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RenormalizationGroupTheoryofEigenMicrostates
TengLiu(刘腾)1†,Gao-KeHu(胡高科)1†,Jia-QiDong(董家奇)2,Jing-FangFan(樊京芳)1,
Mao-XinLiu(刘卯鑫)3,andXiao-SongChen(陈晓松)1*
1SchoolofSystemsScience/InstituteofNonequilibriumSystems,BeijingNormalUniversity,Beijing100875,China
2LanzhouCenterforTheoreticalPhysics,KeyLaboratoryofTheoreticalPhysicsofGansuProvince,
andKeyLaboratoryforMagnetismandMagneticMaterialsofMOE,LanzhouUniversity,
Lanzhou730000,China
3SchoolofScience,BeijingUniversityofPostsandTelecommunications,Beijing100876,China
(Received7June2022;acceptedmanuscriptonline13July2022)
Weproposearenormalizationgroup(RG)theoryofeigenmicrostates,whichareintroducedinthestatistical
ensemblecanbeconsideredasalinearsuperpositionofeigenmicrostateswithprobabilityamplitudesequalto
,thelargesteigenvalueσ1hastwotrivialfixedpointsat
bβ/ν
lowandhightemperaturelimitsandacriticalfixedpointwiththeRGrelationσ1=bσ1,whereβandνare
thecriticalexponentsoforderparameterandcorrelationlength,
dimensions,ithasbeendemonstratedthattheRGtheoryofeigenmicrostatesisabletoidentifythecritical
phenomenabothinequilibriumandnon-equilibriumsystemswithoutconsideringtheHamiltonian,whichisthe
foundationofWilson’sRGtheoryandisabsentformostcomplexsystems.
DOI:-307X/39/8/080503
Insystemsconsistingofmanycomponents(atoms,icalpointshavethesameasymptoticbehaviors.[10]For
molecules,electrons,spins,etc.),oneofthemoststrikingverydifferentsystems(.,fluidsandmagnets)within
phenomenaistheemergenceofdiversetypesofcollectiveagivenclass,thereistheso-calledtwo-scalefactoruni-
behavior,.,phases.[1,2]Underslightchangesofexternalversalitysothattheasymptoticcriticalbehaviorisbe-
conditions,thesystemsmayhavedramaticstructuralre-lievedtobeknowncompletelyprovidedthattwononuni-
arrangements,.,.[11–15]However,itisproved
ischaracterizedbytheso-calledorderparameter,whichbytheRGoftheO(n)symmetricφ4fieldtheorywith
wasintroducedbyLandau.[3]Inassociationwithcontin-thespatiallyanisotropicHamiltonianthatthetwo-scale
uouschangeordiscontinuousjumpoforderparameteratfactoruniversalityisvalidforisotropicsystemslikefluids
thetransitionpoint,thephasetransitioniscontinuousorbutnotforanisotropicsystems.[16]Thisnewfindingisof
Nearthetransitionpointofacontinuousphasetran-realsystems.[17]Thepredicteddependenceofthecritical
sition,thecorrelationlengthξismuchlargerthantheBindercumulantontheanisotropyhasbeenconfirmedby
microscopiclengthscalea
.Hence,a
becomesirrelevanttheMonteCarlostudiesoftheIsingmodels.[18,19]
-Nowadays,weneedtostudyurgentlythecriticalphe-
haviorsofsystemaregovernedbyfluctuationsthatarenomenaofcomplexsystemswithoutknowledgeoftheir
statisticallyself-.[20,21]Recently,weproposedaneigenmi-
ofthisself-similarityandintroducedtheblockspinscalingcrostateapproach(EMA)[22,23]basedonmicrostatesob-
atlengthscalesr≪ξ.[4]Themajorbreakthroughintheensemblecomposedofthemicrostatesduringanobser-
theoryofcriticalphenomenacamewiththerenormaliza-vationtimeperiodisdescribedbyanormalizedmatrix.
tiongroup(RG)theoryofWilson[5,6]andco-workers,[7,8],
whichisbasedonthesimilarityofHamiltonianduringthewecanconsideramicrostateasalinearsuperpositionof
anditstheoreticalpredictionshavebeenverifiedbyexper--
imentsunderthelow-gravityenvironmentofspace.[9]modynamiclimitsindicatesthecondensationofeigenmi-
TheRGtheoryofcriticalphenomenahaselucidatedcrostateandtheemergenceofaphasedescribedbythe
whichstatesthatallphasetransitionscanbedividedintoBose–EinsteincondensationofBosegasandcanbeused
severaldistinctclasses.[10]Withinagivenclass,allcrit-
†Theseauthorscontributedequallytothiswork.
*:******@
©2022ChinesePhysicalSocietyandIOPPublishingLtd
080503-1
ChinesePhysicsLetters39,080503(2022)ExpressLetter
parameterofphasetransitionisidentifiednaturallyasthewhichcorrespondtohightemperaturelimitT=∞and
eigenvalue,whichfollowsafinite-sizescalingscalingformlowtemperaturelimitT=0,respectively.
-sizescalingInaddition,thereisanon-trivialfixedpoint(K=K*,
form,wecancalculatecriticalexponentsofaphasetran-h=0),
=K−K*andδh=h,wehave
appliedsuccessfullytostudytheequilibriumphasetran-theRGtransforms
sitionsoftheIsingmodel[22,23]andtheViscekmodelof
δK=λδK,(8)
thenon-equilibriumcollectivemotion.[24]Inaddition,webK
haveusedEMAtoinvestigateglobaltemperaturefluctua-δhb=λhδh.(9)
tionsoftheEarth[23]andspatialdistributionofozoneat
Thereducedtemperaturet=(K*−K)/K*isrelatedto
differentgeopotentialheights,[25]basedonempiricaldata.
δKast=−δK/K*.
InthisLetter,weproposetheRGtheoryofeigenmi-
Ath=0,thecorrelationlengthapproachesinfinity
,wesummarizetheRGtheoryofWilson−ν
witht→0inpowerlawasξ(t)==0,the
basedontheHamiltonianinthefollowing.
correlationlengthgoestoinfinitywithh→0asξh(h)=
WeconsidertheIsingmodelonad-dimensionallattice−νh
,thesys-
withsizeLandlatticespacinga
.ItsHamiltoniancanbe
temistakenawayfromthecriticalpointwithsmallercor-
writtenas
relationlengthsξ(tb)=ξ(t)/bandξh(hb)=ξh(h)/
1/ν1/νh
∑∑wehavetb=btandhb=-
βH=H
(K,h;{si})=−Ksisj−hsi,(1)
<i,j>iformsofEqs.(8)and(9),weobtainthecriticalexponents
ν=lnb/lnλK,(10)
whereβ=1/kBT,si=±1,
partitionfunctioncanbecalculatedasνh=lnb/lnλh.(11)
∑Werewritethesingularpartoffreeenergydensityas
Z(K,h;L)=exp[−H
(K,h;{si})].(2)
functionsoftandhandhave
{si}
−d1/ν1/νh
Allthermodynamicquantitiescanbeobtainedfromthisfs(t,h;L/a
)=bfs(tb,hb;L/ba
).(12)
,susceptibility,andspecificheatcan
IntheRGtheory,thefirststepistodecreasetheres-
becalculatedbythederivativesoffswithrespecttohand
olutionbychangingtheminimumlengthscalefroma
to
ddtas
ba
,whereb>(ba
)hasNI=bspins
NN1/νh−d(1,h)1/ν1/νh
(t,h;L/a
)=(b)fs(tb,hb;L/ba
),(13)
twogroups,whicharerepresentedbyσIandhavecellspin2/νh−d(2,h)1/ν1/νh
χ(t,h;L/a
)=(b)fs(tb,hb;L/ba
),(14)
SI=±
2/ν−d(2,t)1/ν1/νh
{si}to{SI,σI}.Inthecalculationofpartitionfunction,Cs(t,h;L/a
)=(b)fs(tb,hb;L/ba
).(15)
weconsiderfirstthecontributionsofallσIforasetofcellAftertakingh=0andthelimitL→∞,wechoose
spins{SI}andhaveb∝t−νfort>0andb∝(−t)−νfort<
∑e■obtainthebulkorderparameter
Z(K,h;L)=exp[−H
(Kb,hb;{SI})]
{S}{β
Iam(−t),fort≤0,
e■m(t,0;∞)=(16)
=Z(Kb,hb;L/b),(3)0,fort>0,
whereKbisrelatedtotheinteractionbetweencellspinswithβ=dν−ν/νh,thebulksusceptibility
andhbistheexternalfieldactingoncellspins.{−−γ
Thefreeenergydensitytakestheformaχ(−t),fort≤0,
χ(t,0;∞)=+−γ(17)
aχt,fort>0,
lnZ(K,h;L)lnZe■(K,h;L)
f(K,h;L)=−=−bb,(4)
dwithγ=2ν/ν−dν,andthebulkspecificheat
VbVbh
dd{−−α
whereV=L,Vb=(L/b).Thesingularpartoffreeac(−t),fort≤0,
Cs(t,0;∞)=+−α(18)
energydensityfollowstherelationact,fort>0,
−d
fs(K,h;L/a
)=bfs(Kb,hb;L/ba
),(5)withα=2−dν.
Thecriticalbehaviorsofthermodynamicquantitiesare
whereKbandhbfollowtheRGtransformsessentiallyrealizedbythedivergenceofcorrelationlength
-
K=KRb(K),(6)
b1posedofthecorrelatedcomponentspossessthesizespro-
bd
hb=hR2(K).(7)-
dynamicquantitieswithrespecttoξdaremoreessential.
Thesetransformshavetwotrivialfixedpoints:Theyareβ¯=β/dν,γ¯=γ/dν,andα¯=α/dν,
criticalexponentsofξdareν¯≡dνandν¯≡,
(1)(K=0,h=0),hh
thereducedcriticalexponentsdependonlyonν¯andν¯has
(2)(K=∞,h=0),
β¯=1−1/ν¯h,γ¯=2/ν¯h−1,andα¯=2/ν¯−1.
080503-2
ChinesePhysicsLetters39,080503(2022)ExpressLetter
FortheIsingmodelond-dimensionallattice,ν=,WetakeM>Nmicrostatestocomposeastatisticalen-
(16),[26]=2,3,and4,,whichisdescribedbyanN×MmatrixAwith
Correspondingly,wehaveν¯=,(48),,whichelements
-√
colationonone-andtwo-dimensionallatticeswithlong-Aiτ=Si(τ)/C0,(28)
rangeconnectionprobabilityP(r)∝r−(d+σ)forσ<0,it
[27]∑M∑N2T
wasfoundthatν¯=τ=1i=1Si(τ)sothatTr(A·A)=1.
Now,werewritethesingularpartoffreeenergyasFromthecorrelationmatrixbetweenagentstatesK=
T
AA,wecancalculateeigenvectorsUI,I=1,2,...,N.
d−1¯d1/ν¯d1/ν¯hdT
fs(t,h;L/a
)=(b)fs(t(b),h(b);N/b),(19)ThecorrelationmatrixbetweenmicrostatesC=AA
hasMeigenvectorsVI,I=1,2,...,M.
d
whereN=(L/a
).ThenweobtaintheRGtransforma-Accordingtothesingularvaluedecomposition(SVD),
tionofthethermodynamicfunctionsasfollows:theensemblematrixAcanbedecomposedas
¯
d−β(1,h)d1/ν¯d1/ν¯h
m(t,h;L/a
)=(b)f¯(t(b),h(b);Nb),(20)N
s∑
dγ¯¯(2,h)d1/ν¯d1/ν¯hA=σIUI⊗VI,(29)
χ(t,h;L/a
)=(b)fs(t(b),h(b);Nb),(21)
I=1
dα¯¯(2,t)d1/ν¯d1/ν¯h
Cs(t,h;L/a
)=(b)fs(t(b),h(b);Nb),(22)
∑N2
whereI=1σI=,
dd
whereNb=N/=NandNb=1,wecanreachwecanconsidermicrostateS(τ)asalinearsuperpo-
[28]
thefinite-sizescalingformsitionofeigen√microstatesUIbytherelationS(τ)=
∑N
I=1σIVτIUIC0,whereσIistheprobabilityamplitude
−11/ν¯1/ν¯h
fs(t,h;L/a
)=NY(tN,hN),(23)ofUI.
Forcomplexsystemswithoutdominanteigenmi-
andthethermodynamicquantities
crostate,allσI→0inthelimitsN→∞,M→∞.
¯Withthefinitelimitofσthereisacondensationofeigen
−β1/ν¯1/ν¯hI
m(t,h;L/a
)=NFm(tN,hN),(24)
microstateUI,whichissimilartotheBose–Einsteincon-
γ¯1/ν¯1/ν¯h
χ(t,h;L/a
)=NFχ(tN,hN),(25)
α¯1/ν¯1/ν¯haphasetransitionwiththenewphaseUIandtheorder
Cs(t,h;L/a
)=NFc(tN,hN).(26)
parameterσI.
Aspointedoutabove,theRGtheoryofWilson[5,6]ItisessentialintheEMAtodefinemicrostatesprop-
cannotbeappliedtothesystemswhoseHamiltonianiserlyinordertocatchtherelevantpropertiesofasystem.
-
,microstatesaredefinedbyall
thesesystems,wedevelopheretheRGtheoryofeigen[22,23]
-
[23,24]tems,microstatesaredescribedbytheneighborhoodden-
TheEMAdevelopedbyusfollowstheconcept[22,23]
ofstatisticalensemble,
in1902.[29]Thestatisticalensembleiscomposedofmi-studiesoftheEarthsystemwithEMA,thereducedtem-
peraturefluctuationsandtheozone’svariationsareused
crostatesofasystem,whicharedefinedbythestatesof[23,25]
.
thestatisticalensemblebytheprobabilitydensityofmi-Theprincipalcomponentanalysis(PCA)isaalgo-
thermodynamicconditions,themicrocanonical,canonical,,theinformation
andgrandcanonicalprobabilitydensitiesofmicrostateacrossthefulldatasetiseffectivelycompressedinfewer
weregivenbyGibbs.[29]
equilibriumsystemwithknownenergyfunctioncanbecal