Comment on "Lyapunov statistics and mixing rates for intermittent systems"

0.13 MB·English

by  Roberto Artuso

Comment on “Lyapunov statistics and mixing rates for intermittent systems"
Roberto Artuso1,2,∗ and Cesar Manchein1,3,†
1Center for Nonlinear and Complex Systems and Dipartimento di Scienza e Alta Tecnologia
Via Valleggio 11, 22100 Como (Italy);
2I.N.F.N., Sezione di Milano,
Via Celoria 16, 20133 Milano (Italy);
3Departamento de Física,
Universidade do Estado de Santa Catarina 89219-710 Joinville, (Brazil)
(Dated: January 30, 2013)
In Pires et al. [Phys. Rev. E 84, 066210 (2011)] intermittent maps are considered, and the tight
relationship between correlation decay of smooth observables and large deviations estimates, as for
instance employed in Artuso and Manchein [Phys. Rev. E 80, 036210 (2009)], is questioned. We
try to clarify the problem, and provide rigorous arguments and an analytic estimate that disprove
the objections raised in Pires et al. [Phys. Rev. E 84, 066210 (2011)] when ergodic systems are
considered.
PACSnumbers: 05.45.-a
3
1
0 I. INTRODUCTION dynamicalproperties two regimes haveto be considered:
2
using the same terminology and notation of [1], we have

n In a recently published paper [1], the authors argue an exponential instability regime, for z ∈ (1,2), where
a thatacloserelationshipbetweenpolynomiallargedevia- an invariant probability measure µ exists (i.e. the in-
J
tionsandpowerlawdecayfails,intheformconsideredin variant density is renormalizable), and where we have a
9
2 [2],whichwasinspiredby the rigorousresultsof[3](ear- Lyapunov exponent Λ∞ > 0, and a subexponential in-
lier results, which are relevant to the present discussion stability one, for z > 2, where the invariant measure is
] are [4, 5]). To this end they both consider ergodic and not renormalizable, and no positive Lyapunov exponent
D infinite ergodic systems: the present comment concerns exists: this is one of the prominent examples where in-
C the firstcase,namelythe originalsettingof[2]: wepoint finite ergodic theory applies (see, for instance, [11]). In
. out that the discussion in [2] concerned mixing systems this comment we will consider the exponential instability
n
with a positive Lyapunov exponent (see eq. (1) in [2], regime: this is representative of a wide class of physi-
i
l where the relevant quantity is identically zero in the in- cally relevant systems where, together with exponential
n
[ finite ergodic regime). We also point out that the very instability,longtimetailsforcorrelationsarepresent: ex-
concept of mixing is still under debate in the framework amples are Lorenz type maps [12], area preserving maps
1 of infinite ergodic theorem [6], though some hints that [13],Sinai[14],stadium[15]andmushroom[16]billiards.
v correlationdecayand largedeviations areindeed related In the exponential instability case of Pomeau Manneville
8
4 has recently appeared [7]. We regret that the notion of mapsitiswellknownthat“generic"correlationfunctions
9 “weak chaos" is loosely defined, even though any confu- decay with a well defined power-law exponent (we will
6 sion between ergodic and infinite ergodic system was, in later on comment on the precise meaning of “generic" in
. our opinion, never present, since anunambiguous defini- thiscontext): forexample,byconsideringbothlowerand
1
0 tionofthemainquantitiesemployedin[2]isonlypossible upperbounds,in[17]itisprovedthatthereareLipschitz
3 in the positive Lyapunov - ergodic regime. functions F and G for which
1 To keep the discussion as simple as possible we focus
v: onthemaindynamicalexampleconsideredin[1],namely dµ(F ◦Tn)G− dµF dµG =O(n−(z−11−1));
i the intermittent Pomeau-Manneville map [8], indeed a (cid:12)ZI ZI ZI (cid:12)
X prototype example of weak chaos (in any possible mean- (cid:12) (cid:12) (2)
(cid:12) (cid:12)
r ingofsuchaconcept): thisisdefinedontheunitinterval w(cid:12)hile the upper bound holds for any(cid:12) pair of Lipschitz
a functions, the lower bound cannot share this generality,
I =[0,1] and may be expressed as
and indeed for the non-generic case where dµF =
I
x =T(x )= x +xz| , (1) F(0) the decay is faster [18] (it gains a 1/n factor [19]).
n+1 n n n mod1 R
We point out again that both in [1] and [2] correlation
where, for z > 1, the map presents in x = 0 an indif- decayareconsideredfor sufficiently smoothfunctions: it
ferent fixed point, which influences deeply the dynamics is well known (see for instance [20] in the case of cat
(see, for instance, [9, 10]). From the point of view of maps) that if one considers larger function spaces where
testfunctionsarepickedfrom,onemaygetverydifferent
behavior.
Nowwehavetoconsiderlargedeviation,inordertosee
∗ [email protected] whether their behavior is linked to correlation decay (as
† [email protected] in [2]), or not, as claimed in [1]. More precisely we con-
2
sider (as in [2]) the distribution of finite time Lyapunov concerns η(Λ,n) for Λ < 0, which is identically zero),
exponents while the polynomial case (5) gives exactly
Λ (x)= 1 n−1 lnT′(Tj(x)), (3) MΛ˜(n)=O(n−(z−11−1)) ∀Λ˜ such that 0<Λ˜ <Λ∞,
n n (7)
Xj=0 in perfect agreementwith [2], contrarily to what the au-
thors of [1] claim. Physically the mechanism that leads
where the corresponding probability density is denoted
to the non-generic exponential large deviations for in-
by η(Λ,n) [21], and consider the quantity
termittent dynamics (6) is very simple: while statistical
anomalies are due to long waiting times near the indif-
Λ˜
ferent fixed point, in the fine-tuned case where the value
M (n)= dΛη(Λ,n), (4)
Λ˜ of the observable at the indifferent fixed point coincides
Z0
withthephaseaverageoftheobservable,duringthelam-
for0<Λ˜ <Λ : thisofcourseimpliesthatΛ >0,and inarsequenceBirkhoffsumspickuptherightvalue,con-
∞ ∞
so our considerationsapply to the exponential instability cealing the dynamical anomalies due to sticking. Let us
case, 1<z <2 [22]. In [2] (on the grounds of [3, 4]) it is further add that obviously, as regards finite time Lya-
stated that M (n) has the same polynomialbehaviorof punov exponents, these anomalies manifest themselves
Λ˜
generic correlationfunctions(i.e itvanishesaccordingto in the distribution η(Λ,n) for small Λ, 0 < Λ < Λ ,
∞
a power law with the same exponent), while in the sec- like in [4], whose results are misquoted in [1] (see their
tion II.A of [1] it is claimed that such a quantity decays eq. (4))[27]. Letusfinallyremarkthat,mathematically,
exponentially. results like eq. (5) are upper bounds, and further dis-
This is the crucial point of this Comment: we now cussionisneededasregardstheiroptimalcharacter: this
provethattheargumentsof[1]areincorrect. Theirclaim is indeed discussed in [3], where a lower bound (of the
thatM (n)decaysexponentiallyconsistini.) (theirEq. samepower-lawform)isshowntoholdforaspecialclass
Λ˜
(8)) supposing that in such a case a good rate function of functions, exactly in the case of Pomeau-Manneville
exists for the observable g(x) = lnT′(x), and ii.) that maps.
suchaclaimisconsistentwithatheoremprovedbyPolli- Thoughthemathematicalmisunderstandingsin[1]are
cottandSharp[5]. Thesubjectoflargedeviationsinthe already clear at this point, we can also add an explicit
context of dynamical systems is indeed very important, analytic computation that illustrates the way in which
andthe existenceofnice ratefunctions isnotgenerically polynomial large deviations are attained for Pomeau-
expected unless trajectories enjoy very good statistical Manneville maps. We will obtain a lower bound, that
properties [23, 24]. In the present case, a number of rig- again shows how polynomial large deviations estimates
orous results [3, 5, 25, 26], that apply to the Pomeau- areoptimalfor suchmaps,with a welldefinedexponent.
Manneville map in the exponential instability case, con- In particular we use the Gaspard-Wang piecewice linear
cern exactly polynomial large deviations. As it is cru- approximation, discussed in detail in [10], to which we
cial in the discussion let us recall the theorem proved in referforfulldetails. Theapproximationconsistsinparti-
[5] (the formulation takes into account a sharper result tioningI bythe collectionofsetsA =(ξ ,ξ ), where
n n n−1
obtained in [3]). The theorem applies to any Hölder ob- ξ =1,ξ =a(whereaissuchthatT(a)=a+az =1),
−1 0
servableg(x)withzeroaverage dµg =0. Thetheorem and {ξ } is the decreasing sequence converging to zero,
I n
consists of two large deviation statements: if |g(0)| > ǫ such that T(ξ ) = ξ . Then the new (piecewise lin-
n n−1
R
then ear) map T is constructed such that its slope is con-
L
stant on every A , and T (A ) = A . We denote by
n L n n−1
µ x∈I : 1 n−1g(Tix) ≥ǫ =O(n−(z−11−1)); (5) ∆n =ξn−1−ξn the width of An and by sn =∆n−1/∆n
the slope of T | . The invariant measure density is
( (cid:12)n (cid:12) ) L An
(cid:12) Xi=0 (cid:12) piecewiseconstant,andcanbeeasilyevaluatedbyconsid-
(cid:12) (cid:12)
(cid:12) (cid:12) eringtheequivalentMarkovchain,theinvariantmeasure
(polynomial c(cid:12)ase), while, if(cid:12)|g(0)|<ǫ
weights are µ =µ(A ), and again in the case z ∈(1,2)
n n
we have an invariant probability measure, which shares
n−1
µ x∈I : 1 g(Tix) ≥ǫ =O(e−βn), (6) all the relevantproperties of the correspondingPomeau-
( (cid:12)n (cid:12) ) Manneville invariant probability measure for the same
(cid:12) Xi=0 (cid:12) exponent z [28], moreover the correlation decay rate of
(cid:12) (cid:12)
for some β > 0(cid:12)(cid:12)(exponential (cid:12)(cid:12)case). Now the observ- genericsmoothfunctionsisidenticalto(2)[29]. Forlarge
able we have to consider is gˆ(x) = lnT′(x): while in- n we can easily get the dominant behavior [10]:
deed gˆ(0) = 0, this function has a non-zero average
I dµgˆ = Λ∞, and so, to apply the theorem, we have ∆n ∼(n+1)−z−z1, µn ∼n−z−11. (8)
to consider the observable g˘ = Λ − lnT′(x). Since
∞
R
g˘(0) = Λ the exponential part of the theorem (6) (in- Now we define intervals near the fixed point A =
∞ k
voked in [1]) provides no information about M (n) (it ∪∞ A ; for each y ∈ A the maximal finite time
Λ˜ m=k m n+N
3
Lyapunov exponent we may obtain is now, by using (8), we easily get
N+n
1 1 ∆
Λ(N) (n)= lns = ln N (9)
max n k n ∆
k=XN+1 (cid:18) N+n(cid:19) µ(An+N˜)= ∞ µ(An)∼(n+N˜)−(z−11−1) : (12)
which,forsufficientlyhighN,hasthe followingbehavior
m=Xn+N˜
(see (8)):
1 N +1 n
Λ(N) (n)∼ ln 1+ . (10)
max N +1 n N +1 i.e. polynomial largedeviations with the same exponent
(cid:18) (cid:19) (cid:18) (cid:19)
ruling correlation decay. Though we used the piecewise
Since, away from zero, the function (1/y)ln(1 + y) is linear approximation, the same kind of asymptotics on
bounded, (10) implies that, for any Λ˜ with 0<Λ˜ <Λ interval scaling and measures are known to hold for the
∞
we can fix a value N˜ such that Λ(N˜) (n) < Λ˜ for any n. original map (see for instance [17, 18]).
max
This leads to the following bound:
This work has been partially supported by MIUR–
n−1 PRIN2008projectNonlinearity and disorder in classical
1
µ x∈I : lnT′(Tkx)<Λ˜ ≥µ(A ); (11) and quantum transport processes.
n L L n+N˜
( )
k=0
X
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