Statistical Complexity and Fisher-Shannon Information. Applications

3.3 MB·English

by  Ricardo Lopez-Ruiz

Ricardo Lo´pez-Ruiz, Jaime San˜udo,
Elvira Romera, Xavier Calbet
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Statistical Complexity and
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Fisher-Shannon Information.

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Applications

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– Book Chapter –
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January 13,2012
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Contents
1 StatisticalComplexityandFisher-ShannonInformation.Applications 5
1.1 AStatisticalMeasureofComplexity.SomeApplications ......... 6
1.1.1 Complexityinaphasetransition:coupledmaplattices ..... 11
1.1.2 Complexityversustime:thetetrahedralgas .............. 16
1.2 TheStatisticalComplexityintheContinuousCase............... 29
1.2.1 Entropy/informationanddisequilibrium ................. 30
1.2.2 ThecontinuousversionCˆ oftheLMCcomplexity......... 32
1.2.3 PropertiesofCˆ ...................................... 33
1.3 Fisher-ShannonInformationProduct.SomeApplications ......... 38
1.3.1 Fisher-Shannoninformation:definitionandproperties ..... 38
1.3.2 Fisher-Shannonproductasanelectroniccorrelationmeasure 43
1.3.3 Fisherinformationforasingleparticleinacentralpotential 46
1.4 ApplicationstoQuantumSystems ............................ 47
1.4.1 Formulasinpositionandmomentumspaces.............. 47
1.4.2 TheH-atom......................................... 48
1.4.3 Thequantumharmonicoscillator....................... 52
1.4.4 Thesquarewell...................................... 56
1.4.5 Theperiodictable.................................... 58
1.4.6 Magicnumbersinnuclei.............................. 63
References..................................................... 65
3

Chapter 1
Statistical Complexity and Fisher-Shannon
Information. Applications
AbstractInthischapter,astatisticalmeasureofcomplexityandtheFisher-Shannon
informationproductareintroducedandtheirpropertiesarediscussed.Thesemea-
suresarebasedontheinterplaybetweentheShannoninformation,orafunctionof
it,andtheseparationofthesetofaccessiblestatestoasystemfromtheequiprob-
ability distribution, i.e. the disequilibrium or the Fisher information, respectively.
Differentapplicationsindiscreteandcontinuoussystemsareshown.Someofthem
areconcernedwithquantumsystems,fromprototypicalsystemssuchastheH-atom,
the harmonic oscillator and the square well to other ones such as He-like ions,
Hooke’satomsorjusttheperiodictable.Inallofthem,thesestatisticalindicators
show an interesting behavior able to discern and highlight some conformational
propertiesofthosesystems.
5
6 1 StatisticalComplexityandFisher-ShannonInformation.Applications
1.1 A StatisticalMeasure ofComplexity. SomeApplications
Thiscenturyhasbeentoldtobethecenturyofcomplexity[1].Nowadaystheques-
tion “whatis complexity?”is circulatingoverthe scientific crossroadsofphysics,
biology,mathematicsandcomputerscience,althoughunderthepresentunderstand-
ingoftheworldcouldbenourgenttoanswerthisquestion.However,manydifferent
pointsofviewhavebeendevelopedtothisrespectandhencea lotofdifferentan-
swerscanbefoundintheliterature.Hereweexplainindetailoneoftheseoptions.
Onthemostbasicgrounds,anobject,aprocedure,orsystemissaidtobe“com-
plex” when it doesnotmatch patternsregardedas simple. This soundsrather like
anoxymoronbutcommonknowledgetellsuswhatissimpleandcomplex:simpli-
fiedsystemsoridealizationsarealwaysastartingpointtosolvescientificproblems.
Thenotionof“complexity”inphysics[2,3]startsbyconsideringtheperfectcrystal
and the isolated ideal gas as examplesof simple models and therefore as systems
withzero“complexity”.Letusbrieflyrecalltheirmaincharacteristicswith“order”,
“information”and“equilibrium”.
A perfect crystal is completely ordered and the atoms are arranged following
stringentrulesofsymmetry.Theprobabilitydistributionforthestatesaccessibleto
theperfectcrystaliscenteredaroundaprevailingstateofperfectsymmetry.Asmall
pieceof“information”isenoughtodescribetheperfectcrystal:thedistancesandthe
symmetriesthatdefinetheelementarycell.The“information”storedinthissystem
canbeconsideredminimal.Ontheotherhand,theisolatedidealgasiscompletely
disordered.The system can be found in any of its accessible states with the same
probability.Allofthemcontributeinequalmeasuretothe“information”storedin
theidealgas.Ithasthereforeamaximum“information”.Thesetwosimplesystems
areextremainthescaleof“order”and“information”.Itfollowsthatthedefinition
of“complexity”mustnotbemadeintermsofjust“order”or“information”.
It might seem reasonable to propose a measure of “complexity” by adopting
somekindofdistancefromtheequiprobabledistributionoftheaccessiblestatesof
the system. Defined in this way, “disequilibrium”would givean idea of the prob-
abilistichierarchyofthesystem.“Disequilibrium”wouldbedifferentfromzeroif
thereareprivileged,ormoreprobable,statesamongthoseaccessible.Butthiswould
notwork.Goingbackto thetwo exampleswe beganwith,itis readilyseen thata
perfectcrystal is far from an equidistributionamongthe accessible states because
oneofthemistotallyprevailing,andso“disequilibrium”wouldbemaximum.For
theidealgas,“disequilibrium”wouldbezerobyconstruction.Thereforesuchadis-
tanceor“disequilibrium”(ameasureofaprobabilistichierarchy)cannotbedirectly
associatedwith“complexity”.
InFigure1.1wesketchanintuitivequalitativebehaviorfor“information”H and
“disequilibrium”Dforsystemsrangingfromtheperfectcrystaltotheidealgas.This
graphsuggeststhattheproductofthesetwoquantitiescouldbeusedasameasure
of“complexity”:C=H D.ThefunctionChasindeedthefeaturesandasyntotical
·
propertiesthatonewould expectintuitively:itvanishesfortheperfectcrystaland
for the isolated ideal gas, and it is different from zero for the rest of the systems
1.1 AStatisticalMeasureofComplexity.SomeApplications 7
ofparticles. We will followthese guidelinestoestablish a quantitativemeasureof
“complexity”.
Beforeattemptinganyfurtherprogress,however,wemustrecallthat“complex-
ity”cannotbemeasuredunivocally,becauseitdependsonthenatureofthedescrip-
tion(whichalwaysinvolvesareductionistprocess)andonthescaleofobservation.
Letustakeanexampletoillustratethispoint.Acomputerchipcanlookverydiffer-
entatdifferentscales.Itisanentangledarrayofelectronicelementsatmicroscopic
scalebutonlyanorderedsetofpinsattachedtoablackboxatamacroscopicscale.

INFORMATION = H

C = H*D = COMPLEXITY
DISEQUILIBRIUM = D

CRYSTAL IDEAL GAS
Fig.1.1 Sketchoftheintuitivenotionofthemagnitudesof“information”(H)and“disequilibrium”
(D)forthephysicalsystemsandthebehaviorintuitivelyrequiredforthemagnitude“complexity”.
ThequantityC=H Disproposedtomeasuresuchamagnitude.
·
Weshallnowdiscussameasureof“complexity”basedonthestatisticaldescrip-
tionofsystems.LetusassumethatthesystemhasNaccessiblestates x ,x ,...,x
1 2 N
{ }
whenobservedatagivenscale.WewillcallthisanN-system.Ourunderstandingof
thebehaviorofthissystemdeterminesthecorrespondingprobabilities p ,p ,...,p
1 2 N
(withthecondition(cid:229) N p =1)ofeachstate(p >0foralli).Thenth{eknowledge}
i=1 i i
of theunderlyingphysicallaws atthisscale is incorporatedintoa probabilitydis-
tribution for the accessible states. It is possible to find a quantity measuring the
8 1 StatisticalComplexityandFisher-ShannonInformation.Applications
amountof“information”.Undertothemostelementaryconditionsofconsistency,
Shannon[4]determinedtheuniquefunctionH(p ,p ,...,p )thataccountsforthe
1 2 N
“information”storedinasystem:
N
(cid:229)
H= K p logp , (1.1)
i i
−
i=1
whereK isapositiveconstant.ThequantityH iscalledinformation.Theredefini-
tionof informationH as sometypeof monotonefunctionofthe Shannonentropy
can be also useful in many contexts as we shall show in the next sections. In the
case of a crystal, a state x would be the most probable p 1, and all others x
c c i
∼
wouldbeveryimprobable, p 0i=c.ThenH 0.Ontheotherside,equiprob-
i c
∼ 6 ∼
abilitycharacterizesanisolatedidealgas, p 1/N soH KlogN,i.e.,themax-
i g
∼ ∼
imum of information for a N-system. (Notice that if one assumes equiprobability
andK=k Boltzmannconstant,H isidentifiedwiththethermodinamicentropy,
S=k logN)≡.AnyotherN-systemwillhaveanamountofinformationbetweenthose
twoextrema.
LetusproposeadefinitionofdisequilibriumDinaN-system[5].Theintuitive
notionsuggeststhatsomekindofdistancefromanequiprobabledistributionshould
beadopted.TworequirementsareimposedonthemagnitudeofD:D>0inorderto
haveapositivemeasureof“complexity”andD=0onthelimitofequiprobability.
The straightforward solution is to add the quadratic distances of each state to the
equiprobabilityasfollows:
(cid:229)N 1 2
D= p . (1.2)
i
−N
i=1(cid:18) (cid:19)
According to this definition, a crystal has maximum disequilibrium(for the dom-
inant state, p 1, and D 1 for N ¥ ) while the disequilibriumfor an ideal
c c
∼ → →
gas vanishes (D 0) by construction. For any other system D will have a value
g
∼
betweenthesetwoextrema.
We now introduce the definition of complexityC of a N-system [6, 7]. This is
simply theinterplaybetweenthe informationstoredin the system andits disequi-
librium:
(cid:229)N (cid:229)N 1 2
C=H D= K p logp p . (1.3)
i i i
· − i=1 !· i=1(cid:18) −N(cid:19) !
Thisdefinitionfitstheintuitivearguments.Foracrystal,disequilibriumislargebut
theinformationstoredisvanishinglysmall,soC 0.Ontheotherhand,H islarge
∼
for an ideal gas, but D is small, soC 0 as well. Any other system will have an
∼
intermediatebehaviorandthereforeC>0.
Aswasintuitivelysuggested,thedefinitionofcomplexity(1.3)alsodependson
thescale.Ateachscaleofobservationanewsetofaccessiblestatesappearswithits
correspondingprobabilitydistributionsothatcomplexitychanges.Physicallawsat
eachlevelofobservationallowustoinfertheprobabilitydistributionofthenewset
1.1 AStatisticalMeasureofComplexity.SomeApplications 9
ofaccessiblestates,andthereforedifferentvaluesforH,DandCwillbeobtained.
Thestraightforwardpassagetothecaseofacontinuumnumberofstates,x,canbe
easilyinferred.Thuswemusttreatwithprobabilitydistributionswithacontinuum
support,p(x),andnormalizationcondition +¥¥ p(x)dx=1.Disequilibriumhasthe
limitD= +¥¥ p2(x)dxandthecomplexitycR−ouldbedefinedby:
−
R +¥ +¥
C=H D= K p(x)logp(x)dx p2(x)dx . (1.4)
· − ¥ · ¥
(cid:18) Z− (cid:19) (cid:18)Z− (cid:19)
Asweshallsee,otherpossibilitiesforthecontinuousextensionofCarealsopossi-
ble.
Fig.1.2 Ingeneral,dependenceofcomplexity(C)onnormalizedinformation(H)isnotunivocal:
manydistributions p canpresentthesamevalueofHbutdifferentC.Thisisshowninthecase
i
{ }
N=3.
DirectsimulationsofthedefinitiongivethevaluesofC forgeneralN-systems.
Thesetofallthepossibledistributions p ,p ,...,p whereanN-systemcouldbe
1 2 N
{ }
foundis sampled. Forthe sake of simplicity H is normalizedto the interval[0,1].
ThusH =(cid:229) N p logp/logN.Foreachdistribution p thenormalizedinforma-
i=1 i i { i}
tionH( p ),andthedisequilibriumD( p )(eq.1.2)arecalculated.Ineachcase
i i
{ } { }
thenormalizedcomplexityC=H Disobtainedandthepair(H,C)stored.These
·
twomagnitudesareplottedonadiagram(H,C(H))inordertoverifythequalitative
behaviorpredictedin Figure1.1. For N =2 ananalyticalexpressionforthe curve
10 1 StatisticalComplexityandFisher-ShannonInformation.Applications
C(H)isobtained.Iftheprobabilityofonestateis p =x,thatofthesecondoneis
1
simply p =1 x.Thecomplexityofthesystemwillbe:
2
−
1 x 1 2
C(x)=H(x) D(x)= [xlog +log(1 x)] 2 x . (1.5)
· −log2 1 x − · −2
(cid:18) − (cid:19) (cid:18) (cid:19)
Complexity vanishes for the two simplest 2-systems: the crystal (H =0; p =1,
1
p =0)andtheidealgas(H=1; p =1/2,p =1/2).Letusnoticethatthiscurve
2 1 2
isthesimplestonethatfulfillsalltheconditionsdiscussedintheintroduction.The
largest complexity is reached for H 1/2 and its value is:C(x 0.11) 0.151.
∼ ∼ ∼
ForN>2therelationshipbetweenH andCisnotunivocalanymore.Manydiffer-
ent distributions p store the same informationH but have differentcomplexity
i
{ }
C. Figure 1.2 displays such a behavior for N =3. If we take the maximum com-
plexityC (H)associatedwitheachH acurvesimilartotheonefora2-systemis
max
recovered.Every3-systemwillhaveacomplexitybelowthislineanduppertheline
ofC (H)andalso uppertheminimumenvelopecomplexityC . Theselines
min minenv
will be analytically found in a next section. In Figure 1.3 curvesC (H) for the
max
casesN=3,...,10arealsoshown.Letusobservetheshiftofthecomplexity-curve
peak to smaller values of entropy for rising N. This fact agrees with the intuition
tellingusthatthebiggestcomplexity(numberofpossibilitiesof‘complexification’)
bereachedforlesserentropiesforthesystemswithbiggernumberofstates.
Fig.1.3 Complexity(C=H D)asafunctionofthenormalizedinformation(H)forasystemwith
·
twoaccessiblestates(N=2).Alsocurvesofmaximumcomplexity(Cmax)areshownforthecases:
N=3,...,10.