Large Deviations in Physics: The Legacy of the Law of Large Numbers

323 Pages·2014·5.048 MB·English

by  Angelo Vulpiani, Fabio Cecconi, Massimo Cencini, Andrea Puglisi, Davide Vergni (eds.)

Lecture Notes in Physics 885
Angelo Vulpiani
Fabio Cecconi
Massimo Cencini
Andrea Puglisi
Davide Vergni Editors
Large
Deviations
in Physics
The Legacy of the Law of Large
Numbers
Lecture Notes in Physics
Volume 885
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Angelo Vulpiani (cid:129) Fabio Cecconi (cid:129)
Massimo Cencini (cid:129) Andrea Puglisi (cid:129) Davide Vergni
Editors
Large Deviations in Physics
The Legacy of the Law of Large Numbers
123
Editors
AngeloVulpiani FabioCecconi
DipartimentodiFisica MassimoCencini
UniversitàdiRomaSapienza AndreaPuglisi
Roma IstitutoSistemiComplessi
Italy ConsiglioNazionaledelleRicerche
Roma
Italy
DavideVergni
IstitutoperleApplicazionidelCalcolo
ConsiglioNazionaledelleRicerche
Roma
Italy
ISSN0075-8450 ISSN1616-6361 (electronic)
LectureNotesinPhysics
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Preface
Probabilitytheorywasatitsorigin,almostsolely,centeredaroundgamesofchance
(cardsanddice).WiththeArsConjectandi,JakobBernoulli,whileobtainingthefirst
versionofthetheoremnowknownastheLawofLargeNumbers(LLN),movedthe
theoryofprobabilityawayfrombeingprimarilyavehicleforcalculatinggambling
odds.Thisstephasbeencrucialbyshowingthattheprobabilitytheorymighthave
animportantroleintheunderstandingofavarietyofproblemsinmanyareasofthe
naturalsciencesandhumanexperiences.
In1913,whenthetsarNicholasIIcalledforcelebrationsofthe300thAnniver-
sary of the Romanov rule, the great Russian mathematician Andrei Andreyevich
Markovrespondedbyorganizingasymposiumaimedatcommemoratingadifferent
anniversary.Markov took the occasion to celebrate the bicentenaryof Bernoulli’s
Arsconjectandi:Bernoulliactuallycompletedhisbookby1690,butthebookwas
only published posthumously in 1713 by his nephew Niklaus because of family
quarrels.
Nowadays, one century after Markov,the autocratic tsarist governmentis over
andwecantaketheoccasiontocelebratetheLawofLargeNumberswithnoneed
ofextrascientificpretexts.
The LLN is at the base of a scientific legacy whose relevance cannot be
overestimated. We can start mentioning the great visionary idea of the ergodic
hypothesisbyLudwigBoltzmann.Theergodicityissue,originallyintroducedinthe
contextofthestatisticalmechanicsandthendevelopedasanautonomousbranchof
measure theory, can be seen as the generalizationof the LLN to non-independent
variables. This topic is still an active research field in mathematical physics. In
addition,itconstitutesthestartingpointofthenumericalmethodsusedinstatistical
mechanics,namelythemoleculardynamicsandtheMonteCarlomethods.
The most importantphysicalproperties of macroscopicobjects are determined
bymeanvalues,whosemathematicalbaseisguaranteedbytheLLN.But,inmany
cases, also the fluctuations can be important. The control of “small” fluctuations
around the mean value is provided by the Central Limit Theorem (CLT), whose
generalrelevancewasestablished forthefirst time in 1812with the bookThéorie
Analytique des Probabilités by Pierre Simon Laplace. From a physical point of
v
vi Preface
view, however, even very small fluctuations can be dramatically important. As a
paradigmaticexample,wecanmentionthetreatmentoftheBrownianmotion,which
amongthe manystill in progressapplicationsbroughtconclusiveevidenceforthe
atomichypothesis.
Beyond their conceptualimportance,thanks to their link to response functions
via the fluctuation-dissipationtheorem, fluctuations are becoming more and more
importantinpresent-dayapplications,especiallyviatherecentlyestablishedfluctu-
ation relations. Their relevance is amplified in small (micro- and nano-) systems,
and in materials (as granular matter) where the number of effective elementary
constituentsisnotaslargeasingasesorliquids.Insuchsystemslargeexcursions
from the average cannot be neglected, therefore it is necessary to go beyond the
Gaussian approximation,i.e. beyondthe realm of validity of the CLT. The proper
technicaltooltostudysuchstrongfluctuationsistheLargeDeviationTheory(LDT),
whichgeneralizestheCLT.
ThefirstgeneralmathematicalformulationoftheLargeDeviationTheoryisdue
to Harald Cramér in the 1930s. However, the very first application of LDT can
be ascribed to Boltzmann who, using combinatorial arguments, had been able to
showtherelevanceoftheentropyasabridgebetweenmicroscopicandmacroscopic
levels.
This book encompasses some recent developments of the fundamental limit
theorems – LLN, CLT and LDT – of the probability theory in statistical physics,
in particular: ergodicity breaking, non-equilibrium and fluctuation relations, dis-
ordered systems, computational methods, systems with long-range interactions,
Brownianmotors,chaoticdynamics,anomalousdiffusionandturbulence.
Rome,Italy AngeloVulpiani
December2013 FabioCecconi
MassimoCencini
AndreaPuglisi
DavideVergni
Contents
1 FromtheLaw ofLarge NumberstoLarge Deviation
TheoryinStatisticalPhysics:AnIntroduction.......................... 1
FabioCecconi,MassimoCencini,AndreaPuglisi,Davide
Vergni,andAngeloVulpiani
1.1 Introduction........................................................... 1
1.2 AnInformalHistoricalNote......................................... 3
1.2.1 LawofLargeNumbersandErgodicity.................... 4
1.2.2 CentralLimitTheorems.................................... 9
1.2.3 LargeDeviationTheory.................................... 11
1.3 LDTfortheSumandProductofRandomIndependent
Variables.............................................................. 13
1.3.1 ACombinatorialExample ................................. 13
1.3.2 ProductofRandomVariables.............................. 15
1.4 LargeDeviationTheory:ExamplesFromPhysics ................. 17
1.4.1 EnergyFluctuationsintheCanonicalEnsemble.......... 17
1.4.2 MultiplicativeCascadeinTurbulence..................... 18
1.4.3 ChaoticSystems............................................ 19
1.4.4 DisorderedSystems ........................................ 21
1.4.5 EntropyProductioninMarkovProcesses ................. 24
References.................................................................... 26
2 Ergodicity:HowCanItBeBroken?...................................... 29
GiancarloBenettin,RobertoLivi,andGiorgioParisi
2.1 TheErgodicHypothesis ............................................. 30
2.1.1 TheFundamentalPhysicalIdeas........................... 30
2.1.2 AWell-PosedMathematicalSetting....................... 33
2.2 ErgodicityBreaking.................................................. 36
2.2.1 ErgodicityBreakingintheFermi-Pasta-Ulam
ModelatLowEnergies..................................... 37
2.2.2 ErgodicityBreakingInducedbyBreather
StatesintheDiscreteNonlinearSchrödingerEquation... 47
vii
viii Contents
2.3 ErgodicityBreakingatEquilibrium................................. 53
2.3.1 GeneralConcepts........................................... 53
2.3.2 TheGibbsStates............................................ 54
2.3.3 TheLocalDRLEquilibriumStates........................ 60
2.4 GlassySystem........................................................ 62
2.4.1 AHeuristicConstruction:FiniteVolumeStates .......... 62
2.4.2 TheCaseofManyStates................................... 65
References.................................................................... 68
3 LargeDeviationsinStationaryStates,EspeciallyNonequilibrium... 71
GiovanniJona-Lasinio
3.1 Introduction........................................................... 71
3.2 Assumptions.......................................................... 73
3.3 TheFundamentalFormula........................................... 74
3.4 TimeReversalandItsConsequences................................ 78
3.5 LongRangeCorrelations ............................................ 80
3.6 FluctuationsoftheCurrentandDynamicalPhaseTransitions..... 81
3.7 UniversalityinCurrentFluctuationsandOtherResults............ 83
3.8 NonequilibriumPhaseTransitions .................................. 84
3.9 LargeDeviationsforReaction-DiffusionSystems ................. 85
3.10 ThermodynamicInterpretationoftheLargeDeviation
Functional ............................................................ 85
3.11 ConcludingRemarksandAdditionalReferences................... 89
References.................................................................... 90
4 Fluctuation-DissipationandFluctuationRelations:From
EquilibriumtoNonequilibriumandBack............................... 93
PaoloAdamo,RomanBelousov,andLambertoRondoni
4.1 ConciseHistory ...................................................... 93
4.2 TheBrownianMotionandtheLangevinEquation................. 95
4.3 TheFluctuation-DissipationRelation............................... 99
4.4 EvolutionofProbabilityDistributions .............................. 103
4.4.1 ErgodicityandMixing ..................................... 107
4.5 LinearResponse...................................................... 108
4.6 Onsager-Machlup:ResponsefromSmallDeviations .............. 112
4.7 FluctuationRelations:ResponsefromLargeDeviations........... 118
4.7.1 TheGallavotti-CohenApproach........................... 119
4.7.2 FluctuationRelationsfortheDissipationFunction....... 121
4.7.3 Green-KuboRelations...................................... 125
4.7.4 JarzynskiEquality.......................................... 126
4.8 t-MixingandGeneralResponseTheory ............................ 127
4.9 ConcludingRemarks................................................. 129
References.................................................................... 131
Contents ix
5 LargeDeviationsinDisorderedSpinSystems........................... 135
AndreaCrisantiandLucaLeuzzi
5.1 SomeGeneralResultsonLargeDeviations......................... 135
5.1.1 AnExample:TheMeanFieldIsingModel ............... 139
5.2 Sample-to-SampleFreeEnergyFluctuationsandReplicaTrick... 142
5.2.1 RandomIsingChain........................................ 143
5.2.2 ReplicaTrick................................................ 145
5.2.3 ReplicasintheRandomIsingChain....................... 147
5.2.4 FromSmalltoLargeDeviations........................... 148
5.2.5 RandomDirectedPolymer................................. 151
5.2.6 Mean-FieldSpin-Glass..................................... 151
5.2.7 PositiveLargeDeviations.................................. 153
5.2.8 SphericalSpin-GlassModelandGaussian
RandomMatrices........................................... 154
5.3 Sample-to-SampleFluctuationoftheOverlap...................... 155
5.3.1 BacktotheReplicatedRandomIsingChain.............. 156
5.3.2 RandomFieldIsingModel................................. 158
5.4 AFinalWord ......................................................... 159
References.................................................................... 159
6 LargeDeviationsinMonteCarloMethods.............................. 161
AndreaPelissettoandFedericoRicci-Tersenghi
6.1 Introduction........................................................... 161
6.2 DataReweighting .................................................... 163
6.3 MultipleHistogramMethod......................................... 168
6.4 UmbrellaSamplingandSimulatedTempering ..................... 172
6.4.1 UmbrellaSampling......................................... 172
6.4.2 SimulatedTempering....................................... 173
6.4.3 EquivalenceofSimulatedTemperingand
UmbrellaSampling......................................... 174
6.5 GeneralizingtheUmbrellaMethod:MulticanonicalSampling.... 178
6.6 ParallelTempering ................................................... 184
6.6.1 GeneralConsiderations..................................... 184
6.6.2 SomeGeneralRigorousResults ........................... 186
6.6.3 OptimalChoiceofTemperatures .......................... 187
6.6.4 ImprovingParallelTempering ............................. 189
6.7 Conclusions........................................................... 190
References.................................................................... 190
7 LargeDeviationsTechniquesforLong-RangeInteractions ........... 193
AurelioPatelliandStefanoRuffo
7.1 Long-RangeInteractions............................................. 193
7.2 SomeUsefulResultsofLargeDeviationsTheory.................. 198
7.3 ThermodynamicFunctionsFromLargeDeviationsTheory ....... 199