Equivalence of ensembles for two-species zero-range invariant measures

0.37 MB·English

by  Stefan Grosskinsky

Equivalence of ensembles for two-species
zero-range invariant measures
Stefan Grosskinsky
∗
8 UniversityofWarwick
0
0
2 February 5, 2008

n
a
J

9
Abstract

2
v Westudytheequivalence ofensemblesforstationary measuresofinteracting
9
particle systems with two conserved quantities and unbounded local state space.
2
0 Themainmotivationisacondensation transitioninthezero-rangeprocesswhich
8 has recently attracted attention. Establishing the equivalence of ensembles via
0
convergence inspecificrelativeentropy,wederivethephasediagramforthecon-
6
0 densation transition, which can be understood in terms of the domain of grand-
h/ canonical measures. Of particular interest, also from a mathematical point of
p view,aretheconvergence properties oftheGibbsfreeenergyontheboundary of
-
h that domain, involving large deviations and multivariate local limit theorems of
t subexponential distributions.
a
m
: keywords. zero-rangeprocess;equivalenceofensembles;condensationtransition;
v
relativeentropy
i
X
r
a
1 Introduction
Zero-range processes are interacting particle systems with no restriction on the num-
ber of particles per site, i.e. with unbounded local state space. The jump rate of each
particle depends only on the number of particles at its departure site which leads to
a simple product structure of the stationary measure [1, 2]. These processes have
recently attained muchattentionin thetheoreticalphysicsliterature(see [3]and refer-
ences therein) since they exhibita condensation transitionunder certain conditionson
∗MathematicsInstitute,ZeemanBuilding,UniversityofWarwick,CoventryCV47AL,UK.
email: [email protected]
1
the jump rates [4]. If the particle density exceeds a critical value ρ , the system phase
c
separatesintoahomogeneousbackgroundwithdensityρ andacondensate,wherethe
c
excess particlesaccumulate. Firstrigorousresultson asinglespecies system[5] show
thatthisphasetransitioncanbeunderstoodmathematicallyinthecontextoftheequiv-
alenceofensembles. Thisisaclassicalproblemofmathematicalstatisticalmechanics
[6]whicharisesnaturallyinthecontextofstudyingstationarymeasuresofinteracting
particle systems with conserved quantities, such as energy or the number of particles.
In general, interactingparticlesystemswith severalconservationlawsare currentlyof
particularinterestinnon-equilibriumstatisticalmechanics,sincetheyshowaveryrich
critical behaviour (see [7] and references therein). There are not many general results
for such systems, for example for zero-range processes with more than one particle
species there exist only non-rigorous case studies so far [8]. The motivation of this
paper is to understand the condensation transition in such multi-species processes on
therigorousleveloftheequivalenceofensembles.
Forsimplicityof presentation we focus on systemswith two conserved quantities,
which we interpret as the numberof particles in a two-species system. The local state
space for each species is N = 0,1,... , i.e. the numbers of particles on each lattice
{ }
site are unrestricted. We require that the process has a stationary measure of product
form. Due to the conservation law this induces a family of stationary measures πL,N
with fixed particle numbers N N2 on a lattice of size L, the canonical ensemble.
∈
Another standard family is the grand-canonical ensemble νL, where the numbers of
µ
particles are random variables. The densities ρ (0, )2, the expected numbers
∈ ∞
of particles per site, are controlled by conjugate parameters µ R2, the chemical
∈
potentials. In our case νL is a product measure and is also defined for L , where
µ → ∞
we write ν . Let D R2 denote the maximal domain, such that ν1 is normalizable
µ µ ⊂ µ
withfinitefirst moment.
In the thermodynamic limit N /L ρ as L with densities ρ, one expects
L
→ → ∞
that
∃µ(ρ) ∈ Dµ : πL,NL → νµ(ρ) as L → ∞ . (1.1)
The question of the equivalence of ensembles is for which values of ρ and in what
sense (1.1) holds, and how it has to be modified in the presence of phase separation.
Themainresultsofthispaperare:
1. We establish the equivalence of ensembles (1.1) for all ρ (0, )2 under mild
∈ ∞
regularityassumptionson thestationaryproduct measure.
In the proof we use specific relative entropy (or relative information gain), which is
basedonresultsfrominformationtheory[9]andwaspreviouslyappliedtostudylarge
deviations and the equivalence of ensembles for Gibbsian random fields [10], marked
pointprocesses[11]andweaklydependentmeasures[12]. Acommonfeatureofthese
2
modelsisaboundedHamiltonian,whichcorrespondstoD = R2 intheabovesetting.
µ
In this case, phase separation is a consequence of long-range correlations, leading
to non-differentiability of the Gibbs free energy (or non-convex canonical entropy)
and a first order transition [13, 14]. In our case there are no spatial correlations, but
typicallyD ( R2 duetotheunboundedlocalstatespace,andcondensationisaresult
µ
of large deviation properties of ν1 on the boundary of D , where it turns out to be
µ µ
subexponential.
2. We show how the phase diagram for the condensation transition can be derived
solelyfromtheshapeofD ,andexplainitsrelationtothemodeofconvergence
µ
in (1.1).
The transition is continuous and is characterized by convergence properties of the
GibbsfreeenergyontheboundaryofD . Intheclassificationof[14]thiscorresponds
µ
to thecase ofpartialequivalenceofensembles.
Our results can be directly generalized to any number of particle species with ar-
bitrary discrete local state spaces. We choose to work in a more specific setting for
the simplicity of presentation, since it covers the basic novelties of the paper. From
the point of view of non-equilibrium statistical mechanics these are the first rigorous
resultsonthecondensationtransitioninasystemwithseveralconservationlaws. From
a mathematical point of view, we adapt the theory of the equivalence of ensembles to
study phase separation in systems with unbounded Hamiltonians. Even in the basic
case of stationary product measures the different mathematical origin of the conden-
sation transition leads to interesting new aspects. Our equivalence result involves a
sharp condition on the number of particles and is valid on the (non-empty) boundary
ofD . Theanalysisrequiresresultsonlargedeviations[15,16]andmulti-dimensional
µ
local limit theorems of subexponential distributions [17, 18], as well as convergence
properties of multivariatepower series similar to [19]. In contrast to a previous study
for single-species processes [5], the present paper provides a complete picture of the
mechanismofcondensationin amuchmoregeneral context.
Precise definitions ofthe ensembles and basic properties are givenin thenext sec-
tion. The main results are given in Section 3, including the equivalence of ensembles
and the construction of the phase diagram for the condensation transition. For com-
pleteness, we also include some remarks on fluctuations and the spatial extension of
thecondensate(cf. [5,20]). ProofsaregiveninSection4. Sincethemainresultsapply
forensemblesofmeasuresinageneralcontext,thepaperuptothispointisformulated
without reference to zero-range processes, which are, however, the main motivation
for this study. In Section 5 we explain why these processes provide a natural class
of particle systems for the measures considered in the first sections, and illustrate the
results on the phase diagram by several examples. Some results from convex analysis
needed in theproofoftheequivalenceofensemblesare summarizedintheappendix.
3
2 Preliminaries
2.1 Canonical and grand-canonical measures
ConsiderLindependentidenticallydistributedrandomvectors
η(x) = η (x),η (x) N2 , x Λ , (2.1)
1 2 L
∈ ∈
with some dis(cid:0)crete index s(cid:1)et Λ of size Λ = L. The state space X = (N2)ΛL
L L L
| |
is a measure space with σ-algebra induced by the product topology and the (a-priori)
measure
wL(η) = w η(x) (0, ) for η = η(x) . (2.2)
∈ ∞ x∈ΛL
xY∈ΛL (cid:0) (cid:1) (cid:0) (cid:1)
This should be positive but not necessarily normalized, i.e. w : N2 (0, ) is
→ ∞
arbitrary. SinceX isdiscrete,wesimplifynotationhereandinthefollowingbyusing
L
thesamesymbolsforameasureanditsmass function,i.e. wL(η) = wL η .
{ }
WeinterprettheindexsetΛ asalatticeofsizeLand η X asparticleconfigu-
L L
∈ (cid:0) (cid:1)
rationsofatwo-speciesparticlesystem. Wedonotspecifythegeometryofthelattice,
boundary conditionsordynamicsofthisprocess, theyshouldbe suchthat wL isa sta-
tionary weight, i.e. up to normalization, wL is a stationary distributionof the process.
Genericparticlesystemswiththispropertyare zero-rangeprocesses discussedinSec-
tion5. ApartfromstationarityofwL,theonlyotherrequirementontheparticlesystem
is thatthenumbersofparticles
Σ (η) = Σ1,Σ2 (η) := η(x) N2 (2.3)
L L L ∈
(cid:0) (cid:1) xX∈ΛL
areconservedquantitiesforeachspeciesandthattherearenootherconservationlaws.
Then there existsa family of stationary probabilitymeasures h(Σ )wL which are ab-
L
solutelycontinuouswithrespecttowL,wheretheRadon-Nikodymderivativedepends
onlyontheconservedquantitiesΣ andcanbewrittenasafunctionh : N2 [0, ).
L
→ ∞
Thesetofallstationarymeasuresoftheparticlesystemisconvexandtheextremal
measures are given by choosing h(ΣL) ∝ δΣL,N, i.e. proportional to the Kronecker
delta, fixingthenumberofparticlesto N = (N ,N ) N2. Thefamily
1 2
∈
1
πL,N(η) = ZL,N w η(x) δΣL(η),N , N ∈ N2 (2.4)
xY∈ΛL (cid:0) (cid:1)
is thecanonicalensembleandthemeasures concentrateonfinitesubsets
XL,N = η ΣL(η) = N ( XL (2.5)
(cid:8) (cid:12) (cid:9)
(cid:12) 4
of configurations with fixed particle numbers. The canonical partition function is
ZL,N = wL(XL,N) (0, ), since πL,N = wL . ΣL = N can be written as
∈ ∞ { }
a conditional measure. By assumption, for each fixed L 1 and N N2 the particle
(cid:0) (cid:12) ≥ (cid:1) ∈
system is irreducible on XL,N and πL,N is the uniqu(cid:12)e stationary measure. All other
stationarymeasures onX are convexcombinationsofcanonical measures.
L
Another generic choice is g(Σ ) eµ·ΣL with parameters µ = (µ ,µ ) R2
L 1 2
∝ ∈
called chemical potentials,definingthegrand-canonicalmeasures
1
νL(η) = w η(x) eµ·η(x) . (2.6)
µ z(µ)L
xY∈ΛL (cid:0) (cid:1)
Each νL is supported on X , i.e. Σ is a random variable and the expected value is
µ L L
fixed by thechemicalpotentialsµ, as isdiscussedbelow. Thesemeasures are particu-
larlyconvenientsincetheyareofproductform. Thenormalizing(singlesite)partition
function
z(µ) = w(k)eµ·k (2.7)
k∈N2
X
isaninfinitesum,asopposedtomodelswithboundedlocalstatespace, suchas 0,1
{ }
for latticegases or 1,1 for spin systems. For such systems, z(µ) is defined for all
{− }
µ R2, whereas inourcasethedomainofdefinitionofz willplay acrucial role.
∈
2.2 Properties of grand-canonical measures
Wedefine
D = µ R2 k w(k)eµ·k < fori = 1,2 . (2.8)
µ i
∈ ∞
n (cid:12) kX∈N2 o
(cid:12)
(cid:12)
This implies that for all µ D , z(µ) < and the product measure ν is well de-
µ µ
∈ ∞
fined. Moreover,onD themarginalν1 hasfinitefirstmoments,whichareinterpreted
µ µ
as particledensitiesand givenby
R = (R ,R ) : D (0, )2 , where R (µ) = η , i = 1,2. (2.9)
1 2 µ i i ν1
→ ∞ h i µ
Hereandinthefollowingwewrite .. fortheexpectedvaluewithrespecttomeasure
ν
h i
ν. Note that R (µ) = η (x) independently of the lattice sitex Λ , and that D
i i νµL ∈ L µ
as defined in (2.8) is the maximal domain of definition of R, i.e. D = domR. We
(cid:10) (cid:11) µ
denoteby
D = R D (0, )2 (2.10)
ρ µ
⊂ ∞
(cid:0) (cid:1)
5
the range of R, which characterizes the set of all densities accessible by the grand-
canonical ensemble.
Inthefollowingweassumethat w isexponentiallybounded,i.e.
ξ (0, ) k N2 : w(k) ξ|k| , (2.11)
∃ ∈ ∞ ∀ ∈ ≤
where we write k = k = k2 +k2 1/2. For convenience we further assume that
| | k k2 1 2
thesinglesitemassfunctionw is actuallydefined on [0, )2 with
(cid:0) (cid:1) ∞
w C1 [0, )2,(0, ) , (2.12)
∈ ∞ ∞
which impose(cid:0)sno restrictiono(cid:1)ntherelevantvaluesw(k),k N2.
∈
Lemma 2.1 D = (and thus D = ) if and only if (2.11) is fulfilled. In this case
µ ρ
6 ∅ 6 ∅
D isconvex andcomplete,i.e.
µ
∆(µ∗) := µ µ µ∗, i = 1,2 D whenever µ∗ D . (2.13)
{ | i ≤ i } ⊂ µ ∈ µ
Either D = R2 or the boundary can be characterized in the rotated variables µ˜ =
µ 1
µ µ and µ˜ = µ +µ by ∂D = (µ˜ ,µ˜ (µ˜ )) µ˜ R . Here µ˜ : R R is
1 2 2 1 2 µ 1 2 1 1 2
− ∈ →
continuousand piecewisedifferentiable,with
(cid:8) (cid:12) (cid:9)
(cid:12)
µ˜ (µ˜ ) = limsup 2logw(k)+µ˜ (k k ) (k +k ) . (2.14)
2 1 1 1 2 1 2
− |k|→∞ −
(cid:16) (cid:17)
(cid:14)
All the above properties also hold for domz, the maximal domain of definition of z.
We have D domz and intD = intdomz for the interior, so both sets are equal
µ µ
⊂
or differonlyon theboundary.
Sincethegrand-canonical measures areproduct measuresthepressureisgivenby
1
p(µ) = lim logz(µ)L = logz(µ) , (2.15)
L→∞ L
which is the analogue of the Gibbs free energy. For all µ D the density (2.9) can
µ
∈
bewrittenas
R (µ) = ∂ p(µ) , i = 1,2. (2.16)
i µi
Thederivativesaredefinedone-sidedon∂D D ,whichispossibleduetocomplete-
µ µ
∩
ness ofD and thefollowinglemma.
µ
Lemma 2.2 The single site marginal ν1 has some finite exponential moments if and
µ
only if µ intD . Moreover, p C∞(intD ,R), p C1(D ,R) and p is strictly
µ µ µ
∈ ∈ ∈
6
convex onD .
µ
p andR can beextended continuouslyto
∂1,−∞D = ( ,µ ) µ R : (µ ,µ ) D , (2.17)
µ 2 1 1 2 µ
−∞ ∃ ∈ ∈
(cid:8) (cid:12) (cid:9)
i.e. limitsexistand aregivenby
(cid:12)
∞
0
p( ,µ ) = w(0,k )eµ2k2 and R( ,µ ) = . (2.18)
2 2 2
−∞ −∞ ∂ p( ,µ )
kX2=0 (cid:18) µ2 −∞ 2 (cid:19)
An analogousresultholdsfor∂2,−∞D .
µ
For i = 1,2, if D is bounded in ρ , i.e. for all ρ D , ρ C for some C 0, then
ρ i ρ i
∈ ≤ ≥
D isboundedin µ .
µ i
Since p is strictly convex, R is invertible on D due to (2.16) and we denote the
µ
inverse by M : D D . The entropy density s : (0, )2 R of the grand-
ρ µ
→ ∞ →
canonicalmeasure(2.6)istheconvexconjugateofthepressuregivenbytheLegendre
transform(cf. (A.7))
s(ρ) = p∗(ρ) = sup ρ µ p(µ) . (2.19)
· −
µ∈Dµ
(cid:0) (cid:1)
Thus s, also known as the large deviation rate function, is strictly convex on D and
ρ
convex on (0, )2. For ρ intD it is easy to see that ρ µ p(µ) has a local
ρ
∞ ∈ · −
maximumat M(ρ) and thus
s(ρ) = ρ M(ρ) p M(ρ) and M (ρ) = ∂ s(ρ) , i = 1,2. (2.20)
· − i ρi
(cid:0) (cid:1)
Using convexity of D and p(µ) we can show that there exists a unique maximizer
µ
of the right hand side of (2.19), also for ρ intD . This is the main result of this
ρ
6∈
preliminarysection.
Proposition2.3 For every ρ (0, )2 there exists a unique maximizer M(ρ) D
µ
∈ ∞ ∈
of therighthandsideof(2.19), suchthat
s(ρ) = ρ M(ρ) p M(ρ) . (2.21)
· −
M C (0, )2,R andwe(cid:0)haveM(cid:1) (ρ) = M(ρ)forρ D andM(ρ) ∂D D
ρ µ µ
∈ ∞ ∈ ∈ ∩
forρ D .
ρ
6∈(cid:0) (cid:1)
In particular, D is closed in (0, )2 and ∂D = R(∂D D ), where ∂D denotes
ρ ρ µ µ ρ
∞ ∩
therelativeboundaryof D in(0, )2.
ρ
∞
7
3 Main Results
3.1 Equivalence of ensembles
Considerasequenceofcanonical measuresπL,NL inthethermodynamiclimit,i.e.
N /L ρ asL withdensityρ (0, )2. (3.1)
L
→ → ∞ ∈ ∞
In the following we study the question if the sequence πL,NL converges to a grand-
canonical product measure, and if yes, what is the mode of convergence. To quantify
thedistancebetween themeasuresweusethespecificrelativeentropy
1
hL,N(µ)= L H πL,N νµL , where
H πL,N νµL = (cid:0)πL,N((cid:12)(cid:12)η) (cid:1)log πνLL,N(η(η)) (3.2)
(cid:0) (cid:12) (cid:1) ηX∈XL µ
(cid:12)
is the usual relative entropy, since πL,N is absolutely continuous with respect to νµL.
Usingtherelations
νµL(η)z(µ)L=wL(η)eµ·N forallη ∈ XL,N and
νµL {ΣL = N} z(µ)L=ZL,Neµ·N , (3.3)
which are(cid:0)easilyderiv(cid:1)edfrom (2.4) and(2.6), wecan write
1 µ N 1
hL,N(µ) = −L logνµL {ΣL = N} = p(µ)− L· − L logZL,N , (3.4)
forall L 1, N N2 and µ(cid:0) D . (cid:1)
µ
≥ ∈ ∈
Thesecondpartof(3.4)suggeststhatM(ρ)ofProposition2.3istherightchemical
potential to minimize hL,NL in the thermodynamic limit (3.1). This is the content of
thenexttheoremforwhichweneedafurtherregularityassumptionontheexponential
tail of w, in addition to (2.11) and (2.12). A convenient sufficient condition is that for
all φ [0,π/2]thelimitin theradial directione
φ
∈
1
lim logw(re ) R exists, (3.5)
φ
r→∞ r ∈
andisacontinuousfunctionofφ. Thiscanberelaxedconsiderablyasisdiscussedafter
the proof in Section 4.3. (3.5) holds for example if w is convex, or if w = w + w
1 2
where w isconvexandw has boundedderivative.
1 2
Theorem 3.1 Assume (2.11), (2.12) and (3.5). Then for each particle density ρ
∈
(0, )2 andeverysequenceN asin (3.1)
L
∞
lim hL,NL M(ρ) = 0. (3.6)
L→∞
(cid:0) (cid:1)
8
From this result one can immediately deduce two standard formulations of the equiv-
alence of ensembles, on the level of measures and on the level of thermodynamic
functions. To formulatethefirst version wehave to define all canonical and all grand-
canonical measures on a common state space X = NΛ, where Λ is the (infinite) limit
lattice of an appropriate sequence (Λ ) . The precise construction is deferred to
L L=1,2,..
AppendixB,sinceitisonlynecessarytoformulate(3.7)andhasnofurtherimportance
forourresults.
Corollary3.2 Foreachρ (0, )2 we have
∈ ∞
f f asL , (3.7)
h iπL,NL → h iνM(ρ) → ∞
for all cylinder test functions f C(X,R) with eǫf < for some ǫ > 0. In
∈ h iνM(ρ) ∞
particular,thisincludesallboundedf C (X,R),whichisequivalenttoconvergence
b
∈
in distribution. Moreover,
1
Ll→im∞ L logZL,NL = −s(ρ) . (3.8)
For ρ intD , ν has somefinite exponentialmomentsby Lemma2.1, so in par-
∈ ρ M(ρ)
ticularthecorollary impliesconvergence ofthe local densitiesf(η) = η (x). We note
i
that for a single species with ρ intD convergence is shown even for L2 test func-
ρ
∈
tions in [21], Appendix 2.1. The proof given there relies on rather involved estimates
on the rate of convergence in the local limit theorem, whereas the proof via relative
entropy is muchsimpler(seesection 4.3). Moreover,ourresult coversseveral particle
speciesandcanbegeneralizedtoρ D ,whichisthemainpointofthispaper. Inthis
ρ
6∈
casethenatureoftheconvergencechangesand(3.7)isviolatedforf(η) = η (x)forat
i
least one species i, as will become clear in the next subsection. This difference in the
mode of convergence is a result of the unbounded local state space and is a signature
ofthecondensationtransition.
Forsystemswithboundedlocalstatespace(3.6)impliesconvergenceforallcylin-
der test functionsf C(X,R). But in case ofρ D thelimitingmeasurewouldbe
ρ
∈ 6∈
amixtureofgrand-canonicalmeasures,correspondingtocoexistingdomainswithdif-
ferent distributionsfor large finite systems (see e.g. [13]). This phenomenon is called
phase separation. In analogy to this classical case we interpret our limit result in the
following way: For ρ D the system phase separates into a (homogeneous) back-
ρ
6∈
ground phase with product measure ν given by Theorem 3.1, and a condensate
M(ρ)
or condensed phase which contains the excess particles. According to (3.7), the con-
densate cannot be tested by cylinder functions in the infinite system, its existence is
only a consequenceofthe conservationlaw (in contrast to classical phaseseparation).
The interpretation for large finite systems is that the volume fraction covered by the
condensate domain vanishes as L . In fact, this domain typically concentrates
→ ∞
9
only on a single lattice site, which is proved under additional assumptions in Section
3.3.
Duetotheconservationlaws,thephasespaceoftheparticlesystemis(0, )2,the
∞
setofdensitiesρ. Wesaythattheparticlesystemexhibitsacondensationtransition,if
D ( (0, )2. Asorderparameterofthephasetransitionwechoosethemapping
ρ
∞
R : (0, )2 D , with R (ρ) := R M(ρ) . (3.9)
c ρ c
∞ →
According to the above interpretation, R (ρ) desc(cid:0)ribes t(cid:1)he density of the background
c
phaseina systemwithglobaldensityρ. Notethat byProposition2.3
= ρ , ifρ D
R (ρ) ∈ ρ , (3.10)
c ∂D , ifρ D
ρ ρ
(cid:26)∈ 6∈
so R (D ) = D and R is a projection from (0, )2 onto D . By Lemma 2.2 and
c ρ ρ c ρ
∞
Proposition 2.3, R C (0, )2,D so the transition is continuous (second or-
c ρ
∈ ∞
der), which is directly related to the fact that p C1(D ,R). This is in contrast to
µ
(cid:0) (cid:1) ∈
systems with bounded local state space, where we would have D = R2 and non-
µ
differentiability of p would lead to a first order phase transition with discontinuous
orderparameter [13, 14].
3.2 Phase diagram
In this sectionwe apply standard resultsfrom convexanalysis, which are summarized
in Appendix A, to characterize the phase diagram of the system. By Proposition 2.3,
∂D = R(∂D D ), andthuscondensationoccurs ifand onlyif∂D D = .
ρ µ µ µ µ
∩ ∩ 6 ∅
Theorem 3.3 For every ρ ∂D with µ = M(ρ ), the preimage R−1(ρ ) is given
c ∈ ρ c c c
bythesubgradientδp(µ)as defined in(A.6). Moreover,
ρ +λn λ 0 , ∂D diff’ableinµ
δp(µ) = c µ ≥ µ . (3.11)
ρ +λ+n++λ−n− λ+,λ− 0 , otherwise
(cid:26) c (cid:8) µ µ(cid:12) (cid:9) ≥
(cid:12)
n denotestheno(cid:8)rmalvectorto∂D in(cid:12) µandn+,n(cid:9)− thetwolimitingnormalvectors,
µ µ (cid:12) µ µ
in case∂D isnot differentiableinµ.
µ
Note that by convexity of D , n+ and n− are well defined as the extremal normal
µ µ µ
directions to the set of supporting hyperplanes in µ. In case that the points of non-
differentiability of ∂D (see Lemma 2.1) accumulate in µ, n+ = n− is also possible.
µ µ µ
InthefollowingweuseTheorem3.3toconstructthephasediagramanditsproperties.
By definition(3.9), thepreimage
R−1(ρ ) = ρ (0, )2 R(ρ) = ρ (3.12)
c c ∈ ∞ c
(cid:8) (cid:12) (cid:9)
(cid:12) 10