Colligative properties of solutions: I. Fixed concentrations

0.22 MB·English

by  Kenneth Alexander

ToappearinJournalofStatisticalPhysics
Colligative properties of solutions:
5
I. Fixed concentrations
0
0
2

n Kenneth S. Alexander,1 Marek Biskup,2 and LincolnChayes2
a
J

9

2 Using the formalism of rigorous statistical mechanics, we study the phenom-
v ena of phase separation and freezing-point depression upon freezing of solu-
4
tions. Specifically, we devise an Ising-basedmodelof a solvent-solutesystem
3
and show that, in the ensemble with a fixed amount of solute, a macroscopic
0
7 phaseseparationoccursinanintervalofvaluesofthechemicalpotentialofthe
0 solvent. The boundariesof the phase separation domain in the phase diagram
4 arecharacterizedandshownto asymptoticallyagreewith theformulasusedin
0
heuristicanalysesof freezingpointdepression. The limit of infinitesimalcon-
/
h centrationsisdescribedinasubsequentpaper.
p
-
h
t
a
m 1. INTRODUCTION
:
v 1.1. Motivation
i
X The statistical mechanics of pure systems—most prominently the topic of
r phase transitions and their associated surface phenomena—has been a sub-
a
ject of fairly intensiveresearch in recent years. Several physical principles for
pure systems (the Gibbs phase rule, Wulff construction, etc.) have been put
on amathematicallyrigorousfootingand,ifnecessary,supplementedwithap-
propriate conditions ensuring their validity. The corresponding phenomena in
systemswithseveralmixedcomponents,particularlysolutions,havelongbeen
well-understoodontheleveloftheoreticalphysics. However,theyhavenotre-
ceivedmuchmathematicallyrigorousattentionandinparticularhavenotbeen
derivedrigorouslystartingfromalocalinteraction. Anaturaltaskistousethe
ideas from statistical mechanics of pure systems to develop a higher level of
control for phase transitions in solutions. This is especially desirable in light
(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
oftheimportantrolethatbasicphysicsofthesesystemsplaysinsciences,both
general (chemistry,biology,oceanography)andapplied(metallurgy,etc.). See
e.g. [11,24,27]formorediscussion.
Amongtheperhapsmostinterestingaspectsofphasetransitionsinmixed
systemsisadramaticphaseseparationinsolutionsuponfreezing (orboiling).
A well-known example from “real world” is the formation of brine pockets in
frozen seawater. Here, twoimportantphysicalphenomenaare observed:
(1) Migrationofnearlyallthesaltintowhateverportionofice/watermixture
remainsliquid.
(2) Clear evidenceoffacettingat thewater-ice boundaries.
Quantitativeanalysisalsorevealsthefollowingfact:
(3) Saltedwaterfreezesattemperatureslowerthanthefreezingpointofpure
water. Thisisthephenomenonoffreezingpointdepression.
Phenomenon (1) is what “drives” the physics of sea ice and is thus largely
responsible for the variety of physical effects that have been observed, see
e.g. [17,18]. Notwithstanding, (1–3) are not special to the salt-water system;
they are shared by a large class of the so called non-volatile solutions. A dis-
cussion concerning thegeneral aspects of freezing/boilingof solutions—often
referred to as colligativeproperties—canbefound in[24,27].
Of course, on a heuristic level, the above phenomena are far from mys-
terious. Indeed, (1) follows from the observation that, macroscopically, the
liquid phase provides a more hospitable environment for salt than the solid
phase. Then (3) results by noting that the migration of salt increases the en-
tropiccostoffreezingsotheenergy-entropybalanceforcesthetransitionpoint
toalowertemperature. Finally,concerningobservation(2)wenotethat,dueto
the crystalline nature of ice, the ice-water surface tension will be anisotropic.
Therefore, to describe the shape of brine pockets, a Wulff construction has to
be involvedwiththe caveat that here thecrystallinephaseis on theoutside. In
summary, what is underlying these phenomena is a phase separation accom-
panied by the emergence of a crystal shape. In the context of pure systems,
such topics have been well understood at the level of theoretical physics for
quite some time [12,16,32,33] and, recently (as measured on the above time
scale), also at the level of rigorous theorems in two [2,4,14,22,28,29] and
higher[6,9,10]dimensions.
The purpose of this and a subsequent paper is to study the qualitative
nature of phenomena (1–3) using the formalism of equilibrium statistical me-
chanics. Unfortunately, a microscopically realistic model of salted water/ice
system is far beyond reach of rigorous methods. (In fact, even in pure water,
Colligativepropertiesofsolutions 3
thephenomenonoffreezingissocomplexthatcrystalizationinrealisticmodels
has only recently—and only marginally—been exhibited in computer simula-
tions [26].) Thus we will resort to a simplified version in which salt and both
phases of water are represented by discrete random variables residing at sites
ofaregularlattice. Forthesemodelsweshowthatphaseseparationdominates
a non-trivial region of chemical potentials in the phase diagram—a situation
quiteunlikethepuresystemwherephaseseparation can occuronlyat asingle
value (namely, the transition value) of the chemical potential. The boundary
lines of the phase-separation region can be explicitlycharacterized and shown
to agree with the approximate solutions of the corresponding problem in the
physical-chemistryliterature.
The above constitutes the subject of the present paper. In a subsequent
paper[1]wewilldemonstratethat,forinfinitesimalsaltconcentrationsscaling
appropriatelywiththesizeofthesystem,phaseseparationmaystilloccurdra-
matically in the sense that a non-trivial fraction of the system suddenly melts
(freezes) to form a pocket (crystal). In these circumstances the amount of salt
needed isproportionaltotheboundaryofthesystemwhichshowsthattheon-
set of freezing-point depression is actually a surface phenomenon. On a qual-
itative level, most of the aforementioned conclusions should apply to general
non-volatilesolutionsundertheconditionswhenthesolventfreezes (orboils).
Notwithstanding, throughout this and the subsequent paper we will adopt the
language of salted water and refer to the solid phase of the solvent as ice, to
theliquidphaseas liquid-water,and tothesoluteassalt.
1.2. General Hamiltonian
Ourmodelwillbedefinedonthed-dimensionalhypercubiclatticeZd. Wewill
takethe(formal)nearest-neighborHamiltonianofthefollowingform:
βH = − (αI I +α L L )+κ S I − µ S − µ L . (1.1)
I x y L x y x x S x L x
hXx,yi Xx Xx Xx
Here β is the inverse temperature (henceforth incorporated into the Hamito-
nian), x and y are sites in Zd and hx,yi denotes a neighboring pair of sites.
The quantities I , L and S are the ice (water), liquid (water) and salt vari-
x x x
ables, whichwilltakevaluesin{0,1}withtheadditionalconstraint
I +L = 1 (1.2)
x x
valid at each site x. We will say that I = 1 indicates the presence of ice at x
x
and, similarly, L the presence of liquid at x. Since a single water molecule
x
cannot physically be in an ice state, it is natural to interpret the phraseI = 1
x
as referring to the collective behavior of many particles in the vicinity of x
4 Alexander,BiskupandChayes
which are enacting an ice-like state, though we do not formally incorporate
such aviewpointintoourmodel.
The various terms in (1.1) are essentially self-explanatory: An interac-
tion between neighboring ice points, similarly for neighboring liquid points
(we may assume these to be attractive), an energy penalty κ for a simultane-
ous presence of salt and ice at one point, and, finally, fugacity terms for salt
and liquid. For simplicity(and tractability), there is no direct salt-salt interac-
tion, except for the exclusion rule of at most one salt “particle” at each site.
Additional terms which could have been included are superfluous due to the
constraint (1.2). We will assume throughout that κ > 0, so that the salt-ice
interaction expresses the negativeaffinity of salt to the ice state of water. This
term is entirely—and not subtly—responsible for the general phenomenon of
freezing point depression. We remark that by suitably renaming the variables,
theHamiltonianin(1.1)wouldjustaswelldescribeasystemwithboilingpoint
elevation.
As we said, the variables I and L indicate the presence of ice and liq-
x x
uid water at site x, respectively. The assumptionI +L = 1 guarantees that
x x
somethinghas to be present at x (theconcentration of water in water is unity);
what is perhaps unrealistic is the restriction of I and L to only the extreme
x x
values, namelyI ,L ∈ {0,1}. Suffice it to say that the authors are confident
x x
(e.g., on the basis of [3]) that virtually all the results in this note can be ex-
tended to the cases of continuous variables. However, we will not make any
such mathematical claims; much of this paper will rely heavily on preexisting
technology which, strictly speaking, has only been made to work for the dis-
crete case. A similar discussion applies, of course, to the salt variables. But
here our restriction to S ∈ {0,1} is mostly to ease the exposition; virtually
x
all of ourresultsdirectly extendto thecases whenS takes arbitrary (positive)
x
real valuesaccording to somea prioridistribution.
1.3. Reduction to Ising variables
It is not difficult to see that the “ice-liquid sector” of the general Hamiltonian
(1.1) reduces to a ferromagnetic Ising spin system. On a formal level, this is
achieved by passing to the Ising variables σ = L −I , which in light of the
x x x
constraint(1.2)gives
1+σ 1−σ
L = x and I = x. (1.3)
x x
2 2
By substitutingtheseinto(1.1), wearriveat theinteractionHamiltonian:
1−σ
βH = −J σ σ −h σ +κ S x − µ S , (1.4)
x y x x 2 S x
hXx,yi Xx Xx Xx
Colligativepropertiesofsolutions 5
where thenew parameters J andhare givenby
α +α d µ
J = L I and h = (α −α)+ L. (1.5)
4 2 L I 2
Weremarkthatthethirdsumin(1.4)isstillwrittenintermsof“ice”indicators
sothatH willhaveawelldefinedmeaningevenifκ = ∞,whichcorresponds
to prohibitingsalt entirely at ice-occupied sites. (Notwithstanding,the bulk of
this paper is restricted to finite κ.) Using an appropriate restriction to finite
volumes, the above Hamitonian allows us to define the corresponding Gibbs
measures. Wepostponeanyrelevanttechnicalitiesto Section 2.1.
The Hamiltonian as written foretells the possibility of fluctuations in the
salt concentration. However, this is not the situation which is of physical in-
terest. Indeed, in an open system it is clear that the salt concentration will,
eventually, adjust itself until the system exhibits a pure phase. On the level of
the description provided by (1.4) it is noted that, as grand canonical variables,
the salt particles can be explicitly integrated, the result being the Ising model
at couplingconstantJ andexternal field h , where
eff
1 1+eµS
h = h+ log . (1.6)
eff
2 1+eµS−κ
In this context, phase coexistence is confined to the region h = 0, i.e., a
eff
simple curve in the (µ ,h)-plane. Unfortunately, as is well known [5,19,20,
S
23,30], not much insight on the subject of phase separation is to be gained
by studying the Ising magnet in an external field. Indeed, under (for example)
minus boundary conditions, once h exceeds a particular value, a droplet will
form which all but subsumes the allowed volume. The transitional value of h
scales inversely with the linear size of the system; the exact constants and
the subsequent behavior of the droplet depend on the details of the boundary
conditions.
The described “failure” of the grand canonical description indicates that
thecorrect ensemblein thiscase istheonewithafixed amountofsaltperunit
volume. (The technical definition uses conditioning from the grand canonical
measure; see Section 2.1.) Thisensembleis physicallymorerelevantbecause,
at the moment of freezing, the salt typically does not have enough “mobility”
tobegraduallyreleasedfromthesystem. Itisnotedthat,oncethetotalamount
ofsaltisfixed,thechemicalpotentialµ dropsoutoftheproblem—therelevant
S
parameter is now the salt concentration. As will be seen in Section 2, in our
Ising-based model of the solvent-solute system, fixing the salt concentration
generically leads to sharp phase separation in the Ising configuration. More-
over, this happens for an interval of values of the magnetic field h. Indeed,
the interplay between the salt concentration and the actual external field will
6 Alexander,BiskupandChayes
demand a particular value of the magnetization, even under conditions which
will force a droplet (or ice crystal, depending on the boundary condition) into
thesystem.
Remark 1.1. We finish by noting that, while the parameter h is for-
mallyunrelatedtotemperature, itdoestoalimitedextentplaytheroleoftem-
perature in that it reflects the a priori amount of preference of the system for
watervs ice. Thusthenatural phasediagram tostudyisin the(c,h)-plane.
1.4. Heuristic derivations and outline
Thereasoningwhichledtoformula(1.6)allowsforanimmediateheuristicex-
planation of our principal results. The key simplification—which again boils
down to the absence of salt-salt interaction—is that for any Ising configura-
tion,theamalgamatedcontributionofsalt,i.e.,theGibbsweightsummedover
salt configurations, depends only on the overall magnetization and not on the
details of how themagnetization gets distributedabout the system. In systems
of linear scale L, let Z (M) denote the canonical partition function for the
L
Ising magnet with constrained overall magnetization M. The total partition
functionZ (c,h) at fixedsalt concentrationc can thenbewrittenas
L
Z (c,h) = Z (M)ehMW (M,c), (1.7)
L L L
X
M
where W (M,c) denotes the sum of the salt part of the Boltzmann weight—
L
whichonlydependsontheIsingspinsviathetotalmagnetizationM—overall
salt configurationswithconcentrationc.
As usual, the physical values of the magnetization are those bringing the
dominant contribution to the sum in (1.7). Let us recapitulate the standard
arguments by first considering the case c = 0 (which implies W = 1), i.e.,
L
the usual Ising system at external field h. Here we recall that Z (mLd) can
L
approximatelybewrittenas
Z (mLd) ≈ e−Ld[FJ(m)+C], (1.8)
L
where C is a suitably chosen constant and F (m) is a (normalized) canonical
J
free energy. The principal fact about F (m) is that it vanishes for m in the
J
interval [−m ,m ], where m = m (J) denotes the spontaneous magnetiza-
⋆ ⋆ ⋆ ⋆
tion of the Ising model at coupling J, while it is strictly positive and strictly
convex for m with |m| > m . The presence of the “flat piece” on the graph
⋆
ofF (m)isdirectlyresponsiblefortheexistenceofthephasetransitioninthe
J
Isingmodel: Forh > 0thedominantcontributiontothegrandcanonicalparti-
tion function comes from M & m Ld while for h < 0 the dominant values of
⋆
Colligativepropertiesofsolutions 7
the overall magnetization are M . −m Ld. Thus, once m = m (J) > 0—
⋆ ⋆ ⋆
which happens for J > J (d) with J (d) ∈ (0,∞) whenever d ≥ 2—a phase
c c
transitionoccursat h = 0.
The presence of salt variables drastically changes the entire picture. In-
deed, as we will see in Theorem 2.1, the salt partition function W (M,c)
L
will exhibit a nontrivial exponential behavior which is characterized by
a strictly convex free energy. The resulting exponential growth rate of
Z (M)ehMW (M,c) for M ≈ mLd is thus no longer a function with a flat
L L
piece—instead, for each h there is a unique value of m that optimizes the
corresponding free energy. Notwithstanding (again, due to the absence of
salt-salt interactions) once that m has been selected, the spin configurations
are the typical Ising configurations with overall magnetizations M ≈ mLd.
In particular, whenever Z (c,h) is dominated by values of M ≈ mLd for
L
an m ∈ (−m ,m ), a macroscopic droplet develops in the system. Thus, due
⋆ ⋆
totheone-to-onecorrespondencebetweenhandtheoptimalvalueofm,phase
separation occurs for an interval of values of h at any positive concentration;
seeFig. 1.
Wefinish withan outlineoftheremainderofthispaper and somediscus-
sion of the companion paper [1]. In Section 2 we define precisely the model
ofinterestandstateourmainresultsconcerningtheasymptoticbehaviorofthe
corresponding measure on spin and salt configurations with fixed concentra-
tion of salt. Along with the results comes a description of the phase diagram
and a discussion of freezing-point depression, phase separation, etc., see Sec-
tion 2.3. Our main results are proved in Section 3. In [1] we investigate the
asymptotic of infinitesimal salt concentrations. Interestingly, we find that, in
order to induce phase separation, the concentration has to scale at least as the
inverselinearsizeofthesystem.
2. RIGOROUS RESULTS
2.1. The model
With the (formal) Hamiltonian (1.4) in mind, we can now start on developing
the mathematical layout of the problem. To define the model, we will need
to restrict attention to finite subsets of the lattice. We will mostly focus on
rectangular boxes Λ ⊂ Zd of L × L × ··· × L sites centered at the origin.
L
Our convention for the boundary, ∂Λ, of the set Λ ⊂ Zd will be the collection
of sites outside Λ with a neighbor inside Λ. For each x ∈ Λ, we have the
waterandsaltvariables,σ ∈ {−1,+1}andS ∈ {0,1}. Ontheboundary,we
x x
will consider fixed configurations σ ; most of the time we will be discussing
∂Λ
the cases σ = +1 or σ = −1, referred to as plus and minus boundary
∂Λ ∂Λ
8 Alexander,BiskupandChayes
conditions. Since there is no salt-salt interaction, we may as well set S = 0
x
forall x ∈ Λc.
We will start by defining the interaction Hamiltonian. Let Λ ⊂ Zd be a
finite set. For a spin configuration σ and the pair (σ ,S ) of spin and salt
∂Λ Λ Λ
configurations,welet
1−σ
βH (σ ,S |σ ) = −J σ σ −h σ +κ S x. (2.1)
Λ Λ Λ ∂Λ x y x x
2
X X X
hx,yi x∈Λ x∈Λ
x∈Λ,y∈Zd
Here, as before, hx,yi denotes a nearest-neighbor pair on Zd and the parame-
ters J, h and κ are as discussed above. (In light of the discussion from Sec-
tion1.3thelasttermin(1.4)hasbeenomitted.) Theprobabilitydistributionof
thepair(σ ,S )takes theusualGibbs-Boltzmannform:
Λ Λ
e−βHΛ(σΛ,SΛ|σ∂Λ)
Pσ∂Λ(σ ,S ) = , (2.2)
Λ Λ Λ Z (σ )
Λ ∂Λ
where the normalization constant, Z (σ ), is the partition function. The dis-
Λ ∂Λ
tributions in Λ with the plus and minus boundary conditions will be denoted
L
by P+ and P−, respectively.
L L
Forreasons discussedbefore wewill beinterested in theproblemswitha
fixed salt concentration c ∈ [0,1]. In finite volume, we take this to mean that
thetotalamountofsalt,
N = N (S) = S , (2.3)
L L x
X
x∈ΛL
isfixed. Tosimplifyfuturediscussions,wewilladopttheconventionthat“con-
centration c” means that N ≤ c|Λ | < N + 1, i.e., N = ⌊cLd⌋. We may
L L L L
then define the finite volume Gibbs probability measure with salt concentra-
tion c and plus (or minus) boundary conditions denoted by P+,c,h (or P−,c,h).
L L
In lightof(2.2),theseare givenbytheformulas
P±,c,h(·) = P± · N = ⌊cLd⌋ . (2.4)
L L L
(cid:0) (cid:12) (cid:1)
Both measures P±,c,h depend on the par(cid:12)ameters J and κ in the Hamiltonian.
L
However, we will always regard these as fixed and suppress them from the
notationwheneverpossible.
2.2. Main theorems
In order to describe our first set of results, we will need to bring to bear a few
standard facts about the Ising model. For each spin configuration σ = (σ ) ∈
x
Colligativepropertiesofsolutions 9
{−1,1}ΛL letus definetheoverallmagnetizationinΛ by theformula
L
M = M (σ) = σ . (2.5)
L L x
X
x∈ΛL
Let m(h,J) denote the magnetization of the Ising model with coupling con-
stantJ andexternalfieldh ≥ 0. Asiswellknown,cftheproofofTheorem3.1,
h 7→ m(h,J)continuously(andstrictly)increasesfromthevalueofthesponta-
neousmagnetizationm = m(0,J)tooneashsweepsthrough[0,∞). Inpar-
⋆
ticular, for each m ∈ [m(0,J),1), thereexistsauniqueh = h(m,J) ∈ [0,∞)
such thatm(h,J) = m.
Next we will use the above quantities to define the func-
tion F : (−1,1) → [0,∞), which represents the canonical free energy
J
oftheIsingmodelin(1.8). Asitturnsout—seeTheorem3.1inSection3—we
simplyhave
F (m) = dm′h(m′,J)1 , m ∈ (−1,1). (2.6)
J {m⋆≤m′≤|m|}
Z
Asalreadymentioned,ifJ > J ,whereJ = J (d)isthecriticalcouplingcon-
c c c
stantoftheIsingmodel,thenm > 0andthusF (m) = 0form ∈ [−m ,m ].
⋆ J ⋆ ⋆
(Since J (d) < ∞ only for d ≥ 2, the resulting “flat piece” on the graph
c
of m 7→ F (m) appears only in dimensions d ≥ 2.) From the perspective of
J
the large-deviation theory, cf [13,21], m 7→ F (m) is the large-deviation rate
J
function for the magnetization in the (unconstrained) Ising model; see again
Theorem 3.1.
Let S(p) = plogp + (1 − p)log(1 − p) denote the entropy function of
theBernoullidistributionwithparameterp. (WewillsetS(p) = ∞whenever
p 6∈ [0,1].) Foreach m ∈ (−1,1),each c ∈ [0,1]andeach θ ∈ [0,1],let
1+m 2θc 1−m 2(1−θ)c
Ξ(m,θ;c) = − S − S . (2.7)
2 1+m 2 1−m
(cid:16) (cid:17) (cid:16) (cid:17)
As we will show in Section 3, this quantity represents the entropy of configu-
rationswithfixedsaltconcentrationc,fixedoverallmagnetizationmandfixed
fraction θ of the salt residing “on the plus spins” (and fraction 1 − θ “on the
minusspins”).
Having defined all relevant quantities, we are ready to state our results.
We begin with a large-deviation principle for the magnetization in the mea-
sures P±,c,h:
L
10 Alexander,BiskupandChayes
Theorem 2.1. LetJ > 0andκ > 0befixed. Foreachc ∈ (0,1),each
h ∈ R andeach m ∈ (−1,1),we have
1
lim lim logP±,c,h |M −mLd| ≤ ǫLd
ǫ↓0 L→∞ Ld L L
(cid:0) (cid:1)
= −G (m)+ inf G (m′). (2.8)
h,c h,c
m′∈(−1,1)
Here m 7→ G (m) isgiven by
h,c
G (m) = inf G (m,θ), (2.9)
h,c h,c
θ∈[0,1]
where
G (m,θ) = −hm−κθc−Ξ(m,θ;c)+F (m). (2.10)
h,c J
The function m 7→ G (m) is finite and strictly convex on (−1,1) with
h,c
lim G′ (m) = ±∞. Furthermore,theuniqueminimizerm = m(h,c)of
m→±1 h,c
m 7→ G (m) iscontinuousin bothcand handstrictlyincreasinginh.
h,c
On the basis of the above large-deviation result, we can now character-
izethetypicalconfigurationsofthemeasuresP±,c,h. ConsidertheIsingmodel
L
withcouplingconstantJ andzeroexternalfieldandletP±,J bethecorrespond-
L
ing Gibbs measure in volume Λ and ±-boundary condition. Our main result
L
in thissectionisthenas follows:
Theorem 2.2. LetJ > 0andκ > 0befixed. Letc ∈ (0,1)andh ∈ R,
anddefinetwosequencesofprobabilitymeasuresρ± on[−1,1]bytheformula
L
ρ± [−1,m] = P±,c,h(M ≤ mLd), m ∈ [−1,1]. (2.11)
L L L
(cid:0) (cid:1)
The measures ρ± allow us to write the spin marginal of the measure P±,c,h as
L L
a convex combination of the Ising measures with fixed magnetization; i.e., for
anyset Aofconfigurations(σ ) , wehave
x x∈ΛL
P±,c,h A×{0,1}ΛL = ρ±(dm)P±,J A M = ⌊mLd⌋ . (2.12)
L Z L L L
(cid:0) (cid:1) (cid:0) (cid:12) (cid:1)
(cid:12)
Moreover, if m = m(h,c) denotes the unique minimizer of the function m 7→
G (m) from(2.9), thenthefollowingpropertiesaretrue:
h,c
(1) Given the spin configuration on a finite set Λ ⊂ Zd, the salt variables
on Λ are asymptoticallyindependent. Explicitly, for each finite set Λ ⊂
Zd andanytwo configurationsS ∈ {0,1}Λ andσ¯ ∈ {−1,1}Λ,
Λ Λ
lim P±,c,h S =S σ = σ¯
L Λ Λ Λ Λ
L→∞
(cid:0) (cid:12) (cid:1)
= (cid:12) q δ (S )+(1−q )δ (S ) , (2.13)
σ¯x 1 x σ¯x 0 x
xY∈Λ(cid:8) (cid:9)