The_Chemical_Thermodynamics_for_Growing_Systems

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TheChemicalThermodynamicsforGrowingSystemsYukiSughiyama,∗AtsushiKamimura,∗DimitriLoutchko,andTetsuyaJ.Kobayashi†InstituteofIndustrialScience,TheUniversityofTokyo,4-6-1,Komaba,Meguro-ku,Tokyo153-8505Japan(Dated:January25,2022)Weconsidergrowingopenchemicalreactionsystems(CRSs),inwhichautocatalyticchemicalreactionsareencapsulatedinafinitevolumeanditssizecanchangeinconjunctionwiththereac-tions.ThethermodynamicsofgrowingCRSsisindispensableforunderstandingbiologicalcellsanddesigningprotocellsbyclarifyingthephysicalconditionsandcostsfortheirgrowingstates.Inthiswork,weestablishathermodynamictheoryofgrowingCRSsbyextendingtheHessiangeometricstructureofnon-growingCRSs.ThetheoryprovidestheenvironmentalconditionstodeterminethefateofthegrowingCRSs;growth,shrinkingorequilibration.Wealsoidentifythermodynamicconstraints;onetorestrictthepossiblestatesofthegrowingCRSsandtheothertofurtherlimittheregionwhereanonequilibriumsteadygrowingstatecanexist.Moreover,weevaluatetheentropyproductionrateinthesteadygrowingstate.Thegrowingnonequilibriumstatehasitsoriginintheextensivityofthermodynamics,whichisdifferentfromtheconventionalnonequilibriumstateswithconstantvolume.Theseresultsarederivedfromgeneralthermodynamicconsiderationswithoutassuminganyspecificthermodynamicpotentialsorreactionkinetics;i.e.,theyareobtainedbasedsolelyonthesecondlawofthermodynamics.I.INTRODUCTIONandgrowthshouldaccompanythethermodynamiccost.However,welackatheoreticalbasistoaddressthesefun-Self-replicationisahallmarkoflivingsystemsbydamentalproblemsofgrowingsystems.whichtheyaredifferentiatedfromnonlivingones.SinceInthiswork,weestablishthethermodynamicsforvonNeumann’sformulationofself-reproducingautomatagrowingsystems.Thedifficultyindevelopingitliesin[1,2],thephysicalandchemicalbasisofself-replicationthatthechangeinthevolumeaffectsallreactionsinit.hasbeenpursuedtheoreticallyandexperimentallyinor-Intheconventionaltheoryofchemicalreactions,reactiondertounderstandandsynthesizelivingsystems[3–24].fluxesaredescribedasfunctionsofdensitiesofchemicalsOfthevariouscomponentsnecessaryforself-replication,(concentrations)[43–50],whichpresumesaconstantvol-autocatalyticreactioncycles,thoughtofasthedrivingume.However,ifthevolumechanges,thedensitiescanengine,formacentralpart[25–31].However,thepres-changeeventhoughthenumbersofchemicalsremainun-enceofcyclesisnotsufficientforself-replication.Becausechanged.Hence,itisnecessarytoreturntoathermo-thecyclesshouldbeconfinedinanencapsulatingvolumedynamicformulationinwhichthenumbersofchemicalswhichdefinesthereplicationunit,thesizeofthevolumeandthevolumearetreatedseparately.Inotherwords,shouldalsogrowinaccordancewiththeproductionofwehavetoexplicitlyaccountoftheextensivityofther-chemicalsbythecycles.modynamicfunctions,whichisscaledoutwhentheden-Inspiteoftheactiveinvestigationofautocatalyticre-sitiesaloneareconsidered.Nevertheless,weshouldalsoactioncyclesinthelastdecades[25–31],thegrowthofretainthedensityrepresentationanditsdualrepresenta-volumeanditscouplingwiththeautocatalyticcyclestionbythechemicalpotentialstoappropriatelycharac-havenotbeenthoroughlyinvestigatedsofar.Althoughterizesteadygrowingstatesandtheconditionsimposedtherecentrediscoveryofgrowthlawsofbacteria[32]ini-bytheintensivevariablesoftheenvironment.tiatedasurgeofnewcoarse-grainedautocatalyticmodelsWeclarifythisentangledrelationamongthetriadof[10–12,33–38],thevolumegrowthinthesemodelsiscon-chemicalnumbers,densitiesandpotentialsbyclarifyingsideredonlyheuristically[5,39–42],e.g.,byrepresentingthegeometricstructuretheyform.ThisstructureisbuiltarXiv:2201.09417v1[cond-mat.stat-mech]24Jan2022itwithalinearfunctionofchemicalsinit.ontherecentlydiscoveredHessiangeometricstructureInthelightofchemicalthermodynamics,thechangeinbetweenchemicaldensitiesandpotentialsinaconstantvolumeandtheinfluxandoutfluxofchemicalsdrivenbyvolume[51,52]byadditionallyintroducingthespacethecyclesaremutuallydependentandshouldbether-ofthenumbersofchemicals.Basedonthesecondlawmodynamicallyconsistent.Thisinterdependenceofre-ofthermodynamics,ourtheoryclassifiesthethermody-actionsandvolumeinevitablyconstraintheirpossiblenamicconditionsunderwhichthesystemgrows,shrinksstatesanddynamics.Inaddition,thecyclesthemselvesorequilibrates.Italsorevealstheregioninwhichthemaynotalwaysproceedintheforwarddirection,whichchemicaldensityisconstrainedtoasteadygrowth.Fur-canresultinshrinking,dependingontheenvironmentalthermore,itenablesustoevaluatetheentropyproduc-conditions.Itisnontrivialunderwhatthermodynamictionrate,i.e.,thephysicalcostofthesteadygrowth.Ourconditionsacoherentforwardcycledynamicsandvol-nonequilibriumsystemwithvolumegrowthhasitsoriginumegrowthcanbeachieved.Moreover,asteadycyclingintheextensivityofthermodynamics,whichisdifferent

12fromtheconventionalnonequilibriumsystemswithcon-stantvolume[43–50].Weemphasizethatourderivationisperformedbasedonapurelythermodynamicargument[51–54].Asare-sult,itdoesnotdependonanyparticularformofthermo-dynamicpotentialsorreactionkinetics.Thus,ourtheoryiswidelyapplicableandcontributestounderstandingtheoriginsoflifeandconstructingprotocells[13–24]aswellasseekingtheuniversallawsofbiologicalcells[10–12,32–38].Thispaperisorganizedasfollows.WedevoteSec.IItooutlineourmainresultswithoutshowingthedetailsoftheirderivation.FromSec.IIIonward,westartwiththederivationofourmainresults.InSec.III,weanalyzethebehaviorofthetotalentropyfunctionwithrespecttotimeforchemicalreactiondynamics.WedevoteSec.IVtothepreparationforthegeometricstructureofgrowingsystems.InSec.V,weclassifytheenvironmentalcon-ditionstodeterminethefateofthesystembasedontheformofthetotalentropyfunction.InSec.VI,wecon-siderthesteadygrowingstateandevaluatetheentropyproductionrateinthisstate.WeillustrateourtheoryinSec.VIIontheidealgasasaspecificexampleofthermodynamicpotentials.InSec.VIII,wenumericallyFIG.1.DiagrammaticrepresentationofopenCRSs.TheverifyourtheorybyconsideringaspecificexampleofachemicalreactionsoccurwiththereactionfluxesJ(t)=chemicalreactionsystemcomposedoftheidealgasandr{J(t)},therthreactionofwhichisrepresentedasthechem-obeyingmassactionkinetics.Finally,wesummarizeouricalequationatthebottom.Here,A={Ai}arethelabelsworkwithfurtherdiscussionsinSec.IX.oftheconfinedchemicals,andB={Bm}aretheonesoftheopenchemicalswhichcanmoveacrossthemembranemwiththediffusionfluxesJD(t)={JD(t)}.ThenumbersII.OUTLINEOFTHEMAINRESULTSoftheconfinedandopenchemicalsinthesystemarede-imnotedbyX={X}andN={N},respectively.Also,imA.Thermodynamicsetup(S+)rand(O+)rdenotestoichiometriccoefficientsofthere-imactantsinrthreaction,whereas(S−)rand(O−)raretheonesoftheproducts.ThestoichiometricmatricesaregivenLetusstartwiththepresentationofthesettingoftheiiimmmasSr=(S−)r−(S+)randOr=(O−)r−(O+)r.Fortheo-system(FIG.1).Consideragrowingopenchemicalreac-reticalsimplicity,weignorethetensionofthemembraneandtionsystem(CRS)surroundedbyareservoir.Weassumeassumethatitneverbursts.thatthesystemisalwaysinawell-mixedstate(alocalequilibriumstate),andthereforewecancompletelyde-scribeitbyextensivevariables(E,Ω,N,X).Here,EforthereservoirasΣ˜˜[E,˜Ω˜,N˜],andthereforetheto-T,Π˜,µ˜andΩrepresenttheinternalenergyandthevolume;talentropycanbeexpressedasN={Nm}denotesthenumberofchemicalsthatcanmoveacrossthemembranebetweenthesystemandthehiΣtot=Σ[E,Ω,N,X]+Σ˜˜E,˜Ω˜,N˜,(1)reservoircalledopenchemicals;meanwhile,X=XiT,Π˜,µ˜isthenumberofchemicalsconfinedwithinthesystem;theindicesmandirespectivelyrunfromm=1toNNwhereweusetheadditivityoftheentropy.Furthermore,andfromi=1toNX,whereNNandNXarethenum-duetothehomogeneityoftheentropyfunctionforthebersofspeciesoftheopenandconfinedchemicals.Thesystem,withoutlossofgenerality,wecanwriteitasreservoirischaracterizedbyintensivevariables(T,˜Π˜,µ˜),whereT˜andΠarethetemperatureandthepressure;˜Σ[E,Ω,N,X]=Ωσ[ǫ,n,x],(2)µ˜={µ˜m}isthechemicalpotentialcorrespondingtotheopenchemicals.Also,wedenotethecorrespondingex-tensivevariablesby(E,˜Ω˜,N˜).whereσ[ǫ,n,x]istheentropydensityand(ǫ,n,x):=Inthermodynamics,theentropyfunctionisdefined(E/Ω,N/Ω,X/Ω).Sincethisworkonlytreatsasituationon(E,Ω,N,X)asaconcave,smoothandhomogeneouswithoutphasetransitions,weassumethatσ[ǫ,n,x]isfunctionΣ[E,Ω,N,X].Wewritetheentropyfunctionstrictlyconcave.

23Next,wedefinethedynamicsforthesystemasdEdΩ=JE(t),=JΩ(t),dtdtdNmdXi=OmJr(t)+Jm(t),=SiJr(t),(3)rDrdtdtwhereJ(t),J(t),J(t)={Jm(t)}andJ(t)=EΩDD{Jr(t)}representtheenergy,thevolume,thechemicaldiffusionandthechemicalreactionfluxes,respectively;S=SiandO={Om}denotestoichiometricmatri-rrcesfortheconfinedandtheopenchemicals(seeFIG.1).Theindexrrunsfromr=1toNR,whereNRisthenumberofreactions.Also,inEq.(3),weemployedEin-stein’ssummationconventionfornotationalsimplicity.ThedynamicsofthereservoirisgivenasFIG.2.Diagrammaticrepresentationof(a)isochoricand(b)isobaricsituations.(a)Intheisochoriccase,theexternaldE˜dΩ˜dN˜mpressureΠvariestokeepthevolumeΩconstant.Theinternal˜=−J(t),=−J(t),=−Jm(t).(4)∗EΩDpressureϕ(y),whichalwaysbalanceswithΠ,canconvergeto˜dtdtdt∗EQEQthechemicalequilibriumpressureϕ(y)=Π˜.(b)IntheInthiswork,weassumethatthetimescaleoftheisobariccase,thevolumeΩvariestokeeptheinternalpres-∗reactionsismuchslowerthanthatoftheothers(thatsureϕ(y)alwaysequaltotheconstantexternalpressureΠ.˜∗is,JE(t),JΩ(t),JD(t)≫J(t)).Therefore,ourdynam-Consequently,theinternalpressureϕ(y)=Πmaynotbal-˜∗EQancewiththechemicalequilibriumpressureϕ(y),whichicsiseffectivelygovernedonlybythereactionfluxJ(t)isspecifiedbythechemicalpotentials˜µinthereservoir.This(seeSec.IIIfordetails).Inaddition,weconsiderimbalancedrivesgrowingorshrinkingofthevolume.minimalmotifsofautocatalyticcycles,whichwereiden-tifiedin[26]andcharacterizedbytheregularityofthestoichiometricmatrixSfortheconfinedchemicals,i.e.,x.Thisdualisticrepresentationiscentraltoourtheory.NX=NR=Rank[S](seeAppendixAfordetails).Inaddition,ϕ∗(y)canbeinterpretedasthepressureofthesystematthestateywhosecorrespondingdensityisx=∂ϕ∗(y).B.Thermodynamicpotentials,duality,andtotal∗Ifthevolumeisfixed,theinternalpressureϕ(y)al-entropycharacterizingthegrowingsystemswaysbalanceswiththeexternalpressureΠincurredby˜theboundarytokeepthevolumeΩconstant(seeFIG.˜Withtheabovesetup,weobtainaconjugatepairof2(a)).Furthermore,theinternalpressureϕ∗(y)=Πcon-˜∗thermodynamicpotentials,ϕ(x)andϕ(y),whichplayvergestothepressureϕ∗(yEQ)=Π˜EQatthechemicalpivotalrolesinourtheory.ThepartialgrandpotentialequilibriumstateyEQ.ThestateyEQisdeterminedbydensityϕ(x)=ϕ[T,˜µ˜;x]isdefinedasthechemicalpotentials˜µinthereservoirashino
ϕT,˜µ˜;x:=minǫ−Tσ˜[ǫ,n,x]−µ˜nm,(5)yEQ=−µ˜OmS−1r,(7)ǫ,nmimri−1(seeSec.IVfordetails).Thefunctionϕ∗(y)=whereSistheinverseofthestoichiometricmatrixS.ϕ∗[T,˜µ˜;y]isthefullgrandpotentialdensityobtainedInthedensityrepresentation,thesystemconvergestothechemicalequilibriumstatex=∂ϕ∗(yEQ).bytheLegendretransformationofϕ(x)asEQBycontrast,ingrowingsystemsunderisobariccondi-hiϕ∗T,˜µ˜;y:=maxyixi−ϕ(x).(6)tions,thevolumecanchange.DuetothefasttimescalexofthevolumefluxJ(t),theinternalpressureϕ∗(y)isΩfixedbytheexternal(reservoir)oneΠ(seeFIG.2(b)).˜Inconventionalchemicalthermodynamicswithacon-∗Asaresult,thevolumeatXisvariationallydeterminedstantvolume,ϕ(x)andϕ(y)characterizethesystemascompletely.Theyalsoworkasthedualconvexfunc-tionsinducingtheHessiangeometricstructureofchem-XΩ(X)=argminΩϕ+ΠΩ˜.(8)icalthermodynamics[51,52].Becauseoftheone-to-oneΩΩcorrespondenceoftheLegendretransformationinducedbyϕ(x)andϕ∗(y),wecanequivalentlyspecifyastateAlso,thechemicaldensityxisanonlinearfunctionofthesystemeitherbythedensitiesxorbyitsLeg-ρX(X)ofXasx(X)=X/Ω(X)=:ρX(X).endretransformy=∂ϕ(x).Thethermodynamicinter-Inthiscase,theinternalpressureϕ∗(y)isrestrictedtopretationofyisthecorrespondingchemicalpotentialtotheconstantexternalpressureΠ,whereasthechemical˜

34equilibriumpressureϕ∗(yEQ)isspecifiedbythechemicalpotentials˜µinthereservoir.Ifϕ∗(y)=Πand˜ϕ∗(yEQ)arenotbalanced,thesystemcannotconvergetotheequilibriumstate,andthisimbalancedrivesgrowthorshrinkingofthevolume.Whethergrowthorshrinkingoccursisdeterminedbythesecondlawandthefunctionalformoftotalentropy,whichisrepresentedforgrowingsystemsastotΩ(X)YYΣ(X)=K(ρ(X))+const.,(9)T˜whereKY(y)isdefinedas
KY(y):=ϕ∗yEQ−Π˜−DYyEQ||y.(10)Here,DY[y′||y]istheBregmandivergence[55–57]in-ducedbyϕ∗(y),andy=ρY(X)=∂ϕ(ρ(X))isanon-XlinearmaptoassociatethenumberofchemicalsXwithFIG.3.(a)Agraphrepresentationandchemicalequationsofachemicalpotentialy.aminimalmotifofautocatalyticcycles.Twoconfinedchem-icalsA=(A1,A2)andtwoopenchemicalsB=(B1,B2)undergothetworeactionsR1andR2.(b)Thetimeevo-C.Theconditionsforgrowth,shirinking,andlutionofthevolumeofthesystemfordifferentparameterequilibrationsets(seethecaptioninFIG.8forspecificvaluesofthepa-rameters).Thefateofthesystemisclassifiedbythesign∗EQofϕ(y)−Π.˜(c)ThetimeevolutionofthedensitiesOurfirstclaimprovidestheconditionthatdetermines12(x,x)oftheconfinedchemicals(A1,A2)forthegrowthcasethefateofthesystem,i.e.,growth,shirinkingorequili-ϕ∗(yEQ)−Π˜>0.Theevolutionsareshownfortwodifferentbration.initialconditions1and2.(d)Thetrajectoriesofthesysteminthedensityspace.TheyareconstrainedtotheisobaricXClaim1ThefateofthesystemisclassifiedbythesignmanifoldI(Π˜,µ˜).Inthisexample,thesystemconvergestoofϕ∗(yEQ)−Π˜asfollows:asteadygrowingstatexSG(greensquare),irrespectiveoftheinitialconditions.Suchasteadygrowingstatemustbeinthe
XX∗EQregionR(Π˜,µ˜)⊂I(Π˜,µ˜),highlightedbythereddashed1.Ifandonlyifϕy−Π=0˜,equilibriumstatesrectangle.existandthesystemconvergestooneofthem.2.Ifandonlyifϕ∗(yEQ)−Π˜<0,thesystemeventu-berepresentedasallyshrinksandfinallyvanishes.
R1R2R1R23.Ifandonlyifϕ∗yEQ−Π˜>0,thesystemisgrow-!!ing.A1−11B1−10S=,O=.(11)A22−1B201(seeSec.VandTheorem1fordetails).TheregularityofthematrixSisconfirmedbydet[S]=Thisresultindicatesthatthesystemequilibratesonly−16=0.DenotingthenumberofA=(A1,A2)byX=(X1,X2),thereactiondynamicsfortheconfinedifthepressureΠspecifiedbythereservoirhappensto˜coincidewiththechemicalequilibriumpressureϕ∗(yEQ)chemicalsiswrittenasspecifiedbythereservoirchemicalpotentials˜µ.Other-idXirwise,thesystemshrinksorgrows.=SrJ(t).(12)dtExample1:Togiveanintuitiveillustration,wecon-Inthisexample,weemploymassactionkineticswiththesideraminimalmotifofautocatalyticcycles(seeFIG.localdetailedbalancecondition[43,44,51,52,58]forthe3(a)).Here,twoconfinedchemicalsA=(A1,A2)andreactionfluxJ(t)(seeSec.VIIIfordetails).Further-twoopenchemicalsB=(B1,B2)areinvolvedinthetwomore,weassumetheidealgaspotential:thefunctionalformofϕ∗(y)isobtainedasreactionsR1andR2.Wecanregardtheopenchem-icalsB1andB2asaresourceandwaste,respectively,"#Xyi−νo(T˜)Xµ˜m−µo(T˜)becausetheyareconsumedandproducedwhenthefor-∗imϕ(y)=RT˜eRT˜+eRT˜,wardreactionsprogress.Thestoichiometricmatricescanim

45(seeEq.(59)inSec.VIIforaderivation).ThevolumeΩ(t)canbecalculatedbytheequationofstate,PRT˜XiΩ(X)=i,(13)PΠ˜−RT˜n˜mm(seeEq.(67)inSec.VIIfordetails).InFIG.3(b),weverifiedClaim1bynumericalsimulation.Indeed,theFIG.4.Transitionfromtheshrinkingtothegrowingcase.YY∗EQTheisobaricmanifoldI(Π˜,µ˜)andtheregionZ(˜µ)inthefateofthesystemisclassifiedbythesignofϕ(y)−Π.˜chemicalpotentialspaceareindicatedbythesolidcurveand∗EQthelightpurplecolor,respectively.(a)Whenϕ(y)−Π˜<0holds(i.e.,theshrinkingcase),theintersectiondoesnot∗EQexist.(b)Whenϕ(y)−Π=0holds(i.e.,theequilibriating˜D.Thermodynamicconstraintofisobaricdynamicscase),transitionfromshrinkingtogrowingoccurs.(c)When∗EQϕ(y)−Π˜>0holds(i.e.,thegrowingcase),theintersectionUnderisobaricconditionswithafastvolumefluxRY(Π˜,µ˜)=IY(Π˜,µ˜)∩ZY(˜µ)exists,whichishighlightedbyJΩ(t),thepressureofthesystemshouldbalancewiththecurvedredrectangle.thepressureofthereservoir.Thisconstraintnaturallydefinestheisobaricmanifoldinthechemicalpotentialspace:YYYshouldlieintheregionR(Π˜,µ˜)=I(Π˜,µ˜)∩Z(˜µ).noIYΠ˜,µ˜:=y|ϕ∗(y)−Π=0˜.(14)Here,
ItsLegendretransformIX(Π˜,µ˜):=∂ϕ∗(IY)isahy-ZY(˜µ):=y|ϕ∗yEQ−ϕ∗(y)−DYyEQ||y>0,persurfaceinthedensityspace.Thus,IX(Π˜,µ˜)and(16)IY(Π˜,µ˜)characterizethethermodynamicallyadmissi-designatestheregioninwhichthepositivityofentropyblesubmanifoldsinthedensityandchemicalpotentialproductionisguaranteed.Bytransferringthisconditionspaces,respectively.intothedensityspacebytheLegendretransformation,wehavethefollowingclaimforxSG:Example2:FortheautocatalyticmotifinFIG.3(a),thetimeevolutionofx(t)isshowninFIG.3(c)forthe
Claim2Whenϕ∗yEQ−Π˜>0andasteadygrow-growthcaseinFIG.3(b).ThistimeevolutionisactuallyconstrainedtotheisobaricmanifoldIX(Π˜,µ˜)asshowningstatexSGexists,thestatexSGmustbeintheregionRX(Π˜,µ˜),whereRX(Π˜,µ˜)=∂ϕ∗(RY).TheentropyinFIG.3(d).Sincewehaveassumedidealgaspoten-tials,theisobaricmanifoldIX(Π˜,µ˜)reducestoasimplexproductionrateatthestatexSGisrepresentedasEq.bytheequationofstate(seeSec.VIIfordetails).(15).(SeeSec.VIandTheorem2forthedetails)E.TheconstraintsandthermodynamicpropertiesassociatedwiththesteadygrowingstateExample3:FortheautocatalyticmotifinFIG.3(a),thesteadygrowingstatexSGisindeedlocatedwithintheFinally,weclarifytheadditionalconstraintimposedregionRX(Π˜,µ˜)(seeFIG.3(d)).onthesteadygrowingstatexSG.ThesteadygrowingMoreover,wecanverifythatthetransitionfromstateisdefinedasastatesuchthatthedensityremainstheshrinkingtothegrowingcaseoccurswhenthein-constantwithtimewhereasthevolumekeepsincreasingYYtersectionbetweenI(Π˜,µ˜)andZ(˜µ)appears(see[39–42].FortheautocatalyticmotifshowninFIG.3(a),FIG.4(b)).InFIG.4,theisobaricmanifoldIY(Π˜,µ˜)suchastatexSGexistsandx(t)convergestoasteadyYandtheregionZ(˜µ)areindicatedinthechemi-growingstateasinFIG.3(c,d).calpotentialspace.Fortheshrinkingcase(FIG.Atthisstate,theentropyproductionratecanbeex-4(a)),theintersectionisempty.Bycontrast,forpressedasthegrowingcase(FIG.4(c)),theintersectionexists.Ω(˙t)
Σ˙tot(Ω(t)x)=KYySG,(15)SGT˜Thisconcludestheoutlineofallourmainresults,whichconsistoftheconditionofgrowth,theconstraintsofwhereySGtheLegendretransformofxby∂ϕ.Becausegrowingsystemsandsteadygrowingstates,andtheSGΩ(˙t)>0atthegrowingstate,KY(ySG)shouldbeposi-formsoftotalentropyandentropyproductionatthetivebythesecondlaw.ThisrequirementimpliesthatySGsteadygrowingstate.

56III.THERMODYNAMICSFORGROWING(17)withtheinitialcondition(X0,E˜(0),Ω(0)˜,N˜(0))SYSTEMSisrepresentedasXi(t)=Xi+SiΞr(t),Fromthissectiononward,weworkonthederivationof0rourmainclaimsintroducedinSec.II.Inthissection,weE˜(t)=E˜(0)−EQEQ(X(t)),derivetheformofthetotalentropy,Eq.(1),morespecif-Ω(˜t)=Ω(0)˜−ΩQEQ(X(t)),icallybyemployingtime-scaleseparation.Asaresult,N˜m(t)=N˜m(0)+OmΞr(t)−Nm(X(t)),(21)wewillobtainthetotalentropyfunctionforthereactionrQEQdynamics,Eq.(23).Also,wewillshowthat,giventhewhereΞ(t)={Ξr(t)}istheintegrationofJ(t)withnumberoftheconfinedchemicalsX,thevolumeΩofthetheinitialconditionΞ(0)=0;thisisknownastheex-systemisdeterminedbythevariationalform,Eq.(19),tentofreactioninchemistry.Sincewehaveassumedwiththepartialgrandpotential,Eq.(18).thatSisregular,therearenostoichiometricconstraintsSincewehaveassumedthatJE(t),JΩ(t),JD(t)≫thatrestrictattainablestateofX(t)byitsinitialstateJ(t),wecananalyzethedynamics,Eqs.(3)and(4),X(0);i.e.,thestoichiometriccompatibilityclass[50–52]byseparatingthefastonesJ(t),J(t),J(t)andtheNXEΩDbecomesR>0.Furthermore,byusingtheinversematrixslowoneJ(t).BysolvingthefastdynamicsusingtheS−1,thelastequationinEq.(21)canberewrittenassecondlaw(seeAppendixB),weobtaintheeffectiveslow
rN˜m(t)=OmS−1Xi(t)−Nm(X(t))+const.,dynamics(thereactiondynamics)asriQEQ(22)
rdXidE˜dE(X)wherewesubstituteΞr(t)=S−1Xi(t)−Xiinto=SiJr(t),=−QEQ,i0dtrdtdtthelastequationandabbreviatethetermscomposedofdΩ˜dΩ(X)dN˜mdNm(X)theinitialconditionto“const.”.Therepresentationof=−QEQ,=OmJr(t)−QEQ,rEq.(22)impliesthatourreactiondynamicscanbecom-dtdtdtdtpletelydescribedonlybythetimeevolutionofthecon-(17)finedchemicals,X(t).where(·)representsthevalueattheequilibriumstateNext,weconsiderthetimeevolutionofthetotalen-QEQofthefastdynamics.Wecallthisthequasi-equilibriumtropyduringthereactiondynamics.BysubstitutingEqs.state,becauseitisnottheequilibriumstateoftheslow(21)and(22)intoEq.(1),weobtaindynamics.Byusingthepartialgrandpotential:1Π˜hinoΣtot(X)=Σ(X)−E(X)−Ω(X)QEQQEQQEQΦT,˜µ˜;Ω,X:=minE−T˜Σ[E,Ω,N,X]−µ˜Nm,T˜T˜mE,Nµ˜mmµ˜mm−1
ri(18)+T˜NQEQ(X)−T˜OrSiX+const.thevolumeatthequasi-equilibriumstatewiththenum-nhio=−1ΦT,˜µ˜;Ω,X+ΠΩ˜(X)−yEQXiberoftheconfinedchemicalsXcanbeevaluatedbytheT˜QEQQEQivariationalform:+const.,(23)nhioΩQEQ(X)=argminΦT,˜µ˜;Ω,X+ΠΩ˜.(19)whereweemploytheTaylorexpansionforΣ˜˜andΩT,Π˜,µ˜thepartialgrandpotential,Eq.(18);forsimplicity,weInaddition,theotherextensivevariablescanbecalcu-alsodefinelatedbydifferentiationsofΦ[T,˜µ˜;ΩQEQ,X]asEQ
ry:=−µ˜OmS−1.(24)hiimri∂ΦT,˜µ˜;ΩQEQ,XΣQEQ(X)=−,Accordingtothesecondlaw,thesystemmustclimb∂T˜upthelandscapedeterminedbytheconcavefunctionhiT,˜µ˜;ΩΣtot(X)andfinallyconvergetoitsmaximum,whichis∂ΦQEQ,XNm(X)=−,calledtheequilibriumstate[59],ifitexists.Therefore,QEQ∂µ˜mtoelucidatethefateofthesystem,itisimportanttohihi∂ΦT,˜µ˜;Ω,XanalyzetheformoftheconcavefunctionΣtot(X).WeQEQEQEQ(X)=ΦT,˜µ˜;ΩQEQ,X−T˜canbrieflyclassifytheformofΣtot(X)intothefollow-∂T˜tothiingthreecases:(1)IfΣ(X)isboundedaboveandthe∂ΦT,˜µ˜;ΩQEQ,Xpointsattainingitsmaximumareintheinteriorofthe−µ˜m,(20)domainofX,thatisargmax{Σtot(X)}∈RNX,equi-∂µ˜mX>0libriumstatesexistandthesystemconvergestooneofwhereΣ(X)istheabbreviationofthem.(2)IfΣtot(X)isboundedaboveandthemaxi-QEQΣ[E,Ω,N,X].ThedetailsofthederivationmumofΣtot(X)isatX=0,thevolumeΩ(X(t))QEQQEQQEQQEQareshowninAppendixB.TheformalsolutionofEq.eventuallyshrinksandfinallyvanishes.(3)IfΣtot(X)

67isnotboundedabove,X(t)divergesinthereactiondy-is,wedefinetheprojectivespaceofXasPX.Elementsnamics.Also,thevolumeΩQEQ(X)divergesforX→∞,ofPXaresubsetsr⊂XsuchthatanytwopointsXandbecauseofthehomogeneityofthevolume.Thissitua-X′areinthesamesubset(i.e.,ontherayr[60])ifandtioncorrespondstothegrowthofthesystem.ThemainonlyifX=αX′(α>0)(seeFIG.5(a)).DuetoEq.aimofthisworkistorevealwhatconditiondistinguishes(29),themapρXdescendstoawell-definedmapfromthesethreecases.Intheremainingpartofthispaper,PXtoX:wewilladdressthisproblembyemployingHessianandprojectivegeometry.ρ¯X:r∈PX7→ρ¯X(r)=ρX(X)∈X,forX∈r.(30)Also,Eq.(28)impliesthattheabovemapisinjectiveIV.PREPARATIONFORAGEOMETRIC[61].Therangeofthemap¯ρX,denotedbyRan[¯ρX],isREPRESENTATIONOFISOBARICCHEMICALcalculatedbysolvingthevariationalform,Eq.(27),asREACTIONSYSTEMSfollows.ThecriticalequationofEq.(27)isgivenasXXiXWedevotethissectiontopreparationforthegeometricϕ−∂iϕ+Π=0˜,(31)representationofoursystem.AsmentionedinSec.II,ΩΩΩthehomogeneityofthesystementropyfunctionallowswhere∂ϕ(X/Ω)=∂ϕ(x)/∂xi.Therefore,byustowriteitasix=X/Ωdefiningtheisobaricmanifold:Σ[E,Ω,N,X]=Ωσ[ǫ,n,x],(25)noIXΠ˜,µ˜:=x|ϕ(x)−xi∂ϕ(x)+Π=0˜,(32)iwhereσ[ǫ,n,x]representstheentropydensityand(ǫ,n.x):=(E/Ω,N/Ω,X/Ω);also,wehaveassumedthatweobtainRan[¯ρ]=IX(Π˜,µ˜).Inotherwords,thetimeXσ[ǫ,n,x]isstrictlyconcave.Weintroducethenum-evolutionofthedensityx(t),givenbyEq.(17),iscon-berandthedensityspacesoftheconfinedchemicalsstrainedwithinthissubmanifoldinX.IfwerestrictNXNXasX∈X=R>0andx∈X=R>0,respectively.thecodomainof¯ρtoIX(Π˜,µ˜),themap¯ρ:PX→XXAlso,wedefinethepartialgrandpotentialdensityasIX(Π˜,µ˜)isbijective.Foralateranalysis,wealsode-ϕ(x)=ϕ[T,˜µ˜;x]:=Ω−1Φ[T,˜µ˜;Ω,X]=Φ[T,˜µ˜;1,X/Ω],−1Xfinetheinversemapof¯ρXas¯ρX:I(Π˜,µ˜)→PX,whereweusethehomogeneityofΦ.FromthedefinitionwhichgivesthecorrespondingraytoagivendensityofΦ,Eq.(18),ϕ[T,˜µ˜;x]canberepresentedbyavariantx∈IX(Π˜,µ˜)(seeFIG.5(a)).oftheLegendretransformationofσ[ǫ,n,x]asFinally,weintroducethedualspaceofthedensityhinospaceXasy∈Y=RNX,whichhasthethermody-ϕT,˜µ˜;x=minǫ−Tσ˜[ǫ,n,x]−µ˜nm,(26)mnamicinterpretationasthespaceofchemicalpotentials.ǫ,nAlso,wedefineamapfromXtoYbyusingtheconvexandthereforeϕ(x)isstrictlyconvex.Byusingϕ(x),wefunctionϕ(x)ascanrewriteEq.(19)as∂ϕ∂ϕ:x∈X7→∂ϕ(x)={∂iϕ}=∈Y,(33)X∂xiΩ(X)=ΩQEQ(X)=argminΩϕ+ΠΩ˜.ΩΩ(27)whichyieldsthevaluesofthechemicalpotentialsataFornotationalsimplicity,weomitthesubscript(·),statex.Sinceϕ(x)isstrictlyconvex,themap∂ϕisQEQhereafter.Duetothestrictconvexityofϕ(x),thevolumeinjective.Furthermore,forordinarychemicalreactionsystems,Ran[∂ϕ]isRNX;thus∂ϕisbijective.Tocon-Ω(X)uniquelyexistsforanygivenX(seeAppendixC).Next,wedefineamapfromthenumberspaceXtothestructtheinversemapof∂ϕ,wedefinethestrictlycon-vexfunctionϕ∗(y)onthedualspaceYbytheLegendredensityspaceX:transformation:XiiρX:X∈X7→ρX(X)=ρX(X)=∈X,(28)ϕ∗(y):=maxyxi−ϕ(x),(34)Ω(X)xiwhichgivesthedensitiesoftheconfinedchemicalsatthewhichcorrespondstothefullgrandpotentialdensityandquasi-equilibriumstatewithX.Becauseofhomogeneity,givesthepressureofthesystematastatey.EmployingΩ(αX)=αΩ(X)(α>0),whichisguaranteedbyEq.ϕ∗(y),wecanrepresenttheinversemapas(27),themapρXsatisfies∂ϕ∗∗∗i∗∂ϕ:y∈Y7→∂ϕ(y)=∂ϕ=∈X.(35)ρX(αX)=ρX(X).(29)∂yiToclarifythegeometricinterpretationofthemapρX,Thesetwospaces,XandY,togetherwiththepairofweregardthenumberspaceXasacollectionofrays;thatconvexfunctions,ϕ(x)andϕ∗(y),constitutetheHessian

78geometricstructureofchemicalthermodynamics.Thestructureisfundamentaltocapturethegeometricrela-tionbetweenthetwodualspacesandisusedintensivelyinthesubsequentderivation.TheisobaricmanifoldIX(Π˜,µ˜)inXismappedvia∂ϕtothechemicalpotentialspaceYas
noIYΠ˜,µ˜:=∂ϕIX=y|ϕ∗(y)−Π=0˜,(36)whichisalevelhypersurfaceforthedualconvexfunctionϕ∗(y)(seeFIG.5(a)).Inaddition,wedefinethemapfromXtoIY(Π˜,µ˜)⊂YasρY:=∂ϕ◦ρ,whichalsoXinducesthemap:ρ¯Y:r∈PX7→ρ¯Y(r)=ρY(X)∈IY,forX∈r.(37)Sincethismapisbijective,wedefineinversemapasY
−1−1∗ρ¯=¯ρ◦∂ϕ(seeFIG.5(b)).ThefactthattheXisobaricmanifoldisidenticaltoalevelhypersurfaceforapotentialfunctionisoneofthefundamentalconstituentsintheHessiangeometry.V.FORMOFTHETOTALENTROPYFUNCTIONANDTHEFATEOFTHESYSTEMWiththepreparationintheprevioussection,weareinthepositiontorevealtheformofthetotalentropyfunction,Eq.(23),andpredictthefateofthesystem.Forthispurpose,weintroducetheBregmandivergence[55–57]onY:FIG.5.Diagrammaticrepresentationofthetriadofspaces,Y′∗∗′i∗′′NXD[y||y]:={ϕ(y)−ϕ(y)}−∂ϕ(y){yi−yi},(X,PX),X,andY.(a)Thetopspace,X=R>0,represents(38)thenumberoftheconfinedchemicalsX.WealsodefinethewhichmeasuresthedeviationatthepointybetweensetofraysinXasPX.Anelementr∈PXisasubsetof′∗XsuchthatanytwopointsXandXontherayrsatisfytheconvexfunctionϕ(y)andthetangentplaneatthe′′X=αX(α>0).Thespacesonthebottomleftandbottompointy.Thisdivergencehasthefollowingproperty:NXY′′rightrepresentthedensityspaceX=R>0andthechemicalD[y||y]≥0,theequalityholdsifandonlyify=yNXpotentialspaceY=R,respectively.Arayr∈PXandaandthereforeitactsasanasymmetricdistancefromy′XpointxintheisobaricmanifoldI(Π˜,µ˜)⊂Xaremappedtoy.TheBregmandivergenceisalsooneofthefunda-toeachotherby¯ρ−1X(r)and¯ρ(x).Similarly,arayrandaXmentalconstitutesofHessiangeometry.pointyintheisobaricmanifoldIY(Π˜,µ˜)⊂YaremappedWerewritethetotalentropyfunctionEq.(23)byus-toeachotherby¯ρY(r)andρ¯Y
−1(y).Thespaces,Xandingthedivergenceasfollows.UsingthepartialgrandY,aremappedtoeachotherby∂ϕand∂ϕ∗.(b)Themap−1potentialdensityϕ(x),Eq.(23)canberepresentedasρ¯XfromPXtoXanditsinverse¯ρXcanberepresentedbythecompositionoftwomapsviathespaceY(topline).Ω(X)noYY
−1totEQiSimilarly,themap¯ρfromPXtoYanditsinverseρ¯Σ(X)=−T˜ϕ(ρX(X))−yiρX(X)+Π˜,areexpressedbythecompositionoftwomapsviathespace(39)X(bottomline).wherethemapρXisdefinedinEq.(28)andweneglecttheconstantterm.ThisequationisfurtherrearrangedwhereweuseEq.(32)inthefirstlineandρ=∂ϕ∗◦asX∂ϕ◦ρ=∂ϕ∗◦ρYinthesecondline.Finally,usingthenoXtotΩ(X)EQiYEQΣ(X)=T˜yi−∂iϕ(ρX(X))ρX(X)Bregmandivergencefromρ(X)toy,weobtainΩ(X)nEQo
n
o=y−ρY(X)∂iϕ∗ρY(X),totΩ(X)∗EQYEQYT˜iiΣ(X)=T˜ϕy−Π˜−Dy||ρ(X),(40)(41)

89whereweemployϕ∗(ρY(X))=Π,because˜ρY(X)∈thiscase,yEQdoesnotexistonIY(Π˜,µ˜).ForeveryIY(Π˜,µ˜)(seeEq.(36)).Here,wenotethatthefirsty∈IY(Π˜,µ˜),thecorrespondingrayinXisgivenby∗EQ
Y
−1ABCtwotermsinEq.(41),ϕy−Π,arecalculatedby˜ρ¯(y)(seetheexamples,y,y,yandthecorre-theintensivevariablesofthereservoir,becauseyEQisspondingraysinXintherightpanel).Oneachray,thegivenbyitschemicalpotential˜µasinEq.(24).Inthe
entropyfunctionΣtotincreaseswhenXapproachesthefollowing,wewillshowthatthesignofϕ∗yEQ−Πde-˜originasshownintherightpanelofFIG.6(b).terminesthefateofthesystem.Forconvenience,wealso
denotetermsinthebracketinEq.(41)byFinally,weinvestigatethecaseϕ∗yEQ−Π˜>0,in
whichthegrowthofthesystemisrealized.Inthiscase,Y∗EQYEQK(y):=ϕy−Π˜−Dy||y,(42)YYYaregionR(Π˜,µ˜)⊂I(Π˜,µ˜)existssuchthatK(y)istotYYpositive:thatis,Σ(X)={Ω(X)/T˜}K(ρ(X)).∗EQ
noFirst,letusconsiderthecaseϕy−Π=0,which˜RYΠ˜,µ˜:=y|y∈IYΠ˜,µ˜,KY(y)>0.(43)correspondstothesituationthatequilibriumstatesexistandthesystemconvergestooneofthem.Inthiscase,Theexistenceofthisregionisprovedasfollows.BysinceKY(y)=−DYyEQ||yandΩ(X)>0,theentropytotdefiningthelargerregion,functioninEq.(41)satisfiesΣ(X)≤0,theequalityEQ
holdsifandonlyify=y.Furthermore,fromEq.ZY(˜µ):=y|ϕ∗yEQ−ϕ∗(y)−DYyEQ||y>0,(36),yEQ∈IY(Π˜,µ˜),andthereforeρY(X)canreach(44)EQy.Hence,themaximumoftheentropyfunctionistheregionRY(Π˜,µ˜)canberepresentedbytheintersec-
−1achievedontheraygivenbyρ¯Y(yEQ),whichrepre-tion:sentsasetoftheequilibriumstates.SincethesecondlawimposesthatthetotalentropyfunctionincreasesintheRYΠ˜,µ˜=IYΠ˜,µ˜∩ZY(˜µ).(45)timeevolutionofthesystem,itwillconvergetoapointontheequilibriumray,dependingontheinitialconditionSincetheisobaricmanifoldIY(Π˜,µ˜)inYisthelevelandthefunctionalformofthereactionfluxJ(t)inEq.hypersurfaceofϕ∗(y)specifiedbyΠ(seeEq.(36)),we˜
(17).Weshouldnotethattheequilibriumstateisidenti-obtainRY(Π˜,µ˜)6=φ,onlywhenΠ˜<ϕ∗yEQholds(thefiedbyauniquepointinthedensityspaceX.However,detailsareshowninAppendixD).Duetotheexistence
−1inthenumberspaceX,theequilibriumstatesformarayoftheregion,arayρ¯Y(y)foranyy∈RY(Π˜,µ˜)andtheequilibriumpointthesystemconvergesisoneofY
−1alsoexistsinX;and,oneveryrayρ¯(y),thevaluethepointsontheray.YYK(ρ(X))isapositiveconstant.Furthermore,sinceΩ(X)isanincreasingfunctionontheray,theentropyExample4:ConsidertheautocatalyticmotifshowntotfunctionΣincreaseswhenXdivergesalongtheray.inFIG.3(a)andtheintensivevariablesΠand˜˜µinthe∗EQAccordingly,theentropyfunctionisnotboundedabove,reservoirsatisfyϕ(y)−Π=0.Inthiscase,theiso-˜Yandthesystemisgrowinginthiscase.baricmanifoldI(Π˜,µ˜)inthechemicalpotentialspaceYisshowninFIG.6(a),andyEQliesonIY(Π˜,µ˜).Fur-totExample6:Considertheautocatalyticmotifshowninthermore,themaximumoftheentropyfunctionΣ(X)∗EQ
−1FIG.3(a)andassumethatϕ(y)−Π˜>0.Theregionisachievedontheraygivenbyρ¯Y(yEQ)(seetheZY(˜µ)inYisindicatedbylightpurplecolorintheleftrightpanelofFIG.6(a)).YpanelofFIG.6(c).Then,theregionR(Π˜,µ˜)isgivenYbytheintersectionbetweentheregionZ(˜µ)andtheSecond,weshowthatthesystemeventuallyshrinksY
levelhypersurface(theisobaricmanifold)I(Π˜,µ˜).Forifϕ∗yEQ−Π˜<0.Inthiscase,KY(y)isnegativeanyy∈RY(Π˜,µ˜),thevalueKY(ρY(X))isapositiveY
−1
−1forally∈Y.Thus,onarayinXgivenbyρ¯(y)constant.Thus,onarayρ¯Y(y)inXforeveryy∈foranyy,thevalueKY(ρY(X))isanegativeconstant.RY(Π˜,µ˜),theentropyfunctionΣtotincreaseswhenXInaddition,Ω(X)isanincreasingfunctionontheraydivergesalongtheray.becauseofitshomogeneity.Thus,theentropyfunctionYTheregionZ(˜µ)existsirrespectiveofthesignofΣtotincreaseswhenXapproachestheoriginalongtheϕ∗(yEQ)−ΠasinFIG.6(a,b).However,inthecases˜ray.Accordingly,themaximumoftheentropyfunction∗EQϕ(y)−Π˜≤0,theintersectionwiththeisobaricman-(tobemoreprecise,thesupremumoftheentropyfunc-YifoldI(Π˜,µ˜)doesnotexist.tion)islocatedatX=0;thatis,thesystemeventuallyshrinksandfinallyvanishes.Theabovethreesituationsaresummarizedasfollows:Example5:FortheautocatalyticmotifshowninFIG.Theorem1Ifandonlyifthereservoirconditionsat-
3(a)undertheconditionϕ∗(yEQ)−Π˜<0,theiso-isfiesϕ∗yEQ−Π=0˜,whereyEQ=−µOS˜−1,equi-baricmanifoldIY(Π˜,µ˜)inYisshowninFIG.6(b).Inlibriumstatesexistandthesystemconvergestooneof

910
them.Furthermore,ifandonlyifϕ∗yEQ−Π˜<0,thesystemeventuallyshrinksandfinallyvanishes.By
contrast,ifandonlyifϕ∗yEQ−Π˜>0,thesystemisgrowing.Basedonphysicalintuition,oneexpectsthatthefateofthesystemisclassifiedbya“gradient”inducedbytheintensivevariables(Π˜,µ˜)inthereservoir.Theabovetheoremmakesthisintuitionpreciseinthesensethatϕ∗(yEQ)−Πplaystheroleofthisgradient.Infact,˜ϕ∗(yEQ)−Πisrepresentedbytheintensivevariables˜(Π˜,µ˜),becauseyEQisdeterminedonlybythechemi-calpotential˜µinthereservoirthroughEq.(24).Fur-thermore,whenthegradientisbalanced,i.e.,ϕ∗(yEQ)−Π=0,thesystemconvergestoanequilibriumstate.˜Bycontrast,whenthegradientisnotbalanced,i.e.,ϕ∗(yEQ)−Π˜6=0,thesystemneverreachesanequilibriumstate.Amorepreciseexplanationofthegradientisasfol-lows.Ontheonehand,thechemicalreactionsinthesys-temaimtoachievethestateyEQ,thepressureofwhichisϕ∗(yEQ).Ontheotherhand,theinternalpressureϕ∗(y)ofthesystemalwaysbalanceswithΠ,owingto˜
thefastdynamics.Thegradientϕ∗yEQ−Πrepresents˜
thedifferencebetweenthem.Whenϕ∗yEQ−Π=0,˜FIG.6.FortheautocatalyticmotifshowninFIG.3(a),thetargetpressureϕ∗(yEQ)coincideswiththereservoirwedescribetheisobaricmanifoldIY(Π˜,µ˜)inY(leftpan-pressureΠ.˜Then,thesystemconvergestoanequi-els)andthecorrespondingrays(rightpanels)in
Xgivenbythemapρ¯Y−1=¯ρ−1◦∂ϕ∗.Theheatmapsintherightlibriumstate.Inthecasethatthetargetpressureis
XsmallerthanΠ(i.e.,˜ϕ∗yEQ−Π˜<0),thechemicalpanelsindicatevaluesoftheentropyfunctionΣtot(seethereactionsattempttodecreasetheinternalpressureϕ∗(y)captioninFIG.8forspecificvaluesoftheparameters).(a)∗EQEQYWhenϕ(y)−Π=0,thepoint˜yliesinI(Π˜,µ˜)andfromΠineachtimestep,butthesystemimmediately˜YEQYY∗K(y)=0;fortheothery∈I(Π˜,µ˜),thevalueofK(y)regainsϕ(y)=Π.Thisinfinitesimalandinstantaneous˜totisnegative.Thus,themaximumΣ=0isachievedonpressuregapbetweenthesystemandthereservoirleadsY
−1EQ∗EQtheraygivenbyρ¯(y).(b)Whenϕ(y)−Π˜<0,totheshrinkingandthevanishingofthesystem.ByEQYYthepointydoesnotexistonI(Π˜,µ˜)andK(y)isneg-contrast,ifthetargetpressureislargerthan
Π(i.e.,˜ativeforally∈IY(Π˜,µ˜).Then,onarayinXgivenbyϕ∗yEQ−Π˜>0),bythesameargument,thesystem
ρ¯Y−1(y),thevalueofKY(¯ρY(X))isnegativeandcon-eventuallygrows(expands)ineachtimestepandfinallytotstant.Thus,oneachray,theentropyfunctionΣincreasesdiverges.whenXapproachestheorigin.Asaguide,wedisplaytypi-ABCcalpointsy,yandy,andthecorrespondingraysinX.Y(c)TheregionZ(˜µ)isindicatedbylightpurplecolorin∗EQtheleftpanel.Onlywhenϕ(y)−Π˜>0,theintersec-VI.STEADYGROWINGSTATEYYYYtionR(Π˜,µ˜)=I(Π˜,µ˜)∩Z(˜µ)appears,whereK(y)isYpositiveforanyy∈R(Π˜,µ˜);i.e.,withintherangebetweenInthissection,weconsiderthesteadygrowingstateB1B2YY
−1yandyinI(Π˜,µ˜).Thus,onarayρ¯(y)inXandevaluatetheentropyproductionrateatthestate.Ytotforeveryy∈R(Π˜,µ˜),theentropyfunctionΣincreasesSincethesystemisassumedtogrow,wefocusonthewhenXdivergesalongtheray.WealsoshowthepointsyB1
case:ϕ∗yEQ−Π˜>0.ThesteadygrowingstatexisandyB2atwhichKY(y)=0,andthecorrespondingraysonSGtotdefinedasastatesuchthatthedensityx(t)=X(t)/Ω(t)whichΣ=0.iskeptconstantinthetimeevolutionandΩ(˙t)ispositive,wherethedotdenotesthedifferentiationwithtime.Atthisstate,thenumberofconfinedchemicalsX(t)evolvesX(t)=Ω(t)xSGintoEq.(40),weget−1onlyonaray¯ρX(xSG),becauseX(t)=Ω(t)xSG.noInorderforxSGtobethesteadygrowingstate,thetotΩ(t)EQiΣ(Ω(t)xSG)=T˜yi−∂iϕ(xSG)xSG,(46)entropyproductionrateatthisstatemustbepositive,Σ˙tot(Ω(t)xSG)>0,and,atthesametime,thevol-umemustbeincreasing,i.e.,Ω(˙t)>0.BysubstitutingwhereweuseρX(Ω(t)x)=x.ByrearrangingEq.SGSG

1011(46)asinEq.(41),wehaveTheabovetheoremonlyidentifiestheregionofpos-siblesteadygrowingstates.Theexistenceandunique-Ω(t)
Σtot(Ω(t)x)=KYySG,(47)nessofsuchstatesarenotguaranteed,andwhichstatesSGT˜wouldbechoseninthisregionisnotdetermined.These
whereySG:=∂ϕ(xSG)andKYySGisdefinedinEq.detailscanbeanalyzedanddeterminedoncewespecifyYSG
thefunctionalformofthereactionfluxJ(t).Forexam-(42).SinceKyiskeptconstantwithtime,theple,weassumethatJ(t)oftheCRSgiveninFIG.3(a)entropyproductionratecanberepresentedasobeysmassactionkineticsandobservethatthesteadyΩ(˙t)
growingstateexistsasinFIG.3(d).However,iftheΣ˙tot(Ω(t)x)=KYySG>0.(48)SGT˜functionalformofthekineticlawisdifferentfrommass
action,theexistenceofthesteadygrowingstateisnotBecauseΩ(˙t)>0forthesteadygrowingstate,KYySGguaranteedevenintheCRS.mustbepositive.Accordingly,thechemicalpotentialforByrearrangingEq.(54),weobtaintheconfinedchemicalsatthesteadygrowingstate,ySG,mustlieintheregionRY(Π˜,µ˜)⊂Y(seeEqs.(43),(44)Σ˙tot(t)
and(45)).T˜SG={ϕ∗yEQ−Π˜}−DX[x||x].(55)Ω(˙t)SGEQToclarifytheregionofpossiblexSGinthedensityspaceX,wemaptheregionRY(Π˜,µ˜)toX.First,weThelefthandsideofthisexpressionrepresentsthether-introducetheBregmandivergenceonX:modynamiccostforthevolumegrowth,whereastherightnoDX[x||x′]:={ϕ(x)−ϕ(x′)}−∂ϕ(x′)xi−(x′)i.handsidecanbeinterpretedasfollows.Thefirsttermirepresentstheexternalcontribution,whichisthegradi-
(49)∗EQentϕy−Πinducedbythereservoir.Thesecond˜ThisdivergenceisrelatedtotheoneinY,Eq.(38),termcharacterizestheinternalcontribution,whichistheasDY[y||y′]=DX[∂ϕ∗(y′)||∂ϕ∗(y)].Then,thetermBregmandivergenceDX[x||x]fromtheequilibriumSGEQKY(y)definedbyEq.(42)istransformedasstatexEQtothesteadygrowingstatexSG.Itgivesthe
X∗EQXtotalentropyincrementduringanisochoricrelaxationK(x)=ϕy−Π˜−D[x||xEQ],(50)xSG→xEQ(seeRef.[51]fordetails).Thisfactsuggests
wherex:=∂ϕ∗yEQ.Thus,theregioninthedensitytointerpretthesecondtermastherelaxationcontribu-EQspaceXcanberepresentedastionbythechemicalreactionsinthesystem.Moreover,
intherighthandside,onlythesteadygrowingstatexSGRXΠ˜,µ˜:=∂ϕ∗RYdependsonthereactionfluxJ(t).WhenonedesignsthenoreactionfluxJ(t)tooptimizethethermodynamiccost,=x|x∈IXΠ˜,µ˜,KX(x)>0.(51)theexpression,Eq.(55),mayplayanimportantrole.Rewritingthisregionastheintersectionoftwosubman-ifoldsasinEq.(45),weobtainExample7:FortheexampleshowninFIG.6(c),inwhichϕ∗(yEQ)−Π˜>0holds,theregionZX(˜µ)existsin


RXΠ˜,µ˜:=∂ϕ∗RY=∂ϕ∗IY∩∂ϕ∗ZYX,asindicatedbythelightpurplecolorinFIG.7.Undertheidealgasassumption,theisobaricmanifoldIX(Π˜,µ˜)=IXΠ˜,µ˜∩ZX(˜µ),(52)isasimplexinXaswewillshowinthenextsection.Then,theintersectionRX(Π˜,µ˜)=IX(Π˜,µ˜)∩IX(Π˜,µ˜)XwheretheregionZ(˜µ)inXisrepresentedasexistsasthedashedredrectangleinFIG.7,whereKX(x)ispositiveforanyx∈RX(Π˜,µ˜).IfasteadygrowingZX(˜µ)=x|xi{∂ϕ(x)−∂ϕ(x)}>0.(53)iEQiXstatexSGexists,itmustbeintheregionR(Π˜,µ˜).Theargumentinthissectionissummarizedbythefollowingtheorem:
Theorem2Whenϕ∗yEQ−Π˜>0andasteadygrow-VII.IDEALGASingstatexSGexists,thestatexSGmustlieintheregionRX(Π˜,µ˜).Then,theentropyproductionrateatthestatexSGisrepresentedasInthissection,wedemonstrateourframeworkforCRSsundertheidealgasassumption.Tobemorepre-Ω(˙t)Σ˙tot(t)=KX(x)cise,weassumethatboththesystemandthereservoirSGT˜SGarecomposedofidealgas.Ω(˙t)
Ω(˙t)TowritedownTheorem1inthissituation,wefirst={ϕ∗yEQ−Π˜}−DX[x||x].(54)T˜T˜SGEQevaluatethefullgrandpotentialdensityϕ∗(y).TheformoftheHelmholtzfree-energydensityfortheidealgasis

1112where˜n={n˜m}isthedensityoftheopenchemicalsinthereservoir.Inaddition,fornotationalsimplicity,wedefinethestandarddensityfortheconfinedchemicalsasoi−ν(T˜)/RT˜xo:=ei[62].Then,Eq.(59)isrearrangedtoXX∗iyi/RT˜mϕ(y)=RT˜xoe+RT˜n˜.(61)imNext,wecalculatethegradientϕ∗(yEQ)−ΠinThe-˜orem1.Bydefiningthestandarddensityfortheom−µ(T˜)/RT˜openchemicalsasno:=em,weget˜µm=RT˜log(˜nm/nm).Hence,yEQ=−µOS˜−1inEq.(24)ocanberewrittenas(OS−1)mYnmiEQoyi=RT˜logm.(62)Xn˜FIG.7.TheisobaricmanifoldI(Π˜,µ˜)inX,correspondingmtothecaseinFIG.6(c).IfthesystemiscomposedofidealXXBysubstitutingyEQintoEq.(61),weobtaingas,thenI(Π˜,µ˜)isasimplex.TheregionZ(˜µ)isindicatedXbylightpurplecolor.TheregionR(Π˜,µ˜)istheintersection(OS−1)mXXXYnmibetweenI(Π˜,µ˜)andZ(˜µ),whichisenclosedbythedashedϕ∗(yEQ)−Π=˜RT˜xioredrectangle.Ifasteadygrowingstateexists,itmustbeinon˜mimthisregion.!X−Π˜−RT˜n˜m.(63)knownasmhiXfT˜;n,x=nmµoT˜+RT˜{nmlognm−nm}Here,wenotethatthesecondlineinEq.(63)repre-msentsthepartialpressurethatisproducedbycomposi-mXtionsotherthantheopenchemicalsinthereservoir.For+xiνoT˜+RT˜xilogxi−xi,(56)itheidealgas,Eq.(63)determinesthefateofthesystem.iFinally,wespecifyTheorem2fortheidealgas.TheisobaricmanifoldIX(Π˜,µ˜)inEq.(32)isrewrittenaswhereRrepresentsthegasconstant;µo(T˜)={µo(T˜)}mandνo(T˜)={νo(T˜)}denotethestandardchemicalpo-(!)iXXtentialsoftheopenandconfinedchemicals,respectively.IXΠ˜,µ˜:=x|RT˜xi−Π˜−RT˜n˜m=0,Sincethepartialgrandpotentialdensityϕ[T,˜µ˜;x]canbeimrepresentedbyavariantoftheLegendretransformation:(64)hinhiowhichPimpliesPthe
equationofstate,Π˜=ϕT,˜µ˜;x:=minfT˜;n,x−µ˜nm,(57)RT˜xi+n˜m,anddefinesasimplexinthemimndensityspaceX.Also,byusingEq.(58),theregionwegetZX(˜µ)inEq.(53)canbeexpressedashiX(!)ϕT,˜µ˜;x=ϕ(x)=xiνoT˜+RT˜xilogxi−xiXiiXixEQiZ(˜µ)=x|RT˜xlog>0,(65)Xxi{µ˜o(T˜)}/RT˜i−RT˜em−µm.(58)
EQmi∗EQiy/RT˜wherexEQ=∂ϕy=xoei.NotethatPAlso,fromtheLegendretransformation,Eq.(34),thexilog(xi/xi)canbenegativebecausexandxiEQEQfullgrandpotentialdensityϕ∗(y)canbeexpressedasXarenotnormalized.Thus,theregionR(Π˜,µ˜)isgivenX∗{yo(T˜T˜bytheintersectionbetweenEqs.(64)and(65).Inad-ϕ(y)=RT˜ei−νi)}/Rdition,theBregmandivergenceinthedensityspaceX,iXEq.(49),reducestothegeneralizedKullback-Leiblerdi-{µ˜o(T˜)}/RT˜+RT˜em−µm.(59)vergence[43–45,50]:m"#XxiFurthermore,sincewehaveassumedthatthereservoirDX[x||x]=RT˜xilog−xi−xi.EQiEQalsoconsistsoftheidealgas,thechemicalpotential˜µxEQicanberepresentedas(66)Accordingly,theentropyproductionrateΣ˙tot(t)isevalu-oT˜mSGµ˜m=µm+RT˜log˜n,(60)atedbysubstitutingEqs.(63)and(66)intoEq.(54).To

1213obtaintheentropyproductionrateinEq.(54),westillVIII.NUMERICALVERIFICATIONneedtocalculatethegrowthrateΩ(˙t)andthesteadygrowingstatexSG.Tocomputethem,wemustdeter-Tonumericallyverifyourtheory,wedealwiththemin-minethefunctionalformofthereactionfluxJ(t).WeimalmotifofautocatalyticcyclesasgiveninSec.II,shouldrecallthatTheorem2onlyidentifiestheregionofwhereweassumeidealgasconditionsandmassactionpossiblesteadygrowingstatesxSG.kinetics.Thechemicalequationsofthemotifhavebeenrep-Example8:Thegeometricrepresentationsoftheex-resentedbytworeactionsR1andR2thatinvolvetwoamplesshowninFIG.6and7areobtainedasfollowsconfinedchemicalsA=(A1,A2)andtwoopenchemicalsfortheidealgas.Beforepresentingthegeometry,welistB=(B1,B2):thegivenparameters:(1)thestoichiometricmatricesSR1:A1+B1⇋A2+A2,andO;(2)theintensivevariables(T,˜Π˜,µ˜)inthereser-voir;(3)thestandarddensities{no,xo}andthestandardR2:A2⇋A1+B2.(69)chemicalpotentials{µo(T˜),νo(T˜)}fortheopenandtheAlso,thestoichiometricmatricesareconfinedchemicals,whicharerelatedtoeachotherasoom−µ(T˜)/RT˜i−ν(T˜)/RT˜no=emandxo=ei;(4)thedensityR1R2R1R2n˜fortheopenchemicalsinthereservoir,whichleadsto!!thechemicalpotentialas˜µ=µo(T˜)+RT˜log˜nm.A1−11B1−10mS=,O=.(70)First,wedeterminetheisobaricmanifoldsIX(Π˜,µ˜)A22−1B201andIY(Π˜,µ˜).ByusingthegivenT˜,Πand˜˜n,weobtainXTheregularityofthematrixSischeckedasdet[S]=theisobaricmanifoldI(Π˜,µ˜)inthedensityspaceXfromEq.(64)asthesimplexinFIG.7.Also,wecan−16=0.DenotingthenumberofA=(A1,A2)byX=(X1,X2),thereactiondynamicsfortheconfineddescribetheisobaricmanifoldIY(Π˜,µ˜)inthechemicalchemicalsiswrittenaspotentialspaceYbysubstitutingEq.(61)intoEq.(36),asshownintheleftpanelsofFIG.6.dXi=SiJr(t).(71)Second,wedeterminetheregionsZX(˜µ)andZY(˜µ).rdtByemployingEq.(62),wecancalculateyEQ;andbyapplyingthemap∂iϕ∗(y)=xieyi/RT˜toyEQ,wegetFurthermore,weassumemassactionkineticsforthere-oactionfluxJ(t):xEQ.ThesubstitutionofxEQintoEq.(65)leadstoZX(˜µ)(seethelightpurpleregioninFIG.7).Wealso12111N12XobtainZY(˜µ)bysubstitutingEqs.(61)and(38)intoJ(t)=w+X−w−X,ΩΩEq.(44)(seethelightpurpleregionsintheleftpanels222221NofFIG.6).J(t)=w+X−w−X,(72)ΩThird,wedeterminetheregionRX(Π˜,µ˜)forpossiblewhereN=(N1,N2)denotesthenumberofB=steadygrowingstatesxSGbyEq.(52).Itisgivenby(B,B)inthesystem.TherateconstantswrandwrtheintersectionbetweenIX(Π˜,µ˜)andZX(˜µ),i.e.,the12+−satisfydashedredrectangleinFIG.7.Finally,theentropyfunctionΣtot(X)onXiscalcu-wr1nolog+=−ν0T˜Si+µoT˜Om,(73)latedfromEq.(41).Here,thevolumeΩ(X)isobtainedwrRT˜irmr−fromEq.(27)withEq.(58),i.e.,fromtheequationofstate:whichisknownasthelocaldetailedbalancecondition[43,44,51,52,58].PTosolveEq.(71),weneedtoelucidatethebehaviorofRT˜XiΩ(X)=Pi.(67)NandΩ.Fortheidealgas,thedensityN/ΩoftheopenΠ˜−RT˜n˜mmchemicalsinthesystemcoincideswiththedensity˜ninthereservoir,whichisaconstantintime(seeAppendixAlso,themapρY(X)canbecalculatedasE).Inaddition,ΩisgivenbytheequationofstateasEq.(67).Thus,Eq.(72)canberearrangedasiX22ρY(X)=∂iϕ◦ρX(X)=RT˜log.(68)1111(X)iΩ(X)xiJ(t)=ˆw+X−wˆ−,oΩ(X)J2(t)=ˆw2X2−wˆ2X1,(74)+−TheheatmapsoftherightpanelsofFIG.6areplottedusingtheseequations.whereweabsorbtheconstantdensitiesoftheopenchemi-cals,Nm/Ω,intotherateconstantsasˆwrandˆwr.Then,+−

1314thelocaldetailedbalanceconditioninEq.(73)canbewrittenasrnologwˆ+=−1ν0T˜Si+˜µOm,(75)wˆrirmr−RT˜and,forourspecificexample,itreducestowˆ1x2x2n˜1wˆ2x1n2+=oo,+=oo.(76)wˆ−1x1on1owˆ−2x2on˜2InFIG.8,weshowthetrajectoriesofthesystem,fromtwoinitialconditions1and2,inthespacesX,XandY.Whentheequalityϕ∗(yEQ)−Π=0holds(seeFIG.˜8(a)),thetotalentropyfunctionisincreasingasthesys-temmovesonXandconvergestoapoint,denotedbythesquare,ontheequilibriumray.Thepointdependsontheinitialconditions.InthespacesXandY(FIG.8(b,c)),thesystemmovesontheisobaricmanifoldsIX(Π˜,µ˜)andYFIG.8.TrajectoriesofthesysteminthespacesofthenumberI(Π˜,µ˜),respectively,andconvergestotheequilibriumofconfinedchemicalsX(leftpanel;a,d,g),thedensityspacepointsxandyEQ,irrespectiveoftheinitialconditions.EQX(middlepanel;b,e,h),andthechemicalpotentialspaceYWhenϕ∗(yEQ)−Π˜<0(seeFIG.8(d)),thesystemfirst(rightpanel;c,f,i)fordifferentpressuresΠsatisfying(top;˜convergestoaray,andthenmovesontheraytowardthe∗EQ∗EQa,b,c)ϕ(y)−Π=0,(middle;d,e,f)˜ϕ(y)−Π=˜−5.75,∗EQoriginofX,drivenbytheincreaseoftheentropyfunc-(bottom;g,h,i)ϕ(y)−Π=4˜.25.Foroursimulation,EQtion.InthespacesXandY(FIG.8(e,f)),thesystemxEQ=(9,5.25)andy=(0.118,−0.288).TheparametersXYofthesimulationarefixedasfollows:R=T˜=1,x1=8,movesontheisobaricmanifoldsI(Π˜,µ˜)andI(Π˜,µ˜),o2112212respectively,andconvergestothepointsdenotedbythexo=7,no=2,˜n=1,no=3,˜n=2,ˆw−=ˆw−=121.Therateconstantsofforwardreactions,ˆw+andˆw+,aresquares.ThesepointscorrespondtotherayonwhichcomputedbyEq.(76).Fortheinitialconditions1and2,wethesystemmovestowardtheorigininX.Therefore,theset(X1,X2)=(10,60)and(90,5),respectively.systemfinallyvanishes.Finally,whenϕ∗(yEQ)−Π˜>0(FIG.8(g)),thesys-temfirstconvergestoaray,andthenmovesontherayInthiswork,wehaveassumedthatthestoichiomet-awayfromtheoriginofXwiththeincreaseoftheen-ricmatrixSisregular.Thisimpliesthatthesystemcantropyfunction.InthespacesXandY(FIG.8(h,i)),theXalwaysrelaxtoachemicalequilibriumstatewhenthevol-systemmovesontheisobaricmanifoldsI(Π˜,µ˜)andYumeisfixed,i.e.,intheisochoricsituation[51].InotherI(Π˜,µ˜),respectively,andconvergestopointsxSGandSGwords,thesystemneverreachesastatethatcontinouslyy=∂ϕ(xSG)denotedbythesquares.Thesepointsproducesentropywithconstantvolume,namely,thecon-correspondtotherayonwhichthesystemmovesinX,XYventionalnonequilibriumsteadystate(NESS)[43–50].andareindeedlocatedinR(Π˜,µ˜)andR(Π˜,µ˜)(seeAccordingly,thenonequilibriumstatestreatedhere,no-alsoFIG.6(c)andFIG.7).tablythesteadygrowingstate,arerealizedduetothechangeofthevolume.ThisnonequilibriumstatewithIX.SUMMARYANDDISCUSSIONchangingvolumeoriginatesintheextensivityofthermo-dynamicsandshouldbedistinctfromtheconventionalNESSwithconstantvolume.Wehaveestablishedthethermodynamicsofgrowingchemicalreactionsystems(CRSs)byemployingHessianIfthematrixShasanontrivialrightnullspaceandprojectivegeometry.Inthiswork,wehaveclassi-(dimKer[S]6=0),thesystemmayrelaxtotheNESSfiedtheenvironmentalconditionstodistinguishthefateevenforaconstantvolumesituation.Suchanongrow-oftheCRSs.Furthermore,underthegrowingcondition,ingbutnonequilibriumstateisalsobiologicallyrelevant,wehaveidentifiedtheregioninthedensityspacewhereforexample,thestationaryphaseofcells[12,63–66].Itasteadygrowingstatecanexist.Wehavealsoevalu-isamajorchallengeforthefuturetoclarifyhowtheatedtheentropyproductionrateinthisstate.Itisem-nonequilibriumstatecausedbyvolumegrowthandthephasizedagainthatourresultsarederivedbyageneralconventionalNESSwithoutgrowtharecompatibleandthermodynamicstructurewithoutassuminganyspecificrelatedtoeachother.thermodynamicpotentialsorreactionkinetics;i.e.,theyBycontrast,ifthematrixShasanontrivialleftnullareobtainedbasedsolelyonthesecondlawofthermo-space(dimKer[ST]6=0),thesystemhasconservationdynamics.laws[43,44,51,52].Inourframework,itremainsan

1415openproblemwhethersteadygrowthofthesystemisfinalcondition,thematrixSisreferredtoasminimal,possibleandrealizedwiththeconservationlaws.becauseitdoesnotcontainanysmallermotifssatisfyingInoursetup,wehaveignoredthetensionofthemem-bothproductivityandautonomy.braneandassumedthatitneverbursts(seethecaptionFurthermore,wepreparethefollowingtermsfortheinFIG.1).However,themembranedoeshavetensionproof:Ifaspeciesistheonlyreactantofareaction,weinactualsituations.Evenforsuchcases,ourframeworkcallitthesolitaryreactantofthereaction;otherwise,wecanbeappliedbyeffectivelyincorporatingthetensioncallitacoreactantofthereaction.intothepressureΠ.Furthermore,inbiologicalcells,the˜Theabovedefinitionsimmediatelyleadtothefol-membranemoleculesthemselvesareproducedandsup-lowinglemmas.(Lemma1)Wecanremoveanarbi-pliedbytheintracellularCRS.Inthiscase,thetensionistrarycolumnvectorfromS,whilepreservingautonomy.coupledandchangeswiththeCRS,andthereforetheef-(Lemma2)WecanremoveanarbitraryrowvectorfectiveΠchangeswithtime.Accordingly,ourtheoretical˜fromS,whilepreservingproductivity.(Lemma3)Ifframeworkneedstobeextendedfurther.aspeciesexistssuchthatitisnotthesolitaryreactantOurtheorysurelyservesasthebasisofalltheseex-forallreactionsinS,wecanremovetherowvectorcor-tensions,whichareimportantforconsideringactualandrespondingtothespecies,whilepreservingproductivityexperimentalsituationsofgrowingprotocellsorbiolog-andautonomy.icalcellsandalsoforestablishingthephysicsofself-Withtheabovedefinitionsandlemmas,wenowprovereplicatingsystems.Theorem3.ConsideranautocatalyticcoreSofsizeNX×NRwithrankλ.IfweassumedimKer[S]6=0,wecanremoveacolumnvector,whilepreservingIm[S],ACKNOWLEDGEMENTthatis,preservingproductivity.ThiscontradictstheconditionthatanautocatalyticcoreSisminimal.Thus,TheauthorsthankKentoNakamuraandGentaChibadimKer[S]mustbezero,andthereforewehaveλ=NR.forfruitfuldiscussion.ThisresearchissupportedFurthermore,foreveryspecies,somereactionsexistsuchbyJSPSKAKENHIGrantNumbers19H05799andthatthespeciesisthesolitaryreactantofthereactions.21K21308,andbyJSTCRESTJPMJCR2011andJP-Otherwise,becauseofLemma3,wecanremovearowMJCR1927.vectorandthiscontradictstheconditionagainthatanautocatalyticcoreSisminimal.Thus,wegetNX≤NR.Sinceλ≤NX,NR,itfollowsthatλ=NX=NR.ThisAppendixAmeansthatSisregular.Inthisappendix,weintroducetheconceptofminimalmotifsforgrowingsystemscalled“autocatalyticcores”.AppendixBItwasoriginallyproposedinRef.[26]todeterminewhetherasubnetworkembeddedinalargerchemicalre-Inthisappendix,wederivetheeffectiveslowdynamicsactionnetworkcanbeautocatalytic.Furthermore,thebyemployingthesecondlawofthermodynamics.authorsofRef.[26]haveshownthattheregularityofSincewehaveassumedJE(t),JΩ(t),JD≫J(t),wethestoichiometricmatricesofthemotifsplaysanessen-canignorethereactionfluxJ(t)inEqs.(3)and(4)tialroletoidentifysuchcores,byprovidingthefollowingforthefasttimescale.Then,wegettheeffectivefasttheorem:dynamicsasdEdΩdNmTheorem3Ifachemicalreactionnetworkisanauto-=J(t),=J(t),=Jm(t),EΩDcatalyticcore,itsstoichiometricmatrixSfortheconfineddtdtdtdE˜dΩ˜dN˜mchemicalsmustberegular.m=−JE(t),=−JΩ(t),=−JD(t).(77)dtdtdtInthefollowingpart,wewillbrieflyreviewtheproofTheformalsolutionofEq.(77)withtheinitialconditionoftheirtheorem(Theorem3)withournotations.First,wemathematicallydefineseveralconditionsfor(E0,Ω0,N0,E˜0,Ω˜0,N˜0)canberepresentedasastoichiometricmatrixS.AllofthefollowingdefinitionsE(t)=E0+∆E(t),Ω(t)=Ω0+∆Ω(t),areintroducedinRef.[26].ThematrixSisproductive,Nm(t)=Nm+∆m(t),E˜(t)=E˜−∆(t),NX0N0EifIm[S]∩R>06=φ.ThematrixSisautonomous,ifallΩ(˜t)=Ω˜−∆(t),N˜m(t)=N˜m−∆m(t),(78)columnvectorsofScontainbothstrictlynegativeand0Ω0Nstrictlypositiveelements.ThematrixSisanautocat-where(∆E(t),∆Ω(t),∆N(t))aretheintegralsofthealyticcore,ifSisbothproductiveandautonomous;influxfunctions(JE(t),JΩ(t),JD(t))withtheinitialcon-addition,Ssatisfiesthefollowingcondition:ifweremoveditionJE(0)=JΩ(0)=JD(0)=0.Notethatthenum-aroworacolumnvectorfromS,thereducedmatrixofberoftheconfinedchemicals,X(t),isaconstantinthisSisnotbothproductiveandautonomous.Withthisdynamics.

1516BysubstitutingthissolutionintoEq.(1),wehavetheTheothertwovalues,EQEQandNQEQ,canbecomputedtimeevolutionofthetotalentropyasasfollows.Sincetheequality,hiΣtot(∆E,∆Ω,∆N)=Σ[E0+∆E,Ω0+∆Ω,N0+∆N,X]ΦT,˜µ˜;ΩQEQ,X=−T˜Σ[EQEQ,ΩQEQ,NQEQ,X]1Π˜µ˜m+E−µ˜Nm,(85)−∆−∆+∆m+const.,(79)QEQmQEQT˜ET˜ΩT˜Nholds,thedifferentiationswithrespecttoT˜and˜µleadwhereweusepropertiesofthereservoir;i.e.,∆E(t)≪toE˜0,∆Ω(t)≪Ω˜0,∆N(t)≪N˜0,andtheTaylorex-hipansionforΣ˜˜;wealsousethethermodynamicre-∂ΦT,˜µ˜;ΩQEQ,XT,Π˜,µ˜Σ(X)=−,QEQlations:∂Σ˜T,˜Π˜,µ˜/∂E˜=1/T,∂˜Σ˜T,˜Π˜,µ˜/∂Ω=˜Π˜/T˜andh∂T˜i∂Σ˜˜/∂N˜m=−µ˜m/T˜.Inaddition,weabbreviatethe∂ΦT,˜µ˜;Ω,XT,Π˜,µ˜QEQNm(X)=−,(86)constanttermΣ˜T,˜Π˜,µ˜[E˜0,Ω˜0,N˜0]to“const.”.AccordingQEQ∂µ˜mtothesecondlaw,thesystemmustclimbuptheland-scapedefinedbytheconcavefunctionΣtot(∆,∆,∆)wherewenotethattheimplicitdifferentiationsofEΩNinthetimeevolution,andfinallyconvergetoitsmaxi-Eq.(85)withrespecttoEQEQandNQEQvanish,mum,whichiscalledtheequilibriumstate.Hence,weduetothecriticalequationsforthevariationalforms,getEqs.(82)and(83);alsowedenoteΣQEQ(X)=Σ[EQEQ,ΩQEQ,NQEQ,X].Finally,bysubstitutingEq.(∆E,∆Ω,∆N)→(∆E),(∆Ω),(∆N)(86)intoEq.(85),weobtainQEQQEQQEQhi=argmaxΣtot(∆,∆,∆),hi∂ΦT,˜µ˜;ΩQEQ,XEΩN∆E,∆Ω,∆NE(X)=ΦT,˜µ˜;Ω,X−T˜QEQQEQ(80)hi∂T˜∂ΦT,˜µ˜;ΩQEQ,Xwhere(·)QEQrepresentsthevalueattheequilibriumstate−µ˜m.(87)ofthefastdynamics.However,wecallthisthequasi-∂µ˜mequilibriumstate,becausewelaterconsidertheslowdy-Next,byemployingtheaboveresultsforthefastdy-namics.ByusingtheargumentshiftE=E0+∆E,Ω=namics,wederivetheeffectiveslowdynamics,whichisΩ0+∆Ω,N=N0+∆NandtakingEq.(79)intoaccount,thereactiondynamics.Thetimeevolutionsoftheinter-wegettheextensivevariablesatthequasi-equilibriumnalenergyE(t),thevolumeΩ(t)andthenumberofthestateasopenchemicalsN(t)forthereactiondynamicsareal-readysolved,byusingthetimeevolutionoftheconfined(EQEQ,ΩQEQ,NQEQ)chemicalsX(t)andEqs.(86)and(87),as()1Π˜µ˜mm=argmaxΣ[E,Ω,N,X]−E−Ω+N.E=EQEQ(X),Ω=ΩQEQ(X),N=NQEQ(X).(88)E,Ω,NT˜T˜T˜(81)SubstitutingtheseevolutionsintoEq.(3)andtakingEq.(4)intoaccount,weobtaintheeffectiveslowdynamicsTheabovecharacterizationofthequasi-equilibriumasstatebythevariationalform,Eq.(81),canberear-dXidE˜dE(X)=SiJr(t),=−QEQ,rangedbyintroducingthermodynamicpotentialsasfol-rdtdtdtlows.First,wedefinetheHelmholtzfreeenergyasdΩ˜dΩ(X)dN˜mdNm(X)=−QEQ,=OmJr(t)−QEQ,hinodtdtdtrdtFT˜;Ω,N,X:=minE−T˜Σ[E,Ω,N,X].(82)E(89)Second,byusingtheHelmholtzfreeenergy,weintroducewhichisEq.(17)inthemaintext.thepartialgrandpotential:Beforeclosingthisappendix,wecommentontheex-tensivevariablesinthereservoiratthequasi-equilibriumhinhioΦT,˜µ˜;Ω,X:=minFT˜;Ω,N,X−µ˜Nm.state,(E˜QEQ,Ω˜QEQ,N˜QEQ).Since(∆E)QEQ=EQEQ−mNE0,(∆Ω)QEQ=ΩQEQ−Ω0,and(∆N)QEQ=NQEQ−N0,(83)weget,fromEq.(78),Withtheabovetwothermodynamicpotentials,wecanreformulatethevariationalform,Eq.(81),asE˜QEQ(X)=E˜0−{EQEQ(X)−E0}nhioΩ˜QEQ(X)=Ω˜0−{ΩQEQ(X)−Ω0}ΩQEQ(X)=argminΦT,˜µ˜;Ω,X+ΠΩ˜.(84)N˜m(X)=N˜m−Nm(X)−Nm.(90)ΩQEQ0QEQ0

1617IfwesetaninitialconditionfortheconfinedchemicalsYtoX0,theinitialconditionfortheothervariablesinEq.IYϕ∗(yEQ),µ˜(89),(E˜(0),Ω(0)˜,N˜(0)),isgivenbyEq.(90)withX=X0.yEQAppendixCInthisappendix,foragivenX,weprovetheunique-
nessofthevolumeΩ(X),whichisgivenbyIYΠ˜,µ˜XΩ(X)=argminΩϕ+ΠΩ˜.(91)ΩΩΠ˜′<ϕ∗(yEQ)ThecriticalequationforthisvariationalformisXXiXFIG.9.Illustrationoftheprooffortheexistenceofthein-h(Ω):=ϕ−∂iϕ+Π=0˜,(92)YYYΩΩΩtersectionI(Π˜,µ˜)∩Z(˜µ).TheregionZ(˜µ)isshownby∗∗EQlightpurplecolor.Thesublevelset{y|ϕ(y)<ϕ(y)},where∂ϕ(X/Ω)=∂ϕ(x)/∂xiandwehavede-indicatedbybluecolor,isboundedbythelevelhypersur-ix=X/ΩY∗EQfaceI(ϕ(y),µ˜)(solidcurve).Moreover,theboundary
finedthefunctionh(Ω).Thedifferentiationofh(Ω)isYY∗EQofZ(˜µ)intersectsIϕ(y),µ˜inexactlyonepoint,givenasEQ∗EQYnamelyy.IfΠ˜≥ϕ(y),thehypersurfaceI(Π˜,µ˜)Y(outerdashedcurve)hasnointersectionwithZ(˜µ),whereasdh=Ω−3Xi∂∂ϕXXj.(93)forΠ˜′<ϕ∗(yEQ),thehypersurfaceIY(Π˜′,µ˜)(innerdashedijdΩΩcurve)musthavetheintersectionwithZY(˜µ).Theintersec-tionisindicatedbythesolidblackcurveinsidethecurvedredSinceϕisstrictlyconvex,itsHessian∂i∂jϕispositiverectangle.definite.Thus,thefunctionh(Ω)isastrictlyincreasingfunctionforΩ>0.Accordingly,thecriticalequation,AppendixEEq.(92)hasauniquesolutionforΩ.Therefore,thevolumeΩ(X)isuniquelydeterminedbyagivenX.Intheslowdynamics,thesystemisalwaysinthequasi-equilibriumstate,andthereforethenumberofopenchemicalsN(X)canbeevaluatedinEq.(20)asAppendixDhi∂ΦT,˜µ˜;Ω(X),XInthisappendix,weshowthattheintersection,Nm(X)=−.(95)∂µ˜mRYΠ˜,µ˜=IYΠ˜,µ˜∩ZY(˜µ),(94)DividingbothsidesofthisequationbyΩ(X)yieldshim∂ϕT,˜µ˜;X/Ω(X)∗EQnm(X)=N(X)=,existsonlywhenΠ˜<ϕ(y).SincetheBregmandiver-YEQΩ(X)∂µ˜mgenceDy||yinEq.(44)representsthedeviation{µ˜o(T˜T˜matthepointyEQbetweentheconvexfunctionϕ∗(yEQ)=em−µm)}/R=˜n(96)andtangentplaneatthepointy,apointy′existsinY
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