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ToappearinJournalofStatisticalPhysics Colligative properties of solutions: 5 II. Vanishing concentrations 0 0 2 n Kenneth S. Alexander,1 Marek Biskup,2 and LincolnChayes2 a J 9 2 Wecontinueourstudyofcolligativepropertiesofsolutionsinitiatedinref.[1]. v Wefocusonthesituationswhere,inasystemoflinearsizeL,theconcentration 5 andthe chemicalpotentialscale like c = ξ/Land h = b/L, respectively. We 3 0 find that there exists a critical value ξt such that no phase separation occurs 7 forξ ≤ξ while,forξ >ξ,thetwophasesofthesolventcoexistforaninterval t t 0 of values of b. Moreover, phase separation begins abruptlyin the sense that a 4 macroscopicfractionofthesystemsuddenlyfreezes(ormelts)formingacrystal 0 (or droplet) of the complementary phase when b reaches a critical value. For / h certainvaluesofsystemparameters,under“frozen”boundaryconditions,phase p separation also ends abruptly in the sense that the equilibrium droplet grows - h continuouslywithincreasingbandthensuddenlyjumpsinsizetosubsumethe t entiresystem. Ourfindingsindicatethattheonsetoffreezing-pointdepression a m isinfactasurfacephenomenon. : v i X r 1. INTRODUCTION a 1.1. Overview Inapreviouspaper(ref.[1],henceforthreferredtoasPartI)wedefinedamodel of non-volatile solutions and studied its behavior under the conditions when the solvent undergoes a liquid-solid phase transition. A particular example of interest is the solution of salt in water at temperatures near the freezing point. In accord withPartIwewillrefer tothesoluteassaltand tothetwophasesof solventas iceand liquidwater. AftersomereformulationthemodelisreducedtotheIsingmodelcoupled to an extra collection ofvariables representing thesalt. The (formal) Hamilto- (cid:13)c 2004 by K.S. Alexander, M. Biskup and L. Chayes. Reproduction, by any means, of the entirearticlefornon-commercialpurposesispermittedwithoutcharge. 1DepartmentofMathematics,USC,LosAngeles,California,USA 2DepartmentofMathematics,UCLA,LosAngeles,California,USA. 1 2 Alexander,BiskupandChayes nian isgivenby 1−σ βH = −J σ σ −h σ +κ S x. (1.1) x y x x 2 hXx,yi Xx Xx Here we are confined to the sites of the hypercubic lattice Zd with d ≥ 2, the variable σ ∈ {+1,−1} marks the presence of liquid water (σ = 1) x x and ice (σ = −1) at site x, while S ∈ {0,1} distinguishes whether salt is x x present (S = 1) or absent (S = 0) at x. The coupling between the σ’s is x x ferromagnetic (J > 0), the coupling between the σ’s and theS’s favors salt in liquidwater, i.e.,κ > 0—thisreflects thefact thatthereisanenergeticpenalty forsalt insertedintothecrystalstructureofice. A statistical ensemble of direct physical—and mathematical—relevance is that with fluctuating magnetization (grand canonical spin variables) and a fixed amountofsalt(canonical saltvariables). Theprincipalparameters ofthe systemarethusthesaltconcentrationcandtheexternalfieldh. Aswasshown in Part I for this setup, there is a non-trivial region in the (c,h)-plane where phase separation occurs on a macroscopic scale. Specifically, for (c,h) in this region,adropletwhichtakesanon-trivial(i.e.,non-zeroandnon-one)fraction oftheentirevolumeappears inthesystem. (For“liquid”boundaryconditions, the droplet is actually an ice crystal.) In “magnetic” terms, for each h there is a unique value of the magnetization which is achieved by keeping part of the system in the liquid, i.e., the plus Ising state, and part in the solid, i.e., the minusIsing state. Thisis in sharp contrast to what happens in theunperturbed Ising modelwhere a singlevalueof h(namely, h = 0) correspondsto a whole intervalofpossiblemagnetizations. The main objective of the present paper is to investigate the limit of in- finitesimal salt concentrations. We will take this to mean the following: In a system of linear size L we will consider the above “mixed” ensemble with concentrationcandexternalfieldhscalingtozeroasthesizeofthesystem,L, tendstoinfinity. Thegoalistodescribetheasymptoticpropertiesofthetypical spinconfigurations,particularlywithregardstotheformationofdroplets. The saltmarginalwillnowbeofnointerestbecausesaltparticlesaresosparsethat any localobservablewilleventuallyreport thatthereis nosaltat all. The main conclusions of this work are summarized as follows. First, in a regular system of volume V = Ld of characteristic dimension L, the scal- ing for both the salt concentration and external field is L−1. In particular, we should write h = bL−1 and c = ξL−1. Second, considering such a system withboundaryconditionfavoringtheliquidstateandwithhandcenjoyingthe abovementioned scalings, one of three things will happen as ξ sweeps from 0 to infinity: Colligativepropertiesofsolutions 3 (1) Ifbissufficientlysmallnegative,thesystemisalwaysintheliquidstate. (2) Ifbisofintermediate(negative)values,thereisatransition,atsomeξ(b) from theicestatetotheliquidstate. (3) Most dramatically, for larger (negative) values of b, there is a region— parametrized by ξ (b) < ξ < ξ (b)—where (macroscopic) phase sepa- 1 2 ration occurs. Specifically, the system holds a large crystallinechunk of ice, whose volume fraction varies from unity to some positive amount as ξ varies from ξ (b) to ξ (b). At ξ = ξ (b), all of the remaining ice 1 2 2 suddenlymelts. We obtain analogous results when the boundary conditions favor the ice state, with the ice crystal replaced by a liquid “brine pocket.” However, here a new phenomenon occurs: For certain choices of system parameters, the (growing) volumefractionoccupiedbythebrinepocketremainsboundedawayfromone as ξ increases from ξ (b) to ξ (b), and then jumps discontinuously to one at 1 2 ξ (b). In particular, thereare twodroplettransitions,seeFig. 1. 2 Thus, we claim that the onset of freezing point depression is, in fact, a surface phenomenon. Indeed, for very weak solutions, the bulk behavior of the system is determined by a delicate balance between surface order devia- tionsofthetemperatureandsaltconcentrations. Insomewhatpoeticterms,the predictions of this work are that at the liquid-ice coexistence temperature it is possible to melt a substantial portion of the ice via a pinch of salt whose size is only of the order V1−d1. (However, we make no claims as to how long one wouldhavetowait inorderto observethisphenomenon.) The remainder of this paper is organized as follows. In the next section wereiteratethebasicsetupofourmodelandintroducesomefurtherobjectsof relevance. The main results are stated in Sections 2.1-2.3; the corresponding proofs comein Section 3. In order to keep the section and formula numbering independentofPart I; wewillprefix thenumbersfromPart Iby “I.” 1.2. Basic objects We beginby a quick reminder of the model; further details and motivationare tobefoundinPartI.LetΛ ⊂ Zd beafinitesetandlet∂Λdenoteits(external) boundary. For each x ∈ Λ, we introduce the water and salt variables, σ ∈ x {−1,+1}andS ∈ {0,1};on∂Λwewillconsiderafixedconfigurationσ ∈ x ∂Λ {−1,+1}∂Λ. Thefinite-volumeHamiltonianisthenafunctionof(σ ,S )and Λ Λ theboundaryconditionσ thattakes theform ∂Λ 1−σ βH (σ ,S |σ ) = −J σ σ −h σ +κ S x. (1.2) Λ Λ Λ ∂Λ x y x x 2 hXx,yi Xx∈Λ Xx∈Λ x∈Λ,y∈Zd 4 Alexander,BiskupandChayes ξ ice b.c. liquid liquid ξ phase phase separation ice separation ice b b liquid b.c. Fig. 1. The phase diagram of the ice-water system with Hamiltonian (1.1) and fixed salt concentration c in a Wulff-shaped vessel of linear size L. The left plot corresponds to the systemwithplusboundaryconditions,concentrationc = ξ/Landfieldparameterh = b/L, theplotontherightdepictsthesituationforminusboundaryconditions. Itisnotedthatasξ rangesin(0,∞)withbfixed,threedistinctmodesofbehavioremerge,intheL → ∞limit, dependingonthevalueofb. Thethickblacklinesmarkthephaseboundarieswhereadroplet transition occurs; on the white lines the fraction of liquid (or solid) in the system changes continuously. Here, as usual, hx,yi denotes a nearest-neighbor pair on Zd and the parame- tersJ,κandhrepresentthechemicalaffinityofwatertowater,negativeaffin- ity of salt to ice and the difference of the chemical potentials for liquid-water and ice, respectively. The a priori probability distribution of the pair (σ ,S ) takes the usual Λ Λ Gibbs-Boltzmann form Pσ∂Λ(σ ,S ) ∝ e−βHΛ(σΛ,SΛ|σ∂Λ). For reasons ex- Λ Λ Λ plained in Part I, we will focus our attention on the ensemble with a fixed totalamountofsalt. Therelevantquantityis defined by N = S . (1.3) Λ x Xx∈Λ Themainobject ofinterestinthispaperis thentheconditionalmeasure Pσ∂Λ,c,h(·) = Pσ∂Λ · N = ⌊c|Λ|⌋ , (1.4) Λ Λ Λ (cid:0) (cid:12) (cid:1) (cid:12) where |Λ| denotes the number of sites in Λ. We will mostly focus on the situ- ations when σ ≡ +1 or σ ≡ −1, i.e., the plus or minus boundary condi- ∂Λ ∂Λ tions. In thesecases we denotetheabovemeasureby P±,c,h, respectively. Λ Colligativepropertiesofsolutions 5 The surface nature of the macroscopic phase separation—namely, the caseswhentheconcentrationscalesliketheinverselinearscaleofthesystem— indicatesthatthequantitativeaspectsoftheanalysismaydependsensitivelyon theshapeofthevolumeinwhichthemodelisstudied. Thus,tokeepthiswork manageable, we will restrict our rigorous treatment of these cases to volumes ofaparticularshapeinwhichthedropletcostisthesameasininfinitevolume. The obvious advantage of this restriction is the possibility of explicit calculations; the disadvantage is that the shape actually depends on the value of the coupling constant J. Notwithstanding, we expect that all of our findings are qualitativelycorrecteveninrectangularvolumesbutthatcannotbeguaranteed withoutafairamountofextrawork; see[17]foran example. Let V ⊂ Rd be a connected set with connected complement and unit Lebesgue volume. We will consider a sequence (V ) of latticevolumes which L are justdiscretizedblow-upsofV byscalefactor L: V = {x ∈ Zd: x/L ∈ V}. (1.5) L (The sequence of L × ··· × L boxes (Λ ) from Part I is recovered by let- L ting V = [0,1)d.) The particular “shape” V for which we will prove the macroscopic phase separation coincides with that of an equilibrium droplet— the Wulff-shapedvolume—whichwe will define next. We will stay rathersuc- cinct;detailsand proofs can befound instandardliteratureonWulffconstruc- tion ( [2,4,5,7,8,12] or the review [6]). Readers familiar with these concepts may considerskippingtherest ofthissection and passingdirectly to thestate- mentsofourmainresults. Consider the ferromagnetic Ising model at coupling J ≥ 0 and zero external field and let P±,J denote the corresponding Gibbs measure in finite vol- Λ ume Λ ⊂ Zd and plus/minus boundary conditions. As is well known, there exists a number J = J (d), with J (1) = ∞ and J (d) ∈ (0,∞) if d ≥ 2, c c c c suchthatforeveryJ > J theexpectationofanyspininΛwithrespecttoP±,J c Λ isboundedawayfromzerouniformlyinΛ ⊂ Zd. Thelimitingvalueofthisex- pectation in the plus state—typically called the spontaneous magnetization— will be denoted by m = m (J). (Notethat m = 0 for J < J whilem > 0 ⋆ ⋆ ⋆ c ⋆ forJ > J .) c Nextwewillrecall thebasicsetupfortheanalysisofsurfacephenomena. For each unit vector n ∈ Rd, we first define the surface free energy τ (n) J in direction n. To this end let us consider a rectangular box V(N,M) ⊂ Rd with “square” base ofsideN and heightM oriented such that n is orthogonal to the base. The box is centered at the origin. We let Z+,J denote the Ising N,M partition function in V(N,M) ∩ Zd with plus boundary conditions. We will alsoconsidertheinclinedDobrushinboundaryconditionwhichtakesvalue+1 at the sites x of the boundary of V(N,M) ∩ Zd for which x · n > 0 and −1 6 Alexander,BiskupandChayes at theothersites. Denotingthecorrespondingpartitionfunctionby Z±,J,n, the N,M surface free energy τ (n) isthen defined by J 1 Z±,J,n τ (n) = − lim lim log N,M . (1.6) J M→∞N→∞ Nd−1 Z+,J N,M The limit exists by subadditivity arguments. The quantity τ (n) determines J thecostofan interface orthogonalto vectorn. As expected, as soon as J > J , the function n 7→ τ (n) is uniformly c J positive [14]. In order to evaluate the cost of a curved interface, τ (n) will J havetobeintegratedoverthesurface. Explicitly,wewillletJ > J and,given c a bounded set V ⊂ Rd with piecewise smooth boundary, we define the Wulff functionalW by theintegral J W (V) = τ (n)dA, (1.7) J J Z ∂V where dA is the (Hausdorff) surface measure and n is the position-dependent unit normal vector to the surface. The Wulff shape W is the unique minimizer (modulo translation)of V 7→ W (V) among bounded sets V ⊂ Rd with piecewise smooth boundary and unit Lebesgue volume. We let (W ) denote the L sequenceofWulff-shaped latticevolumesdefined from V = W via(1.5). 2. MAIN RESULTS We are now in a position to state and prove our main results. As indicated before, wewill focus on the limitofinfinitesimalconcentrations (and external fields) where c and h scale as the reciprocal of the linear size of the system. Our resultscome in fourtheorems: In Theorem 2.1 we statethebasic surfaceorder large-deviationprinciple. Theorems 2.2 and 2.3 describe theminimizers of the requisite rate functions for liquid and ice boundary conditions, respectively. Finally, Theorem 2.4 provides some control of the spin marginal of the correspondingGibbsmeasure. 2.1. Large deviation principle for magnetization Thecontroloftheregimeunderconsiderationinvolvesthesurface-orderlarge- deviation principle for the total magnetization in the Ising model. In a finite set Λ ⊂ Zd, thequantityunderconsiderationsis givenby M = σ . (2.1) Λ x Xx∈Λ Colligativepropertiesofsolutions 7 Unfortunately, the rigorous results available at present for d ≥ 3 do not cover allofthecasestowhichouranalysismightapply. Inordertoreducetheamount of necessary provisos in the statement of the theorems, we will formulate the relevantpropertiesas an assumption: Assumption A Letd ≥ 2 andletusconsiderasequenceofWulff-shape volumesW . LetJ > J andrecallthatP±,J denotestheGibbsstateofthe L c WL IsingmodelinW ,with±-boundaryconditionandcouplingconstantJ. Let L m = m (J)denotethespontaneousmagnetization.Thenthereexistfunctions ⋆ ⋆ M : [−m ,m ] → [0,∞)suchthat ±,J ⋆ ⋆ 1 lim lim logP±,J |M −mLd| ≤ ǫLd = −M (m) (2.2) ǫ↓0 L→∞ Ld−1 L L ±,J (cid:0) (cid:1) holdsforeachm ∈ [−m ,m ]. Moreover,thereisaconstantw ∈ (0,∞) ⋆ ⋆ 1 suchthat m ∓m d−1 M (m) = ⋆ d w (2.3) ±,J 1 2m (cid:16) ⋆ (cid:17) istrueforallm ∈ [−m ,m ]. ⋆ ⋆ Thefirst part of AssumptionA—thesurface-order large-deviationprinciple (2.2)—has rigorously been verified for square boxes (and magnetizations near ±m ) in d = 2 [8,12] and in d ≥ 3 [5,7]. The extension to Wulff- ⋆ shape domains for all m ∈ [−m ,m ] requires only minor modifications in ⋆ ⋆ d = 2 [16]. For d ≥ 3 Wulff-shape domains should be analogously control- lablebutexplicitdetailshavenotappeared. Thefact(provedin[16]ford = 2) that the rate function is given by (2.3) for all magnetizations in [−m ,m ] is ⋆ ⋆ specifictotheWulff-shapedomains;forotherdomainsoneexpectstheformula tobetrueonlywhen|m ∓m|issmallenoughtoensurethattheappropriately- ⋆ sized Wulff-shape droplet fits inside the enclosing volume. Thus, Assump- tionA isa provenfact ford = 2, and itisimminentlyprovableford ≥ 3. Theunderlyingreason why(2.2)holdsistheexistenceofmultiplestates. Indeed,toachievethemagnetizationm ∈ (−m ,m )onedoesnothavetoalter ⋆ ⋆ the local distribution of the spin configurations (which is what has to be done form 6∈ [−m ,m ]);itsufficestocreateadropletofonephaseinsidetheother. ⋆ ⋆ Thecostisjustthesurfacefreeenergyofthedroplet;thebestpossibledropletis obtained by optimizingtheWulfffunctional(1.7). This is thecontent of (2.3). However,thedropletisconfinedtoafinitesetand,onceitbecomessufficiently large,theshapeoftheenclosingvolumebecomesrelevant. Ingenericvolumes the presence of this additional constraint in the variational problem actually makes the resulting cost larger than (2.3)—which represents the cost of an unconstrained droplet. But, in Wulff-shape volumes, (2.3) holds regardless of 8 Alexander,BiskupandChayes the droplet size as long as |m| ≤ m . An explicit formula for M (m) for ⋆ ±,J square volumes has been obtained in d = 2 [17]; the situation in d ≥ 3 has been addressed in[10,11]. Onthebasisoftheaboveassumptions,wearereadytostateourfirstmain result concerning the measure P±,c,h with c ∼ ξ/L and h ∼ b/L. Using θ WL to denote the fraction of salt on the plus spins, we begin by introducing the relevantentropyfunction 2θ 2(1−θ) Υ(m,θ) = −θlog −(1−θ)log . (2.4) 1+m 1−m We remark that if we write a full expression for the bulk entropy, Ξ(m,θ;c), see formula (3.5), at fixed m, c and θ, then, modulo some irrelevant terms, the quantity Υ(m,θ) is given by (∂/∂c)Ξ(m,θ;c) at c = 0. Thus, when we scalec ∼ ξ/L,thequantityξΥ(m,θ)representstherelevant(surfaceorder)en- tropyofsaltwithmandθfixed. ThefollowingisananalogueofTheoremI.2.1 from Part Iforthecaseat hand: Theorem 2.1. Let d ≥ 2 and let J > J (d) and κ > 0 be fixed. Let c m = m (J) denote the spontaneous magnetization of the Ising model. Sup- ⋆ ⋆ pose that (2.2) in Assumption A holds and let (c ) and (h ) be two sequences L L suchthatc ≥ 0 forallLand thatthelimits L ξ = lim Lc and b = lim Lh (2.5) L L L→∞ L→∞ exist andarefinite. Then forallm ∈ [−m ,m ], ⋆ ⋆ 1 lim lim logP±,cL,hL |M −mLd| ≤ ǫLd ǫ↓0 L→∞ Ld−1 WL L (cid:0) (cid:1) = −Q± (m)+ inf Q± (m′), (2.6) b,ξ |m′|≤m⋆ b,ξ where Q± (m) = inf Q± (m,θ) with b,ξ θ∈[0,1] b,ξ Q± (m,θ) = −bm−ξκθ−ξΥ(m,θ)+M (m), (2.7) b,ξ ±,J Various calculations in the future will require a somewhat more explicit expression for the rate function m 7→ Q± (m) on the right-hand side b,ξ of (2.6). To derive such an expression, we first note that the minimizer ofθ 7→ Q± (m,θ) isuniquelydeterminedbytheequation b,ξ θ 1+m = eκ. (2.8) 1−θ 1−m Colligativepropertiesofsolutions 9 PluggingthisintoQ± (m,θ) tellsus that b,ξ Q± (m) = −bm−ξg(m)+M (m), (2.9) b,ξ ±,J where 1−m 1+m g(m) = log +eκ . (2.10) (cid:18) 2 2 (cid:19) Clearly, g isstrictlyconcaveforanyκ > 0. 2.2. Macroscopicphaseseparation—“liquid”boundaryconditions While Theorem I.2.1 of Part I and Theorem 2.1 above may appear formally similar,thesolutionsoftheassociatedvariationalproblemsareratherdifferent. Indeed,unlikethe“bulk”ratefunctionG (m)ofPartI,thefunctionsQ± (m) h,c b,ξ arenotgenericallystrictlyconvexwhichinturnsleadstoapossibilityofhaving morethan oneminimizingm. We considerfirst thecase ofplus(thatis, liquid water) boundaryconditions. Let d ≥ 2 and let J > J (d) and κ > 0 be fixed. To make our formulas c manageable, foranyfunctionφ: [−m ,m ] → R let ususetheabbreviation ⋆ ⋆ φ(m )−φ(−m ) D⋆ = ⋆ ⋆ (2.11) φ 2m ⋆ fortheslopeofφ between −m and m . Further, let usintroducethequantity ⋆ ⋆ w ξ = 1 g′(−m )−D⋆ −1 (2.12) t 2m d ⋆ g ⋆ (cid:0) (cid:1) and thepiecewiselinearfunctionb : [0,∞) → R whichis defined by 2 − w1 −ξD⋆, ξ < ξ b (ξ) =  2m⋆ g t (2.13) 2 −d−1 w1 −ξg′(−m ), ξ ≥ ξ. d 2m⋆ ⋆ t  Ournextresultis as follows: Theorem 2.2. Let d ≥ 2 and let J > J (d) and κ > 0 be fixed. Let c the objects Q+ , ξ and b be as defined above. Then there exists a (strictly) b,ξ t 2 decreasing and continuous function b : [0,∞) → R with the following prop- 1 erties: (1) b (ξ) ≥ b (ξ)forallξ ≥ 0,andb (ξ) = b (ξ)iffξ ≤ ξ. 1 2 1 2 t (2) b′ is continuous on [0,∞), b′(ξ) → −g′(m ) as ξ → ∞ and b is 1 1 ⋆ 1 strictlyconvexon [ξ,∞). t 10 Alexander,BiskupandChayes (3) Forb 6= b (ξ),b (ξ),thefunctionm 7→ Q+ (m)isminimizedbyasingle 1 2 b,ξ numberm = m (b,ξ) ∈ [−m ,m ]which satisfies + ⋆ ⋆ = m , if b > b (ξ), ⋆ 1 m (b,ξ)∈ (−m ,m ), if b (ξ) < b < b (ξ), (2.14) +  ⋆ ⋆ 2 1  = −m , if b < b (ξ). ⋆ 2   (4) The function b 7→ m (b,ξ) is strictly increasing for b ∈ [b (ξ),b (ξ)], + 2 1 is continuous on the portion of the line b = b (ξ) for which ξ > ξ and 2 t has a jump discontinuity along the line defined by b = b (ξ). The only 1 minimizers at b = b (ξ) and b = b (ξ) are the corresponding limits 1 2 ofb 7→ m (b,ξ). + Thepreviousstatementessentiallycharacterizesthephasediagramforthe cases described in (2.5). Focusingontheplusboundaryconditionwehavethe following facts: For reduced concentrations ξ exceeding the critical value ξ, t thereexistsarangeofreducedmagneticfieldsbwhereanon-trivialdropletap- pearsinthesystem. Thisrangeisenclosedbytwocurveswhicharethegraphs offunctionsb andb above. Forbdecreasingtob (ξ),thesystemisinthepure 1 2 1 plus—i.e.,liquid—phasebut,interestingly,atb amacroscopicdroplet—anice 1 crystal—suddenly appears in the system. As b further decreases the ice crystal keeps growing to subsume the entire system when b = b (ξ). For ξ ≤ ξ 2 t no phase separation occurs; the transition at b = b (ξ) = b (ξ) is directly 1 2 from m = m tom = −m . ⋆ ⋆ Itisnotedthatthesituationforξ nearzerocorrespondstotheIsingmodel with negative external field proportional to 1/L. In two-dimensional setting, the latter problem has been studied in [16]. As already mentioned, the gener- alizationstorectangularboxeswillrequireanon-trivialamountofextrawork. For the unadorned Ising model (i.e., c = 0) this has been carried out in great detail in [17] for d = 2 (see also [13]) and in less detail in general dimen- sions[10,11]. It is reassuring to observe that the above results mesh favorably with the corresponding asymptotic of Part I. For finite concentrations and external fields, there are two curves, c 7→ h (c) and c 7→ h (c), which mark the + − boundaries of the phase separation region against the liquid and ice regions, respectively. Thecurvec 7→ h (c)is givenby theequation + 1 1−q h (c) = log +, (2.15) + 2 1−q − where (q ,q ) isthe(unique)solutionof + − q q 1+m 1−m + = eκ − , q ⋆ +q ⋆ = c. (2.16) + − 1−q 1−q 2 2 + −