dynamic phase transition theory in pvt systemsfluid/paper/79.pdf · dynamic phase transition theory...

Dynamic Phase Transition Theory inPVT Systems

TIAN MA & SHOUHONG WANG

Dedicated to Ciprian Foias on the occasion of his 75th birthdaywith great admiration and respect. We hope that this paper is in Ciprian’s spirit of

applying advanced mathematical concepts and methods to problems influid mechanics and other areas of sciences.

ABSTRACT. In this article, we initiate a program of research to studydynamic phase transitions in nonlinear sciences. The main objective ofthis article is two-fold. First, we introduce some general principles onphase transition dynamics, including a new dynamic transition classi-fication scheme, and a Ginzburg-Landau theory for modeling equilib-rium phase transitions. Second, we apply the general principles andthe recently developed dynamic transition theory to study dynamicphase transitions of physical vapor transport (PVT) systems. In par-ticular, we establish a new time-dependent Ginzburg-Landau model,whose dynamic transition analysis is carried out. The analysis for thePVT systems in this article leads to a few physical predictions, whichare otherwise unclear from the physical point of view.

1. INTRODUCTION

The main objectives of this article are:

(1) to introduce a new dynamic phase transition classification scheme for non-linear systems,

(2) to present a new time-dependent Ginzburg-Landau model for general equi-librium phase transitions, and

(3) to study the dynamic phase transitions for physical vapor transport (PVT)systems.

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Indiana University Mathematics Journal c , Vol. 57, No. 6 (2008), Special Issue

2862 TIAN MA & SHOUHONG WANG

First, as we know, there are several classification schemes for phase transitions.The one used most often is the classical Ehrenfest classification scheme, which isbased on the order of differentiability of the energy functional. The new dynamicphase transition classification scheme we propose here is based on the dynamictransition theory developed recently by the authors [11, 17], which has been usedto study numerous problems in sciences and engineering, including in particularproblems in classical and geophysical fluid dynamics, biology and chemistry, andphase transitions. The new scheme introduced in this article classifies phase tran-sitions into three categories: Type-I, Type-II and Type-III, corresponding respec-tively to the continuous, the jump and the mixed transitions in the mathematicaldynamic transition theory.

We shall see from the applications to PVT systems that this new classificationscheme is more transparent. In addition, it can be used to identify high-order (inthe Ehrenfest sense) transitions, which are usually hard to derive both theoreticallyand experimentally.

Second, we introduce a general principle, leading to a unified approach toderive Ginzburg-Landau type of time-dependent models for equilibrium phasetransitions. This principle is based on the le Chatelier principle and some generalcharacteristics of pseudo-gradient systems. In view of this new dynamic modelingand the dynamic transition theory, we shall see that the states after transition ofteninclude not only equilibria, but also some transient states, which are physicallyimportant as well.

Third, with the aforementioned principles at our disposal, we derive a newdynamic model for PVT system, whose dynamic transition can be explicitly ob-tained, leading to some specific physical predictions.

One important ingredient of the model is based on a new form of the Gibbsfree energy, where the mechanical coupling term p is adjusted by adding a coef-ficient α, leading to (4.4) as the first two terms in the Taylor expansion. This isphenomenologically required by the van der Waals equation (4.2) and the mathe-matical analysis based on the new dynamical transition theory.

As we know, a PVT system is a system composed of one type of molecules,and the interaction between molecules is governed by the van der Waals law. Themolecules generally have a repulsive core and a short-range attraction region out-side the core. Such systems have a number of phases: gas, liquid and solid, anda solid can appear in a few phases. The most typical example of a PVT system iswater. In general, the phase transitions of PVT systems mainly refer to the gas-liquid, gas-solid and liquid-solid phase transitions. These transitions are all firstorder in the Ehrenfest sense (i.e., discontinuous) and are accompanied by a latentheat and a change in density.

A PT-phase diagram of a typical PVT system is schematically illustrated byFigure 1.1, where point A is the triple point at which the gas, liquid, and solidphases can coexist. Point C is the Andrews critical point at which the gas-liquidcoexistence curve terminates [20]. The fact that the gas-liquid coexistence curvehas a critical point means that we can go continuously from a gaseous state to

Dynamic Phase Transition Theory in PVT Systems 2863

a liquid state without ever meeting an observable phase transition, if we choosethe right path. The critical point in the liquid-solid coexistence curve has neverbeen observed. It implies that we must go through a first-order phase transitionin going from the liquid to the solid state (Type-II or III in the sense of dynamicclassification scheme).

liquid

gas

solid

A

B

C

p

T

FIGURE 1.1. Coexistence curves of a typical PVT system: A isthe triple point, C is the critical point, and the dashed cure is amelting line with negative slope.

The phase diagram in the PV-plane is given by Figure 1.2, where the dashedcurves represent lines of constant temperature. In the region of coexistence ofphases, the isotherms (dashed lines) are always flat, indicating that in these regionsthe change in volume (density) occurs for constant pressure p and temperature T .

liquid

- solidsolidgas - liquid

gas - solid

gasAA A

T < T

T = T

T > TCB B

p

c

c

c

V

FIGURE 1.2. PV-phase diagram, the dashed line represent isotherms.

One of the main objectives of this article is to study the dynamic phase tran-sitions beyond the Andrews point C, and to make some physical predictions.

For this purpose, first, by both mathematical and physical considerations,a proper form of the Gibbs free energy functional is given, and then the time-dependent Ginzburg-Landau type of model for PVT systems is derived naturallyusing the general principle as mentioned above. Then, using the dynamical tran-sition theory, we derive a system of critical parameter equations to determine the


Andrews point C = (TC,pC) given by (6.6). Then we show that the gas-liquidtransition is first-order before the Andrews point C, and third order after the pointC, as demonstrated in Physical Conclusion 6.1.

In the gas-solid and liquid-solid transitions, there also exist metastable states.Metastable states corresponds to those stable states whose basins of attraction areseparated by some unstable stables; this often happens when the hysteresis phe-nomenon appears as well. For the gas-solid case, the metastable states correspondto the superheated solid and supercooled liquid, and for the liquid-solid case, themetastable states correspond to the superheated solid and supercooled liquid; seePhysical Conclusion 6.2.

Based on the analysis for the PVT systems, we observe that the symmetryof fluctuations for general thermodynamic systems may not be universally true.In other words, in some systems with multi-equilibrium states, the fluctuationsnear a critical point occur only in one basin of attraction of some equilibriumstates, which are the ones that can be physically observed. We have observedsimilar asymmetry for ferromagnetic systems [18], and we believe this is a universalphenomenon, which needs to be seriously considered.

This article is organized as follows. The new dynamic classification schemeand the Ginzburg-Landau model for equilibrium phase transitions are given inSections 2 and 3. Section 4 introduces a time-dependent model for PVT systems,which is analyzed in Section 5. Section 6 provides physical consequences of theanalysis and some new physical predictions, leading to a conjecture for the asym-metry of fluctuations in Section 7. Some dynamical transition theorems are givenin Appendix A for convenience.

2. GENERAL PRINCIPLES OF PHASE TRANSITION DYNAMICS

In this section, we introduce a new dynamic transition theory, leading to a dy-namic phase transition classification scheme for both equilibrium and non-equilibrium phase transitions.

2.1. Dynamic transition theory. In sciences, nonlinear dissipative systemsare generally governed by differential equations, which can be expressed as anabstract evolution equation:

(2.1)dudt= Lλu+G(u,λ), u(0) =ϕ,

where X and X1 are two Banach spaces, and X1 ⊂ X is a compact and dense inclu-sion, u : [0,∞) → X is unknown function, and λ ∈ R1 is the system parameter.

Assume that Lλ : X1 → X is a parameterized linear completely continuousfield depending continuously on λ ∈ R1, which satisfies

(2.2)

Lλ = −A+ Bλ a sectorial operator,

A : X1 → X a linear homeomorphism,

Bλ : X1 → X a linear compact operator.


In this case, we can define the fractional order spaces Xσ for σ ∈ R1. Then wealso assume that G(· , λ) : Xα → X is Cr(r ≥ 1) bounded mapping for some0 ≤ α < 1, depending continuously on λ ∈ R1, and

(2.3) G(u,λ) = o(‖u‖Xα) ∀λ ∈ R1.

Hereafter we always assume the conditions (2.2) and (2.3), which representthat the system (2.1) has a dissipative structure.

A state of the system (2.1) at λ is usually referred to as a compact invariant setΣλ. In many applications, Σλ is a singular point or a periodic orbit. A state Σλ of(2.1) is stable if Σλ is an attractor; otherwise Σλ is called unstable.

Definition 2.1. We say that the system (2.1) has a phase transition from a stateΣλ at λ = λ0 if Σλ is stable on λ < λ0 (or on λ > λ0) and is unstable on λ > λ0 (oron λ < λ0). The critical parameter λ0 is called a critical point. In other words, thephase transition corresponds to an exchange of stable states.

An important class of phase transition is associated with the attractor bifur-cation introduced by the authors [7, 8, 11, 12]. The attractor bifurcation theoryhas been applied to various problems in nonlinear sciences, including e.g. classicalfluid dynamics [9, 13–16], geophysical fluid dynamics [5, 6]. Of course, we notethat bifurcation and transition are two related but different concepts.

Let βj(λ) ∈ C | j ∈ N be the eigenvalues (counting multiplicity) of Lλ,and assume that

Re βi(λ)

< 0 if λ < λ0,

= 0 if λ = λ0,

> 0 if λ > λ0,

∀1 ≤ i ≤m,(2.4)

Re βj(λ0) < 0, ∀j ≥m+ 1.(2.5)

The following theorem is a basic principle of transitions from equilibriumstates, which provides sufficient conditions and a basic classification for transitionsof nonlinear dissipative systems. This theorem is a direct consequence of the centermanifold theorems and the stable manifold theorems; we omit the proof.

Theorem 2.2. Let the conditions (2.4) and (2.5) hold true. Then, the system(2.1) must have a transition from (u, λ) = (0, λ0), and there is a neighborhoodU ⊂ X of u = 0 such that the transition is one of the following three types:

(1) CONTINUOUS TRANSITION: There exists an open and dense set Uλ ⊂ Usuch that for any ϕ ∈ Uλ, the solution uλ(t,ϕ) of (2.1) satisfies

limλ→λ0

lim supt→∞

‖uλ(t,ϕ)‖X = 0.


(2) JUMP TRANSITION: For any λ0 < λ < λ0 + ε with some ε > 0, there is anopen and dense set Uλ ⊂ U such that for any ϕ ∈ Uλ,

lim supt→∞

‖uλ(t,ϕ)‖X ≥ δ > 0 for some δ > 0 is independent of λ.

(3) MIXED TRANSITION: For any λ0 < λ < λ0 + ε with some ε > 0, U canbe decomposed into two open sets Uλ1 and Uλ2 (U

λi not necessarily connected):

U = Uλ1 + Uλ2 , Uλ1 ∩ Uλ2 = ∅, such that

limλ→λ0

lim supt→∞

‖u(t,ϕ)‖X = 0, ∀ϕ ∈ Uλ1 ,

lim supt→∞

‖u(t,ϕ)‖X ≥ δ > 0, ∀ϕ ∈ Uλ2 .

An important aspect of the transition theory is to determine which of thethree types of transitions given by Theorem 2.2 occurs in a specific problem. Acorresponding dynamic transition has been developed recently by the authors forthis purpose; see [17,19]. We refer interested readers to these references for detailsof the theory; for the sake of convenience, the theorems from this theory to beused here in the analysis of the PVT systems will be given in the Appendix.

2.2. New dynamic classification scheme. The concept of phase transitionoriginates from the statistical physics and thermodynamics. In physics and chem-istry, “phase” means the homogeneous part in a heterogeneous system. The phasein the dynamic transition theory here refers to a stable state in systems fromphysics, chemistry, biology, ecology, economics, fluid dynamics and geophysicalfluid dynamics, etcetera. Hence, the content of phase transition is treated in moregeneral sense. In fact, the phase transition dynamics introduced here can be ap-plied to a wide variety of topics involving the universal critical phenomena of statechanges in nature in a unified mathematical viewpoint and manner.

If the system (2.1) possesses a gradient-type structure, then the phase transi-tions are called equilibrium phase transitions; otherwise they are called the non-equilibrium phase transitions.

Classically, there are several ways to classify phase transitions. The one mostused is the Ehrenfest classification scheme, which groups phase transitions basedon the degree of non-analyticity involved. First-order phase transitions are alsocalled discontinuous, and higher-order phase transitions (n > 1) are called con-tinuous.

With Theorem 2.2 at our disposal, we are in a position to give a new dynamicclassification scheme for both equilibrium and non-equilibrium phase transitions.

Definition 2.3 (Dynamic Classification of Phase Transitions). The phase tran-sitions for (2.1) at λ = λ0 are classified using their dynamic properties: continuous,jump, and mixed, as given in Theorem 2.2, which are called Type-I, Type-II, andType-III, respectively.


The main characteristics of Type-II phase transitions are that there is a gapbetween the basic state Σ1

λ = (0, λ) and the transition state Σ2λ at the critical

point λ0. In thermodynamics, the metastable states correspond in general to thesuper-heated or super-cooled states, which have been found in many physical phe-nomena.

In a Type-I phase transition, the two states Σ1λ and Σ2

λ mentioned above meetat λ0, i.e., the system undergoes a continuous change from Σ1

λ to Σ2λ.

In a Type-III phase transition, there are at least two different stable states, Σλ2and Σλ3 at λ0, and the system undergoes a continuous transition to Σλ2 or a jumptransition to Σλ3 , depending on the fluctuations.

For equilibrium phase transitions, it is clear that a Type-II phase transition ofgradient-type systems must be discontinuous or of zero order because there is agap between Σ1

λ and Σjλ (2 ≤ j ≤ K). For a Type-I phase transition, the energyfunction is continuous, and consequently, it is an n-th order transition in theEhrenfest sense for some n ≥ 2. A Type-III phase transition is indefinite: for thetransition from Σ1

λ to Σ2λ it may be continuous, i.e., dF−/dλ = dF+2 /dλ, and for

the transitions from Σ1λ to Σ3

λ it may be discontinuous, i.e., dF−/dλ ≠ dF+3 /dλat λ = λ0.

Finally, we recall the physical meaning of the derivatives of F(u,λ) on λ instatistical physics. In thermodynamics, the parameter λ generally stands for tem-perature T and pressure p: λ = (T ,p). The energy functional F(u, T ,p) is thethermodynamic potential, and u is the order parameter.

For the PVT system, the first-order derivatives include:(1) the entropy: S = −∂F/∂T , and(2) the phase volume: V = ∂F/∂p.

The second-order derivatives are:(1) the heat capacity in the constant pressure: Cp = −T(∂2F/∂T 2);(2) the compression coefficient κ = −(1/V)(∂2F/∂p2); and(3) the thermal expansion coefficient α = (1/V)(∂2F/(∂T ∂p)).

Thus, for the first-order phase transition, the discontinuity of ∂F/∂p at thecritical point λ0 = (T0, p0) implies the discontinuity of phase volume:

(2.6) ∆V = V 2 − V 1 = ∂F+

∂p− ∂F

−

∂p,

and the discontinuity of ∂F/∂T implies that there is a gap between both phaseentropies:

(2.7) ∆S = S2 − S1 = −∂F+

∂T+ ∂F

−

∂T

Physically, ∆S is a non-measurable quantity, hence, we always use latent heat∆H = T∆S to determine the first-order phase transition, and ∆H stands for the


absorbing heat for the transition from phase Σ1λ to phase Σ2

λ. In the two-phasecoexistence situation, ∆H and ∆V are related by the Clausius-Clapeyron equation:

dpdT

= ∆HT∆V .

For second-order phase transitions, the variance of heat capacity (or specificheat), compression coefficient, and the thermal expansion coefficient at the criticalvalue are important static properties, which are measurable physical quantities.

3. A UNIFIED GINZBURG-LANDAU MODEL FOREQUILIBRIUM PHASE TRANSITIONS

In this section, a unified time-dependent Ginzburg-Landau model for equilib-rium phase transitions is derived based on the le Chatelier principle and on somemathematical insights on pseudo-gradient systems.

3.1. Gradient-type systems. We start with a general definition of gradient-type operators. Let H1 and H be two Hilbert spaces, L : H1 → H a sectorialoperator, and G : Hθ → H a Cr (r ≥ 1) mapping for some 0 < θ < 1. LetY ⊂ Hα (0 < α < 1) be a Banach space.

Definition 3.1. A mapping L+G : H1 → H is called a gradient-type operator ifthere exist a C1 function F : Y → R1 and a constant C > 0 and β ≤ 1 such that

DF(u) ∈ H ∀u ∈ Hβ,

−〈DF(u), Lu+Gu〉H ≥ C‖DF(u)‖2H ∀u ∈ H1,

DF(u0) = 0 a Lu0 +Gu0 = 0 .

In this case, F is called the energy function of L+G.

Obviously, a gradient operator L+G = −DF is of the gradient-type structure,but converse statement is not true. If L+G : H1 → H is a gradient-type operator,then in general L+G is in the form

(3.1) L+G = ADF + B : H1 → H

where A : H1 → H is a negative definite linear operator, and

(3.2)

〈Au,u〉H ≤ −C‖u‖2H for some C > 0,

〈Bu,DF(u)〉H ≤ 0 ∀u ∈ H1,

DF(u0) = 0⇒ Bu0 = 0.


It is worth pointing out that we can study the global dynamics of the gradient typetime-dependent system, including the existence and structure of global attractors.In particular, we can prove that the global attractors will consist of steady statesand their unstable manifolds [19]; see also Foias and Temam [1, 3] for globalattractors of the Navier-Stokes equations.

3.2. Dynamic model for equilibrium phase transitions. Consider a ther-mal system with a control parameter λ. The classical le Chatelier principle saysthat if an external stress is applied to a system at equilibrium, the system willadjust itself to minimize that stress. It allows us to predict the direction a reac-tion will take when we perturb the equilibrium by changing the pressure, volume,temperature, or component concentrations.

By the mathematical characterization of gradient-type systems and thele Chatelier principle, for a system with thermodynamic potential H (u, λ), thegoverning equations are essentially determined by the functionalH (u, λ). Whenthe order parameters (u1, · · · , um) are nonconserved variables, i.e., the integers∫

Ωui(x, t)dx = ai(t) ≠ constant.

then the time-dependent equations are given by

(3.3)

∂ui∂t= −βi δδui

H (u, λ)+ Φi(u,∇u,λ) for 1 ≤ i ≤m,

∂u∂n

∣∣∣∂Ω = 0 (or u|∂Ω = 0) ,

where δ/δui are the variational derivative, βi > 0 and Φi satisfy

(3.4)∫Ω∑iΦi δδuiH (u, λ)dx = 0.

The condition (3.4) is required by the Le Chatelier principle. We remark herethat following the le Chatelier principle, one should have an inequality constraint.However, physical systems often obey most simplified rules, as many existing mod-els for specific problems are consistent with the equality constraint here. Thisremark applies to the constraint (3.10) below as well.

In view of (3.1), the right-hand side of the first equation in (3.3) correspondsto L+G = ADF+B, withA given by the coefficients βi, and with B given by Φi. Inpractical applications of the unified model above, the terms Φi can be determinedby physical laws and by (3.4); see the applications to superconductivity below.

When the order parameters are the number density and the system has nomaterial exchange with the external, then uj (1 ≤ j ≤m) are conserved, i.e.,

(3.5)∫Ωuj(x, t)dx = constant.


This conservation law requires a continuity equation

(3.6)∂uj∂t= −∇ · Jj(u, λ),

where Jj(u, λ) is the flux of the component uj . In addition, Jj satisfies

(3.7) Jj = −kj∇(µj −∑i≠jµi),

where µl is the chemical potential of the component ul,

(3.8) µj −∑i≠jµi =

δδuj

H (u, λ)−φj(u,∇u,λ),

andφj(u,λ) is a function depending on the other components ui (i ≠ j). Thus,from (3.6)-(3.8) we obtain the dynamical equations as follows:

(3.9)

∂uj∂t= βj∆

[δδuj

H (u, λ)−φj(u,∇u,λ)]

for 1 ≤ j ≤m,

∂u∂n

∣∣∣∂Ω = 0,

∂∆u∂n

∣∣∣∂Ω = 0

where βj > 0 are constants, and φj satisfies

(3.10)∫Ω∑j∆φj · δ

δujH (u, λ)dx = 0.

As the highest order derivative term in(δ/(δuj)

)H (u, λ) is often a Laplacianof u, we have put the boundary conditions as given in (3.9). Of course, otherboundary conditions are possible, and depend on the physical problem.

When m = 1, i.e., the system is a binary system, consisting of two com-ponents A and B, then the term φj = 0. The model above covers the classicalCahn-Hilliard model. In fact, for the binary system, the classical Cahn-Hilliardequation can be generalized to include the effect of entropy, leading to a coupledsystem of equations for the order parameter u and for the entropy. The analysisof both the classical and generalized Cahn-Hilliard equations lead to some spe-cific new physical predictions. This will be reported in two forthcoming papers.It is worth mentioning that for multi-component systems, these φj play an im-portant rule in deriving time-dependent models. We shall address this issue in aforthcoming paper.

If the order parameters (u1, · · · , uk) are coupled to the conserved variables(uk+1, · · · , um), then the dynamical equations are


(3.11)

∂ui∂t= −βi

δδui

H (u, λ)+ Φi(u,∇u,λ) for 1 ≤ i ≤ k,

∂uj∂t= βj∆

[δδuj

H (u, λ)−φj(u,∇u,λ)]

for k+ 1 ≤ j ≤m,

∂ui∂n

∣∣∣∂Ω = 0 (or ui|∂Ω = 0) for 1 ≤ i ≤ k,

∂uj∂n

∣∣∣∂Ω = 0,

∂∆uj∂n

∣∣∣∂Ω = 0 for k+ 1 ≤ j ≤m.

Here Φi and φj satisfy (3.4) and (3.10), respectively.The model (3.11) we derive here gives a general form of the governing equa-

tions to thermodynamic phase transitions, and will play a crucial role in studyingthe dynamics of equilibrium phase transition in statistic physics.

Finally, we remark that the steady state solutions of (3.11) are simply the crit-ical points of the energy functional H, satisfying the following steady state equa-tions:

(3.12)

βiδδui

H (u0, λ) = 0 ∀1 ≤ i ≤ k,

βj∆ δδuj

H (u0, λ) = 0 ∀k+ 1 ≤ j ≤m,

∂ui∂n

∣∣∣∂Ω = 0 (or ui|∂Ω = 0) ∀1 ≤ i ≤ k,

∂uj∂n

∣∣∣∂Ω = 0,

∂∆uj∂n

∣∣∣∂Ω = 0 ∀k+ 1 ≤ j ≤m.

Hence the unified model (3.11) is consistent with classical models in the steadystate case.

To derive the steady state equations above, we only have to show that a steadystate solution u0 of (3.11) satisfies

(3.13)Φi(u0,∇u0, λ) = 0 ∀1 ≤ i ≤ k,

∆φj(u0,∇u0, λ) = 0 ∀k+ 1 ≤ j ≤m,

To see this, let u0 satisfy that

(3.14)

βiδδui

H (u0, λ)− Φi(u0,∇u0, λ) = 0 ∀1 ≤ i ≤ k,

βj∆ δδuj

H (u0, λ)−∆φj(u0,∇u0, λ) = 0 ∀k+ 1 ≤ j ≤m.


Multiplying Φi(u0,∇u0, λ) andφj(u0,∇u0, λ) on the first and the second equa-tions of (3.14) respectively, and integrating them, then we infer from (3.4) and(3.10) that

∫Ω∑iΦ2i (u0,∇u0, λ)dx = 0,

∫Ω∑j|∇φj(u0,∇u0, λ)|2 dx = 0,

which imply that (3.13) holds true.

3.3. Application to superconductivity. In practical applications of the uni-fied model above, the terms Φi can be determined by physical laws and (3.4). Tosee this and to show how one applies the unified model to a specific problem, weconsider as an example the time-dependent Ginzburg-Landau equations for su-perconductivity with an applied field. We note that dynamic transitions for theGinzburg-Landau model of superconductivity was conducted by the authors in[10], and further studies will be reported elsewhere.

As we know, the time-dependent Ginzburg-Landau model for superconduc-tivity was first proposed Schmid [21] and subsequently validated by Gor’kov andEliashberg [4] in the context of the microscopic Bardeen-Cooper-Schrieffer (BCS)theory of superconductivity. Because of gauge invariance, the generalization isnontrivial. In addition to the order parameter and the vector potential, a thirdvariable is needed to complete the description of the physical state of the systemin a manner consistent with the gauge invariance.

With the unified model in our disposal, we can derive the model for super-conductivity in a much simpler and transparent fashion.

We start with the Ginzburg-Landau free energy G, corresponding to notationH used before for the free energy, given by

(3.15) G =∫Ω[

12ms

|(−ih∇− eScA)ψ|2 + a

2|ψ|2 + b

4|ψ|4 + H

2

8π− HHa

4π

],

where Ω ⊂ Rn (n = 2,3) is a bounded domain, the complex valued functionψ : Ω→ C stands for the order parameter, the vector valued function H : Ω→ Rn

stands for the magnetic field, Ha is the applied field, h is the Planck constant, eSand ms are the charge and mass of a Cooper pair, c is the speed of light, A is themagnetic potential satisfyingH = curl A, and the parameters a = a(T), b = b(T)are coefficients; see de Gennes [2]. The order parameter ψ describes the localdensity ns of superconducting electrons: |ψ|2 = ns .


By (3.15), the equation (3.3) can be written as

k1∂ψ∂t+φ1(ψ,A) = −

12ms

(ih∇+ escA)2ψ− aψ− b|ψ|2ψ,(3.16)

k2∂A∂t+φ2(ψ,A) = −

14π

curl2A+ 14π

curl Ha(3.17)

− e2s

msc2 |ψ|2A−esh

2msci(ψ∗∇ψ−ψ∇ψ∗),

where k1, k2 > 0 are constants, φi(ψ,A) are two operators to be determined byphysical laws and (3.4) in several steps as follows:

First, the Maxwell equations give that

(3.18)σc2∂A∂t+σ∇φ+ 1

4πcurl2A− 1

4πcurl Ha = Js,

where σ is the conductivity, φ is the electric potential, and Js is the supercurrentdensity. On the other hand, by Quantum Mechanics, the supercurrent density Jsis given by

(3.19) Js = −e2s

msc2 |ψ|2A−esh

2msci(ψ∗∇ψ−ψ∇ψ∗).

In comparison with (3.17), (3.18) and (3.19), we have

(3.20) φ2(ψ,A) = σ∇φ.Second, by direct computation, it is easy to show that by (3.16), (3.17) and

(3.4), φ1 is uniquely determined from (3.20) by:

φ1(ψ,A) = iαφψ,where α is a coefficient.

Third, according to the physical dimensional balance, also due to P. L. Gork’ov,the coefficients k1 and α are given by k1 = h2/2msD and α = hes/2msD, whereD is the diffusion coefficient. Thus, we deduce the following TDGL equations:

h2

2msD

(∂∂t+ ieshφ)ψ = − 1

2ms

(ih∇+ es

cA)2ψ− aψ− b|ψ|2ψ,

σc2∂A∂t+ σ∇φ = − 1

4πcurl2A+ 1

4πcurlHa −

e2s

msc2 |ψ|2A(3.21)

− esh2msc

i(ψ∗∇ψ−ψ∇ψ∗).


4. TIME-DEPENDENT MODEL FOR PVT SYSTEMS

In this section, we use the general principles derived in the last section to derive atime-dependent Ginzburg-Landau model for PVT systems, which will be used tocarry out dynamic transition analysis and physical predictions in this article.

4.1. van der Waals equation and Gibbs energy The classical and the sim-plest equation of state which can exhibit many of the essential features of thegas-liquid phase transition is the van der Waals equation:

(4.1) v3 −(b + RT

p

)v2 + a

pv − ab

p= 0,

where v is the molar volume, p is the pressure, T is the temperature, R is theuniversal gas constant, b is the revised constant of inherent volume, and a is therevised constant of attractive force between molecules. If we adopt the molardensity ρ = 1/v to replace v in (4.1), then the van der Waals equation becomes

(4.2) −(bp + RT)ρ + aρ2 − abρ3 + p = 0.

Now, we shall apply thermodynamic potentials to investigate the phase transi-tions of PVT systems, and we shall see later that the van der Waals equation can bederived as a Euler-Langrange equation for the minimizers of the Gibbs free energyfor PVT systems at gaseous states.

Consider an isothermal-isopiestic process. The thermodynamic potential istaken to be the Gibbs free energy. In this case, the order parameters are the molardensity ρ and the entropy density S, and the control parameters are the pressurep and temperature T . The general form of the Gibbs free energy for PVT systemsis given as

(4.3) G(ρ, S, T ,p)

=∫Ω[µ1

2|∇ρ|2 + µ2

2|∇S|2 + g(ρ, S, T ,p)− ST −α(ρ, T ,p)p

]dx,

where g and α are differentiable with respect to ρ and S, Ω ⊂ R3 is the container,and αp is the mechanical coupling term in the Gibbs free energy, which can beexpressed by

(4.4) α(ρ, T ,p)p = ρp − 12bρ2p,

where b = b(T ,p) depends continuously on T and p. In fact, this mechanicalcoupling term should be p. In view of the van der Waals equation (4.2) and themathematical analysis based on the new dynamical transition theory, phenomeno-logically we need to adjust the term by adding a coefficient α, leading to (4.4) as


the first two terms in the Taylor expansion. Although the van der Waals equationworks for gaseous sates only, by choosing the dependence of the coefficient b onthe temperature and the pressure, the energy applies to the liquid and solid statesas well. This is a very subtle term from the physical point of view to derive afeasible free energy.

Based on both the physical and mathematical considerations, we take the Tay-lor expansion of g(ρ, S, T ,p) on ρ and S as follows

(4.5) g = 12α1ρ2 + 1

2β1S2 + β2Sρ2 − 1

3α2ρ3 + 1

4α3ρ4,

where αi (1 ≤ i ≤ 3), β1 and β2 depend continuously on T and p, and

(4.6) β1 = β1(T ,p) > 0, αi = αi(T ,p) > 0, i = 2,3.

4.2. Dynamical equations for PVT systems. In a PVT system, the orderparameter is u = (ρ, S),

ρ = ρ1 − ρ0, S = S1 − S0,

where ρi and Si (i = 0,1) represent the density and entropy, ρ0, S0 are referencepoints. Hence the conjugate variables of ρ and S are the pressure p and thetemperature T . Thus, by the standard model (3.3), we derive from (4.3)–(4.5) thefollowing general form of the time-dependent equations governing a PVT system:

(4.7)

∂ρ∂t= µ1∆ρ − (α1 + bp)ρ +α2ρ2 −α3ρ3 − 2β2ρS + p,

∂S∂t= µ2∆S − β1S − β2ρ2 + T.

Although the domain Ω depends on T and p, we can still take the Neumannboundary condition

(4.8)∂ρ∂n= 0,

∂S∂n= 0 on ∂Ω.

An important special case for PVT systems is that the pressure and temper-ature functions are homogeneous in Ω. Thus we can assume that ρ and S areindependent of x ∈ Ω, and the free energy (4.3) with (4.4) and (4.5) can beexpressed as

(4.9) G(ρ, S, T ,p) = α1

2ρ2+ β1

2S2+β2Sρ2−α2

3ρ3+α3

4ρ4+ bρ

2p2−ρp−ST .


From (4.9) we get the dynamical equations as

(4.10)

dρdt= −(α1 + bp)ρ +α2ρ2 −α3ρ3 − 2β2Sρ + p,

dSdt= −β1S − β2ρ2 + T.

Because β1 > 0 for all T and p, we can replace the second equation of (4.10) by

(4.11) S = β−11 (T − β2ρ2).

Then, (4.10) are equivalent to the following equation

(4.12)dρdt= −(α1 + bp + 2β−1

1 β2T)ρ +α2ρ2 − (α3 − 2β22β−11 )ρ

3 + p.

It is clear that if α1 = 0,2β−11 β2 = R,α2 = a, (α3 − 2β2

2β−11 ) = ab, then the

steady state equation of (4.12) is referred to the van der Waals equation.

5. PHASE TRANSITION DYNAMICS FOR PVT SYSTEMS

In this section we use (4.12) to discuss dynamical properties of transitions for PVTsystems, and remark that similar results can also derived using the more generalform of the time-dependent model (4.7).

Let ρ0 be a steady state solution of (4.12). We take the transformation

ρ = ρ0 + ρ′.

Then Equation (4.12) becomes (drop the prime)

(5.1)dρdt= λρ + a2ρ2 − a3ρ3.

where

λ = 2α2ρ0 − 3a3ρ20 −α1 − bp − 2β2β−1

1 T,

a2 = α2 − 3a3ρ0,

a3 = α3 − 2β22β−11 .

In the PT-plane, near a non-triple point A = (T∗, p∗) (see Figure 1.1), thecritical parameter equation

λ = λ(T ,p) = 0 in |T − T∗| < δ, |p − p∗| < δ


for some δ > 0, defines a continuous function T = φ(p), such that

(5.2) λ

< 0 if T > φ(p),

= 0 if T = φ(p),

> 0 if T < φ(p).

The main result in this section is the following dynamic transition theoremfor PVT systems.

Theorem 5.1. Let T0 = φ(p0) and a3 > 0. Then the system (5.1) has atransition at (T ,p) = (T0, p0), and the following assertions hold true:

(1) If the coefficient a2 = a2(T ,p) in (5.1) is zero at (T0, p0), i.e., a2(T0, p0) =0, then the transition is of Type-I, as schematically shown in Figure 5.1 (a) and(b).

(2) If a2(T0, p0) ≠ 0, then the transition is of Type-III (i.e., the mixed type), andthe following assertions hold true:(a) There are two transition solutions near (T0, p0) as

(5.3) ρ±(T ,p) = 12a3

(a2 ±

√a2

2 + 4a3λ).

(b) There is a saddle-node bifurcation at (T1, p1), where T1 > T0 and p1 < p0if φ′(p0) > 0, and p0 > p1 if φ′(p0) < 0.

(c) For φ′(p0) > 0, when a2(T0, p0) > 0 the transition diagrams are illus-trated by Figure 5.2 (a)–(b), where ρ+ is stable for all (T ,p) near (T0, p0),and ρ = 0 is stable, ρ− a saddle for T0 < T < T1 and p1 < p < p0, andρ− is stable, ρ = 0 a saddle for T < T0, p > p0;

(d) When a2(T0, p0) < 0, the transition diagrams are illustrated by Figure 5.3(a)–(b), where ρ− is stable for all (T ,p) near (T0, p0), and ρ = 0 is stable,ρ+ a saddle for T0 < T < T1, p1 < p < p0, and ρ+ is stable, ρ = 0 asaddle for T < T0, p0 < p.

Proof. By (5.1), the theorem follows from the transition theorems (TheoremsA.1 and A.2) and the singularity separation theorem (Theorem A.3). We omit thedetailed routine analysis.

6. PHYSICAL CONCLUSIONS AND PREDICTIONS

To discuss the physical significance of Theorem 5.1, we recall the classical pV -phase diagram given by Figure 1.2. We take ρ = 1/V to replace volume V , thenthe gas-liquid coexistence curve in the ρ-P plane is illustrated by Figure 6.1.

According to the physical experiments, in the gas-liquid coexistence region inthe ρ − p phase diagram, there exist metastable states. Mathematically speaking,


TT0 0

(a)

ρ

pp0

(b)

ρ

FIGURE 5.1. Continuous transition for the case whereφ′(p0) > 0

TT0 0

(a)

T1

ρ

ρ _

+

ρ _

ρ

p0

ρρ

ρ _

+

ρ _

p p01

(b)

FIGURE 5.2. Type-III (mixed) transition for a2 > 0.

TT0

ρ

0 T1

ρ +

ρ _

(a)

p0

ρ

0p

1

ρ

p

+

ρ _

(b)

FIGURE 5.3. Type-III (mixed) transition for a2 < 0.

the metastable states are the attractors which have a small basin of attraction. InFigure 6.1, the dashed lines aa′ and bb′ represent the metastable states, and thepoints in aa′ correspond to super-heated liquid, the points in bb′ correspond tosuper-cooled gas. The ρ − p phase diagram shows that along the isothermal lineT > TC where C = (TC, PC) is the Andrews critical point, when the pressure pincreases, the density ρ varies continuously from gaseous to liquid states. Howeveralong the isothermal line T < TC when the pressure p increases to p0 the densityρ will undergo an abrupt change, and a transition from gaseous state a to a liquid


state b accompanied with an isothermal exothermal process to occur. Likewise,such processes also occur in the gas-solid and liquid-solid transitions.

T > T

T = TT < T

a

a’

b

b’

p

ρ

p0

c

c

c

gas

gas - liquid

liquid

C

FIGURE 6.1. The gas-liquid coexistence curve in ρ − p plane:point C is the Andrews critical point, the dashed lines aa′ andbb′ stand for the metastable states.

We now return to discuss Theorem 5.1. The steady state solution ρ0 =ρ0(T ,p) of (4.12) can be taken to represent a desired state in investigating dif-ferent transition situation. For example, for studying the gas-liquid transition wecan take ρ0 as the gas density near the transition temperature and pressure, and forthe liquid-solid transition we take ρ0 as the liquid density, i.e., the lower state den-sity. For convenience, in the following we alway consider the gas-liquid transition,and take ρ0 as the gas density.

Let T0 = φ(p0), and φ′(p0) > 0. Then Theorem 5.1 implies the followingphysical conclusions:

First, near (T0, p0), there are three stable equilibrium states of (4.12) for eachof the three cases: a2 > 0, a2 = 0, or a2 < 0:

(a) If a2 < 0, they are

(6.1)

ϕ+ = ρ0 + ρ+ for |T − T0| < δ and |p − p0| < δ,ϕ0 = ρ0 for T0 < T or p < p0,

ϕ− = ρ0 + ρ− for T < T0 and p > p0;

(b) If a2 = 0, they are

(6.2)ϕ± = ρ0 + ρ± for T < T0 and p > p0,

ϕ0 = ρ0 for T > T0 or p < p0,

where ρ± = ±√λ/a3;(c) If a2 < 0, they are

(6.3)

ϕ+ = ρ0 + ρ+ for T < T0 and p > p0,

ϕ0 = ρ0 for T > T0 or p < p0,

ϕ− = ρ0 + ρ− for |T − T0| < δ and |p − p0| < δ.


Here the state ϕ+ represents the liquid density, ϕ0 the real gas density, and ϕ−the underlying state density.

Second, putting S = β−11 (T − β2ρ2) into (4.9), the free energy becomes

G(ρ, T ,p) = −12β−1

1 T2 + 1

2(α1 + bp + 2β−1

1 β2T)ρ2

− 13α2ρ3 + 1

4(α3 − 2β2

2β−11 )ρ

4 − pρ.

The values of G at equilibrium states of (5.1) are given by

G(ρ0 + ρ±) =G(ρ0)− λ2 (ρ±)2 − a2

3(ρ±)3 + a3

4(ρ±)4(6.4)

=G(ρ0)−λ4(ρ±)2 − a2

12(ρ±)3.

By λ = 0 at (T0, p0), it follows from (6.1)-(6.4) that for (T ,p) near (T0, p0) wehave

(6.5)

G(ϕ+) < G(ϕ0) < G(ϕ−) for a2 > 0,

G(ϕ+) = G(ϕ−) < G(ϕ0) for a2 = 0,

G(ϕ−) < G(ϕ0) < G(ϕ+) for a2 < 0.

Third, according to Theorem 5.1 and the transition diagrams (Figures 5.1-5.3), there are two types of transition behaviors characterized by some functionsΦ = ρ(T ,p) near (T ,p) = (T0, p0), which are called the transition functions.Here, for simplicity we fixed p = p0 and consider Φ = ρ(T) as function of T .

If a2 > 0, the transition functions are

Φ+(T) = ρ0(T) for T > T∗,

ρ0(T)+ ρ+(T) for T < T∗,

Φ−(T) = ρ0(T) for T ≥ T0,

ρ0(T)+ ρ− for T < T0,

for some T0 ≤ T∗ < T1. Φ+(T) has a finite jump at T = T∗ as shown in Figure6.2(a), and Φ−(T) is continuous, but has a discontinuous derivative at T = T0;see Figure 6.2(b).


If a2 = 0, the functions are

Φ+(T) = ρ0(T) for T ≥ T0,

ρ0(T)+ ρ+(T) for T < T0,

Φ−(T) = ρ0(T) for T ≥ T0,

ρ0(T)+ ρ−(T) for T < T0,

and Φ+ and Φ− are continuous with discontinuous derivatives at T = T0; seeFigure 6.3(a) and (b).

If a2 < 0, the functions are

Φ+(T) = ρ0(T) for T ≥ T0,

ρ0(T)+ ρ+(T) for T < T0,

Φ−(T) = ρ0(T) for T > T∗,

ρ0(T)+ ρ−(T) for T < T∗,

for some T0 ≤ T∗ < T1, Φ+ is continuous with a discontinuous derivative atT = T0 as shown in Figure 6.4(a), and Φ− is discontinuous at T = T∗ as shown inFigure 6.4(b).

Fourth, based on physical facts, near the gas-liquid transition point (T0, p0)the density ρ is a decreasing function of T , therefore the transition process Φ+ isrealistic, and Φ− is unrealistic. From the transition function Φ+(T), the gas-liquidtransition can be well understood.

ρ

ρ ρρ

TT *

+

0

0+

(a)

ρρ ρ

ρ

TT0

00+

_

(b)

FIGURE 6.2. The transition functions for a2 > 0: (a) the curveof ρ = Φ+, (b) the curve of ρ = Φ−.

Fifth, we now explain the gas-liquid transition by using the transition functionΦ+(T). Let the two curves

(6.6) λ = λ(T ,p) = 0 and a2 = a2(T ,p) = 0


ρ

ρ ρ

ρ

TT

+

0

0+

0

(a)

ρ

ρ

TT

0

0

(b)

FIGURE 6.3. The transition functions for a2 = 0: (a) the curveof ρ = Φ+, (b) the curve of ρ = Φ−.

ρ

ρ ρ

ρ

TT

+

0

0+

0

(a)

ρ

ρ ρ

ρ

TT

0

0+

_

*

(b)

FIGURE 6.4. The transition functions for a2 < 0: (a) the curveof ρ = Φ+, (b) the curve of ρ = Φ−.

intersect at C = (TC,pC) in PT-plane (see Figure 6.5), and the curve segment ABof λ = 0 is divided into two parts AC and CB by the point C such that

a2(T ,p) > 0 for (T ,p) ∈ AC,a2(T ,p) < 0 for (T ,p) ∈ CB.

We shall see that the point C = (TC,pC) is the Andrews critical point, and thecurve segment AC is the gas-liquid coexistence curve, as shown in Figure 1.1.

In fact, when (T0, p0) ∈ AC,a2(T0, p0) > 0 and the transition functionΦ+(T) has a jump at T = T∗ (see Figure 6.1(a)):

ρ0 → ρ0 + ρ+, ρ+ ' a2

a3> 0.

Hence the system undergoes a transition from a gaseous state to a liquid state withan abrupt change in density. On the other hand, by (6.5) there is an energy gapbetween the gaseous and liquid states:

∆E = G(ρ0 + ρ+)−G(ρ0) < 0 at T = T∗.


This energy gap |∆E| stands for a latent heat, and∆E < 0 shows that the transitionfrom a gaseous state to a liquid state is an isothermal exothermal process, and froma liquid state to gaseous state is an isothermal endothermal process.

B

C

Aa 0=

λ = 0

2

T

p

FIGURE 6.5. The point C = (TC,pC) is the Andrews critical point.

When (T0, p0) = (TC,pC), a2(TC,pC) = 0 and the transition function Φ+(T)is continuous as shown in Figure 6.3(a). Near T = TC ,

Φ+(T) = ρ0(T)+√λa3

for T < TC.

By the Landau mean field theory, we have

(6.7) λ(T) = α(TC − T) (α > 0 a constant).

Thus we infer from (6.4) and (6.7) that

G(Φ+(T)) = G(ρ0)−α2

4a3(T − TC)2 for T < TC.

The difference of the heat capacity at T = TC is

∆C = −TC ∂2

∂T 2

(G(Φ+(T+C ))−G(ρ0)

) = α2

2a3TC > 0.

Namely the heat capacity has a finite jump at T = TC , therefore the transition atT = TC is of the second order.

When (T0, p0) ∈ CB, a2(T0, p0) < 0 and the transition function Φ+(T) iscontinuous as shown in Figure 6.4(a). Near T = T0,

Φ+(T) = ρ0(T)+ 1|a2|

λ(T)+ o(|λ|) , T ≤ T0.

Notice that (5.3) implies that

ρ+ = λa2+ o(|λ|).


Then we deduce from (6.4) and (6.7) that if T > T0,

G(Φ+(T)) = G(ρ0(T)),

and if T ≤ T0,

G(Φ+(T)) = G(ρ0(T))− α3

6|a2|2(T0 − T)3 + o(|T0 − T |3).

Namely, the free energy G(Φ+(T)) is continuously differentiable up to thesecond order at T = T0, and the transition is of the third order. It implies that as(T0, p0) ∈ CB the Type-I transition at (T0, p0) can not be observed by physicalexperiments. Therefore we can derive the following physical conclusion:

Physical Conclusion 6.1. The point C = (TC,pC) satisfying (6.6) correspondsto the Andrews critical point, at which the gas-liquid transition is of the second or-der. In addition, when λ(T0, p0) = 0, a2(T0, p0) > 0, the gas-liquid transitionat (T0, p0) is of the first order accompanied with a latent heat to occur, and whenλ(T0, p0) = 0, a2(T0, p0) < 0, the transition at (T0, p0) is of the third order.

Sixth, as λ(T0, p0) = 0 and a2(T0, p0) > 0, the gas-liquid transition point(T∗, p∗) is in the range T0 ≤ T∗ < T1 and p1 < p∗ ≤ p0; see Figure 5.2(a)-(b). In fact, in the region T0 ≤ T < T1 and p1 < p < p0, the two stable statesϕ0 = ρ0(T ,p) and ϕ+ = ρ0(T ,p) + ρ+(T ,p) are attractors, each possessinga small basin of attraction. Therefore they correspond to metastable states, andϕ0 can be considered as a super cooled gas, while ϕ+ can be considered a superheated liquid.

Finally, likewise, we can also discuss the gas-solid and liquid-solid transitions,and derive the following physical conclusion:

Physical Conclusion 6.2. In the gas-solid and liquid-solid transitions, there alsoexist metastable states. For the gas-solid case, the metastable states correspond to thesuperheated solid and supercooled liquid, and for the liquid-solid case, the metastablestates correspond to the superheated solid and supercooled liquid.

7. ASYMMETRY OF FLUCTUATIONS

The discussions above suggest that for PVT systems, there are two possible phasetransition behaviors near a critical point, and theoretically each of them has someprobability to take place, however only one of them can appear in reality. For theferromagnetic systems we again see this situation; see [18].

One explanation of such phenomena is that the symmety of fluctuation neara critical point is not generally true in equilibrium phase transitions. To make thestatement more clear, we first introduce some concepts.

Let G(u,λ) be free energy of a thermodynamic system, u = (u1, . . . , un) bethe order parameter, and λ = (λ1, . . . , λm) the control parameter (n, m ≥ 1).


Assume that u is defined in the function space L2(Ω,Rn) and λ ∈ Rm. Then thespace

X = (u, λ) | u ∈ L2(Ω,Rn), λ ∈ Rm

is called the state space of the system.Let (u0, λ0) ∈ X be a stable equilibrium state of the system; namely (u0, λ0)

is a locally minimal state of G(u,λ). We say that the system has a fluctuation at(u0, λ0) if it deviates randomly from (u0, λ0) to (u, λ) with

‖u−u0‖ + |λ− λ0| > 0.

In this case, (u, λ) is called a state of fluctuation.The so called symmetry of fluctuation means that for given r > 0, all states

(u, λ) of fluctuation satisfying

‖u−u0‖ + |λ− λ0| = r , (u, λ) ∈ X,

have the same probability to appear in real world. Otherwise, we say that thefluctuation is asymmetric.

The observations in both the PVT systems and the ferromagnetic systemsstrongly suggest the following physical conjecture, regarding to the uniqueness ofatransition behaviors.

Physical Conjecture (Asymmetry of Fluctuations). The symmetry of fluctua-tions for general thermodynamic systems may not be universally true. In other words,in some systems with multi-equilibrium states, the fluctuations near a critical pointoccur only in one basin of attraction of some equilibrium states, which are the onesthat can be physically observed.

APPENDIX A. DYNAMIC TRANSITION THEOREMS

We recall here a few dynamic transition theorems which are used in this article toanalyze the transitions for the PVT systems; see Ma and Wang [17,19] for details.

Let e1(λ) and e∗1 (λ) be the eigenvectors of Lλ and L∗λ respectively correspond-ing to β1(λ) with

Lλ0e1 = 0, L∗λ0e∗1 = 0, 〈e1, e∗1 〉 = 1.

Let Φ(x, λ) be the center manifold function of (2.1) near λ = λ0, and assume that

(A.1) 〈G(xe1 + Φ(x, λ0), λ0), e∗1 〉 = αxk + o(|x|k),

where k ≥ 2 is an integer and α ≠ 0 is a real number.


Theorem A.1. Let the conditions (2.4) and (2.5) with m = 1 and (A.1) holdtrue. If k=odd and α ≠ 0 in (A.1), then the following assertions hold true:

(1) If α > 0, then (2.1) has a jump transition from (0, λ0), and bifurcates onλ < λ0 to exactly two saddle points vλ1 and vλ2 with the Morse index one, asshown in Figure A.1.

v vu = 0

1 2λλ

(a)

u = 0

(b)

FIGURE A.1. Topological structure of Type-II transition of(2.1) when k=odd and α > 0: (a) λ < λ0; (b) λ ≥ λ0. Herethe horizontal line represents the center manifold.

(2) If α < 0, then (2.1) has a continuous transition from (0, λ0), which is anattractor bifurcation as shown in Figure A.2.

u = 0

(a)(b)

u = 0v

v v

v

1

1 2

2

λ

λ

λ

λ

FIGURE A.2. Topological structure of Type-I transition of (2.1)when k=odd and α < 0: (a) λ ≤ λ0; (b) λ > λ0.

(3) The bifurcated solutions vλ1 and vλ2 in the cases above can be expressed as:

vλ1,2 = ±∣∣∣∣β1(λ)α

∣∣∣∣1/(k−1)e1(λ)+ o(|β1|1/(k−1)).

Theorem A.2. Let the conditions (2.4) and (2.5) with m = 1 and (A.1) holdtrue. If k=even and α ≠ 0, then we have the following assertions:

(1) (2.1) has a mixed transition from (0, λ0). More precisely, there exists a neigh-borhood U ⊂ X of u = 0 such that U is separated into two disjoint open sets Uλ1and Uλ2 by the stable manifold Γλ of u = 0 satisfying the following properties asshown in Figure A.3:(a) U = Uλ1 +Uλ2 + Γλ,


(a)

v u = 0λ

(b)

u = 0

u = 0

v

v

v1 2λλ

λ

(c)

FIGURE A.3. Topological structure of Type-III transition of(2.1) when k=even and α ≠ 0: (a) λ < λ0; (b) λ = λ0; (c)λ > λ0. Here Uλ1 is the unstable domain, and Uλ2 the stabledomain.

(b) the transition in Uλ1 is jump, and(c) the transition in Uλ2 is continuous.

(2) (2.1) bifurcates in Uλ2 to an attractor vλ on λ > λ0:

limt→∞

‖u(t,ϕ)− vλ‖X = 0 for ϕ ∈ Uλ2 .

(3) (2.1) bifurcates on λ < λ0 to a unique saddle point vλ with the Morse indexone.

(4) The bifurcated singular point vλ can be expressed as

vλ = −(β1(λ)α

)1/(k−1)e1 + o(|β1|1/(k−1)).

Now we consider (2.1) defined on Hilbert spaces X = H and X1 = H1. LetLλ = −A + λB. For Lλ and G(· , λ) : H1 → H, we assume that A : H1 → H issymmetric, and

〈Au,u〉H ≥ c‖u‖2H1/2,(A.2)

〈Bu,u〉H ≥ c‖u‖2H,(A.3)

〈Gu,u〉H ≤ −c1‖u‖pH + c2‖u‖2H,(A.4)

where p > 2, c, c1, c2 > 0 are constants.


Theorem A.3. Assume (2.3), (2.4) and (A.2)–(A.4) hold true, then (2.1) has atransition at (u, λ) = (0, λ0) with the following properties:

(1) If u = 0 is an even-order nondegenerate singular point of Lλ + G at λ = λ0,then (2.1) has a singular separation of singular points at some (u1, λ1) ∈ H ×(−∞, λ0).

(2) If m = 1 and G satisfies (A.1) with α > 0 if k=odd and α ≠ 0 if k=even,then (2.1) has a saddle-node bifurcation at some singular point (u1, λ1) withλ1 < λ0.

Acknowledgment. The work was supported in part by the Office of NavalResearch and by the National Science Foundation.

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TIAN MA:Department of MathematicsSichuan UniversityChengdu, P.R. ChinaE-MAIL: [email protected]

SHOUHONG WANG:Department of MathematicsIndiana UniversityBloomington, IN 47405, U.S.A.

and

State Key Laboratory of Numerical Modeling for Atmospheric Sciences andGeophysical Fluid Dynamics (LASG)Institute of Atmospheric PhysicsChinese Academy of SciencesBeijing, P.R. ChinaE-MAIL: [email protected]

KEY WORDS AND PHRASES: PVT system, Ginzburg-Landau theory, dynamic transitions, dynamicclassification scheme of phase transitions.

2000 MATHEMATICS SUBJECT CLASSIFICATION: 82B26, 37G35.

Received : March 11th, 2008; revised: June 21st, 2008.Article electronically published on September 19th, 2008.

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