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2 WHAT ENTROPY AT THE EDGE OF CHAOS? ∗
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e MARCELLO LISSIA,MASSIMO CORADDUANDROBERTO TONELLI
m Ist. Naz. Fisica Nucleare (I.N.F.N.), Dipart. di Fisica dell’Universit`a di Cagliari,
- INFM-SLACS Laboratory, I-09042 Monserrato (CA), Italy
t E-mail: [email protected]
a
t
s
.
t
a Numerical experiments support the interesting conjecture that statistical methods be
m applicablenotonlytofully-chaoticsystems,butalsoattheedgeofchaosbyusingTsal-
lis’ generalizations of the standard exponential and entropy. In particular, the entropy
-
d increases linearly and the sensitivity to initial conditions grows as a generalized expo-
n nential. We show that this conjecture has actually a broader validity by using a large
o classofdeformedentropiesandexponentials andthelogisticmapastestcases.
c
[
2 Chaotic systems at the edge of chaos constitute natural experimental labora-
v tories for extensions of Boltzmann-Gibbs statistical mechanics. The concept of
9
generalized exponential could unify power-law and exponential sensitivity to ini-
9
1
2 tial conditions leading to the definition of generalized Liapounov exponents : the
1 sensitivity ξ ≡ limt→∞lim∆x(0)→0∆x(t)/∆x(0) ∼ exp(λt), where the generalized
0 exponential exp(x) = exp (x) = [1+(1−q)x]1/(1−q); the exponential behavior for
5 q
0 the fully-chaoticregimeisrecoveredforq →1: limq→g1expq(λqt)=exp(λt). Analo-
/ gously,agengeralizationoftheKolmogoroventropyshoulddescribetherelevantrate
t
a oflossofinformation. AgeneraldiscussionoftherelationbetweentheKolmogorov-
m
Sinai entropy rate and the statistical entropy of fully-chaotic systems can be found
-
in Ref. 2: asymptotically and for ideal coarse graining, the entropy grows linearly
d
n withtime. ThegeneralizedentropyproposedbyTsallis3S =(1− N pq)/(q−1),
q i=1 i
o with p the fraction of the ensemble found in the i-th cell, reproduces this picture
c i P
at the edge of chaos; it growslinearly for a specific value of the entropic parameter
:
v
i q =qsens =0.2445 in the logistic map: limt→∞limL→0Sq(t)/t=Kq. The same ex-
X ponent describes the asymptotic power-law sensitivity to initial conditions 4. This
r conjecture includes an extension of the Pesin identity K = λ . Numerical evi-
a 1 q q
denceswiththe entropicformS existforthelogistic andgeneralizedlogistic-like
q
5
maps .
6
Renormalization group methods yield the asymptotic exponent of the sensi-
tivity to initial conditions in the logistic and generalized logistic maps for specific
∗This work was partially supported by MIUR (Ministero dell’Istruzione, dell’Universita` e della
Ricerca)underMIUR-PRIN-2003project“Theoreticalphysicsofthenucleusandthemany-body
systems”.
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initial conditions on the attractor; the Pesin identity for Tsallis’ entropy has been
7
also studied .
Sensitivity and entropy production have been studied in one-dimensional dis-
8
sipative maps using ensemble-averaged initial conditions and for two simpletic
9
standard maps : the statistical picture has been confirmed with a different
q = qave = 0.35 8. The ensemble-averaged initial conditions is relevant for the
sens
relation between ergodicity and chaos and for practical experiments.
Thepresentstudydemonstratesthebroaderapplicabilityoftheabove-described
picturebyusingtheconsistentstatisticalmechanicsarisingfromthetwo-parameter
family 10,11,12,13 of logarithms
ξα−ξ−β
ln(ξ)≡ . (1)
α+β
Physicalrequirements14 ontheeresultingentropyselect15 0≤α≤1and0≤β <
12 16
1. All the entropies of this class: (i) are concave , (ii) are Lesche stable , and
15
(iii) yield normalizable distributions ; in addition, we shall show that they (iv)
yield a finite non-zero asymptotic rate of entropy production for the logistic map
with the appropriate choice of α.
We have considered the whole class, but we shall here report results for three
interesting one-parameter cases:
3
(1) the original Tsallis proposal (α=1−q, β =0):
ξ1−q −1
ln(ξ)=ln (ξ)≡ ; (2)
q
1−q
(2) Abe’s logarithm e
ξ1/qA−1−ξqA−1
ln(ξ)=ln (ξ)≡ , (3)
A
1/q −q
A A
where q = 1/(1+α)eand β = α/(1+α), which has the same quantum-group
A
symmetry of and is related to the entropy introduced in Ref. 17;
(3) and Kaniadakis’ logarithm, α = β = κ, which shares the same symmetry
18
group of the relativistic momentum transformation
ξκ−ξ−κ
ln(ξ)=ln (ξ)≡ . (4)
κ
2κ
The sensitivity to initial conditions and the entropy production has been
e
studied in the logistic map x = 1 − µx2 at the infinite-bifurcation point
i+1 i
µ∞ = 1.401155189. The generalized logarithm ln(ξ) of the sensitivity, ξ(t) =
(2µ)t t−1|x | for 1 ≤ t ≤ 80, has been uniformly averaged by randomly choos-
i=0 i
ing 4×107 initial conditions −1<x <1. Analogoeusly to the chaotic regime, the
Q 0
deformed logarithm of ξ should yield a straight line ln(ξ(t))=ln(exp(λt))=λt.
Following Ref. 8, where the exponent obtained with this averaging procedure,
indicated by h···i, was denoted qav for Tsallis’ enteropy, eacheogf the generalized
sens
logarithms, hln(ξ(t))i, has been fitted to a quadratic function for 1≤t≤80 and α
e
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hasbeenchosensuchthatthe coefficientofthe quadratictermbe zero: we callthis
value αav .
sens
Statisticalerrors,estimatedby repeatingthe wholeprocedurewithsub-samples
ofthe4×107initialconditions,andsystematicuncertainties,estimatedbyincluding
different numbers of points in the fit, have been quadratically combined.
We find that the asymptotic exponent αav = 0.650±0.005 is consistent with
sens
the valueofRef.8: qav =1−αav ≈0.36. Theerroronαav isdominatedbythe
sens sens sens
systematic one(choice ofthe number ofpoints)due tothe inclusionofsmallvalues
of ξ which produces 1% discrepancies from the common asymptotic behavior.
Figure 1 shows the straight-line behavior of ln(ξ) for all formulations when α=
αav (right frame); the corresponding slopes λ (generalized Lyapunov exponents)
sens
are 0.271±0.004 (Tsallis), 0.185±0.004 (Abee) and 0.148±0.004 (Kaniadakis).
While α is a universal characteristic of the map, the slope λ strongly depends on
the choice of the logarithm.
24
8
Tsallis
20 Abe
Tsallis Kaniadakis
Abe
16 Kaniadakis 6
)
ξ)og(12 <S>(t4
l
8
2
4
0 0
0 2 0 4 0 6 0 t 80 0 5 1 0 1 5 2 0 2 5 t 30
Figure 1. Generalized logarithm of the sensitivity to initial conditions (left) and generalized
entropy(right)asfunctionoftime. Fromtoptobottom,Tsallis’,Abe’sandKaniadakis’logarithms
(entropies) for α = αasevns. In the left frame the slopes λ (generalized Lyapunov exponents) are
0.271±0.004, 0.185±0.004 and 0.148±0.004; in the right frame the slopes K (generalized
Kolmogoroventropies)are0.267±0.004,0.186±0.004and0.152±0.004.
The entropy has been calculated by dividing the interval (−1,1) in W = 105
equal-size boxes, putting at the initial time N = 106 copies of the system with a
uniform random distribution within one box, and then letting the systems evolve
according to the map. At each time p (t)≡ n (t)/N, where n (t) is the number of
i i i
systems found in the box i at time t, the entropy of the ensemble is
N 1 N p1−α(t)−p1+β(t)
S(t)≡ p (t)ln( ) = i i (5)
i
p (t) α+β
* i + * +
i=1 i=1
X X
where h···i is an averageover 2e×104 experiments, eachone starting from one box
randomly chosen among the N boxes. The application of the MaxEnt principle to
the entropy (5) yields as distribution the deformed exponential that is the inverse
−1 15
function of the corresponding logarithm of Eq. (1): exp(x)=ln (x) .
g e
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15 10
9
8
10 7
<S( t)> <s(t)>56
4
5
3
2
1
00 5 10 15 20 25 30 00 5 10 15 20 25 30
t t
8
7
6
5
>
S(t)4
<
3
2
1
00 5 10 15 20 25 30
t
Figure2. Tsallis’(topleft),Abe’s(topright)andKaniadakis’(bottom)entropiesasfunctionof
time for α=1−q =0.80, 0.74, 0.64, 0.56, 0.52 (from top to bottom). Straight lines are guides
fortheeyes whenα=0.64≈αasevns.
Analogously to the strong chaotic case, where an exponential sensibility (α =
β =0)is associatedto a linear risingShannonentropy,whichis defined interms of
the usual logarithm (α=β =0), we use the same values α and β of the sensitivity
for the entropy of Eq. (5). Fig. 1 shows (right frame) that this choice leads to
entropies that grow linearly: the corresponding slopes K (generalized Kolmogorov
entropies) are 0.267±0.004 (Tsallis), 0.186±0.004 (Abe) and 0.152±0.004 (Ka-
niadakis). This linear behavior disappears when α 6= αav as shown in Fig. 2 for
sens
Tsallis’, Abe’s and Kaniadakis’ entropies.
Inaddition,thewholeclassofentropiesandlogarithmsverifiesthePesinidentity
K =λ confirming what was already knownfor Tsallis’formulation1,8. The values
ofλ andK for the specific Tsallis’,Abe’s andKaniadakisformulationsare givenin
the caption to Fig. 1 as important explicit examples of this identity.
An intuitive explanation of the dependence of the value of K on β and details
on the calculations can be found in Ref. 19.
In summary, numerical evidence corroborates and extends Tsallis’ conjecture
that, analogously to strongly chaotic systems, also weak chaotic systems can be
described by an appropriate statistical formalism. In addition to sharing the same
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asymptotic power-lawbehaviorto correctlydescribe chaoticsystems,extended for-
malisms should verify precise theoretical requirements.
Theserequirementsdefine a largeclassofentropies;within this classwe use the
two-parameter formula (5), which includes Tsallis’s seminal proposal. Its simple
power-law form describes both small and large probability behaviors. Specifically,
the logistic map shows:
(a)apower-lowsensitivitytoinitialconditionswithaspecificexponentξ ∼t1/α,
whereα=0.650±0.005;thissensitivitycanbedescribedbydeformedexponentials
with the same asymptotic behavior ξ(t)=exp(λt) (see Fig. 1, left frame);
(b) a constant asymptotic entropy production rate (see Fig. 1, right frame) for
trace-form entropies that go as p1−α in theglimit of small probabilities p, where α
is the same exponent of the sensitivity;
(c) the asymptotic exponent α is related to parameters of known entropies:
3
α=1−q,whereq istheentropicindexofTsallis’thermodynamics ;α=1/q −1,
A
17
where q appears in the generalization (3) of Abe’s entropy ; α = κ, where κ is
A
18
the parameter in Kaniadakis’ statistics ;
(d) Pesin identity holds S /t → K = λ for each choice of entropy and cor-
β β β
responding exponential in the class, even if the value of K = λ depends on the
β β
19
specific entropy and it is not characteristic of the map as it is α ;
(e) this picture is not valid for every entropy: an important counterexample is
theRenyientropya,S(R)(t)= (1−q)−1log N pq(t) ,whichhasanon-linear
q i=1 i
behavior for any choice of the pDarameter q =h1−α (see FiiEg. 3).
P
2
>
S(t)
<
1
10 20 30
t
Figure3. Renyi’sentropyfor0.1≤α=1−q≤0.95(fromtoptobottom).
We gratefully thank S. Abe, F. Baldovin, G. Kaniadakis, G. Mezzorani,
P. Quarati, A. Robledo, A. M. Scarfone, U. Tirnakli, and C. Tsallis for sugges-
aAcomparisonofTsallis’andRenyi’sentropiesforthelogisticmapcanalsobefoundinRef.20.
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tions and comments.
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