j ( s ) with the eigenvalue k = k j ,is H j ( q, p ) = (cid:12)(cid:12)(cid:12)(cid:12)Z ∂ B c ( q,p ) ,k j ( s ) u j ( s ) ds (cid:12)(cid:12)(cid:12)(cid:12) , (19)which is positive definite by construction. In the semiclassical limit j → ∞ , and k j → ∞ , we shall observe that the Poincar´e Husimi function is concentrated on theclassical invariant regions, which can be an invariant torus, a chaotic component, or theentire Poincar´e surface of section ( s, p ) if the motion is ergodic. This is a consequence ofPUSC, bearing in mind that the Husimi function is just a Gaussian smoothed Wignerfunction, where in the semiclassical limit the width of the smoothing Gaussian becomesless and less important. Therefore, we expect that in the semiclassical limit the Poincar´eHusimi functions will directly correspond to either the classical regular regions or to theclassical chaotic regions, with the exceptions having measure zero. We can then use thePoincar´e Husimi functions to classify and thus also to separate the regular and chaoticeigenstates, and thereby also separate the regular and the chaotic spectral subsequencesof the energy eigenvalues E j = k j . This is what we do in the next section 3.
3. Separating the regular and chaotic eigenstates and subspectra
We consider the billiard (11) with λ = 0 .
15. As mentioned, the value of the classicalparameter is ρ = 0 . k ∈ [2000 , k = 2000 is estimatedby the Weyl rule as about 1.000.000. In Appendix A we show that the semiclassicalcondition of sufficiently large ratio of the classical transport time t T and the Heisenbergtime t H is well satisfied, satisfying the inequality k ≪ N T /
2, where N T is the classicaltransport time in units of the number of collisions, and is equal to N T ≈ . Then,when calculating the Poincar´e Husimi functions, the momentum p is rescaled by theeigenvalue k j , such that p = 1 corresponds to the original p = k j . For each eigenstate thePoincar´e Husimi function was calculated as follows. We have set up a grid of 400 × q, p ), thus reduced due to the symmetries(reflection symmetry and time reversal symmetry). The grid points are defined as( q i , p j ) = (∆ q/ i ∆ q, ∆ p/ j ∆ p ), where ∆ q = L /
800 and ∆ p = 1 / q ∈ [0 , L /
2] and p ∈ [0 , q ∆ p . The integrationmethod used to evaluate (19) is a simple trapeze rule with the step ds ∝ λ B /
20, where λ B = 2 π/k j is the de Broglie wavelength. The important point is now that the valuesof the Poincar´e Husimi functions on the grid are normalized in such a way that theirsum is equal to one. ynamical localization in chaotic systems Figure 1.
Examples of chaotic (left) and regular (right) states in the Poincar´e-Husimirepresentation. k j ( M ) from top down are: chaotic: k j ( M ) = 2000.0021815 (0.978),2000.0181794 (0.981), 2000.0000068 (0.989), 2000.0258600 (0.965); regular: k j ( M ) =2000.0081402 (-0.987), 2000.0777155 ( -0.821), 2000.0786759 ( -0.528), 2000.0112417 (-0.829). The gray background is the classically chaotic invariant component. We showonly one quarter of the surface of section ( s, p ) ∈ [0 , L / × [0 , Some examples of the Poincar´e Husimi functions are shown in figure 1.Now the classification of eigenstates can be performed by their projection onto theclassical surface of section. As we are very deep in the semiclassical regime we do expectwith probability one that either an eigenstate is regular or chaotic, with exceptionshaving measure zero, ideally. To automate this task we have ascribed to each point onthe grid a number A i,j whose value is either +1 if the grid point lies within the classicalchaotic region or − collisions, enough for the convergence (within certain very smalldistance). Each visited cell ( i, j ) on the grid has then been assigned value A i,j = +1,the remaining ones were assigned the value -1.The Poincar´e Husimi function H ( q, p ) (19) (normalized) was calculated on the gridpoints and the overlap index M was calculated according to the definition M = X i,j H i,j A i,j . (20)In practice, M is not exactly +1 or −
1, but can have a value in between. The reasons ynamical localization in chaotic systems /k j ). If so, the question is, where to cut the distribution of the M -values,at the threshold value M t , such that all states with M < M t are declared regular andthose with M > M t chaotic.There are two natural criteria: (I) The classical criterion: the threshold value M t ischosen such that we have exactly ρ fraction of regular levels and ρ = 1 − ρ of chaoticlevels. (II) The quantum criterion: we choose M t such that we get the best possibleagreement of the chaotic level spacing distribution with the Brody distribution (6), whichis expected to capture the dynamical localization effects of the chaotic eigenstates.
4. Results
To begin with we first look at the total energy spectrum E j = k j , for k j ∈ [2000 , N = 587653 level spacings together with thebest fitting BRB distribution (2), with E r being Poissonian and E c being the Brody gapprobability (8), derived from (6), with the classical ρ = 0 . β = 0 .
45. For the reference the BR distribution (5), the Poisson and the GOE levelspacing distributions (3) are shown. In figure 2b we show the U -function representationof the level spacing distribution, as introduced by Prosen and Robnik [23] and definedin the Appendix B. It was the U -function which was used in finding the best fittingdistributions. Whilst in the first case the agreement is perfect, in the U -plot we see thatthe quantally adjusted parameter ρ = 0 .
19 instead of its classical value 0 .
175 leads toeven better agreement. In this case β = 0 .
47. Please observe that the deviations hereare already extremely small, so the significance of the best fit is of extreme importance.Let us now separate the regular and chaotic eigenstates and the correspondingeigenvalues, after unfolding, according to the method described in section 3, using theclassical criterion (I). The corresponding threshold value of the index M is found to be M t = 0 . β = 0 .
444 for the chaotic levels and almost pure Poissonfor the regular levels.In order to make our analysis deeper and more refined we show first the histogramof the M -values in figure 4a, where we see the two different threshold values M t , namelythe classical one at M t = 0 . ρ = 0 . M t = 0 .
75 in which case we get ρ = 0 . R = Z ∞ P ( S ) (( P ( S ) − P F ( S )) dS, (21)where P ( S ) describes the data, whilst P F ( S ) denotes the theoretical distribution fittingthe data, namely F stands for Brody or Izrailev. At each chosen threshold value M t we consider the set of chaotic levels for which by definition M ≥ M t . By performing ynamical localization in chaotic systems . . . . . . Full spectrum S P PoissonGOEBRBRBData − . . . . Full spectrum S U - U B R B Figure 2. (a; left) The level spacing distribution for the entire spectrum afterunfolding for N = 587653 spacings, with k j ∈ [2000 , ρ = 0 .
175 and β = 0 .
45. In the U -function plot (b; right), we show U ( data ) − U ( BRB ) as a function of S , and it isclearly seen that the BRB distribution with the quantally determined ρ = 0 .
19 and β = 0 .
47 is even better fit to the data (dashed, denoted by BRBq). The belt aroundthe data curve indicates the expected statistical ± one-sigma error. . . . . . . Chaotic levels S P PoissonGOEBrodyData 0 1 2 3 4 . . . . . . Regular levels S P PoissonGOEData
Figure 3.
Separation of levels using the classical criterion M t = 0 . β = 0 . ynamical localization in chaotic systems Histogram of M classification measure M f r equen cy −1.0 −0.5 0.0 0.5 1.0 classical thresholdquantum threshold −1.0 −0.5 0.0 0.5 1.0 − . − . − . − . − . − . Fit quality threshold M l og R classical threshold IzrailevBrodyPoisson
Figure 4. (a; left) Distribution of the index M with the locations of the thresholdvalues of M , the classical M t = 0 .
431 and the quantum one M t = 0 .
75. In the firstcase classical ρ = 0 . ρ = 0 . R , defined in (21), for the Brody andIzrailev distributions for chaotic levels M > M t , versus M t , and the same for Poissondistribution for the regular levels with M < M t . the best fit at such M t (= threshold M ) both for Brody and Izrailev, we calculate thefit deviation measure (21) and plot its decadic logarithm as a function of M t in figure4b. We see that at low M t Izrailev is somewhat better than Brody, but this is theunphysical domain of much too small M t . Near the classical threshold M t = 0 .
431 theybecome comparably good, but at increasing the M t Brody exhibits a very sharp andnarrow minimum, whilst Izrailev curve increases. This sharp minimum of R for Brodyis at the value of M which by definition we called the quantum threshold M t = 0 . R quantity (21) for the Poisson distributioin vs. M t , wherewe see also a deep sharp minimum at M t ≈ .
8, thus almost at the quantum threshold M t = 0 .
75. The conclusion of this analysis is that by varying the threshold value M t there is a point M t = 0 .
75 at which the Brody distribution for all chaotic levels with
M > M t is globally the best and also better than Izrailev fit at any M t . For logicalconsistency, it is important that at (almost) the same value of M t the fit of the Poissondistribution for the regular levels with M < M t is globally the best, as is evident fromthe R plot in figure 4b.We show the U -function plots for the regular levels and the chaotic levels in figure5, using the Brody distribution, for the classical and quantum criterions, correspondingto the two different values of M t explained above. In both cases we plot the difference U ( data ) − U ( ideal ), so that in case of perfect agreement the line would coincide with theabscissa. We clearly see that the quantum criterion yields noticable better agreementthan the classical criterion. Again, it must be emphasized that the agreement isextremely good, and the deviations of U ( data ) from U ( ideal ) are very small numbers.Finally, we should comment on the relevance of the Brody distribution. The Brody ynamical localization in chaotic systems − . − . . . Regular levels S U - U P o i ss on − . . . . Chaotic levels S U - U B r od y Figure 5.
The U -function plots as differences U ( data ) − U ( ideal ) for the regular andchaotic levels, for both criteria, the classical one and the quantum one. The beltsaround the data lines indicate the expected statistical ± one-sigma errors. distribution [43, 44] (6) still has no theoretical foundation, but it definitely capturescorrectly the effects of dynamical (Chirikov) localization of chaotic eigenstates. Thishas been recently confirmed by Manos and Robnik [39] in case of the kicked rotator,namely for the quasienergies, and clearly is demonstrated in the present work for theautonomous Hamilton system, exemplified by the 2D billiard that we have chosen forthis analysis. It remains as an open theoretical problem to derive the Brody distributionin this context.Since the Brody distribution is not known theoretically to be the preferred and/or“the right one”, we have considered again also the Izrailev distribution [35, 38, 34] (10),studied very recently also in [39]. Entirely in line with the findings in [39] we found thatBrody is much better model of the level spacing distribution than the Izrailev’s one.This is clearly demonstrated in figure 6.
5. Conclusions
We have used a billiard system of the mixed type (11), with λ = 0 .
15, as introducedin [46]-[47], and have shown that using the Poincar´e Husimi functions we can separatethe regular and chaotic eigenstates. The successful separation of course also entirelyconfirms the Berry-Robnik picture [5] of separating the regular and chaotic levels inthe semiclassical limit, where the tunneling effects can be neglected. With great andunprecedented statistical significance we have shown that the chaotic levels exhibitBrody level spacing distribution, whilst the regular levels obey Poissonian statistics.This analysis not only confirms the Berry-Robnik picture [5] of conceptually separatingthe regular and chaotic levels, based on the PUSC and embodied in formula (1), butalso demonstrates that the dynamical localization effects of the chaotic eigenstates arevery well captured by the Brody distribution, in analogy with the same finding in ynamical localization in chaotic systems − . − . . . Best fitting Brody S U - U I z r a il e v − . − . . . Best fitting Izrailev S U - U I z r a il e v Figure 6.
We show the U -function plot for the chaotic levels, clearly showing thatBrody distribution (dashed) is much better than Izrailev distribution. On the left weshow the best fitting Izrailev distribution at the point where Brody is globally thebest fit using the quantum threshold M t = 0 .
75. On the right we show the Izrailev fitat the point M t = − . M t = − .
5, showing that also there the Brody fit is better than Izrailev. Thisis impressive because the effects are small, and the statistical significance very high.The belt around the data line indicates the expected statistical ± one-sigma error. the Floquet systems, in particular the kicked rotator [34, 39], where the quasienergyspectra are analyzed. One educated guess for the occurrence of fractional power lawlevel repulsion (meaning 0 < β <
1) in dynamically localized but classically fully chaoticperiodic systems is Izrailev’s observation (see e.g. [34] and the references therein), thatthe joint probability distribution for a Circular Orthogonal Ensemble (COE), which sofar as the level spacings are concerned is also GOE, should be generalized in the senseof Dyson for noninteger β . Of course, this is just a hypothesis, and it certainly cannotindicate theoretically whether Brody or Izrailev distribution should be ”the right one”,leaving us with the empirical studies and conclusions of this work. The analogies andconnections between the billiard problem and the Floquet problem have been discussedin more detail in the recent work [63].Whilst in the kicked rotator the relationship between the localization measure ofthe eigenstates and the spectral level repulsion (Brody) parameter β exists, as proposedby Izrailev, and confirmed by Manos and Robnik, in the time independent Hamiltonsystems, like the one discussed in the present work, such relationship is lacking and isopen for the future work. It involves great numerical efforts.The theoretical derivation of the Brody level spacing distribution for thedynamically localized eigenstates is thus also an open problem for the future. Thebilliard systems are not just nice theoretical toy models, but are suitable also for theexperimental applications, like in quantum dots, and microwave cavities introduced andstudied extensively over decades by H.-J. St¨ockmann [1]. We also propose to study from ynamical localization in chaotic systems Acknowledgements
Financial support of the Slovenian Research Agency ARRS under the grants P1-0306and J1-4004 is gratefully acknowledged.
Appendix A: The semiclassical condition
Here we calculate the Heisenberg time and the classical transport time for the billiarddomain B defined in equation (11) with λ = 0 .
15. According to the leading order ofthe Weyl formula, which is in fact just the simple Thomas-Fermi rule, we have for thenumber of levels N ( E ) below and up to the energy E of a Hamiltonian H ( q , p ) N ( E ) = 1(2 π ~ ) Z H ( q , p ) ≤ E d q d p . (22)Since H = p / (2 m ), with constant zero potential energy inside B , where m is the massof the billiard point particle, and H is infinite on the boundary ∂ B , we get at once N ( E ) = 2 π A mE (2 π ~ ) . (23)The density of levels is ρ ( E ) = 1 / (∆ E ) = dN ( E ) /dE = A m/ (2 π ~ ) and thus theHeisenberg time is t H = 2 π ~ ρ ( E ) = A m ~ . (24)The classical transport time is denoted by t T , and in units of the number of collisions N T can be written as t T = ¯ lN T v = ¯ lN T p E/m , (25)where ¯ l is the mean free path of the billiard particle and v = p E/m is its speed atthe energy E . Thus for the ratio α = t H /t T we get α = t H t T = A kN T ¯ l (26)where k = p mE/ ~ . Taking into account that ¯ l ≈ π A / L (this is so-called Santalo’sformula, see e.g. [70]), we have α = t H t T = L kπN T . (27) ynamical localization in chaotic systems L ≈ π and we arrive at thefinal estimate α = 2 kN T . (28)Thus the condition for the occurrence of dynamical localization α ≤ k ≤ N T . (29)As our levels are in the interval k ∈ [2000 , N T is estimated as N T ≈ ,we see that the semiclassical condition (29) is very well satisfied. In figure 7 we explicitlyshow the growth of the second moment of p , namely h p i , for an ensemble of 2000 initialconditions uniformly distributed in the chaotic component on the interval s ∈ [0 , L / p = 0, where the averaging is taken over the ensemble and the time. We see thatindeed about N T ≈ collisions are necessary to reach the saturation value of h p i .More detailed study of the relevant transport properties will be published elsewhere[71]. Appendix B: The U-function representation of the level spacingdistribution
First we estimate the expected fluctuation (error) of the cumulative (integrated) levelspacing distribution W ( S ), which contains N s objects. At a certain S we have theprobability W that a level is in the interval [0 , W ] and 1 − W that it is in the interval[ W, P ( k ) of having k levels in the firstand N s − k levels in the second interval we have P ( k ) = N s ! k !( N s − k )! W k (1 − W ) N s − k . (30)Then the average values are equal to < k > = N s W, < k > = N s W + N s ( N s − W , (31)and the variance V ( k ) = < k > − < k > = N s W (1 − W ) . (32)But the probability W is estimated in the mean as k/N s . Its variance is V ( W ) = V (cid:18) kN s (cid:19) = 1 N s V ( k ) = W (1 − W ) N s (33)and therefore the estimated error of W (standard deviation, the square root of thevariance) is given by δW = p V ( W ) = s W (1 − W ) N s . (34) ynamical localization in chaotic systems . . . . . . log10(n_collision) p ^ Figure 7.
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