Table Of ContentOn universality of algebraic decays in Hamiltonian systems
G. Cristadoro1,2 and R. Ketzmerick3,4
1 Max Planck Institute for the Physics of Complex Systems, N¨othnitzer Str. 38, 01187 Dresden, Germany
2 Center for Nonlinear and Complex Systems, Via Valleggio 11, 22100 Como, Italy
3 Institut fu¨r Theoretische Physik, Technische Universita¨t Dresden, 01062 Dresden, Germany
4 Kavli Institute for Theoretical Physics, UCSB, Santa Barbara, CA93106
(Dated: February 2, 2008)
Hamiltoniansystemswithamixedphasespacetypicallyexhibitanalgebraicdecayofcorrelations
andofPoincar´erecurrences,withnumericalexperimentsoverfinitetimesshowingsystem-dependent
power-lawexponents. Weconjecturetheexistenceofauniversalasymptoticdecaybased onresults
for a Markov tree model with random scaling factors for the transition probabilities. Numerical
simulationsfordifferentHamiltoniansystemssupportthisconjectureandpermitthedetermination
8
of theuniversal exponent.
0
0
PACSnumbers: 05.45-a
2
n
a The phase space of Hamiltonian systems with two
J
degrees of freedom generally shows an intricate mix-
7
ture of chaotic and regular structures. Regular regions
1
consisting of ‘islands’ of quasi-periodic motion (KAM
tori) appear hierarchically interspersed in a chaotic sea.
]
D The character of the motion in the irregular component
C is dominated by stickiness of trajectories close to the
. boundary circles separating regular from irregular re-
n
gions. Stickiness crucially affects statistical quantities,
i
l likecorrelations,thedistributionofPoincar´erecurrences
n
[ or anomalous diffusion, which show related algebraic
behaviors. Finite-time numerical experiments revealed
1
system-dependent values for the power-law exponent z
v
6 of the decay of the distribution of Poincar´e recurrences,
5 P(t)∼t−z,thatistheprobabilitytoreturntoagivenre-
7 gion after a time larger than t. Great effort was devoted
2
over the last two decades to understand if there is an
. FIG. 1: (Color online) (a) Survival probability PI(t) for 100
1 asymptoticuniversaldecayforP(t)andhowitstemporal
realizations of random scaling factors (thin lines) and their
0 behavior is related to the hierarchicalphase-space struc- logarithmic average (thick line). Inset: binary tree with root
8
ture near regular boundaries [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. s¯and absorbing site Ds¯. (b) Rescaled survival probabilities
0 In particular, renormalization techniques were used to PI(t)tzI with zI from Eq. (10), showing thesame asymptotic
v: derivescalingrelationsfortheself-similarstructureclose power-lawexponents. (c)Varianceσ2(t)ofln[PI(t)tzI]show-
i ing saturation.
X to the critical circle with golden mean frequency [6, 9]
or for trapping in exact self-similar island-around-island
r
a structures [8]. More recently it was demonstrated [10]
numerical support. As a consequence the average of
thatbothscenariosareusuallypresentinanHamiltonian
P(t) over different Hamiltonian systems is a meaningful
system, thus strengthening the validity of a Markov tree
quantity and allows for an efficient determination of the
approach [7]. While the self-similar Markov tree model
asymptotic universal decay. We numerically extract a
succeeds to reproduce a power-law decay, it is based on
universal exponent z ≃ 1.57 for Poincar´e recurrences in
the approximation of exact self-similarity [7, 11]. It re-
Hamiltonian systems.
mainsanopenquestionhowdeviationsfromexactscaling
affect the algebraic decay and the issue of universality.
We consider a Markov model on a binary tree. For
In this paper we study the role of random scaling an arbitrary site s, we call Ds the site obtained by go-
factors in a Markov tree model. Initially, different ing up one level in the tree towards the root site s¯ (see
realizationsofthe randomnessgiverise toquite different Fig. 1a, inset). The dynamics on the tree is defined by
power-law behaviors (see Fig. 1). In contrast, we show the probabilities to move in one time step to one of the
that asymptotically the exponent of the algebraic decay neighboring sites. As we are interested in the survival
isthesameforallrealizations. Wearguethatthisisrep- probability we put Ds¯as an absorbing site and we fix a
resentative for Hamiltonian systems and present strong non-zero probability of leaving the tree ps¯→Ds¯ [12]. The
2
transition probabilities for all sites s below s¯ are then
chosen according to
pDs→s/ps→Ds = As (1)
ps→Ds/pDs→DDs = Bs,
where the scaling factors A and B are positive ran-
s s
domvariableswith the constraints(i) hAi<1/2and(ii)
B < 1. The constraint (ii) ensures that the transition
probabilitiesdecreasemovingawayfromthe roots¯. The
constraint (i) allows to define an initial probability den-
sity ρ (t = 0), with normalization ρ (0) = 1, that
s s s
wouldbe invariantfor the closedsystPem(ps¯→Ds¯=0) by FIG.2: (Coloronline)Power-lawexponentzI vs.hAiaccord-
ing to analytic estimate Eq. (10) (line) and from fitting the
satisfying detailed balance ρs(0)ps→Ds = ρDs(0)pDs→s.
asymptotic decay of the numerically computed exact evolu-
Duetothenon-zerotransitionprobabilityps¯→Ds¯ofleav- tion,Eq.(2),forasinglerealizationofscalingfactors(circles).
ing the tree towards the absorbing site Ds¯there is a de-
ThedistributionforthescalingfactorAs(Bs)ischosentobe
caying survival probability uniform on an interval centered around hAi (hBi=0.3) with
width 0.12 (0.1) and is independent from Bs (As).
P (t):= ρ (t), (2)
I s
Xs
The products run over the sites Dis encountered on the
ofbeingstillinsidethetreeattimet,withthesubscriptI
direct path from s to the root s¯and n(s) is the number
standing for the “invariant” initial condition.
of levels ascended.
In order to quantify the decay of the survival proba-
AsweareinterestedintheasymptoticbehaviorofP (t)
bility we make use of the following observation: In the I
fort→∞,weneedtofindtherightboundaryofthestrip
sum, Eq. (2), the density ρ (t) at a site s gives its most
s of convergence of Eq.(4). The Gamma function has no
importantcontributionwhileitisdecayingexponentially,
poleswithpositiverealvalueandthusweconcentrateon
witharatethatisdominatedbytheprobabilityps→Dsto the factor containing the sum over the sites s. Fixing z
escape in upward direction. At later times, even though
real,theinfinite sumofproductsofrandomvariables A
itsdecaybecomesalgebraicallyslow,itscontributioncan Bz
diverges with probability one [15] if β(z)>1/2, where
be neglected [13]. The survival probability can thus be
approximated by a sum of decaying densities [14] A σ
β(z)= min . (7)
PI(t)≈ ρs(0)e−ps→Dst. (3) 06σ61(cid:28)(cid:18)Bz(cid:19) (cid:29)
Xs Usingtheconstraints(i)and(ii)ofourrandomvariables
Numerically,we findthis to be agoodapproximation,in AandB, onefinds thatβ(z) ismonotonicallyincreasing
from β(z = 0) < 1/2 to β(z → ∞) = 1. This implies
the sense that the ratio with the exact decay of Eq. (2)
that there is a critical z with
is found to be asymptotically constant. I
Power-law asymptotics can be conveniently extracted
β(z )=1/2. (8)
using singularity analysis of the Mellin transform. An I
algebraic decay P(t) ∼ t−α is transformed into a simple Thus the survival probability asymptotically decays as
pole of its Mellin transform P∗(z) = dttz−1P(t) at
z =α. In particular we find using Eq. (3R) PI(t)∼C(t)t−zI, (9)
∗ ρs(0) withC(t)growingatmostlogarithmically[16]. Westress
P (z) = Γ(z)
I Xs pzs→Ds that zI depends on the probability distributions of the
scaling factors A and B only and not on the specific val-
n(s)−1
= Γ(z) ρs¯(0) ADis (4) ues appearing in a given tree. The power-law exponent
pzs¯→Ds¯ Xs Yi=0 BDzis zI is thus realization independent.
For not too wide distributions of A and B, e.g.
where Γ(z) is the Gamma function and where we used h( A )2i < 2 h A i2, one can show that the minimum
BzI BzI
in Eq.(7) appears at σ =1 and Eq. (8) simplifies to
n(s)−1
ρs(0) = ρs¯(0) ADis (5) A
=1/2. (10)
iY=0 (cid:28)BzI(cid:29)
n(s)−1
A numericalexample is shownin Fig.1 for randomvari-
ps→Ds = ps¯→Ds¯ BDis. (6)
iY=0 ablesAandB independentanduniformlydistributedon
3
anintervalinlogarithmicscalewithlnA∈[−4.21,−3.17] as it was suggested in Ref. [7].
and lnB ∈ [−2.02,−0.98], giving z = 1.88. It clearly Our proposed model with random scaling factors,
I
demonstrates that, even though initially quite different Eq. (1), is the simplest of this kind. It is not aimed
power laws emerge, asymptotically all realizations have atgivingaquantitativedescription,butitelucidates the
the same power-law exponent. In addition, one observes qualitativeconsequencesofrandomnessinthescalingfac-
that the averagedcurve shows this asymptotic exponent tors,namelythepresenceofawell-definedasymptotical-
beginning at much smaller times. gebraic decay with a realization-independent power-law
Moreover, in this case of not too wide distributions, exponent. We do not expect qualitative changes of this
one can show that fluctuations around the asymptotic scenario by further refinements of the model, like partly
algebraic decay vanish for each realization, as observed fixing scaling factors for the modeling of level-scaling or
inFig.1b. This is due to anincreasingnumber ofsitess by changing the number of branches [15].
contributingsignificantlytoEq.(3)astimeincreases. We In order to draw conclusions from our model about
have numerically demonstrated this by finding that the Hamiltoniansystems,wemakeuseofthehypothesisthat
inverse participation ratio s[ρs(t)/P(t)]2 of the con- thereexistsauniversalmechanismgeneratingthehierar-
tributing densities decreasesPlike a power law as a func- chical fine-scale structure of Hamiltonian systems, lead-
tionoftime(notshown). Itstressesthefactthatthecon- ing to a universal distribution of scaling factors. In par-
tributionto P(t) atlargetimes arisesfromanincreasing ticular, we consider different Hamiltonian systems, with
number of paths [10]. respect to their fine-scale structure, as different realiza-
In Fig. 2 we show numerical calculations of the tionsfromonestatisticalensemble,justasthisisthecase
power-law exponent fitted from the exact evolution for a specific realization of the random scaling factors A
Eq.(2)forasingle realizationofscalingfactors,showing and B in the tree model. The immediate consequence of
good agreementwith the analyticalprediction, Eq. (10). thishypothesisandoftherealizationindependenceofthe
This agreement gives also an indirect confirmation of power-law decays found in our model is, that all Hamil-
the validity of the approximation used in Eq. (3). We toniansystems should exhibit the same universalpower-
note, that Eq. (10) reproduces the power-law exponent lawdecay. Thenumericallyobserveddifferentpower-law
of deterministically chosen scaling factors A and B in decaysoverfinite timesaremerelyrealization-dependent
Ref. [7]. fluctuations around the universal decay.
Asanotherimportantconsequence,wearenowableto
We now want to argue why this model is able to cap- effectively address the practical question of how to nu-
ture the essential features of transport in Hamiltonian merically determine the universal power-law exponent.
systemswithamixedphasespace. Thediscretenatureof The straightforward approach is to extract it from P(t)
theMarkovmodelisjustifiedbythepresenceofinvariant of a single Hamiltonian system determined up to very
structuresinphasespacethatactaspartialbarriers(e.g. large times. However, in order to have a reliable power-
cantoriwithminimumflux)andthateffectivelyseparate law exponent from a single P(t) one would need much
regions where the trajectory spends a long time before larger times than in previous studies (which would go
moving to the adjacent one. The specific set of transi- beyond current computational resources). The reason
tionprobabilitiesdepends onthe precisestructureofthe is that the trajectories of the chosen Hamiltonian sys-
phase space, e.g., the transition probability for a trajec- tem musthave exploreda statistically significantsample
tory to leave a regionof area a leaking througha partial of the fine structure of its phase-space in order to sam-
barrier with flux ∆w is given by ∆w/a. Area conserva- ple the universal distribution of scaling factors. Instead,
tion assures that the flux is the same in both directions we propose to average P(t) over different Hamiltonian
and thus the scaling factors A in Eq. (1) are related to systems. This procedure averages out fluctuations and
s
the scaling of areas of the connected regions, while the should give a much better defined power-law decay al-
factors B are related to the ratio of escape rates. ready for smaller times, see e.g. Refs. [1, 20]. It is im-
s
A typical mixed phase space is hierarchically orga- portant to stress, that due to the results of our model
nized [17]. While approaching the boundary circle of a such an averaging has a clear meaning: It is not an av-
givenislandofregularmotion(level-renormalization[18]) erage over different power laws, but it corresponds to a
therearesecondaryislandchainsencirclingthemainone fasterreductionoftherealization-dependentfluctuations
(class-renormalization [19]). This picture repeats on all aroundtheuniversaldecay,obtainedbyanincreasedsta-
scalesadinfinitum. Usingthesefeaturesofamixedphase tistical sampling of the universal distribution of scaling
spaceMeissandOtt[7]modeledthe transportinHamil- factors.
tonian systems as a random walk on a binary tree with Guided by this idea we study anensemble ofareapre-
fixed scaling factors for level- and class-scaling, respec- serving maps of the form p = p −V′(q ); q =
n+1 n n n+1
tively. However, typical scaling factors in class-scaling q +p withV(q)= aq2+bq3 for100pairsofparam-
n n+1 2 3
and,atleastinitially, inlevel-scalingdovaryandamore eters(1<a,b<2). Thesemapshaveastablefixedpoint
realistic model must take into account these variations, at(0,0)andanunstable fixedpointat(−a,0). The fine
b
4
calinvestigationssupportthishypothesisandsuggestthe
universal exponent z ≃ 1.57±0.03. Future studies will
concentrateona directextractionofthe universaldistri-
butionsofareaandescape-ratescalingsingenericHamil-
toniansystems (bothnumericallyandanalytically),that
willeventuallyleadtoafullytheoreticalderivationofthe
universal exponent z.
WethankE.G.Altmannforusefuldiscussions. Thisre-
searchwassupportedinpartbytheDFGundercontract
KE537/3-3andtheNSFunderGrantNo. PHY05-51164.
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FIG. 3: (Color online) (a) Survivalprobabilities P(t) for 100 [3] B.V.ChirikovandD.L.Shepelyansky,PhysicaD13,395
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(1984)
their logarithmic average (thick line). Inset: phase space of
[4] R. S.MacKay, J. D. Meiss and I.C. Percival Phys. Rev.
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P(t)t with the fitted exponent z =1.57 from (a). (c) Vari-
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ance σ2(t) of ln[P(t)tz] showing saturation. [6] J.D.Hanson,J.R.CaryandJ.D.MeissJ.Stat. Phys.39
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nian system to a uniform initial distribution just in the [12] Theprobabilityps¯→Ds¯setsanoveralltimescalethatdoes
chaotic component. Numerically, this initial condition is not affect the power-law exponent. By choosing ps¯→Ds¯
not achievable. Instead, we choose an initial condition smallenoughoneensuresthat0<ps→s′ <1foralls,s′.
[13] It can beshown that theasymptotic power-law decay of
in a box in the chaotic region far away from the island,
ρs(t) is faster (by a factor 1/t) than that of the survival
which changes the exponent by one, z =z +1 [21].
I probability PI(t).
The averaged curve in Fig. 3 presents a well de-
[14] In the case of an exact self-similar tree (delta distribu-
fined power-law decay with exponent z ≃ 1.57± 0.03, tions for As and Bs) this is equivalent to an expansion
where the error describes the variation for different fit- in terms of Pollicott-Ruelle resonances. For an example
ting ranges. Quite importantly, the variance of the fluc- in one dimension see F. Barra and T. Gilbert. J. Stat.
tuations aroundthis averagedpower-lawsaturates. This Phys., 109 777 (2002).
[15] M.J.Kearney,V.M.DwyerandP.C.BressloffJ.Phys.A:
agreeswiththe predictionofourmodel(see Fig.1c)and
Math. Gen. 30 L409-L414 (1997)
it gives direct evidence that all individual power-law de-
[16] Apolewithmultiplicityk+1bringsafactoroftheform
cays are consistent with a single universal decay. We k
(lnt) . Imaginary parts of other poles contribute to an
find similar results with the same exponent for an en- oscillatingbehavioroftheprefactorC(t).Furtherconsid-
semble of kicked double-harmonic potentials. Motivated erations are necessary in the case of an infinite number
by our work, Altmann [22] studied an ensemble of mod- of poles with the same real part.
ified standard maps, confirming our results qualitatively [17] J.D. Meiss Rev. Mod. Phys. 64 795 (1992)
[18] R. S.MacKay Physica D 7 283 (1983)
and quantitatively.
[19] J.D. Meiss Phys. Rev. A 34 2375 (1986)
In conclusion,we conjecture a universalpower-lawde- [20] F. Christiansen and P. Grassberger, Phys. Lett. A 181,
cay P(t) ∼ t−z from the analysis of a Markov model
47 (1993)
with random scaling factors and from the hypothesis of [21] J.D. Meiss Chaos 7 139 (1996)
universality in the statistical properties of the fine-scale [22] E.G. Altmann (Ph.D. thesis, Universit¨at Wuppertal,
structure of typical Hamiltonian systems. Our numeri- 2007).
