Research ArticleRelative Entropies and Jensen Divergences in the Classical Limit

A M Kowalski12 and A Plastino1

1La Plata National University and Argentinarsquos National Research Council (IFLP-CCT-CONICET)-C C 7271900 La Plata Argentina2Comision de Investigaciones Cientıficas (CIC) Argentina

Correspondence should be addressed to A M Kowalski kowalskifisicaunlpeduar

Received 30 September 2014 Revised 21 December 2014 Accepted 11 January 2015

Academic Editor Jos De Brabanter

Copyright copy 2015 A M Kowalski and A Plastino This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Metrics and distances in probability spaces have shown to be useful tools for physical purposes Here we use this idea withemphasis on Jensen Divergences and relative entropies to investigate features of the road towards the classical limit A well-knownsemiclassical model is used and recourse is made to numerical techniques via the well-known Bandt and Pompe methodology toextract probability distributions from the pertinent time-series associated with dynamical data

1 Introduction

Many problems of quantum and statistical mechanics canbe formulated in terms of a distance between probabilitydistributions A frequently used quantity to compare twoprobability distributions which arose in information theoryis the Kullback-Leibler (KL) Divergence [1] Given twoprobability distributions119875 and119876 it provides an estimation ofhowmuch information119875 contains relative to119876 andmeasuresthe expected number of extra bits required to code samplesfrom 119875 when using a code based on 119876 rather than using acode based on 119875 [2] Some of them can also be regarded asentropic distances

It is well-known that systems that are characterizedby either long-range interactions long-term memories ormultifractality are best described by a formalism [3] calledTsallisrsquo 119902-statistics The basic quantity is the entropy [119902 isin R(119902 = 1)]

119878119902 =1

(119902 minus 1)

119899

sum

119894=1

119901119894 [1 minus 119901119894119902minus1

] (1)

with 119901119894 being probabilities associated with the 119899 differentsystem-configurations The entropic index (or deformation

parameter) 119902 describes the deviations of Tsallis entropy fromthe standard Boltzmann-Gibbs-Shannon (BGS) one One has

119878 = minus

119899

sum

119894=1

119901119894 ln119901119894 (2)

The BGS entropy works best for systems composed ofeither independent subsystems or interacting via short-rangeforces whose subsystems can access all the available phasespace [4ndash7] For systems exhibiting long-range correlationsmemory or fractal properties Tsallisrsquo entropy becomes themost convenient quantifier [4ndash17] Tsallis relative entropiesstudied in [18] are NOT distances in probability space Analternative tool is the Jensen ShannonDivergence introducedby Lamberti et al [2]

How good are these quantifiers to statistically describecomplex scenarios To answer we will apply the above-mentioned quantifiers to a well-known semiclassical systemin its path towards the classical limit [19 20] The pertinentdynamics displays regular zones chaotic ones and otherregions that although not chaotic possess complex featuresThe system has been investigated in detail from a purelydynamic viewpoint [20] and also from a statistical one [21ndash23] For this a prerequisite emerges how to extract informa-tion from a time-series (TS) [24] The data at our disposalalways possess a stochastic component due to noise [25 26]

Hindawi Publishing CorporationAdvances in StatisticsVolume 2015 Article ID 581259 8 pageshttpdxdoiorg1011552015581259

2 Advances in Statistics

so that different extraction-procedures attain distinct degreesof quality We will employ Bandt and Pompersquos approach[27] that determines the probability distribution associatedto time-series on the basis of the nature of the underlyingattractor (see [27] for the mathematical details)

In this paper we employ the normalized versions of Tsallisrelative entropy [10 18 28] to which we add the JensenDivergences associated to it We will compare via Jensen(i) the probability distribution functions (PDFs) associatedto the systemrsquos dynamic equationrsquos solutions in their routetowards the classical limit [20] with (ii) the PDF associatedto the classical solutions Our present Jensen way will be alsocompared to the 119902-Kullback-Leibler analyzed in [18]

The relative entropies and Jensen Divergences mentionedabove are discussed in Section 2 which briefly recapitu-lates notions concerning the Tsallis relative entropy andthe Kullback-Leibler relative one The Jensen Shannon andgeneralized Jensen Divergences are also discussed in thisSection A simple but illustrative example is considered inSection 21 In Section 22 we consider normalized formscorresponding to the information quantifiers mentioned inSection 21 As a test-scenario the semiclassical system and itsclassical limit are described in Section 3 and the concomitantresults presented in Section 4 Finally some conclusions aredrawn in Section 5

2 Kullback and Tsallis RelativeEntropies Jensen Shannon andGeneralized Jensen Divergences

The relative entropies (RE) quantify the difference betweentwo probability distributions 119875 and 119876 [29] The best rep-resentative is Kullback-Leiblerrsquos (KL) one based on theBGS canonical measure (2) For two normalized discreteprobability distribution functions (PDF)119875 = (1199011 119901119899) and119876 = (1199021 119902119899) (119899 gt 1) one has

119863KL (119875 119876) =119899

sum

119894=1

119901119894 ln(119901119894

119902119894

) (3)

with 119863KL(119875 119876) ge 0 119863KL(119875 119876) = 0 if and only if 119875 = 119876 Oneassumes that either 119902119894 = 0 for all values of 119894 or that if one119901119894 = 0 then 119902119894 = 0 as well [30] In such an instance peopletake 00 = 1 [30] (also 0 ln 0 = 0 of course)

KL can be seen as a particular case of the generalizedTsallis relative entropy [10 18 28]

119863119902 (119875 119876) =1

119902 minus 1

119899

sum

119894=1

119901119894 [(119901119894

119902119894

)

119902minus1

minus 1] (4)

when 119902 rarr 1 [10 18 28] 119863119902(119875 119876) ge 0 if 119902 ge 0 For 119902 gt 0

one has 119863119902(119875 119876) = 0 if and only if 119875 = 119876 For 119902 = 0 one has119863119902(119875 119876) = 0 for all 119875 and 119876

The two entropies (3) and (4) provide an estimation ofhow much information 119875 contains relative to 119876 [29] Theyalso can be regarded as entropic distances alternative meansfor comparing the distribution119876 to 119875 Our two entropies arenot symmetric in 119875 minus 119876

So as to deal with the nonsymmetric nature of the KLDivergence Lamberti et al [2] proposed using the followingquantity

119869119878 (119875 119876) =1

2[119863KL (119875

119875 + 119876

2) + 119863KL (119876

119875 + 119876

2)] (5)

as a measure of the distance between the probability distribu-tions119875 and119876 119869119878(119875 119876) = 0 if and only if119875 = 119876This quantityverifies 119869119878(119875 119876) = 119869119878(119876 119875)Moreover its square root satisfiesthe triangle inequality [2] In terms of the Shannon entropyexpression (5) can be rewritten in the form

119869119878 (119875 119876) = 119878 (119875 + 119876

2) minus

1

2119878 (119875) minus

1

2119878 (119876) (6)

We can obtain a generalization of Jensenrsquos Divergence byusing the relative entropy (4) instead of the KL Divergencethat is

119869119863119902 (119875 119876) =1

2[119863119902 (119875

119875 + 119876

2) + 119863119902 (119876

119875 + 119876

2)] (7)

21 An Illustrative Example A simple scenario will nowillustrate on the behavior of our quantifiers Let us evaluate119863119902(119875 119876) for the certainty versus the equiprobability case thatis (i) 119875 = (1199011 119901119899) for 119901119896 = 1 and 119901119894 = 0 if 119894 = 119896 and (ii)119876 = (1199021 119902119899) with 119902119894 = 1119899 forall119894 In this case (4) adopts theform

119863119902 (119875 119876) =119899119902minus1

minus 1

119902 minus 1 (8)

if 119902 gt 1 For 119902 = 1 that is Kullback-Leibler entropy (3)instance we obtain 119863KL(119875 119876) = ln 119899 For these same PDFsconsider now 119869119863119902(119875 119876) given by (7) We find

119869119863119902 (119875 119876)

=1

119902 minus 1(1 + 119899

119902) (1 + 119899)

(1minus119902)+ (119899 minus 1)

2(2minus119902)119899minus 1

(9)

if 119902 gt 1 In the Jensen Shannon Divergence-case (119902 = 1)given by (6) one has

119869119878 (119875 119876)

= minus1

2(

119899 + 1

119899) ln (119899 + 1) minus 2 ln (2119899) + ln (119899)

(10)

Let us discuss the behavior of these quantities for large 119899when 119899 rarr infin We ascertain that 119863119902(119875 119876) rarr infin (and119863KL(119875 119876) rarr infin) Instead the Jensen Divergences attain anasymptotic value

119869119863119902 (119875 119876) 997888rarr2119902minus1

minus 1

119902 minus 1 (11)

and 119869119878(119875 119876) rarr ln 2 Comparing (11) and (8) one notes that119869119863119902(119875 119876) for large 119899 behaves like119863119902(119875 119876) for 119899 = 2 (119869119878(119875 119876)behaves like 119863KL(119875 119876)) What are the consequences on the

Advances in Statistics 3

0

n

60 120 180 240 300 360 420 480 540 600 660 720

800E + 010

700E + 010

600E + 010

500E + 010

400E + 010

300E + 010

200E + 010

100E + 010

000E + 000

Gen

eral

ized

rela

tive e

ntro

py

(a)

45

40

35

30

25

20

15

10

05

00

Gen

eral

ized

Jens

en D

iver

genc

e

0

n

60 120 180 240 300 360 420 480 540 600 660 720

(b)

Figure 1 (a) 119863119902(119875 119876) and (b) 119869119863

119902(119875 119876) versus 119899 for the certainty versus the equiprobability cases (see Section 21) for 119902 = 5 For large 119899

when 119899 rarr infin we ascertain that 119863119902(119875 119876) rarr infin Instead the Jensen Divergences attain an asymptotic value The maximum 119899 was chosen

to coincide with the number of states 119899 of the semiclassical system described in Figures 4 5 and 6 This fact facilitates comparison

45

40

35

30

25

20

15

10

05

10 15 20 25 30 35 40 45 50

q

Gen

eral

ized

rela

tive e

ntro

py

(a)

14

12

10

08

06

04

02

00

10 15 20 25 30 35 40 45 50

q

Gen

eral

ized

Jens

en D

iver

genc

e

(b)

Figure 2 (a) 119863119902(119875 119876) and (b) 119869119863

119902(119875 119876) versus 119902 for 119899 = 2 for the same case described by Figure 1 The behavior of 119869119863

119902(119875 119876) resembles

that of119863119902(119875 119876)

119902-dependence Given reasonable 119902-values for small 119899 (119899 sim

2) the behavior of 119869119863119902(119875 119876) resembles that of 119863119902(119875 119876) (seeFigure 2) For 119899 ≫ 2 the 119902-dependence is quite different (seescales in Figure 3) The asymptotic behavior for both large 119899and large 119902 is such that 119863119902(119875 119876) sim 119899

119902119902 while 119869119863119902(119875 119876) sim

2119902119902 Thus scale differences may become astronomicIn real-life statistical problems 119899 ≫ 2 on the basis

of our simple but illustrative example we expect quitedifferent behaviors for 119863119902(119875 119876) and 119869119863119902(119875 119876) regarding the119902-dependence for large 119902 119902-variations aremuch smoother for119869119863119902(119875 119876) than for119863119902(119875 119876)

In this work we consider (see Sections 3 and 4) asystem representing the zerothmode contribution of a strongexternal field to the production of chargedmeson pairs morespecifically the roald lading to the classical limitThe ensuingdynamics is much richer and more complex However wewill see that the pertinent difference in the 119902-behaviors of

our quantifier is the one of our simple example The twoquantities (a) maximum 119899 of Figure 1 and (b) the 119899-valuein Figure 3 were both chosen so as to coincide with oursemiclassical systemrsquos number of states for this case whichfacilitates comparison

22 Normalized Quantities It is convenient to work witha normalized version for our information quantifiers forthe sake of a better comparison between different results[18] In this way the quantifierrsquos values are restricted tothe [0 1] interval via division by its maximum allowablevalue Accordingly the example of the preceding sectioncan be useful as a reference If we divide 119863KL(119875 119876) by ln 119899expression (3) becomes

119863119873

KL (119875 119876) =1

ln 119899

119899

sum

119894=1

119901119894 ln(119901119894

119902119894

) (12)

4 Advances in Statistics

800E + 010

700E + 010

600E + 010

500E + 010

400E + 010

300E + 010

200E + 010

100E + 010

000E + 000

minus100E + 01010 15 20 25 30 35 40 45 50

q

Gen

eral

ized

rela

tive e

ntro

py

(a)

45

40

35

30

25

20

15

10

05

10 15 20 25 30 35 40 45 50

q

Gen

eral

ized

Jens

en D

iver

genc

e

(b)

Figure 3 (a) 119863119902(119875 119876) and (b) 119869119863

119902(119875 119876) versus 119902 as in Figure 2 but for 119899 = 720 The 119899-value was chosen to coincide with the semiclassical

systemrsquos number of states We observe different behaviors for119863119902(119875 119876) and 119869119863

119902(119875 119876) regarding the 119902-dependence for large 119902 119902-variations are

much smoother for 119869119863119902(119875 119876) than for119863

119902(119875 119876)

with 0 le 119863119873

KL le 1 If one wishes to employ a normalized119863119902(119875 119876) version for comparison purposes one must con-sider two cases So as to normalize the expression of119863119902(119875 119876)given by (4) (a) if 119902 ge 1 we divide by 119863119902(119875 119876) given by(8) and (b) if 0 le 119902 lt 1 we divide by (119899minus119902 minus 1)(119902 minus 1)when computing 119863119902(119875 119876) for the equiprobability versus thecertainty example We obtain

119863119873

119902(119875 119876) =

1

119899119902minus1 minus 1

119899

sum

119894=1

119901119894 [(119901119894

119902119894

)

119902minus1

minus 1]

119902 ge 1

(13a)

119863119873

119902(119875 119876) =

1

119899minus119902 minus 1

119899

sum

119894=1

119901119894 [(119901119894

119902119894

)

119902minus1

minus 1]

0 le 119902 lt 1

(13b)

with 0 le 119863119873

119902(119875 119876) le 1 To normalize the Jensen Shannon

Divergence (5) we divide by 119869119878(119875 119876) (see (10)) so as toobtain a normalized 119869119878119873(119875 119876) For 119869119863119902(119875 119876) we use only thenormalization procedure corresponding to the case 119902 ge 1Wedivide by 119869119863119902(119875 119876) (see (9)) so as to obtain the normalized119869119863119873

119902(119875 119876) Of course 119863119873KL(119875 119876) = 0 119863119873

119902(119875 119876) = 0

119869119878119873(119875 119876) = 0 and 119869119863119873

119902(119875 119876) = 0 hold if and only if 119875 = 119876

We will work below with these normalized quantities

3 Classical-Quantum Transition A Review

This is a really important issueThe classical limit of quantummechanics (CLQM) can be viewed as one of the frontiers ofphysics [31ndash35] being the origin of exciting discussion (seeeg [31 32] and references therein) An interesting subissueis that of ldquoquantumrdquo chaotic motion Recent advances madeby distinct researchers are available in [36] and referencestherein Another related interesting subissue is that of thegeneralized uncertainty principle (GUP) ([37 38]) Zurek

[33ndash35] and others investigated the emergence of the classicalworld from quantum mechanics

We are interested here in a semiclassical perspectivefor which several directions can be found the historicalWKB Born-Oppenheimer approaches and so forthThe twointeracting systems considered by Bonilla and Guinea [39]Cooper et al [19] andKowalski et al [20 40] constitute com-positemodels in which one system is classical and the other isquantalThismakes sense when the quantum effects of one ofthe two systems are negligible in comparison to those of theother one [20] Examples encompass Bloch equations two-level systems interacting with an electromagnetic field withina cavity and collective nuclear motion We are concernedbelow with a bipartite system representing the zeroth modecontribution of a strong external field to the production ofcharged meson pairs [19 20] whose Hamiltonian reads

=1

2(1199012

119898119902

+1198751198602

119898cl+ 119898119902120596

21199092) (14)

119909 and 119901 above are quantum operators while 119860 and 119875119860 areclassical canonical conjugate variables The term 120596

2= 1205961199022+

11989021198602 is an interaction one introducing nonlinearity with

120596119902 a frequency 119898119902 and 119898cl are masses corresponding tothe quantum and classical systems respectively As shownin [40] when faced with (14) one has to deal with theautonomous system of nonlinear coupled equations

119889 ⟨1199092⟩

119889119905=

⟨⟩

119898119902

119889 ⟨1199012⟩

119889119905= minus119898119902120596

2⟨⟩

119889 ⟨⟩

119889119905= 2(

⟨1199012⟩

119898119902

minus 1198981199021205962⟨1199092⟩)

Advances in Statistics 5

119889119860

119889119905=

119875119860

119898cl

119889119875119860

119889119905= minus1198902119898119902119860⟨119909

2⟩

= 119909119901 + 119901119909

(15)

derived from Ehrenfestrsquos relations for quantum variables andfrom canonical Hamiltonrsquos equations for classical ones [40]To investigate the classical limit one needs also to considerthe classical counterpart of the Hamiltonian (14) in whichall variables are classical In such case Hamiltonrsquos equationslead to a classical version of (15) One can consult [40] forfurther details The classical equations are identical in formto (15) after replacing quantum mean values by classicalvariables The classical limit is obtained considering the limitof a ldquorelative energyrdquo [20]

119864119903 =|119864|

11986812120596119902

997888rarr infin (16)

where 119864 is the total energy of the system and 119868 is an invariantof the motion described by the system (15) 119868 relates to theuncertainty principle

119868 = ⟨1199092⟩ ⟨1199012⟩ minus

⟨⟩2

4geℎ2

4

(17)

To tackle this system one appeals to numerical solution Thepertinent analysis is affected by plotting quantities of interestagainst 119864119903 that ranges in [1infin] For 119864119903 = 1 the quantumsystem acquires all the energy 119864 = 119868

12120596119902 and the quantal

and classical variables get located at the fixed point (⟨1199092⟩ =

11986812119898119902120596119902 ⟨119901

2⟩ = 119868

12119898119902120596119902 ⟨⟩ = 0 119860 = 0 119875119860 = 0)

[40] Since 119860 = 0 the two systems become uncoupled For119864119903 sim 1 the system is of an almost quantal nature with a quasi-periodic dynamics [20]

As 119864119903 augments quantum features are rapidly lost and asemiclassical region is entered From a given value 119864119903

cl themorphology of the solutions to (15) begins to resemble thatof classical curves [20] One indeed achieves convergence of(15)rsquos solutions to the classical ones For very large 119864119903-valuesthe system thus becomes classical Both types of solutionscoincide We regard as semiclassical the region 1 lt 119864119903 lt 119864119903

clWithin such an interval we highlight the important value119864119903 = 119864119903

P at which chaos emerges [40] We associate to ourphysical problem a time-series given by the 119864119903-evolution ofappropriate expectation values of the dynamical variables anduse entropic relations so as to compare the PDF associated tothe purely classical dynamic solution with the semiclassicalones as these evolve towards the classical limit

4 Numerical Results

In our numerical study we used 119898119902 = 119898cl = 120596119902 = 119890 = 1For the initial conditions needed to tackle (15) we employed119864 = 06 Thus we fixed 119864 and then varied 119868 in order to

determine the distinct 119864119903rsquos We employed 55 different valuesfor 119868 Further we set ⟨119871⟩(0) = 119871(0) = 0 119860(0) = 0 (for thequantum and for the classical instances) while 1199092(0) ⟨1199092⟩(0)take values in the intervals (0 2119864) (119864minusradic1198642 minus 119868 119864+radic1198642 minus 119868)with 119868 le 119864

2 respectively Here 119864119903P

= 33282 and 119864119903cl=

2155264 At 119864119903119872 we encounter a maximum of the quantifier

called statistical complexity and 119864119903119872= 8 0904

Remember that our ldquosignalrdquo represents the systemrsquos stateat a given 119864119903 Sampling this signal we can extract severalPDFs using the Bandt and Pompe technique [27] for whichit is convenient to adopt the largest 119863-value that verifies thecondition 119863 ≪ 119872 [27] Such value is 119863 = 6 because wewill be concerned with vectors whose components compriseat least119872 = 5000 data-points for each orbit For verificationpurposes we also employed 119863 = 5 without detectingappreciable changes

In evaluating our relative entropies our 119875-distributionsare extracted from the time-series for the different 119864119903rsquos while119876 is associated to the classical PDF

Figure 4 represents the Kullback Divergence 119863119873KL(119875 119876)and the generalized relative entropy 119863

119873

119902(119875 119876) (ie the

pseudodistances (psd) between PDFs) for 119902 = 35 that werecomputed in [18] All curves show that (i) the maximal psdbetween the pertinent PDFs is encountered for 119864119903 = 1corresponding to the purely quantal situation and (ii) thatthey grow smaller as 119864119903 augments tending to vanish for119864119903 rarr infin as they should One sees that the results dependupon 119902 We specially mark in our plots special 119864119903-valueswhose great dynamical significance was pointed aboveTheseare 119864119903

P (chaos begins) 119864119903cl (start of the transitional zone)

and 119864119903119872 known to be endowed with maximum statistical

complexity [41] As a general trend when 119902 grows the size ofthe transition region diminishes and that of the classical zonegrows For the Kullback Divergence the size of the transitionregion looks overestimated if one considers the location of119864119903

cl (Figure 4(a)) As found in [18] we can assert that ourdescription is optimal for 15 lt 119902 lt 25 The descriptioninstead lose quality for 119902 ge 25 (Figure 4(b))

In Figures 5 and 6 we plot the Jensen Shannon Diver-gence 119869119878(119875 119876) and the generalized Jensen Divergences119869119863119902(119875 119876) for different values of 119902 All curves show that themaximal distance between the pertinent PDFs is encounteredfor119864119903 = 1 and that the distance grows smaller as119864119903 augmentssave for oscillations in the transition zone We note thatthe three regions are well described (which is not the casefor 119863119873KL(119875 119876)) for any reasonable 119902-value Our probability-distance tends to vanish for 119864119903 rarr infin but in slower fashionthan for119863119873KL(119875 119876) and119863

119873

119902(119875 119876)

The 119902-dependence is quite different in the case of sym-metric versus nonsymmetric relative entropies as can be alsoseen in the simple example of Section 21 notwithstandingthe fact the this difference is significantly reduced for small119902 As 119902 grows 119869119863119873

119902(119875 119876) changes in a much smoother

fashion than 119863119873

119902(119875 119876) which casts doubts concerning the

appropriateness of employing119863119873119902(119875 119876)

6 Advances in Statistics

06

04

02

00

0 10 20 30 40 50 60 70 80 90 100

Kullb

ack

rela

tive e

ntro

py

ErM

ErEr

clEr119979

(a)

10

8

6

4

2

0

times10minus5

0 10 20 30 40 50 60 70 80 90 100Er

M

ErEr

cl

Gen

eral

ized

rela

tive e

ntro

py

Er119979

(b)

Figure 4 (a) The normalized Kullback Divergence 119863119873KL(119875 119876) is plotted versus 119864119903 (b) Normalized generalized relative entropy 119863119873

119902(119875 119876)

versus 119864119903for 119902 = 35 As noted in [18] (i) the maximal distance (pseudodistance) between the pertinent PDFs is encountered for 119864

119903= 1

corresponding to the purely quantal situation and (ii) that distance (pseudodistance) grows smaller as 119864119903augments tending to vanish for

119864119903rarr infin For the Kullback Divergence description the size of the transition region looks overestimated if one considers the location of 119864

119903

clIn (b) the transition region almost disappears

05

04

03

02

01

000 10 20 30 40 50 60 70 80 90 100

ErM

ErEr

cl

Jens

en S

hann

on D

iver

genc

e

Er119979

(a)

05

04

03

02

01

000 10 20 30 40 50 60 70 80 90 100

ErM

ErEr

cl

Gen

eral

ized

Jens

en D

iver

genc

e

Er119979

(b)

Figure 5 We plot (a) the Jensen Shannon Divergence versus 119864119903and (b) the generalized Jensen Divergence versus 119864

119903for 119902 = 15 We observe

that the behavior is similar for both curves As in previous figures here the maximal distance is encountered for 119864119903= 1 and distance grows

smaller as 119864119903augments The description of the three dynamical regions is optimal

Themaximum statistical complexity value [41] is attainedat 119864119903119872 In Figures 4 5 and 6 we detect shape-changes before

and after 119864119903119872 Figures 5 and 6 also display there a local

minimum for the Jensen Divergence

5 Conclusions

Thephysics involved is that of a special bipartite semiclassicalsystem that represents the zeroth mode contribution of astrong external field to the production of charged meson

pairs [19 20] The system is endowed with three-dynamicalregions as characterized by the values adopted by theparameter 119864119903 We have a purely quantal zone (119864119903 ≃ 1)a semiclassical one and finally a classical sector The twolatter ones are separated by the value 119864119903

cl (see Section 3) Inevaluating (i) the normalized relative entropies ((12) (13a)and (13b)) [18] and (ii) normalized versions of the JensenDivergences given by (6) and (7) the 119875-PDFs are extractedfrom time-series associated to different 119864119903-values while 119876 isalways the PDF that describes the classical scenario We are

Advances in Statistics 7

05

04

03

02

01

000 10 20 30 40 50 60 70 80 90 100

ErM

ErEr

cl

Gen

eral

ized

Jens

en D

iver

genc

e

Er119979

(a)

05

04

03

02

01

000 10 20 30 40 50 60 70 80 90 100

ErM

Er

Gen

eral

ized

Jens

en D

iver

genc

e

Ercl

Er119979

(b)

Figure 6 We plot (a) the Jensen Divergence and (b) the generalized Jensen Divergence both versus 119864119903 for (i) 119902 = 3 and (ii) 119902 = 5 We

observe that the behavior of both curves resembles that of Figure 5 The description of the three dynamical regions is optimal

thus speaking of ldquodistancesrdquo between (a) the PDFs extractedin the path towards the classical limit and (b) the classicalPDF

For the normalized Jensen Divergences all curves showthat the maximal distance between the pertinent PDFs isencountered for 119864119903 = 1 and that the distance grows smalleras 119864119903 augments save for oscillation in the transition zone

Our three regions are well described for any reasonable119902 (which is not the case with 119863

119873

119902(119875 119876)) The vanishing of

the distance referred to above for 119864119903 rarr infin acquires nowa much slower pace than that of 119863119873

119902(119875 119876) (and 119863

119873

KL(119875 119876))These results are to be expected given the behavior ofthe concomitant statistical quantifiers in the example ofSection 21

We conclude that the Jensen Divergences 119869119863119873

119902(119875 119876)

(and 119869119878119873(119875 119876)) improve on the description provided by

the corresponding nonsymmetric relative entropies119863119873119902(119875 119876)

(and119863KL(119875 119876))

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

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[2] P W Lamberti M T Martin A Plastino and O A RossoldquoIntensive entropic non-triviality measurerdquo Physica AmdashStatistical Mechanics and its Applications vol 334 no 1-2 pp119ndash131 2004

[3] C Tsallis ldquoPossible generalization of Boltzmann-Gibbs statis-ticsrdquo Journal of Statistical Physics vol 52 no 1-2 pp 479ndash4871988

[4] R Hanel and S Thurner ldquoGeneralized Boltzmann factors andthe maximum entropy principle entropies for complex sys-temsrdquo Physica AmdashStatistical Mechanics and its Applications vol380 no 1-2 pp 109ndash114 2007

[5] G Kaniadakis ldquoStatistical mechanics in the context of specialrelativityrdquo Physical Review E Statistical Nonlinear and SoftMatter Physics vol 66 no 5 Article ID 056125 17 pages 2002

[6] M P Almeida ldquoGeneralized entropies from first principlesrdquoPhysica A Statistical Mechanics and its Applications vol 300no 3-4 pp 424ndash432 2001

[7] J Naudts ldquoDeformed exponentials and logarithms in general-ized thermostatisticsrdquo Physica A Statistical Mechanics and itsApplications vol 316 no 1ndash4 pp 323ndash334 2002

[8] P A Alemany and D H Zanette ldquoFractal randomwalks from avariational formalism for Tsallis entropiesrdquo Physical Review Evol 49 no 2 pp R956ndashR958 1994

[9] C Tsallis ldquoNonextensive thermostatistics and fractalsrdquo Fractalsvol 3 p 541 1995

[10] C Tsallis ldquoGeneralized entropy-based criterion for consistenttestingrdquo Physical Review E vol 58 no 2 pp 1442ndash1445 1998

[11] S Tong A Bezerianos J Paul Y Zhu and N Thakor ldquoNonex-tensive entropy measure of EEG following brain injury fromcardiac arrestrdquo Physica A Statistical Mechanics and Its Applica-tions vol 305 no 3-4 pp 619ndash628 2002

[12] C Tsallis C Anteneodo L Borland and R Osorio ldquoNonexten-sive statisticalmechanics and economicsrdquo Physica AmdashStatisticalMechanics and its Applications vol 324 no 1-2 pp 89ndash1002003

[13] O A Rosso M T Martın and A Plastino ldquoBrain electricalactivity analysis using wavelet-based informational tools (II)Tsallis non-extensivity and complexity measuresrdquo Physica Avol 320 pp 497ndash511 2003

[14] L Borland ldquoLong-range memory and nonextensivity in finan-cialmarketsrdquoEurophysics News vol 36 no 6 pp 228ndash231 2005

[15] H Huang H Xie and Z Wang ldquoThe analysis of VF and VTwith wavelet-based Tsallis informationmeasurerdquo Physics LettersA vol 336 no 2-3 pp 180ndash187 2005

8 Advances in Statistics

[16] D G Perez L ZuninoM TMartınMGaravaglia A Plastinoand O A Rosso ldquoModel-free stochastic processes studied withq-wavelet-based informational toolsrdquoPhysics Letters A vol 364pp 259ndash266 2007

[17] M Kalimeri C Papadimitriou G Balasis and K EftaxiasldquoDynamical complexity detection in pre-seismic emissionsusing nonadditive Tsallis entropyrdquo Physica AmdashStatisticalMechanics and its Applications vol 387 no 5-6 pp 1161ndash11722008

[18] AMKowalsiM TMartin andA PlastinoPhysicaA In press[19] F Cooper J Dawson S Habib and R D Ryne ldquoChaos in time-

dependent variational approximations to quantum dynamicsrdquoPhysical Review EmdashStatistical Physics Plasmas Fluids andRelated Interdisciplinary Topics vol 57 no 2 pp 1489ndash14981998

[20] A M Kowalski A Plastino and A N Proto ldquoClassical limitsrdquoPhysics Letters A vol 297 no 3-4 pp 162ndash172 2002

[21] A M Kowalski M T Martın A Plastino and O A RossoldquoBandt-Pompe approach to the classical-quantum transitionrdquoPhysica D Nonlinear Phenomena vol 233 no 1 pp 21ndash31 2007

[22] A M Kowalski M T Martin A Plastino and L ZuninoldquoTsallisrsquo deformation parameter 119902 quantifies the classical-quantum transitionrdquo Physica AmdashStatistical Mechanics and itsApplications vol 388 no 10 pp 1985ndash1994 2009

[23] A M Kowalski and A Plastino ldquoBandt-Pompe-Tsallis quan-tifier and quantum-classical transitionrdquo Physica A StatisticalMechanics and Its Applications vol 388 no 19 pp 4061ndash40672009

[24] A Kowalski M T Martin A Plastino and G Judge ldquoOnextracting probability distribution information from timeseriesrdquo Entropy vol 14 no 10 pp 1829ndash1841 2012

[25] H Wold A Study in the Analysis of Stationary Time SeriesAlmqvist amp Wiksell Upsala Canada 1938

[26] J Kurths and H Herzel ldquoProbability theory and related fieldsrdquoPhysica D vol 25 no 1ndash3 pp 165ndash172 1987

[27] C Bandt and B Pompe ldquoPermutation entropy a natural com-plexity measure for time seriesrdquo Physical Review Letters vol 88Article ID 174102 2002

[28] L Borland A R Plastino and C Tsallis ldquoInformation gainwithin nonextensive thermostatisticsrdquo Journal of MathematicalPhysics vol 39 no 12 pp 6490ndash6501 1998 Erratum in Journalof Mathematical Physics vol 40 p 2196 1999

[29] A M Kowalski R D Rossignoli and E M F CuradoConceptsand Recent Advances in Generalized Information Measures andStatistics Bentham Science Publishers 2013

[30] httpenwikipediaorgwikiKullback-Leibler divergence[31] J J Halliwell and J M Yearsley ldquoArrival times complex poten-

tials and decoherent historiesrdquo Physical Review A vol 79 no 6Article ID 062101 17 pages 2009

[32] M J Everitt W J Munro and T P Spiller ldquoQuantum-classicalcrossover of a field moderdquo Physical Review A vol 79 no 3Article ID 032328 2009

[33] H D Zeh ldquoWhy Bohmrsquos quantum theoryrdquo Foundations ofPhysics Letters vol 12 no 2 pp 197ndash200 1999

[34] W H Zurek ldquoPointer basis of quantum apparatus into whatmixture does the wave packet collapserdquo Physical Review DmdashParticles and Fields vol 24 no 6 pp 1516ndash1525 1981

[35] W H Zurek ldquoDecoherence einselection and the quantumorigins of the classicalrdquoReviews ofModern Physics vol 75 no 3pp 715ndash775 2003

[36] A M Kowalski M T Martin A Plastino and A N ProtoldquoClassical limit and chaotic regime in a semi-quantumHamilto-nianrdquo International Journal of Bifurcation and Chaos in AppliedSciences and Engineering vol 13 no 8 pp 2315ndash2325 2003

[37] A Tawfik ldquoImpacts of generalized uncertainty principle onblack hole thermodynamics and Salecker-Wigner inequalitiesrdquoJournal of Cosmology and Astroparticle Physics vol 2013 article040 2013

[38] E El Dahab and A Tawfik ldquoMeasurable maximal energy andminimal time intervalrdquo Canadian Journal of Physics vol 92 no10 pp 1124ndash1129 2014

[39] L L Bonilla and F Guinea ldquoCollapse of the wave packet andchaos in a model with classical and quantum degrees offreedomrdquo Physical Review A vol 45 no 11 pp 7718ndash7728 1992

[40] A M Kowalski M T Martın J Nunez A Plastino and A NProto ldquoQuantitative indicator for semiquantum chaosrdquoPhysicalReview A vol 58 no 3 pp 2596ndash2599 1998

[41] A M Kowalski and A Plastino ldquoThe Tsallis-complexity of asemiclassical time-evolutionrdquo Physica A Statistical Mechanicsand Its Applications vol 391 no 22 pp 5375ndash5383 2012

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