Physics and Complexity: a brief spin glass perspective

0.08 MB·English

by  David Sherrington

Chapter 1
Physics and Complexity: a brief spin glass
perspective
2 DavidSherrington
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- Abstract Complex macroscopic behaviour can arise in many-body systems with
s
i onlyverysimpleelementsasaconsequenceofthecombinationofcompetitionand
d
inhomogeneity.This paperattempts to illustrate howstatistical physicshas driven
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at this recognition,hascontributednew insightsandmethodologiesofwide applica-
m tioninfluencingmanyfieldsofscience,andhasbeenstimulatedinreturn.
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o 1.1 Introduction
c
[

Manybodysystemsofevenverysimplemicroscopicconstituentswithverysimple
1
interactionrulescanshownovelemergenceintheirmacroscopicbehaviour.When
v
2 theinteractions(andanyconstraints)arealsomutuallyincompatible(frustrated)and
5 thereismacroscopicallyrelevantquencheddisorder,thentheemergentmacroscopic
8 behaviourcan be complex(in waysto bediscussed) andnotsimply anticipatable.
1
Recentyearshaveseenmajoradvancesinunderstandingsuchbehaviour,inrecog-
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1 nizingconceptualubiquitiesacrossmanyapparentlydifferentsystemsandinforg-
0 ing,transferringandapplyingnewmethodologies.Statistical physicshasplayeda
2
majorpartin drivinganddevelopingthe subjectandin providingnew methodsto
1
studyandquantifyit.Thispaperisintendedtoprovideabriefbroadbrushintroduc-
:
v
tion.
i
X A key part of these developments has been the combination of minimalist
r modelling, developmentof new concepts and techniques, and fruitful transfers of
a the knowledge between different systems. Here we shall concentrate on a simple
paradigmicmodel,demonstrateitsubiquitousnessamongseveraloftenverydiffer-
DavidSherrington
RudolfPeierlsCentreforTheoreticalPhysics,1KebleRd.,OxfordOX13NP,UK,andSantaFeIn-
stitute,1399HydeParkRd.,SantaFe,NM87501,USA.e-mail:[email protected]
1
2 DavidSherrington
entsystems,problemsandcontexts,andintroducesomeoftheusefulconceptsthat
havearisen.
1.2 The Dean’s Problem andSpin Glasses
The genesis for the explosion of interest and activity in complexity within the
physics community was in an attempt to understand a group of magnetic alloys
knownasspinglasses1[1].Buthereweshallstartwith aproblemthatrequiresno
physicstoappreciate,theDean’sproblem[2].
A College Dean is faced with the task of distributing N students between two
dormitoriesasamicablyaspossiblebutgiventhatsomepairsofstudentspreferto
bein thesame dormwhileotherpairswantto beseparated.Ifanyoddnumberof
studentshaveanoddnumberofantagonisticpairwisepreferencesthentheirprefer-
encescannotallbesatisfiedsimultaneously.Thisisanexampleoffrustration.The
Dean’s Problem can can be modelled as a mathematical optimization problem by
definingacostfunctionHthatistobeminimiized:
H=−(cid:229) J s s ;s =±1 (1.1)
ij i j
(ij)
wherethei,jlabelstudents,s =±1indicatesdormA/Bandthe{J =J }denote
ij ji
thesignandmagnitudeoftheinter-studentpairpreferences(+= prefer).We shall
further concentrate on the situation where the {J } are chosen randomly and in-
ij
dependentlyfromasingle(intensive)distributionP(J)ofzeromean2,therandom
Dean’sProblem.Thenumberofcombinationsofpossiblechoicesgrowsexponen-
tially in N (as2N). Thereis also, in general,no simple localiterative modeofso-
lution.Hence,ingeneral,whenN becomeslargetheDean’sproblembecomesvery
hard,inthelanguageofcomputerscience,NP-complete[3].
Infact,thecostfunctionoftherandomDean’sProblemwasalreadyintroduced
in 1975 as a potentially soluble model for a spin glass; there it is known as the
Sherrington-Kirkpatrick (SK) model [4]. In this model H is the Hamiltonian (or
energyfunction),thei,j labelspins,thes theirorientation(up/down)andthe{J}
aretheexchangeinteractionsbetweenpairsofspins.
In the latter case one was naturally interested in the effectsof temperatureand
ofphasetransitionsasitisvaried.InthestandardprocedureofGibbsianstatistical
mechanics,inthermalequilibriumtheprobabilityofamicrostate{s }isgivenby
P({s })=Z−1exp[−H ({s })/T] (1.2)
{Jij}
whereZisthepartitionfunction
1Spinglasseswereoriginallyobservedasmagneticalloyswithunusualnon-periodicspinorder-
ing.Theywerealsolaterrecognizedashavingmanyotherfascinatingglassyproperties
2Thisrestrictionisnotessentialbutrepresentsthepotentiallyhardestcase.
1 PhysicsandComplexity:abriefspinglassperspective 3
Z= (cid:229) exp[−H ({s })/T]; (1.3)
{Jij}
{s }
thesubscript{J }hasbeenaddedtoH tomakeexplicitthatitisfortheparticular
ij
instanceofthe(random)choiceof{J }.Againinthespiritofstatisticalphysicsone
ij
mayusefullyconsidertypicalphysicalpropertiesoverrealizationsofthequenched
disorder,obtainablebyaveragingthemoverthosechoices3.
SolvingtheSKmodelhasbeenagreatchallengeandhasledtonewandsubtle
mathematical techniques and theoretical conceptualizations, backed by new com-
puter simulational methodologies and experimentation, the detailed discussion of
whichisbeyondthescopeofthisbriefreport.Howeverabriefsketchwillbegiven
ofsomeoftheconceptualdeductions.
Letusstartpictorially.Acartoonofthesituationisthatofahierarchicallyrugged
landscapetodescribetheenergy/costasafunctionofpositioninthespaceofmicro-
scopiccoordinatesandsuchthatforanylocalperturbationsofthemicroscopicstate
thatallowonlydownhillmovesthesystemrapidlygetsstuckanditisimpossibleto
iterate to the true minimumor evena state close to it. Addingtemperatureallows
alsouphillmoveswithaprobabilityrelatedtoexp[−d H/T]whered Histheenergy
change. But still for T <T the system has this glassy hindranceto equilibration,
g
a non-ergodicitythat shows up, for example, in differences in response functions
measuredwithdifferenthistoricalprotocols.
Theoretical studies of the SK model have given this picture substance, clarifi-
cation and quantification,partly by introductionof new conceptsbeyondthose of
conventionalstatisticalphysics,especiallythroughtheworkofGiorgioParisi[1,5].
Letusassumethat,atanytemperatureofinterest,oursystemhaspossiblyseveral
essentiallyseparatemacrostates,whichwelabelbyindices{S}.Ausefulmeasure
ofsimilarityoftwomicrostatesS,S′isgivenbytheir‘overlap’,definedas
qSS′ =N−1(cid:229) hs iiShs iiS′. (1.4)
i
wherehs i measuresthethermalaverageofs inmacrostateS.
i S i
Thedistributionofoverlapsisgivenby
P{Jij}(q)=(cid:229) WSWS′d (q−qSS′), (1.5)
S,S′
whereW istheprobabilityoffindingthesysteminmacrostateS.
S
In general, the macrostates can depend on the specific choice of the {J } but
ij
for the SK model the average of P (q) can be calculated, as also other more
{Jij}
complicated distributions of the q , such as the correlation of pairwise overlap
SS′
distributionsfor3macrostatesS,S′,S′′.
Forasimple(non-complex)systemthereisonlyonethermodynamicallyrelevant
macrostateandhenceP(q)hasa single deltafunctionpeak;atq=0 fora param-
3Thisisincontrastwithtraditionalcomputersciencewhichhasbeenmoreconcernedwithworst
instances.
4 DavidSherrington
agnet(in theabsenceof anexternalfield)andatq=m2 fora ferromagnet,where
misthemagnetizationperspin.Incontrast,inacomplexsystemP(q)hasstructure
indicating many relevantmacrostates4. This is the case for the SK model beneath
acriticaltemperatureandforsufficientfrustration,asmeasuredbytheratioofthe
standarddeviationofP(J)comparedwithitsmean.Furthemoreothermeasuresof
theq-distributionindicateahierarchicalstructure,ultrametricityandaphylogenic-
treestructureforrelatingoverlapsofmacrostates,chaoticevolutionwithvariations
ofglobalparameters,andalsonon-self-averagingofappropriatemeasures.
These observationsand others give substance to and quantify the rugged land-
scapepicturewithmacrostatebarriersimpenetrableontimescalesbecominginfinite
withN.Forfinite-rangedspinglassesthispicturemustberelaxedtohaveonlyfinite
barriers,butstillwithanon-trivialphasetransitiontoaglassystate.
The macroscopicdynamicsin the spin glass phase also shows noveland inter-
estingglassybehaviour5,neverequilibratingandhavingsignificantdeviationsfrom
theusualfluctuation-dissipationrelationship6.
Abriefintroductiontothemethodolgiestoarriveattheseconclusionsisdeferred
toalatersection.
1.3 Transfers andExtensions
Theknowledgegainedfromsuchspinglassstudieshasbeenappliedtoincreasing
understandingofseveralotherphysicallydifferentsystemsandproblems,viamath-
ematical and conceptualtransfers and extensions. Conversely these other systems
havepresentedinterestingnewchallengesforstatisticalphysics.Inthissectionwe
shallillustratethisbrieflyviadiscussionofsomeofthesetransfersandstimulating
extensions.
Instatic/thermodynamicextensionsthereexistseveraldifferentanaloguesofthe
quenchedandannealedmicroscopicvariables,{J}and{s }above,andoftheinten-
sivecontrols,suchasT.Naturally,indynamicsofsystemswithquencheddisorder
theannealedvariables(suchasthe{s }above)becomedynamical,butalsoonecan
considercasesinwhichthepreviouslyquenchedparametersarealsodynamicalbut
withslowerfundamentalmicroscopictimescales7.
4Theoverlineindicatesanaverageoverthequencheddisorder.
5 There are several possible microscopic dynamics thatleads tothesame equilibrium/Gibbsian
measure,butallsuchemployinglocaldynamicsleadtoglassiness.
6Insteadonefindsamodifiedfluctuation-dissipationrelationwiththetemperaturenormalizedby
theinstantaneousauto-correlation.
7 Sometimesonespeaksoffastandslowmicroscopicvariablesbutitshouldbeemphasisedthat
theserefertotheunderlyingmicroscopictime-scales.Glassinessleadstomuchslowermacroscopic
timescales.
1 PhysicsandComplexity:abriefspinglassperspective 5
1.3.1 Optimizationand satisfiability
Alreadyinsection(1.2),oneexampleofeqn.(1.1)asanoptimizationproblemwas
given (the Dean’s Problem). Another classic hard computer science optimization
problemisthatofequipartitioningarandomgraphsoastominimisethecross-links.
In thiscase the costfunctionto minimisecan be againbe written asin eqn.(1.1),
nowwiththe{i}labellingverticesofthegraph,the{J }equalto1onedges/graph-
ij
linksbetweenverticesandzerowherethereisnolinkbetweeniand j,the{s =±}
i
indicating whether vertices i are in the first or second partition and with the frus-
tratingconstraint(cid:229) s =0imposingequipartitioning.Withouttheglobalconstraint
i i
thisisarandomferromagnet,butwithitthesystemisinthesamecomplexityclass
asaspinglass.
Another classic hard optimization problem that extends eqn. (1.1) in an ap-
parently simple way but in fact leads to new consequence is that of random K-
satisfiability (K-SAT) [11]. Here the object is to investigate the simultaneous sat-
isfiability of many,M, randomlychosen clauses, each made up of K possible mi-
croscopicconditionsinvolvingalargenumber,N,ofbinaryvariables.Labellingthe
variables{s }={±1}andwritingx toindicates =1andx toindicates =−1,
i i i i i
aK-clausehastheform
(y ory or....y ); i=1,..M (1.6)
i1 i2 iK
wherethey arex orx .InRandomK-Satthe{i }arechosenrandomlyfromthe
ij ij ij j
N possibilitiesand the choiceof y =x orx is also random,in bothcases then
ij ij ij
quenched.Inthiscaseonefinds,forthethermodynamicallyrelevanttypicalsystem,
thattherearetwotransitionsastheratioa =M/N isincreasedinthelimitN→¥ ;
fora >a itisnotpossibletosatifyalltheclausessimultaneously(UNSAT),for
c1
a <a theproblemissatisfiableinprinciple(SAT),butfora <a <a itisvery
c1 c2 c1
difficulttosatisfy(inthesensethatallsimplelocalvariationalalgorithmsstick)and
thisregionisknownasHARD-SAT.Thesedistinctionsareattributabletoregionsof
fundamentallydifferentfractionationofthespaceofsatisfiability,differentlevelsof
complexity.
1.3.2 K-spinglass
Infact,againtherewasastimulatingprecursorofthisK-SATdiscoveryina“what-
if” extensionof the SK model[6, 7] in whichthe 2-spininteractionsof eqn. (1.1)
arereplacedbyK-spininteractions:
H =− (cid:229) J s s ....s i (1.7)
K i1,i2...iK i1 i2 K
(i1,i2...iK)
6 DavidSherrington
in which the J are again chosen randomly and independently from an in-
i1,i2...iK
tensive distribution of zero mean. In this case, two different phase transitions are
observedasafunctionoftemperature,alowerthermodynamictransitionandady-
namicaltransitionthatisataslightlyhighertemperature,bothtocomplexspinglass
phases.Thethermodynamictransitionrepresentswhatisachievableinprinciplein
asituationinwhichallmicrostatescanbeaccessed;thedynamicaltransitionrepre-
sentsthesituationwherethesystemgetsstuckandcannotexploreallthepossibili-
ties,analoguesofHARDSAT-UNSATandSAT-HARDSAT.
TheK-spinglassisalsocomplexwithanon-trivialoverlapdistributionfunction
P(q) but now the state first reached as the transitions are crossed has a different
structurefromthatfoundforthe2-spincase.Now
P(q)=(1−x)d (q−q )+xd (q−q ); (1.8)
min max
incontrastwiththeSKcasewherethereiscontinuousweightbelowthemaximum
q .Thetwodeltafunctionsdemonstratethatthereisstillthecomplexityofmany
max
equivalentbutdifferentmacrostates,butnowwithequalmutualorthogonalities(as
comparedwiththe2-spinSKcasewherethereisacontinuousrangeofmacrostates).
ThissituationturnsouttobequitecommoninmanyextensionsbeyondSK.
1.3.3 Statics,dynamicsand temperature
Atthispointitisperhapsusefultosayafewmorewordsaboutthedifferencesbe-
tweenstatics/thermodynamicsanddynamicsinstatisticalphysics,andabouttypes
ofmicro-dynamicsandanaloguesoftemperature.
Inaphysicalsystemoneoftenwishestostudythermodynamicequilibrium,as-
sumingallmicrostatesareattainableifonewaitslongenough.Inoptimizationprob-
lemsonetypicallyhastwotypesofproblem;thefirstdeterminingwhatisattainable
inprinciple,thesecondconsideringhowtoattainit.Theformeristheanalogueof
thermodynamicequilibrium,thelatterofdynamics.
Inaphysicalsystemthetruemicroscopicdynamicsisgivenbynature.However,
inoptimizationstudiestheinvestigatorhastheopportunitytodeterminethemicro-
dynamicsthroughthecomputeralgorithmsheorshechoosestoemploy.
Temperatureenters the statistical mechanicsof a physicalproblem in the stan-
dardBoltzmann-Gibbsensemblefashion,orasameasureofthestochasticnoisein
the dynamics. We have already noted that it can also enter an optimization prob-
lem in a very similar fashion if there is inbuilt uncertainty in the quantity to be
optimized. But stochadtic noise can also usefully be introduced into the artificial
computeralgorithmicdynamicsused to tryto find thatoptimum.Thisis the basis
of the optimization technique of simulated annealing where noise of variance T
A
is deliberately introduced to enable the probabalistic scaling of barriers, and then
graduallyreducedtozero[8].
1 PhysicsandComplexity:abriefspinglassperspective 7
1.3.4 Neural networks
Thebrainismadeupofaverylargenumberofneurons,firingatdifferentratesand
extents,interconnectedbyanevenmuchlargernumberofsynapses,bothexcitory
and inhibitory. In a simple model due to Hopfield [9] one can consider a cartoon
describable again by a control function of the form of eqn. (1.1). In this model
theneurons{i}areidealizedbybinaryMcCullough-Pittsvariables{s =±1},the
i
synapsesby{J },positiveforexcitatoryandnegativeforinhibitory,withstochastic
ij
neural microdynamics of effective temperature T emulating the width of the
neural
sigmoidalresponseofaneuron’soutputtothecombinedinputfromallitsafferent
synapses,weightedbythecorrespondingactivityoftheafferentneurons.
Thesynapsesaredistributedoverbothsigns,yieldingfrustrationandapparently
random at first sight. However actually they are coded to enable attractor basins
related to memorized patterns of the neural microstates {x m }; m =1,...p=a N.
i
Thesimilarityofaneuralmicro-statetoapatternm isgivenbyanoverlap
mm =N−1(cid:229) hs i x m . (1.9)
i S i
i
Retrievalofmemorym istheattractorprocessinwhichasystemstartedwithasmall
m m
m iteratestowardsalargevalueofm .
InHopfield’soriginalmodelhetookthe{J }tobegivenbytheHebb-inspired
ij
form
J =p−1(cid:229) x m x m (1.10)
ij i j
m
with randomlyquenched{x } 8. For a less than a T -dependentcritical value
i neural
a (T )patternscanberetrieved.Beyonditonlyquasi-randomspinglassmin-
c neural
ima unrelatedto the memorisedpatternsremain( and still onlyfor T nottoo
neural
large).However,other{J }permitaslightlylargercapacity(asalsocanoccurfor
ij
correlatedpatterns).
Again the landscape cartoon is illustratively useful. It can be envisaged as one
for H as a functionof the neuralmicrostates(of allthe neurons),with the dy-
{Jij}
namicsoneofmotioninthatlandscape,searchingforminimausinglocaldeviation
attempts. The memory basins are large minima. Clearly one would like to have
many different retrievable memories. Hence frustration is necessary. But equally,
toomuchfrustrationwouldleadtoaspin-glass-likestatewithminimaunrelatedto
learnedmemories.
This cartoon also leads immediately to the recognition that learning involves
modifying the landscape so as to place the attractor minima around the states to
be retrieved. This extension can be modelled minimally via a system of coupled
dynamicsofneuronswhosestatedynamicsisfast(attemptingretrievalorgeneral-
ization) and synapses that also vary dynamically but on a much slower timescale
andinresponsetoexternalperturbations(yieldinglearning)[10].
8i.e.uncorrelatedpatterns
8 DavidSherrington
1.3.5 Minoritygame
More examples of many-body systems with complex macrobehaviour are to be
found in social systems, in which the microscopic units are people (or groups of
peopleorinstitutions),sometimesco-operating,oftencompeting.Hereexplicitdis-
cussionwillberestrictedtoonesimplemodel,theMinorityGame[12],devisedto
emulate some featuresof a stockmarket.N ‘agents’ play a gamein which at each
time-stepeachagentmakesoneoftwochoiceswiththeobjectivetomakethechoice
whichisintheminority9.Theyhavenodirectknowledgeofoneanotherbut(inthe
originalversion)make their choices based on the commonly-availableknowledge
of the historical actual minority choices, using their own individual stategies and
experience to make their own decisions. In the spirit of minimalism we consider
all agents (i) to have the same ‘memories’, of the minority choices for the last m
time-steps,(ii)toeachhavetwostrategiesgivenbyrandomlychosenandquenched
Booleanoperatorsthat,actingonthem-stringofbinaryentriesrepresentingthemi-
noritychoicesforthelastmsteps,outputabinaryinstructiononthechoicetomake,
(iii)usingapersonal‘point-score’tokeeptallyofhowtheirstrategieswouldhave
performedifused,increasingthescoreeachtimetheywouldhavechosentheactual
minority,and(iv)usingtheirstrategywiththelargerpoint-score.Frustrationisrep-
resentedintheminorityrequirement,whilequencheddisorderarisesintherandom
choiceofindividualstrategies.
Simulationalstudiesofthe‘volatility’,thestandarddeviationoftheactualminor-
itychoice,shows(i)adeviationfromindividuallyrandomchoices,indicatingcor-
relationthroughthecommoninformation,(ii)acusp-minimumatacriticalvaluea
c
oftheratiooftheinformationdimensiontothenumberofagentsa =D/N=2m/N,
suggestingaphasetransitionata ,(iii)ergodicityfora >a butnon-ergodicde-
c c
pendenceonthepoint-scoreinitializationfora <a ,indicatingthatthetransition
c
representstheonsetofcomplexity.Thisisreminiscentofthecuspandtheergodic-
nonergodic transition observed in the susceptibilities of spin glass systems as the
temperatureisreducedthroughthespinglasstransition.
Furthermore, this behaviour is essentially unaltered if the ‘true’ history is re-
placedbyafictitious‘random’historyateachstep,withallagentsbeinggiventhe
samefalsehistory,indicatingthatitprincipallyrepresentsacarrierforaneffective
interaction between the agents. Indeed, generalising to a D-dimensional random
history information space, consideringthis as a vector-space and the strategies as
quenched D-vectors of components{Rs,m }; s=1,2, m =1,..D in that space, and
i
averaging over the stochastically random ‘information’,one is led to an effective
control function analagous to those of eqn. (1.1) and eqn. (1.10) with p replaced
by a , now {x = (R1−R2)/2}, an extra multiplicative minus sign on the right
i i i
hand side of eqn. (1.10), and also a random-field term dependent upon the {x }
i
and{w =(R1+R2)/2}.Asnoted,thereisanergodic-nonergodictransitionatacr-
i i i
9Thephilosophyisthatonegetsthebestpricebysellingwhenmostwanttobuyorbuyingwhen
mostwanttosell.
1 PhysicsandComplexity:abriefspinglassperspective 9
titicala ,butnowthepictureisoneofthe{x }asrepellersratherthantheattractors
i
oftheHopfieldmodel10.
Thetypicalbehaviourofthissystem,asforthespinglasses, canbestudiedus-
ing a dynamical generating functional method [13], averaged over the choice of
quenchedstrategies,inamanneroutlinedbelow.Theaveragedmany-bodysystem
can then be mapped into an effective representative agent ensemble with memory
andcolourednoise,withboththenoisecorrelationsandthememorykerneldeter-
minedself-consistentlyovertheensemble.Notethatthisisincontrastto(andcor-
rects) the common assumption of a single deterministic representative agent. The
phasetransitionfromergodictonon-ergodicismanifestbyasingularityinthetwo-
timepoint-signcorrelationfunction
C(t,t′)=N−1(cid:229) sgn(p(t)sgn(p(t′)=hsgn(p(t)sgn(p(t′)i (1.11)
i i ens
i
wherethefirstequalityreferstothemany-bodyproblemandtheseconditsequiva-
lenceintheeffectiveagentensemble.
1.4 Methodologies
For systems in equilibrium, physical observables are given by lnZ evaluated for
the specific instance of any quenched parameters, or strictly the generalized gen-
eratingfunctionlnZ({l })wherethe {l }aregeneratingfieldsto betakento zero
afteranappropriateoperation(suchas¶ /¶l )isperformed.Hencetheaverageover
quenched disorder is given by lnZ. One would like to perform the average over
quencheddisorderexplicitlytoyieldaneffectivesystem.However,sinceZisasum
overexponentialsofafunctionofthevariables,lnZ isdifficulttoaveragedirectly
soinsteadoneusestherelation
lnZ=Lim n−1(Zn−1) (1.12)
n→0
and interprets the Zn as corresponding to a system whose variables have extra
‘replica’ labels, a =1,..n, for which one can then average the partition function,
an easier operation, at the price of needing to take the eventual limit n→0. The
relevant‘orderparameters’arethencorrelationsbetweenreplicas
qab =N−1(cid:229) hs a s b i (1.13)
i i T
i
10 Onecan make themodel even more minimalby allowingeach agent onlyone strategy {xi}
which(s)heeitherfollowsifitspoint-score ispositiveoractsoppositelytoifthepoint-score is
negative.Thisremovestherandom-fieldtermandalsothecuspinthetabularasavolatility,but
retainstheergodic-nonergodictransition[14].
10 DavidSherrington
whereh..i referstoathermalaverageintheeffectivepost-averagingsystem.This
T
orderparameterisnon-zerointhepresenceoffrozenorder,butmoreinterestingly
(andsubtly)alsoexhibitsthefurtherremarkablefeatureofspontaneousreplicasym-
metrybreaking,indicatingcomplexity.Afterfurthersubtletiesbeyondthescopeof
thisshortintroduction,thereemergesanorderfunctionq(x);x∈[0.1]fromwhich
theaverageoverlapfunctionisobtainedby
P(q)=dx/dq (1.14)
FordynamicstheanalogueofthepartitionfunctionZisageneratingfunctional,
whichmaybewrittensymbolicallyas
Z = (cid:213) d (microscopiceqns.ofmotion)exp({lf }) (1.15)
dyn Z
allvariables,alltimes
where the f symbolize the microscopic variables and a Jacobian is implicit. Av-
eraging over the quenched disorder now induces interaction between epochs and
integrating out the microscopic variables results in the effective single agent en-
sembleformulation,as wellasemergentcorrelationandresponsefunctionsasthe
dynamic order parameter analogues of the static inter-replica overlaps, exhibiting
non-analyticityataphasetransitiontonon-ergodicity.
1.5 Conclusion
Abriefillustrationhasbeenpresentedofhowcomplexco-operativebehaviourarises
inmanybodysystemsduetothecombinationoffrustrationanddisorderinthemi-
croscopicsofevenverysimplyformulatedproblemswithveryfewparameters.Such
systemsarenotonlyexamplesofAnderson’sfamousquotation“Moreisdifferent”
butalsodemonstratethatfrustrationanddisorderinmicroscopicscanleadtocom-
plexity in macroscopics; i.e. many and complexly related differents. Furthermore,
thiscomplexityarisesinsystemswithverysimplefew-valuedmicroscopicparam-
eters;complexityisnotthesameascomplicationanddoesnotrequireit.
Therehasalsobeendemonstratedvaluabletransfersbetweensystemsthatappear
verydifferentatfirstsight,throughthemediaofmathematicalmodelling,conceptu-
alizationandinvestigatorymethodologies,asituationreminiscentofthesuccessful
useoftheRosettastoneinlearninganunknownlanguagescriptbycomparisonwith
anotherthatcarriesthesamemessageinadifferentformat.
Theperspectivetakenhasbeenofstatisticalphysics,butitmustbeemphasised
that the stimulation has been both from and to physics, since many of these com-
plex systems have interesting features in their microscopic underpinning that are
richer than those in the physics of conventionaldictionary definition and provide
newchallengestothephysicist.
Also of note is how a blue skies attemptto understandsome obscuremagnetic
alloys through soluble but, for the experimental alloys, unphysical modelling has