Correlation decay and large deviations for mixed systems

0.12 MB·

by  Roberto Artuso

Correlation decay and large deviations for mixed systems
R.Artuso
DipartimentoSAT,andCenterforNonlinearandComplexSystems
Universita`degliStudidell’Insubria,
Como,22100,Italy,
IstitutoNazionalediFisicaNucleare,SezionediMilano
Milano,20133,Italy
E-mail:[email protected]
6 C.Manchein∗andM.Sala
1
DepartamentodeF´ısica,UniversidadedoEstadodeSantaCatarina
0
Joinville,89219-710,Brazil
2
E-mail:[email protected]
∗
n
a
J

3
1
Abstract

]
D
We consider low–dimensionaldynamical systems with a mixed phase space and discuss the typical appearance of
C slow,polynomialdecayofcorrelations:inparticularweemphasizehowthismixingrateisrelatedtolargedeviations
. properties.
n
i
l Keywords: Mixing,largedeviations,correlationfunctions,finite-timeLyapunovexponents.
n
[

2 1. Introduction 2. Weakchaos
v
1 Hereweprovideanumberofexamplesofthekindof
2
systems our analysis is devoted to: simplest examples
9
involveonedimensionaldynamics. Heretheprototype
2
0 caseisrepresentedbyPomeau-Mannevillemap[9],
In this contribution we plan to provide a short re-
.
1 view of results concerning mixing properties of de-
0 terministic dynamical systems. In particular our em- xn+1 =TPM(xn)= xn+xznmod1, (1)
6 phasis will be on systems enjoying ergodicity in the (cid:12)(cid:12)
1 whichpresentsanextremelyreachbe(cid:12)haviourasthein-
: conventionalsense[1,2], andnotinfiniteergodicity[3], termittency parameter z is varied. At z = 1 it coin-
v wherethe veryconceptofmixingturnsouttobe quite
i cideswithBernoullimap,thestandardexampleoffully
X delicate[4, 5]. The general idea is that when some
chaotic(uniformlyhyperbolicdynamics),butassoonas
deterministic dynamics presents a mixed phase space,
r z > 1the0fixedpointbecomesmarginal,theinvariant
a namelycoexistenceofachaoticseawith(evenarbitrar-
measuredevelopsasingularityattheorigin,andanoma-
ily small) regular structures, sticking of a typical tra-
lousfeaturesappear[10,11].Whenz 2thesingularity
jectoryclose to regularzonesenhancesseverelycorre- ≥
of the invariantmeasure at the origin becomesnon in-
lations, and generically the speed of mixing becomes
tegrable,and(1)providesoneofthesimplestexamples
polynomial, where instead, for a fully chaotic system
of infiniteergodicity[12, 13]. Inthe regime1 < z < 2
we expect exponential decay of correlations. In what
thesystemisergodic(theinvariantprobabilitymeasure
followswewilldenotebyweaklychaoticsystemsthose
willbedenotedbyµ),anddisplayspower-lawdecayof
for which a power-law decay of correlations is indeed
correlation functions: in particular Hu[14] provedthat
present.Weemphasisethat,beyondthesimple“mathe-
thereexistLipschitzfunctionsF andGsuchthat
matical”exampleswewillmentioninthenextsection,
thegeneralobservationsstillapplytonontrivialphysi-
dµ(F Tn )G dµF dµG =O(n ξ), (2)
calsettings,notablyinfluiddynamics[6–8]. (cid:12)(cid:12)ZI ◦ PM −ZI ZI (cid:12)(cid:12) −
(cid:12) (cid:12)
Preprintsubmittedtopublication (cid:12)(cid:12) (cid:12)(cid:12)January14,2016
wheretheexponentξ is determinedby theintermit- structure of regular regions, where typical orbits stick
PM
tencyparameterzthrough for a long time, causing slow correlation decay, is ex-
tremelyrich(seeforexample[27–29]). Thereisaclass
1
ξ = 1. (3) of differentarea-preservingmapswith a simpler phase
PM
z 1 −
− spacestructure:
In particular one notices that ξ diverges in the
Bernoulli limit z 1+, while foPrMz 3/2 correla- L : yn+1 =yn+ f(xn) mod2π, (7)
tions are not integr→able, the standard ce≥ntral limit the- (ε,γ) ( xn+1 = xn+yn+1 mod2π,
oremdoesnotholdandproperlyrenormalizedBirkhoff
where f(x )isdefinedby
n
averagesconvergetoaLe´vystablelaw[15]. Weremark
thatin this examplethe “regular”regionamountsonly f(x )=[x (1 ε)sin(x )]γ. (8)
n n n
ontheindifferentfixedpointattheorigin. − −
Anotherwell knownexample is providedby Pikovsky Suchamapwasintroducedin[30],forε=0andγ=1,
mapT [16](see [17]formoredetailedbibliography), andgeneralizedin[31,32],whereatransitiontopoly-
P
whichisimplicitlydefinedby nomialdecayof correlationswasobservedasε 0+.
→
Asamatteroffactforε>0themapisfullyhyperbolic
1
[1+T (x)]z, 0< x<1/(2z), (anddisplaysexponentialcorrelationsdecay),while,for
P
x= T2Pz(x)+ 21z[1−TP(x)]z, 1/(2z)< x<1; (4)iεng=a0rothleeafinxaelodgpoouisnttoatth(e0o,0ri)gbinecinomPoemspeaarua-bMoalincn,epvlailyle-
whilefornegativevaluesof x [ 1,0],themapisde- maps (and γ is a sort of intermittency exponent)2. By
TfineidsathsaTt,P(w−hxi)le=ret−aiTnPin(xg).indA∈iffre−ermenatrkfiaxbeledfpeoaitnutrse(oaft cinon[3s2id]eirsinwgasthaergduyendatmhaictsthaelocnogrrtehlaetiuonnstdaebclaeymfoarnεifo=ld0,
P ispolynomial,with
x= 1)withanintermittencyparameterz,theinvariant
±
probabilitymeasure µ is the Lebesgue measure (it is a 3(γ+1)
ξ = ; (9)
simpleexerciseto verifythatbywritingdownthe cor- L 3γ 1
respondingPerron-Frobeniusoperator):thisisobtained −
bylettingtheinstabilityunboundedclosetotheorigin. thisinparticularpredictsanexponentξL = 3whenγ =
Againwehaveapolynomialdecayrateforcorrelations, 1: forsucha parametervaluethere isa rigorouslower
with bound[34]ξL 2.
1 ≥
ξ = ; (5)
P
z 1
− 3. Indirectapproachtocorrelations:recurrences
here any value of z > 1 is allowed, and correlations
becomenonintegrable(withageneralizedcentrallimit Direct numerical investigations of correlation func-
theorem[17])forz 2. tionsarenotoriouslyhardtoaccomplish,since, evenif
≥
Examplesofsimilarbehaviourinhamiltoniandynamics theergodicinvariantmeasureisknown(likeinthecase
include billiard tables in two dimensions, like the sta- of area-preserving maps), a Monte Carlo computation
diumorSinaibilliard(seeforinstance[18]),thelimitof ofcorrelationfunctionswithN pointsinvolvesaner-
MC
kissing discs fordiamondbilliards[19, 20] 1, or mush- rorproportionalto1/√N ,whichcauseshugefluctu-
MC
roombilliards[21]. ationsaftermoderatetimes.
Anotherpopularcontextisthatofarea-preservingmaps, Anindirectapproach,whichhasbeenwidelyusedinthe
where the prototype example is the so called standard lastdecades,involvesrecurrencetime(Poincare´)statis-
map[22,23]: tics. Such an approach was pioneered in [24, 25, 35],
S(K) :( yxnn++11==yxnn−+Kyn+si1n(xn) mmoodd22ππ., (6) abnrideflityaddemsictrsibaeriagosriomupslebavseisr[s3io6n] (osefeita[l2s4o]:[37su])p.poWsee
we partition the phase space into two disjoint sub-
Despite the simple structure of (6), rigorous analysis sets, labelled 0 and 1: from a long trajectory of the
of the standard map is extremely difficult[26], and the system we extract the sequence of residence times
n ,n ,...,n ,... (for simplicity we are considering a
1 2 k
1Diamond billiards are in this context particularly interesting,
sincebyvarying ageometrical parameter -the radius ofthe bound-
ingdiscs-weareabletoturnthecorrelationspeedfromexponential 2Noticethatwhenεbecomesnegativethefixedpointattheorigin
topowerlaw. becomeselliptic[33].
2
discretetimedynamics)onasinglesubset(namelythe butionsoffinitetimeaverages:inparticularmanystud-
waitingtimesbeforecrossingtheborder).Thisleadsto ieshaveconcernedfeaturesofdistributionfunctionsof
a probabilitydistribution for residence times ℘(n): we finite-timeLyapunovexponents[27,42–46]. Arigorous
alsosupposethattheaverageresidencetimeν = n = approachwasproposedin[47](seealso[48,49]):takea
h i
k ℘(k)isfinite. Nowwemakethe(severe)hypoth- one-dimensionalmapT,thenfinite-timeLyapunovex-
·
Pesisthatcrossingtheborderleadstoacompletedecor- ponentsaredefinedas
relationofthedynamics[38,39]: inthisway[40],ifwe
1 dT(n)(x)
consider the autocorrelation function of an observable λ (x )= ln . (13)
G: CGG(m)=hG(m0+m)G(m0)i−hGi2, (10) Suchfinitetimenes0timatnesg(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)enerdicxally(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)xd0(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)ependuponthe
we have that G(m0 + m)G(m0) = G2 if no cross- initialconditionx0,sotheyarecharacterizedbyaproba-
ing takes plache between m0 andim0 +h miand G(m0 + bilitydistributionPn(λn),which,forergodicandchaotic
m)G(m ) = G 2 otherwise. IfwedenotebyΠh(m)the systems, collapsestoa delta distributioninthe asymp-
0
probabiliitythhatinocrossingtookplaceinthetimelapse toticlimit
[m0,m0+m],then nlim Pn(λn)=δ(λ−λ), (14)
→∞
C (m)=Π(m) G2 +(1 Π(m)) G 2 G 2 =Π(m) G2 Gwh2e.reλistheLyapunovexponentofthemapT. Ifwe
GG h i − h i −h i h i−h fiix a threshold λ˜ < λ, and compute the sub-threshold
((cid:16)11) (cid:17)
Now it is easy to expressΠ(m) in termsof ℘(k), since weights:
λ˜
theprobabilitythatapointchosenatrandomisthefirst (n)= dλ P (λ ) (15)
pointofaresidencesequenceoflengthkis℘(k)/νwhile Mλ˜ Z n n n
−∞
theprobabilitythatapointchosenatrandomisthefirst, wehavethatsuchquantitiesvanishinthelargetimelim-
orthesecond,ofak+1residencesequenceis℘(k+1)/ν: its, howevertheirdecayisrelatedtocorrelationdecay:
thus: morepreciselytherigorousresultin[47]statesthat,in-
dipendentlyofthechoiceofthethresholdλ˜,if
Π(m)= 1(℘(m)+2 ℘(m+1)+3 ℘(m+2)+ )= 1 ∞ ∞ ℘(i). 1
ν · · ··· ν (n) , (16)
Xj=m Xi=j Mλ˜ ∼ nσ
(12)
thencorrelationfunctionsofsmoothobservablessatisfy
Inthisway,anexponentialdecayof℘(m)yieldsasim-
thebound
ilarexponentialdecayforcorrelationfunctions,whilea
1
power-lawdecay℘(m) m α correspondstoapolyno- C(m) , (17)
mialcorrelationdecay,∼with−ξ = α 2: such a quanti- ≤ mσ−1
− i.e. ξT σ 1. Such a bound cannot be how-
tative correspondence has been scrutinized in [20] for ≥ −
everoptimal,asforPomeau-Mannevillemapsasimple
a number of billiard systems, in full agreement with argument[50]showsthatξ =σ.
PM
known rigorousresults (see for instance [41]). We re-
We notice that an assessment like (16) is a (non-
mark again that the crucial approximation involved in
exponential) large deviation estimate (see for instance
theabovereasoningconsistsinassumingdecorrelation
[51]), and as a matter of fact the most general
at each crossing: for a different kind of time statistics
results[52–54]can bestatedasfollows: if we consider
(flight times between collisions in a Lorentz gas with
amapT,suchthatξ isthepowerlawmixingrate,than
T
infinite horizon) such an hypothesis has been investi- Birkhoffsumsofanobservableψsatisfythefollowing
gatedin detail[40]: for shorttimes it obviouslymisses
estimate:
features of real correlations (for instance multiple col-
lriespiorondsubceetswteheenanseyimghpbtootuicrirneggdimisec.s),whileitaccurately µx:(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)1nXnj=−01ψ(T(j)(x))−Z dµψ(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)>ǫ≤Cψ,ǫn1ξT(1;8)
4. Indirect approach to correlations: large devia- thatispo(cid:12)lynomiallargedeviations,w(cid:12)ithanexponentin-
tions dependentofthethreshold,andcoincidingwiththeone
ruling correlations decay3. In one dimension (18) in-
The fundamental idea, at a qualitative level, is that cludesthecaseoffinite-timeLyapunovexponents(15),
thesamedynamicalmechanismthatslowsdowncorre-
lationdecayisresponsibleforanomalous(broad)distri- 3In[52]suchaboundhasbeenshowntobeoptimal.
3
with ψ(x) = ln T (x), while in higher dimensions the phase space, where sticking manifests in slow, poly-
′
| |
leadingfinite-time Lyapunovexponentcannotbe writ- nomial mixing rates. Together with the popular use
tenasasimpleBirkhoffsum: neverthelessitstillrepre- ofPoincare´ recurrences,weemphasizenewtechniques
sentsanaturalindicator. basedonlargedeviationsproperties: we presentin the
Such a technique has been used in a variety of last section novel calculations that corroborate the ef-
context[55]: fromonedimensionalmapsas(4),toarea fectivenessofsuchamethod,bycomputingpolynomial
preservingmaps (7), and it has been also employedto mixingratesofaweaklychaoticarea-preservingmaps
corroborate universality claims[56] for correlation de- withveryhighprecision.
cay of area preserving maps with mixed phase space.
More recently this method was also used in exploring 0.16 n=2×103 (a)
mixingpropertiesofcoupledintermittentmaps[57]. n=5×103
0.12 n=1×104
)
5. Amodelexample n
λ
To provide an illustration of the technique we de- P(0.08
scribed in the former section, we provide new numer- 1
−
ical experiments on the family (7) of area-preserving 00.04
1
maps. Inparticularwewanttoemphasizetwoaspects:
the transitionfromchaoticε > 0 to intermittentε = 0
behavior,andhow(18)offersanefficientnumericaltool 0.00
tocomputeexact(polynomial)mixingratesinthelatter 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15
λ
case. n
1.0
In Fig. (1) we plot Pn(λn) for different cases: the hy- n=2×103 n=1×104 (b)
perbolic case (a) has been obtained by 5 106 initial n=5×103 n=5×104
· 0.8
condition, while the intermittent case (b) refers to 107
) 10−1
initial conditions. One almost obvious feature is that λn0.6
whenwegotoε = 0casethedistributionslookasym- ( 10−3
P
metric,duetothefrequentappearanceoflowestimates 1 0.4 10−5
forLyapunovexponentsduetostickingtotheparabolic −
0
fixedpoint4. Thisleadsto a quantitativeanalysisonce 10.2 10−7
10−3 10−2 10−1 100
we estimate how the subthreshold tail of the distribu-
tionsshrinkstozero(15):inthehyperboliccase(Fig.(2 0.0
(a)),weobserveapurelyexponentialdecay(noticethat 0.0 0.4 0.8 1.2
the data referring to the case γ = 2 have been shifted λn
toavoidoverlapping),asexpectedforafullyhyperbolic
case, while in the parabolic ε = 0 case, we observe a
powerlawdecay,which,inviewof(18)shouldcoincide Figure1:Distributionoffinite-timemaximalLyapunovexponentsfor
themap(7)with(a)ε=1andγ=2and(b)ε=0andγ=2.(Inset:
with the mixing rate. A regression analysis yields for
distributions of finite-time maximal Lyapunov exponents plotted in
aγn=d2fo,rMγλ˜(=n)3∼n−1(.n80)±0.01n,w1.h49ile0.(094,)wprheidleict(s9)ξLpr=ed9i/c5ts, logarithmscale.)
ξ =3/2,witneMssiλn˜gho∼wla−rge±deviationanalysisturns
L
tobeanextremelypowerfultoolinthenumericalanal-
Acknowledgments
ysisofquantitativemixingpropertiesofdynamicalsys-
temswithweakchaoticproperties.
R.A. thanksXavier LeonciniandSandroVaientifor
many interesting discussions about such topics along
6. Conclusions many years. C.M. thanks CNPq (Brazil), and M.S.
thanksCAPES(Brazil)forfinancialsupport.
We have reviewed indirect methods for the investi-
gations of correlation decay for systems with mixed
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