Stability of equilibrium under constraints: Role of. second-order constrained derivatives

Transcription

1 arxiv: Stability of equilibrium uner constraints: Role of secon-orer constraine erivatives amás Gál Department of eoretical Pysics University of Debrecen 4 Debrecen Hungary aress: galt@pys.unieb.u bstract: In te stability analysis of an equilibrium given by a stationary point of a functional free energy functional e.g. te secon erivative of plays te essential role. If te system in equilibrium is subject to te conservation constraint of some etensive property e.g. volume material or energy conservation te Euler equation etermining te stationary point corresponing to te equilibrium alters accoring to te meto of Lagrange multipliers. Here te question as to ow te effects of constraints can be taken into account in a stability analysis base on secon functional erivatives is eamine. It is sown tat te concept of constraine secon erivatives incorporates all te effects ue to constraints; terefore constraine secon erivatives provie te proper tool for te stability analysis of equilibria uner constraints. or a pysically important type of constraints it is emonstrate ow te presente teory works. urter te rigorous erivation of a recently obtaine stability conition for a special case of equilibrium of ultratin-film binary mitures is given presenting a guie for similar analyses.

2 I. Introuction onservation of some etensive quantity is neee to be accounte for in many pysical teories as in te case of flui-ynamical moels containing conserve orer parameters e.g. e ynamics of various systems is often governe by te erivative of some functional of te ynamical variables escribing te motion suc as te free-energy functional in flui ynamics. Since in general may ave pysical relevance only over te omain of 's obeying te given conservation constraints te equations of motion containing as to be invariant uner te replacement of by anoter functional tat equals for 's satisfying te constraint. erefore te erivative of as to be moifie accoring to te constraints leaing to te appearance of constraine functional erivatives µ ; introuce in 3. In many cases see tose in e.g. tis moification of an is cancelle in te equations of motion for e.g. ue to a operator; in oter wors te form of te equations of motion itself ensures te above-mentione invariance. However tis cannot be epecte in general especially not for comple constraints for eample constraints coupling te variables of te given. is is sown also by te fluiynamical moel propose by larke 45 for te escription of simultaneous ewetting an pase separation in tin-film binary mitures 6. In 5 terefore te meto of constraine ifferentiation 3 is utilize to set up te equations of motion for tis moel. See Ref. for an analysis of tat application.

3 esie first-orer erivatives secon-orer erivatives also play an essential role in pysics for eample in stationary-point analysis incluing te stability analysis of equilibria see e.g. 47- in flui ynamics or in nonlinear response teory. Hence te question immeiately arises ow secon erivatives moify uner constraints. In tis paper te proper moification of secon erivatives will be given an it will be sown tat tese constraine secon erivatives provie te necessary tool for pysics for te analysis of equilibria uner conservation constraints. s an eample a pysically important type of constraints of volume an material conservation will be consiere sowing also ow te stability conition obtaine in 4 for a special case of equilibrium emerges. Since te appearance of te present work on arxiv te teory presente in te following as been applie in te stability analysis of roplet growt in supercoole vapors in te special case of a norm-conserving constraint accounting for particle number conservation. e stability conition of Eq.3 in is a straigt consequence of Eq.74b below. e paper is organize as follows: Sec.II gives te necessary backgroun for te present work escribing te concept of constraine functional erivatives. urter Sec.II introuces te problem of ow to efine constraine secon erivatives raising tree routes to efine tem. In Sec.IV te answer to tis question is given; it is sown tat one of te efinitions propose in Sec.II incorporates all effects to be taken into account wen using secon functional erivatives uner constraints. Sec.III presents a preliminary to Sec.IV by analyzing a concrete pysical eample of equilibrium analysis tat can be treate in a simplifie way at te same time it alreay sows well te effects to account for in a general treatment of constraints. 3

4 II. onstraine secon erivatives onstraine functional erivatives In matematics te erivative of a functional at some 3 is efine as a functional D ; tat is linear in an gives for all 's were D ; o ; a o lim ;. is is te so-calle récet erivative. lternatively te erivative of a functional can be efine as a continuous linear functional D G ; in tat gives te Gâteau ifferential te irectional erivative in te irection for any D G ε ; lim ε ε. b is erivative is calle te Gâteau erivative. e following teorem trows ligt upon te connection between te two efinitions: If te récet erivative eists at a ten te Gâteau erivative eists tere as well an te two erivatives are equal. is means tat te récet efinition is a stronger efinition. In pysical equations a functional erivative D ; appears in a form efine by D ; for all 's. 3 4

5 can be obtaine from D ; formally by writing ; toug note tat can be consiere as a function only in a generalize sense in fact it is a istribution. s mentione in te Introuction te proper treatment of constraints in a pysical teory in general requires te moification of te erivative. In 3 te formula f f f 4 for tat moification as been propose for constraints of te form f. 5 f is an invertible function wic may ave an eplicit -epenence as well an f enotes its first erivative. We will call a -constraine or -conserving erivative. constraine erivative given by Eq.4 fulfils two essential conitions 4: i two functionals tat are equal over a omain of 's etermine by Eq.5 a -restricte omain soul ave equal -conserving erivatives over tat omain -equality conition an ii for a functional tat is inepenent of of in te sense tat f λ f were λ is an arbitrary real number -inepenence conition. rom conition i 6 f u f 7 5

6 follows wit u an arbitrary function tat integrates to one wile conition ii fies u as f u. or linear constraints wit linear i.e. for constraints f g L -inepenence means tat λ for any scalar λ i.e. is omogeneous of egree zero from wic 8 giving inee Eq.6. In tis section in te following te arguments will be presente consiering te special case of te normalization conservation 9 for simplicity in presentation; owever teir generalization for arbitrary an for more comple constraints is straigtforwar. or tis case te formula Eq.4 gives. n essential feature of -conserving erivatives is tat tey eliver te ifferential for an arbitrary tat correspons to te -conserving cange of te functional variable ; tat is for all 's. a n unconstraine is projecte to an -conserving component ~ wit ~ via. b 6

7 o ave Eq.a te projection Pˆ coul be replace by any ˆ P { u } were u integrates to one 4; i.e. u. We coose Eq.b in orer to ave for -inepenent functionals for wic λ 4. ote tat te essential criterion P ˆPˆ Pˆ i.e. Pˆ is trivially fulfille since. How to efine constraine secon erivatives? e secon erivative D ; of a functional is efine as te erivative of te first erivative D ; of wic is a symmetric bilinear functional. In pysics it is usually represente by efine by D ; for all 's. Higer-orer erivatives of a functional can be efine similarly. ote tat iger Gâteau erivatives will give te iger Gâteau ifferentials or variations D n G n n ; t n in any irection. Wit iger erivatives t t ten a functional can be given by a aylor epansion wit remainer n n...!.. 3 7

8 It is wort noting tat an avantage of te Gâteau concept of a functional erivative is tat te Gâteau ifferential i.e. te irectional erivative is efine witout te Gâteau erivative an e.g. a aylor-like epansion along a irection may be given by te nt Gâteau ifferentials n witout eisting nt Gâteau erivatives wile te récet ifferential an te récet erivative are attace concepts. yiels aking te -conserving erivative of te first -conserving erivative of a functional. 4 Eq.4 transforms a symmetric in an ' into an asymmetric secon erivative 3 ue to a single term. However altoug secon erivatives D emerge as te erivatives of first erivatives it is not necessarily te proper way of efining. ote tat even for efine by Eq. justifie. as to be In 5 anoter possible way to efine constraine secon an iger erivatives is given. ollowing te iea of Eq. for all 's 5 arises wic yiels te formula 8

9 . 6 is satisfies te -equality conition for secon erivatives since te secon erivatives of two functionals tat are equal on a omain of 's may iffer only by some µ µ on tat omain wic ifference is cancelle in Eq.6. On te oter an of Eq.6 oes not satisfy te -inepenence conition for secon erivatives. or a egree-zero omogeneous 7 wic comes from ifferentiating Eq.8 wit respect to. It can be seen tat Eq.6 oes not become for egree-zero omogeneous s: te terms besie in Eq.6 o not vanis. false logic coul make one say tat te -inepenence conition is satisfie by Eq.6: If is egree-one omogeneous ence ols for its secon erivative following from ten is egree-zero omogeneous; consequently 's -conserving secon erivative soul reuce to wic it inee oes accoring to te relation above. However in tis case it is of course wat soul give back te "unconstraine secon" erivative wic it oes giving back. 9

10 Directly following te route 4 tat efines te -conserving erivative of a functional over a omain of 's of a given as te unconstraine erivative of 's egree-zero omogeneous etension : 8 wit owever yiels a efinition of tat satisfies also te -inepenence conition. Defining an -constraine secon erivative as : 9 leas to. Eq. inee fulfils te -inepenence conition an also te -equality conition of secon erivatives wic can be seen by using Eq.7. otice tat similarly to Eq.6 Eq. preserves te symmetry of a symmetric in an '; owever instea of simply cancelling te "unesirable" term Eq. symmetrizes it. Having te tree above possible efinitions Eqs.4 6 an of an - conserving secon erivative te question is wic one we soul coose. ote tat te tree formulae yiels te same result at a stationary uner Eq.9 were but

11 for a non-stationary an for oter constraints tis is not so in general. Sall it be Eq.4 wic may be te straigtest in origination but tere is no conceptual reason bein it? Or Eq. wic fulfils te -inepenence conition? Or Eq.6 not containing te first erivative of wic unoubtely seems to be a reasonable feature? is question will be answere in te following troug consiering te concrete case of a pysically important comple constraint taken from te tin liqui film moel 4 propose by larke te ynamics of wic was given later 5 by te application of constraine ifferentiation. o gain insigt into te effects of constraints tat ave to be taken into account in a stability analysis of equilibrium first in te net section we will re-consier larke s treatment 4 of a special case of equilibrium in is moel an clarify its matematical aspects. III. case of equilibrium uner constraint coupling te variables of te free-energy functional In 4 larke eamines te stability of equilibrium of a moel of tin-film binary mitures escribe by two variables te eigt an te composition. e eamine equilibrium wit an flat omogeneous istribution correspons to a stationary point of te free-energy functional built from an uner te constraint of volume an material conservation an respectively wit an enoting te area of te film. e emerging Euler-Lagrange equations are

12 µ µ 3a an µ 3b wic utilizing tat is taken in a form f give f µ µ 4a an f µ 4b for te constant. o obtain te conition of instability of te above state te problem is trace back in 4 to te stability analysis of simple two-variable functions g y y. In tat case te sufficient conition for a local maimum at a given stationary point i.e. an instable stationary point in te case of te free energy is well-known to be g y y y g y y y g y y yy <. 5 e epression on te left sie in Eq.5 is te eterminant of te Hessian matri of g y y. o account for te constraints an larke makes use of 6 an 7 coming from Eqs. an respectively in te aylor epansion of te free-energy

13 3 iger-orer terms 8 taken at f f f f... f 9 to obtain f f µ... f 3 e variations an are ten anle by teir ourier series epansions... cos sin b a π π 3a an... cos sin b a π π 3b respectively were

14 4 3 ue to Eq.6. or simplicity is taken to be of one imension instea of te two imensions in 4. e integrals in Eq.3 tus become b a b a a a a a 34 an a a ; 35 consequently... a f a a f µ a f. 36 inally an account for te constraints as regars te variations i.e. Eq.3 an... b b a a 37 coming from Eq.7 wit Eq.34 an an Eq.3 as to be mae. Eq.37 can be inserte into Eq.35 an into Eq.34 in a case Eq. is not require in te place of. e obtaine term being of orer iger tan two tis yiels... a f

15 f µ a Since a... an a... f a... iger-orer terms. 38 can now be varie freely by fiing all but a pair of i an j as Eq.38 gives a two-variable secon-orer variational problem yieling a f f f f < 39 as te criterion of instability on te basis of Eq.5. larke as obtaine tis result in a somewat more complicate fasion actually te matematical essence of wic owever is just tat escribe above. ote also tat in 4 f is ivie into two parts f f f wic as no relevance ere. b of s e stability conition Eq.39 as been verifie also troug te equations of motion set up in 5. However two essential questions arise wit respect to te above erivation in te contet of more general situations. irst wy soul Eq.7 wit Eq.6 be utilize in te way utilize above obtaining Eq.3 cancelling te term completely? Secon oes not te occurrence of te new iger-orer terms ue to te use of Eq.37 matter at all wit respect to te stability analysis? ote tat tis actually leas to te interesting fact tat formally even a free variation of /... /... a an a in Eq.36 yiels te result Eq.39; toug of course te constraint Eq.7 is alreay taken into account in Eq.36 in some way. e answers to tese questions are reassuring. or in te proof of te teorem from wic te criterion Eq.5 in te case of two-variable orinary functions also follows tat te secon erivative D ; of a functional is nonnegative for all 's at a local minimum of an reverse for a maimum 63 i.e. 5

16 for all 's 4 two essential elements are tat i te first erivative D ; vanises an ii igertan-secon orer terms become zero. Wit strict inequalities more precisely wit p > p < in Eq.4 an wit some restriction on s see below Eq.74b te conition becomes a sufficient conition for a local minimum maimum. or more comple situations especially in te case of nonlinear constraints like Eqs. an too owever anoter important issue soul be accounte for. Eq.4 as alreay been generalize for te case of constraints present base on Ljusternik s teorems 3. It can be written as i.e. D ; µ D ; for all 's 4a µ for all 's 4b wit strict inequalities in te sufficient version; see note below Eq.74b. µ is te Lagrange multiplier corresponing to te constraint wic is given by µ at te stationary. is suc tat ; 4 tat is oes not ave to satisfy te constraint itself but te first erivative of te constraint wic is te same only for linear s. ifficulty wit te use of Eq.4 is tat te variations are not free; te fulfillment of Eq.4 as to be ensure someow. s will be sown in te net section tis problem can be solve by te use of constraine secon erivatives in te place of unconstraine secon erivatives by wic te constraint 6

17 on te variations can be eliminate. urter constraine secon erivatives incorporate te Lagrange multiplier in Eq.4 too proviing a natural general treatment of constraints in te stability analysis of equilibria. IV. e use of constraine secon erivatives in stability analysis of equilibrium wit constraints aking te constraints an into account in te variations an in Eq.8 was possible to be one in te way one in Sec.III because of te constancy of te first erivatives in an te form const. of te secon erivatives secon erivative kernels. ese simplifying circumstances came from te simple nature of te consiere equilibrium namely an are constant an tat f of f in te consiere moel epens on only via te functional variables an. In tis section we wis to treat te general situation. We will consier a general constraint 43 wic may be multi-component an may enote many variables.. Relaing te constraint on varying te functional variable by te use of constraine erivatives onsier te aylor epansion of te functional above a omain etermine by some constraint Eq.43 7

18 ! ! 44 In te case is an n-component variable te secon erivative above will be an n n matri te tir erivative an n n n matri etc. satisfies te constraint tat is y epaning into its aylor epansion tis gives! !. 46 If is a stationary point of uner te constraint Eq.43 it satisfies te Euler-Lagrange equation µ 47 were te Lagrange multiplier µ is constant wit respect to. Wit te use of Eqs.46 an 47 te first-orer term in Eq.44 can be eliminate obtaining! µ 3! 3 3 µ In orer to free te variations from te constraint we introuce a mapping wit te following properties: i maps any onto a wic satisfies 8

19 te constraint Eq.43 an ii becomes an ientity for s i.e. ~ ~. or an arbitrary cange of wit a given tis ten gives a -conserving cange of via. is yiels an epansion of in terms of unconstraine variations 49 Wit te use of te efinition m m : m m 5 following Eqs.8 an 9 Eq.49 can be written as 5 e prime of enotes tat te -constraine erivative of Eq.5 is not require to fulfill a conition like te -inepenence conition in aition to te -equality conition; tat is oes not ave to be "omogeneous" 4 of egree zero. is means tat tere is some freeom in coosing to obtain a constraine erivative via Eq.5. In te following owever we will write instea of for simplicity but will still stan for a -constraine erivative efine via a egree-zero -omogeneous i.e.. Insertion of Eq.5 into Eq.48 gives wit! µ iger-orer terms in

20 Eq.53 is noting else tan te constraine secon erivative efine accoring to Eq.. e * is to istinguis tis efinition from te one given by Eq.5. Eq.5 gives us µ for all 54 as te necessary conition for a local minimum maimum in te place of Eq.4. Similar to Eq.4 tis becomes a sufficient conition if we replace / wit p > / p <. is is justifie by te fact tat in Eq.4 can be written wit te elp of as 55 an te secon-orer term in Eq.5 as been obtaine via te first term of Eq.5. o prove tat Eq.55 is inee a variation satisfying Eq.4 just insert it into Eq.4 an use an e latter equation is a trivial generalization of 34 a straigt consequence of te -equality conition. o illustrate te above troug a pysical eample we will now give te corresponing epressions for larke s moel i.e. for a two-variable functional wit te constraint of Eqs. an. is constraint as a general enoug form to ave a general relevance an be wort giving for it te corresponing constraine erivatives. It is also relatively simple but comple enoug to trow ligt onto te caracter of constraine secon erivatives. e variational form of Eq. corresponing to Eqs.44 an 46 is

21 58a or. 58b e corresponing epression for Eq. as alreay been given; see Eq.6. Wit te use of te Euler-Lagrange equations 3 an Eqs.6 an 58 te aylor epansion of Eq.8 can be written over te constraine omain as µ iger-orer terms. 59 e ine enotes tat te variations obey Eqs.6 an 58 tis notation was not use in Eqs.8-3 an 36 for simplicity. e epansions for an to be inserte into Eq.59 can be given as... 6a an... 6b were te erivatives are calculate as te unconstraine first an secon erivatives of

22 . 6 Eq.6 is te egree-zero -omogeneous etension of satisfying Eqs. an see. Eq.6 is noting else tan te aylor epansion of Eq.6. ote tat terms containing vanis in Eq.6a since actually oes not ave a epenence on. We mention ere tat te first-orer variations an i.e. an satisfying 6 instea of te full constraint Eq.58 are given by 63a an. 63b Inserting Eqs.6 into Eq.59 te linear operators acting on namely 64a an 64b

23 3 can be carrie over to te kernels in Eq.59. We empasize ere tat any can be taken in te form Eq.6 ue to te construction of. y tis we obtain an epression were te variations an are unconstraine: µ µ iger-orer terms 65 wit 66 67

24 an. 68 e emerging iger-orer terms in Eq.65 ue to te full epansion Eq.6 are irrelevant wit respect to equilibrium analysis just as te original iger-orer terms as pointe out previously. s can be observe te general epression Eq.65 reuces significantly in te case of te special equilibrium consiere in Sec.III since e.g. It will give back Eq.38. ote tat te term containing. 69 e.g. is wat ensures te cancellation of an of te urier epansions of an. It as to be unerline tat in te general case µ appears not only besie but also besie.. ccounting for all effects ue to constraints troug constraine erivatives s we ave seen freeing te variations from te constraints naturally leas to te appearance of constraine erivatives as efine accoring to Eqs. an 53. However tere is more in teir concept regaring te analysis of functionals uner constraints. Observe tat constraine secon erivatives efine accoring to Eq.5 can be written as 4

25 5. 7 pplying a furter ifferentiation wit respect to in te secon line of Eq.7 gives te tir-orer constraine erivative an so on. In Eq.7 te first term on te rigt sie is efine by n n n n n. 7 ote tat for Eq.56 ols; tat is. ow insert Eq.5 into s aylor epansion 44 an collect te terms of same orer in to fin wit te elp of Eqs.7 an 7!... 3! 3 73

26 y calculating te epressions corresponing to Eqs.7 an 7 for iger-orer constraine erivatives it is easy to see tat Eq.73 can be continue for iger-orer terms. t a stationary or critical point 4; tus te first term in Eq.73 vanises. or te secon-orer term of Eq.73 we can establis a result analogous to Eq.4 wit te elp of Eq.54. eorem necessary conition for a local etremum. If at a critical point of a functional tere is a local minimum maimum of for all 's. 74a Wit strict inequalities we obtain a sufficient conition for te eistence of local etremum: eorem sufficient conition for a local etremum. If at a critical point of a functional p > p < for all* nonzero 's 74b ten tere is a local minimum maimum of. p is an arbitrarily small in absolute value positive negative constant inepenent of. In oter wors te infimum supremum of te constraine secon ifferential as to be greater less tan zero to ave a local minimum maimum. is is similarly so in te unconstraine case an also in te case of Eq.4 3. e presence of p in Eq.74b is important; it is to rule out cases of sequences of positive negative secon ifferentials tening to zero wit. ecause of tis te omain of all nonzero s as to be restricte in Eq.74b enote by all* to avoi te secon ifferential tening to zero wit wic coul give a zero infimum supremum. is can be acieve formally e.g. by restricting to be of. ut note tat tis is not a rastic restriction since any can be obtaine by multiplying 6

27 a of norm one by some positive constant; tat is we o not ave to account for a constraint on te variations. Proof. Inserting Eq.47 into Eq.7 gives µ. 75 ompare Eq.74 wit Eq.75 inserte wit Eq.54 to see tat all we ave to prove is. 76 Eq.76 can be prove by inserting te epansion Eq.5 into Eq.46 an noticing tat since te variation is now unconstraine terms of te same orer in ave to cancel eac oter. Eq.76 emerges from te secon-orer terms cancelling eac oter. lternatively may also be ifferentiate to obtain Eq.76. It is wort mentioning ere tat epressions of iger-orer similar to Eq.76 can be obtaine in te way Eq.76 as been obtaine. In practice Eq.74b can be applie in te way Eq.4 is usually applie by eamining te eigenvalue spectrum of te secon erivative 67. In te presence of constraints te eigenvalue equation becomes λ. 77 If all te eigenvalues are greater less tan some positive negative number p arbitrarily close to zero tere is a local minimum maimum at te eamine stationary point. s eplaine below Eq.74b p is neee to eclue zero being a cluster point of te λ s. is can be prove on te basis of Eq.74b completely analogously to te unconstraine case. local minimum usually represents a stable equilibrium wile te oter cases imply an 7

28 8 instable or metastable equilibrium. is meto as been applie by Uline an orti in in te stability analysis of roplet growt in supercoole vapors. Recently tere as been muc interest in te problem of accounting for constraints in te analysis of stationary points bot in te infinite-imensional 8- an in te finiteimensional case. It as been establise tat in te presence of constraints te eigenvalues of λ µ give us a tool to etermine if at a critical point tere is a local minimum or maimum or neiter of tem. However altoug it is true tat if p > λ for all λ at a given critical point ten tere is a local minimum tere can still be a local minimum if tere is only one negative λ 9. or tis case Vogel 9 as prove a criterion to etermine weter tere inee is a local minimum or not. Eamining te eigenvalue spectrum of Eq.77 presents an alternative way to ecie about te nature of a critical point. urning to te concrete eample of a two-variable functional wit constraints an te corresponing constraine secon erivatives emerge as 78 79

29 9 an. 8 e terms in Eqs an 8 are wort giving eplicitly: 8a 8b 8a 8b an 83a. 83b e unconstraine secon erivatives of course vanis for tese special cases wic are obtaine also if one substitutes an into Eqs e conition of total instability of te equilibrium state in te case of te freeenergy functional will be

30 3 < for all s. 84 Wen applying tis result for te special case of equilibrium consiere in 4 an in te preceing section many of te terms of Eqs.78 8 vanis giving f 85 f µ 86 an f 87 wit µ f. en inserting te urier epansions Eqs.3 into te above ifferentials an applying Eq.84 give back Eq.39. It can be seen tat relying on constraine secon erivatives irectly leas to te final result witout any furter consierations regaring te proper account for constraints. ere is no nee to consier aitional terms coming from te first ifferential ue to te constraints an to account for constraine variations. It can be conclue tat te constraine secon erivatives efine accoring to Eq.5 provie te proper groun for te analysis of functionals incluing te analysis of stationary points uner constraints. is answers te question raise in Section II too. On te basis of Eq.73 it may be not too bol to epect tis fining to be vali also for iger-

31 3 orer constraine erivatives efine by Eq.5. s seen eventually tis result as been obtaine on te basis of te iea bein Eq.5. e reason tat iea gave a efinition in Sec.II tat is ifferent from Eq.9 in te case of te -conservation constraint is te special nature of linear constraints a kin of egeneracy manifeste also wit respect to te issue of simultaneous constraints 3 an wit respect to te origination of -constraine erivatives as Gâteau erivatives along -conserving pats 4. or given by Eq.b itself fulfils te constraint Eq.9 leaing to a ifferent -constraine secon erivative tan Eq.9. However taking te full epansion Eq.5 for into account formally also yiels te same efinition as Eq.9. Having mentione te Gâteau kin of efinition of erivatives along -conserving pats 4 it is wort giving te corresponing formula for te nonlinear constraint of Eqs.- ε ε ε ε ε ε lim ; D G. 88 It is interesting to eamine te secon erivatives obtaine by two successive - constraine ifferentiations i.e. te case of Eq.4. e secon -constraine erivatives for te constraint of Eqs.- will be 89 9

32 3 9 an. 9 Eqs.89-9 sow tat not only o te erivatives 89 an 9 not retain te symmetry in of a symmetric an respectively but even te orer of ifferentiation wit respect to an becomes relevant. inally we mention tat te arguments of tis stuy can be applie to finiteimensional vector spaces were constraine erivatives can be introuce too 3. In tat case Eq.74a e.g. takes te form... j i n j i j i n a. 93 V. Summary It as been sown tat constraine secon erivatives efine accoring to Eq.5 incorporate all secon-orer effects ue to constraints; consequently constraine secon erivatives provie te proper tool for pysics for te stability analysis of equilibria uner conservation constraints. In te presence of constraints Eq.74 gives te proper generalization of te well-known conition Eq.4 for te eistence of a local etremum. More generally it can be conclue on te basis of Eq.73 tat uner constraints te unconstraine erivatives of orer m ave to be replace by te corresponing constraine erivatives 5 in problems base on te aylor epansion of te functional in question. or te pysically important type of constraints of Eqs.- for wic te constraine secon erivatives are given in Eqs.78-8 it as been emonstrate ow te presente

33 teory works sowing also ow te stability conition obtaine by larke for a special case of equilibrium in is tin-film ynamical moel 45 emerges. ppeni: On te coice of te mapping e most general form for a constraine first erivative in te case of te normconserving constraint 9 is 4 u were u is an arbitrary function tat integrates to one. Eq. can be obtaine from 3 u inserte into Eq.5. e constraine erivatives emerging wit te use of Eq. fulfil te -equality conition an can be use in Eqs.73 an 74. s pointe out in 4 te coice u of 3 yiels te intuitively appealing property of for -inepenent functionals. In a ynamical teory tat oes not irectly emerge from a variational principle like te moel evelope in 5 te coice of u may ave relevance; owever in establising Eq.73 an in particular Eq.74 u can be cosen arbitrarily an in practice tat coice can be base on pragmatic consierations. In tis ppeni we will i give te constraine secon erivatives emerging from Eq. ii eamine te special coice of u an iii sow tat Eq. inee leas to te most general form of iger-orer constraine erivatives tat can be applie in Eq.73. Differentiating yiels ~ ~ u u. 3 33

34 Eq.3 gives Eq. by insertion of wit gives u u u u u u. Differentiation of Eq.3 u u u. 4 is ten yiels te constraine secon erivative use of it u u u u u. 5 ormally te simplest coice for u is te Dirac elta function. Wit te 6 emerges as a constraine first erivative 4. It inee fulfils te -equality conition because any -inepenent constant ae to te erivative cancels in Eq.6. It is interesting to recognize tat in ensity functional teory D 45 tis form of constraine erivatives appears eplicitly in te basic Euler-Lagrange equation of te teory. In D an energy ensity functional uner te constraint E v v v v n n n r v r r is efine wose minimum n v v r r elivers te groun-state energy of te -electron system in te scalar eternal potential vr v. is minimum principle leas to te Euler-Lagrange equation 34

35 n v v v r µ 7 n r for te etermination of te groun-state nr v. Eq.7 also gives te eternal potential as a functional of te groun-state ensity v n in accorance wit te first Hoenberg-on teorem 4 wic establises a one-to-one corresponence between vr v wit te arbitrary aitive constant fie by some coice an nr v. or electronic potentials vr v v. is gives n µ ; tat is n v n n v r n v n r n. 8 Eq.8 ten gives v n uniquely. It also sows tat v n is a constraine erivative of te ensity functional n v n v r n v 9 n r accoring to Eq.6 wit r v. obtaine: rom Eq.5 te constraine secon erivative wit u can be easily. is epression seems to be very simple an appealing; owever tere are two problems wit te use of it. irst in te case of functionals f... wic form frequently appears in pysical applications te elta function enters te secon erivative; tis means tat te last term in Eq. will contain. Secon in practical calculations ue to teir usually approimate nature relying on te value of at one given point may strongly effect te accuracy of te result. or D were te v n 35

36 v v mapping is efine wit n r r fie we mention tat te inverse ensity response function can be given as v r v n v v v n r n r n r on te basis of Eq.9. Since te eact n is muc more complicate tan simply aving v v v a form f r n r... r te above-mentione problem oes not occur in te eact teory. v ote tat if we ave an etension of v n from te r n omain te -conserving constraint on te ifferentiation in Eq. can be relae. We now eamine te question as to weter Eq. inee embraces all possible s tat give goo constraine erivatives of all orer wic fulfill te -equality conition. otice tat tere is a wie range of possible s i.e. wic give a for any an become an ientity for s. ese can even involve pysics yieling complicate possibly non-analytical forms. e key for answering tis question is to observe tat. Differentiation of tis relation wit respect to gives. ow Eq. can be use in to obtain. 3 Since ue to it can be seen tat te coice u 4 36

37 gives back Eq.. o justify tat leas to te same secon-orer constraine erivative as te one given by Eq.5 wit te coice Eq.4 ifferentiate Eq. fully. 5 en use Eq.5 in te secon erivative of wit respect to see te secon line of Eq.7. is will give te same epression as Eq.5 wit Eqs.4 an u 6 inserte. e iger-orer cases can be similarly verifie; tus we ave sown tat from Eq. wit an arbitrary u integrating to one any proper constraine erivative of any orer can be erive. s an eample te mapping n n propose in 6 to give an etension of te D functional n from te ensity omain of n v v r r may be mentione. In 6 n n is efine troug an ensemble generalization of v n for noninteger s as n v n n r v n. n r v v is te groun-state -electron ensity in te potential vr v assuming non-egeneracy. or tis n n v v n r v u r is te ukui function an important cemical reactivity ine. rom Eq. we ave v v v n r v v r n v v v n r v v v r r r v r n r 7 v v instea of a simple r r on te rigt sie. ote tat if we put an -conserving constraint on te ifferentiation n r v v will vanis in Eq.7 but at te same time an v u r will appear in its place or if for te given application te use of an ambiguous restricte 37

38 ifferentiation suffices an arbitrary function g r v wic in te en can be cosen to be zero. inally we give te generalization of Eq.. 8 at is u of u 9 4 can be given as u. Eq. inee integrates to one as can be seen by an application of te cain rule of ifferentiation. In te case of a constraint 5 Eq. can be generalize too yieling f u f f. e corresponing mapping for te comple constraint of Eqs. an is given by an u a u. b cknowlegements: is work was supporte by te grant D48675 from O. References P.. Hoenberg. I. Halperin Rev. Mo. Pys Gál J. Mat. Pys arxiv:pysics/745 38

39 3. Gál Pys. Rev Gál J. Pys larke Eur. Pys. J. E larke Macromolecules R. Yerusalmi-Rozen. erle J. lein Science H. Wang R. J. omposto J. em. Pys H. J. ung R. J. omposto Pys. Rev. Lett Y. Liao Z. Su Z. Sun. Si L. n Macromol. Rapi ommun H. P. iscer W. Dieteric Pys. Rev. E Sarma J. Mittal Pys. Rev. Lett S.. Punnatanam D. S. orti J. em. Pys Pototsky M. esteorn D. Merkt U. iele Pys. Rev. E See e.g. D. J. Evans an G. P. Morriss Statistical Mecanics of onequilibrium Liquis caemic Lonon 99 M. J. Uline D. S. orti J. em. Pys P. lancar E. rüning Variational Metos in Matematical Pysics: Unifie pproac Springer-Verlag erlin Gál J. Mat. em arxiv:mat-p/ Gál Pys. Lett ote tat in Eqs.5 an 6 an irrelevant factor is missing before te secon erivative. 6 M. R. Hestenes alculus of Variations an Optimal ontrol eory Wiley ew York H.. Davis Statistical Mecanics of Pases Interfaces an in ilms Wiley ew York J. H. Maocks rc. Rat. Mec. nal

40 9. I. Vogel Pacific J. Mat L. Greenberg J. H. Maocks.. Hoffman Mat. acr. 9 9 D. Spring mer. Mat. Montly Hassell E. Rees mer. Mat. Montly Gál J. Pys arxiv:pysics/639 4 P. Hoenberg W. on Pys. Rev R. G. Parr W. Yang Density unctional eory of toms an Molecules Ofor University Press ew York Gál P. Geerlings Pys. Rev arxiv: