Superfluidity Phase Transitions for Liquid 4He System

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DISCRETE AND CONTINUOUS doi:10.3934/dcdsb.2019045 DYNAMICAL SYSTEMS SERIES B Volume 24, Number 9, September 2019 pp. 5107–5120

SUPERFLUIDITY PHASE TRANSITIONS FOR LIQUID 4HE SYSTEM

Jiayan Yang School of Medical Informatics and Engineering Southwest Medical University Luzhou, Sichuan 646000, China Dongpei Zhang∗ Department of Mathematics Sichuan University Chengdu, Sichuan 610064, China

(Communicated by Shouhong Wang)

Abstract. The main objective of this paper is to investigate the superﬂuidity phase transition theory-modeling and analysis-for liquid 4He system. Based on the new Gibbs free energy and the potential-descending principle proposed recently in [18, 25], the dynamic equations describing the λ-transition and solid- liquid transition of liquid 4He system are derived. Further, by the dynamical transition theory, the two obtained models are proven to exhibit Ehrenfest second-order transition and ﬁrst-order transition, respectively, which are well consistent with the physical experimental results.

1. Introduction. Superfluidity in liquid 4He is a phase of matter, which was first described by Einstein [8] in 1925 and discovered by P. L. Kapitsa [14], J. F. Allen and D. Misener [1] in 1937, at which the liquid’s viscosity becomes zero. More pre- cisely, liquid 4He with viscosity exhibits superfluid behavior, when the temperature decreases under the critical value Tc = 2.17K. Besides, the superfluid phase in liquid 4He (called Bose-liquid), is a consequent of Bose-Einstein statistics obeyed by 4He atoms.

Figure 1. T -p phase diagram of 4He

2010 Mathematics Subject Classiﬁcation. Primary: 34C23, 35Q99, 37G35; Secondary: 37L10. Key words and phrases. 4He system, superﬂuidity, Gibbs free energy, phase transition. ∗ Corresponding author: Dongpei Zhang.

5107 5108 JIAYAN YANG AND DONGPEI ZHANG

It is well known that 4He has four phases, which are solid phase, normal liquid phase (known as He I), superfluid phase (known as He II) and gas phase, at low temperatures. And the phase diagram of 4He is shown schematically in Fig. 1, where T represents temperature, p is pressure. Note that, in Fig. 1, there are two phase transitions relating to superfluid phase, which are λ-transition (BC-line is the critical line) and solid-liquid transition (EC-curve is the critical curve). The main objectives of this article are 1) to obtain the dynamic models for the λ-transition and solid-liquid transition. 2) to analysis the two phase transitions and derive the theoretical phase diagram of 4He, basing on the obtained dynamic models. As a major facet of the study of quantum hydrodynamics, the superfluidity has been described through microscopic and phenomenological theories. The two theories are complementing to each other. For microscopic theories, the basis is the concept of macroscopic occupation of the ground state developed by London [19] and the theory that the transition to superfluid in liquid 4He is an example of Bose-Einstein condensate proposed by London and Tisza [30]. There have been amounts of studies on the superfluidity based on the microscopic theories, we refer readers to [17, 26, 28] and the references therein. For phenomenological theories, the basis is two-fluid model and mean field theory developed by Landau [12, 16]. Based on Landau’s idea, there have been many researches on the transition to superfluid. For examples, Fabrizio presented a Ginzburg-Landau model of 4He and characterized superfluidity as a second order phase transition in [9]. Fabrizio and Mongiov`ıin [10] formulated a hydrodynamical model of superfluid and by using this model, they described the phase transition and obtained the pressure-temperature phase diagram which represents the transition, the thermodynamic restrictions and a maximum theorem for the phase field. Moreover, Ma and Wang established a time-dependent Ginzburg-Landau model for liquid 4He system and investigated its dynamic phase transitions in [23]. For more examples, we refer [2, 24] and so on. The superfluidity phase transition theory in this paper follows the Landau’s approach using the thermodynamic potential, but is not a mean field theory approach. And the thermodynamic potential (Gibbs free energy) applied in this paper is proposed very recently in [18], which is derived basing on first principles. For first principles, we refer [18] and the reference therein. Based on the thermodynamic potential and potential-descending principle [25], the dynamic models for the λ-transition and solid-liquid transition (see Fig. 1) of 4He system are derived. By using the dynamical phase transition developed by Ma and Wang [21, 22], the superfluidity phase transition theorems for the derived models are given, which show the models exhibit second-order transition and first- order transition, i.e., continuous transition and random transition with dynamic classification, respectively. Additionally, the corresponding T -p theoretical phase diagram is obtained, which agrees well with experiments. Note that for 4He system, the existence and uniqueness of the solutions, the existence of global and exponential attractors, and the asymptotic behavior of the solutions have been widely and deeply investigated by many authors, see [3,4,5,6, 13] and the references therein. Moreover, there are many researchers investigating the critical behavior, the shear viscosity, the critical exponents and vortices of 4He, see [7,9, 11, 15, 27, 29] and the references therein. SUPERFLUIDITY PHASE TRANSITIONS FOR LIQUID 4HE SYSTEM 5109

The paper is organized as follows. The preliminaries, including the potential- descending principle and the dynamical transition theory, are given in Section 2. In Section 3, we introduce the new Gibbs free energy and the general dynamic model for liquid 4He system. Section 4 is devoted to investigating the superﬂuidity phase transitions of liquid 4He system.

2. Preliminaries. In this section, we focus on introducing the preliminaries used in this paper. First, we introduce the potential-descending principle [25], which is crucial for the derivation of the dynamic equations of 4He system. Then we introduce the dynamical transition theory [21, 22], which provides the basic mathematical method for the research of the superﬂuidity phase transitions.

2.1. The potential-descending principle. For a non-equilibrium thermodynamic system, let λ = (λ1, ··· , λN ) be the control parameters and u = (u1, ··· , ul) be the order parameters of the thermodynamic system. Assume that the thermodynamic potential of the system can be expressed as F = F (u, λ). (1) The potential-descending principle [25] is described as follows, which is a funda- mental principle for the non-equilibrium thermodynamic systems. Principle 2.1. (Potential-Descending Principle) For each thermodynamic system, there are order parameters u = (u1, ··· , ul), control parameters λ = (λ1, ··· , λN ), and the thermodynamic potential functional F (u; λ). For a non-equilibrium state u(t; u0) of the system with initial state u(0; u0) = u0, we have the following proper- ties: (1) The potential F (u(t; u0); λ) is decreasing, i.e., d F (u(t; u ); λ) < 0, for any t > 0. dt 0

(2) the order parameters u(t; u0) have a limit lim u(t) = u. t→∞ (3) there is an open and dense set O of initial data in the space of state functions, such that for any u0 ∈ O, the corresponding u is a minimum of F , which is called an equilibrium of the thermodynamic system δF (u, λ) = 0. It is well known that the governing equations of the system are essentially determined by its thermodynamic potential (1). Based on the Principle 2.1, the following uniﬁed dynamical model for non-equilibrium thermodynamic systems can be derived, see [20, 25]. In general, the order parameter u can be divided into two parts, i.e., u = (v, w). Thus the functional (1) can be rewritten as F = F (v, w, λ). (2) Consequently, the uniﬁed dynamical model for the non-equilibrium thermodynamic system is given by (see [20]) 5110 JIAYAN YANG AND DONGPEI ZHANG

 ∂v δ  k = −a F (v, w, λ) + Φ (v, w, λ), 1 ≤ k ≤ l,  ∂t k δv k  k  δ F (v, w, λ) = 0, (3)  δw  Z  δ  F (v, w, λ) · Φk(v, w, λ)dx = 0, Ω δvk l δ where ak > 0 are constant coeﬃcients, v = (vk)k=1, δ· F represents the variation of F at an element of the order parameter u. We refer interested readers to [20, 25] for details.

2.2. Dynamical transition theory. In thermodynamic phase transitions, for a given basic equilibrium state u = 0, the deviation from this basic state is generally governed by differential equations with dissipative structure, which can be expressed in the following abstract form. Let H and H1 be two Hilbert spaces, H1 ⊂ H is a dense and compact inclusion. The abstract form of a dissipative dynamical system is given by  du  = L u + G(u, λ), dt λ (4)  u(0) = ϕ, where u is order parameter and λ is control parameter. Lλ and G(u, λ) satisfy (1) Lλ : H1 → H is a parameterized linear completely continuous field depending continuously on λ, which satisfies  Lλ = −Aλ + B is a sectorial operator,  Aλ : H1 → H is a linear homeomorphism, (5)   B : H1 → H is a compact operator. r (2) G(·, λ): Hσ → H is a C (r ≥ 1) bounded mapping for some 0 ≤ σ < 1, where Hσ is the fractional order space, and

G(u, λ) = o(kukHσ ). (6) Hereafter, we always assume the conditions (5) and (6) hold true, which imply that the system (4) has a dissipative structure. For the system (4), we ﬁrst introduce the mathematical deﬁnition of transition [22] from the basic equilibrium state u = 0.

Deﬁnition 2.1. We say that the system (4) has a transition from (u, λ) = (0, λ0) at λ = λ0 if the following two conditions hold true: (1) if λ < λ0, u = 0 is locally asymptotically stable for (4); (2) if λ > λ0, there exists a neighborhood U ⊂ H of u = 0 independent of λ, such that for any ϕ ∈ U\Γλ the solution uλ(t, ϕ) of (4) satisﬁes that  lim sup kuλ(t, ϕ)kH ≥ δ(λ) > 0,  t→∞  lim δ(λ) = δ ≥ 0, λ→λ0 where Γλ is the stable manifold of u = 0, with codim Γλ ≥ 1 in H for λ > λ0.

It should be pointed out that λ0 is the critical control parameter. Then we introduce the following general principle of phase transition dynamics [20], which oﬀers a guiding principle for studying phase transitions. SUPERFLUIDITY PHASE TRANSITIONS FOR LIQUID 4HE SYSTEM 5111

Principle 2.2. (Principle of Phase Transition Dynamics) Phase transitions of all dissipative systems can be classiﬁed into three categories:

(1) continuous transition if limλ→λ0 uλ = 0;

(2) catastrophic transition if limλ→λ0 uλ 6= 0; and

(3) random transition if both limλ→λ0 uλ = 0 and limλ→λ0 uλ 6= 0 occur. Here u = 0 is the basic equilibrium state, and uλ are the transition states (physically, transition states correspond to local attractors). The validity of the above principle is ensured by the dynamic transition theorem(see [22] Theorem 2.1.3). It is well known that for the system (4), as soon as the linear instability occurs, the system always undergoes a dynamical transition. It means that the eigenvalues of the linear operator Lλ are important for the researching of phase transition. Let {βj(λ) ∈ C|j ∈ N} be the eigenvalues (counting multiplicity) of Lλ, and assume that  < 0, λ < λ0,  Reβi(λ) = 0, λ = λ0, 1 ≤ i ≤ m,  (7)  > 0, λ > λ0,

Reβj(λ0) < 0, for any j ≥ m + 1, where m ≥ 1 is the multiplicity of the first eigenvalue β1. Physically, the above assumption (7) is called principle of exchange of stabilities (PES). And the following lemma [20] shows that (7) is closely related to the phase transition of the system (4) and all dynamic transitions can be classified into the three categories. Lemma 2.2. Assume (7) holds true, then the system (4) always undergoes a dynamic transition to one of the three types of dynamic transitions: continuous, catastrophic and random, as λ crosses the critical threshold λ0. For the type of phase transition, it is worth mentioning that the following lemma [20] shows that there are only first, second and third order transitions for the phase transition of a thermodynamic system, and provides the relationship between the Ehrenfest and dynamic classifications. Lemma 2.3. For the phase transition of a thermodynamic system, there exist only first-order, second-order and third-order phase transitions. Moreover the following relations between the Ehrenfest classification and the dynamical classification hold true: second-order ⇐⇒ continuous; first-order ←− catastrophic; either first or third-order ←− random; first-order −→ either catastrophic or random; third-order −→ random with asymmetric fluctuations.

3. The Gibbs free energy for liquid 4He system. In this section, we ﬁrst introduce the Gibbs free energy for liquid 4He system proposed in [18] and then the general dynamic equations for liquid 4He system we derived.

3.1. The Gibbs free energy of liquid 4He system. The order parameter describing superﬂuidity is characterized by a non vanishing complex valued function 5112 JIAYAN YANG AND DONGPEI ZHANG

n ψ = ψ1 + iψ2 :Ω → C, where Ω ⊂ R (1 ≤ n ≤ 3) is a bounded open domain. From the two ﬂuid point of view, the density of liquid 4He can be given by

ρ = ρn + ρs, where ρs is the superfluid density and ρn is the normal fluid density. The square 2 2 |ψ| is proportional to ρs, and without loss of generality, we take ψ as |ψ| = ρs. In addition, ρ satisfies the following conservation law: Z 2 [ρn + |ψ| ]dx = C, Ω where C > 0 is a constant. It implies the total number of particles in the system is conserved. The Gibbs free energy of liquid 4He proposed recently in [18] is expressed as Z 2 α 2 1 2 2 ~ 2 F (ψ, ρn,S) = |∇ρn| + (b p + A1T )ρn − bpρn − µρ + |∇ψ| Ω 2 2 2m g − g |ψ|2 + 1 |ψ|4 + g bρ |ψ|2 + A T (1 + bρ) ln(1 + bρ) (8) 0 2 2 n 2 2 2 2 − η0TS + η1T Sρn − η2TS|ψ| − ST dx.

Here α > 0 is a constant, ~ is the Planck constant, m is the mass of 4He atom, b is the Van der Waals constant. µ is chemical potential, which also plays the role of Lagrangian multiplier. g0 > 0, g1 > 0 are interaction constants, where g0 is the binding potential of condensates and g1 is the coupling constant of particles. T represents temperature, p is pressure, S is entropy. A1,A2 and η0 > 0, η1 > 0, η2 > 0 are parameters depending on T, p. In physics, g2 is the interaction potential between normal liquid particles and superﬂuid particles and satisﬁes < 0, for attractive interaction, g 2 > 0, for repulsive interaction.

4 Note that, g2 > 0 in He system. 3.2. The general dynamical model of liquid 4He system. For 4He system, the order parameters are u = (ψ, ρn,S), the control parameters are λ = (T, p). By the potential-descending principle (i.e., Principle 2.1) and the Gibbs free energy (8), we can derive the following general time-dependent dynamic equations governing the superﬂuidity of liquid 4He:  2 2 ∂ψ ~ η2T η2T 2  = ∆ψ + (g0 − )ψ − g2bρnψ − (g1 + )|ψ| ψ  ∂t 2m 2η0 2η0   η η T  + 1 2 ρ2 ψ − A T bψ ln(1 + bρ + b|ψ|2),  2η n 2 n 0 (9) ∂ρ η T η η T  n =α∆ρ − (b2p + A T − 1 )ρ − g b|ψ|2 + 1 2 ρ |ψ|2  ∂t n 1 η n 2 η n  0 0  2  η1T 3 2  − ρn + bp − A2T b ln(1 + bρn + b|ψ| ), η0 where µ = A2T b. We may supplement (9) with the following initial-boundary value conditions

ρn = (ρn)0, ψ = ψ0, (10) SUPERFLUIDITY PHASE TRANSITIONS FOR LIQUID 4HE SYSTEM 5113

∂ρn ∂ψ = 0, = 0. (11) ∂n ∂Ω ∂n ∂Ω

4. Superfluidity phase transitions for liquid 4He system with homogeneous distribution. In the lab scale, the diffusion effect is negligible, which means ∇ρn = 0 and ∇ψ = 0. Thus, in this section, we focus on the superfluidity phase transition theory for liquid 4He system with homogeneous distribution. Furthermore, the components for the whole section include the following aspects: (1) the potential functionals and the dynamic models, (2) PES for dynamical models and theoretical T -p phase diagram, (3) the superfluid phase transition theorems for the liquid 4He system.

4.1. The potential functionals and the dynamic models. For the superﬂuid phase transition of liquid 4He, the transition behaviour near BC-line (λ-transition) is diﬀerent from that near EC-curve (solid-liquid transition), see Fig. 1. It inspires us to discuss the phase transition respectively.

4.1.1. The potential functional and the dynamic model near BC-line. In order to investigate the phase transition near the BC-line in Fig. 1, we ﬁrst need study the thermodynamic potential and propose a proper dynamic model near the BC-line. Note that, for the liquid 4He system with homogeneous distribution, we have ∇ρn = 0 and ∇ψ = 0. Moreover, in the vicinity of BC-line (see Fig. 1), the temperature T is ultra low. Therefore, A2T (1 + bρ) ln(1 + bρ) ' 0 in (8). Thus, the thermodynamic potential of liquid 4He near the BC-line in Fig. 1 can be expressed as

1 2 2 g1 4 2 F (ψ, ρn, S, T, p) = aρ − pρn − g0|ψ| + |ψ| + g2ρn|ψ| 2 n 2 (12) 2 2 − η0TS − η2TS|ψ| − ST, where the coeﬃcients a, b, g0, g1, g2, η0, η2 are both positive constants. By the Principle 2.1, we take the following dynamic model near BC-line in Fig. 1  dψ ∂  = − F (ψ, ρn, S, T, p),  dt ∂ψ∗   dρn ∂ = − F (ψ, ρn, S, T, p) = 0, (13)  dt ∂ρn   ∂  F (ψ, ρn, S, T, p) = 0. ∂S Combining with (12) and (13), we can derive the following dynamic equation dψ = λ ψ − α |ψ|2ψ, (14) dt 1 1 where

g2 η2 λ1 = g0 − p − T, (15) a 2η0 2 2 η2 g2 α1 = g1 + T − , (16) 2η0 a and α1 is assumed to be positive. 5114 JIAYAN YANG AND DONGPEI ZHANG

Remark 1. Here, we need to make a further explanation for the thermodynamic potential (12) and the dynamic model (13). 2 (1) In physics, the term η1T Sρn in (8) represents the thermal energy generated by the collision of two 4He particles. In the vicinity of BC-line in Fig. 1, the thermal energy generated by the collision is very small because of the ultra low temperature 2 and low pressure. Therefore, the term η1T Sρn can be omitted in (12). (2) For dynamic model (13), in an area sufficiently close to BC-line in Fig. 1, the change of ρn is very small. Consequently, we take the state steady equation ∂ F = 0 with respect to ρn, which is different from (9). ∂ρn 4.1.2. The potential functional and the dynamic model near EC-curve. Near the EC-curve in Fig. 1, the thermodynamic potential of 4He system can be given as 1 g F (ψ, ρ , S, T, p) = (γ p − γ T )ρ2 − pρ − g |ψ|2 + 1 |ψ|4 + g ρ |ψ|2 n 2 0 1 n n 0 2 2 n 2 (17) T 2 2 − S + η1T Sρn − ST, α0 where the coefficients γ0, γ1, g0, g1, g2, α0, η1 are both positive constants. By the Principle 2.1, we take the dynamic model near the EC-curve in Fig. 1 as the following:  dρ ∂  n = − F (ψ, ρ , S, T, p),  dt ∂ρ n  n  dψ ∂ = − F (ψ, ρn, S, T, p) = 0, (18)  dt ∂ψ∗   ∂  F (ψ, ρn, S, T, p) = 0. ∂S Combined (17) and (18), the following governing equations can be derived

dρ n = b ρ − b ρ3 + p, (19) dt 0 n 1 n 1 ρs = (g0 − g2ρn), (20) g1 2 where ρs = |ψ| represents the density of superﬂuid and

b0 = α0η1 − γ0p + γ1T > 0, (21) 2 b1 = α0η1. (22) Remark 2. Here, a further explanation for the thermodynamic potential (17) and the dynamic model (18) is given. 2 (1) In physics, the term −η2TS|ψ| in (8) represents the thermal energy reduced by the condensation of 4He particles. Near the EC-curve in Fig. 1, the thermal energy reduced by the condensation of 4He particles is very small. Consequently, 2 we omit the term −η2TS|ψ| in (17). (2) For the dynamic model (18), similarly, the change of ψ is very small in an area suﬃciently close to EC-curve in Fig. 1. Therefore, we take the state steady ∂ equation ∂ψ∗ F = 0 with respect to ψ. SUPERFLUIDITY PHASE TRANSITIONS FOR LIQUID 4HE SYSTEM 5115

4.2. PES for the dynamic models and theoretical T -p phase transition diagram. According to the dynamical transition theory, the PES is crucial for the phase transition. Thus, in this section, we study the critical parameters equations and PES. Since superﬂuid phase transition is the transition between superﬂuid phase and another phase (solid phase or normal liquid phase), we need to provide two stable phases (i.e. steady-state solution) for the two dynamic equations (14) and (19) respectively. Near the BC-line in Fig. 1, obviously, ψ = 0 is the steady-state solution for (14), which implies the normal liquid phase is one phase for (14). Near the EC-curve in Fig. 1, we consider the solid phase of (19). For b1 b0 in (19), 4He is in solid phase. Therefore, we have the following approximate steady state solution of (19)

0 p ρn = , (23) α0η1 − γ0p + γ1T which represents a solid phase of 4He. 0 0 For the convenience of discussion, we take the translation ρn = ρn − ρn in (19). Then the equation (19) can be rewritten as (omitting the primes) dρ n = λ ρ − 3b ρ0 ρ2 − b ρ3 , (24) dt 2 n 1 n n 1 n where 0 2 λ2 = α0η1 − γ0p + γ1T − 3b1(ρn) . (25) Then it suﬃces to study the phase transition of (24) at the steady state solution ρn = 0 instead of (19) at (23). As a matter of fact, phase transition is determined by the critical-crossing of the eigenvalues of the linear part. For this purpose, we need to get the critical-crossing conditions for the eigenvalues λ1 in (15) and λ2 in (25). Let λ1 = 0. It is not diﬃcult to obtain the following equation that critical control parameters (Tc, pc) in λ1 satisfy g2 η2 pc + Tc − g0 = 0. (26) a 2η0

Combining (23) with (25) and letting λ2 = 0, it is easy to derive the following equation that critical control parameters (Tc, pc) in λ2 satisfy 1 2 γ0pc + (3b1) 3 (pc) 3 − γ1Tc − α0η1 = 0. (27) where b1 in (22). Obviously, the critical parameter equation (26) gives a simple line l1 in the T p- 2 plane R+, see Fig. 2(1). Also, the equation (27) determines a curve l2 in the 2 T p-plane R+, see Fig. 2(2). Before the discussion, the following condition which the coeﬃcients in (26) and (27) satisfy is proposed

2 3 ag0γ0 1 ag0 2 3 + (3α0η1) − α0η1 > 0. (28) g2 g2

Under the condition (28), the line l1 intersects with the curve l2 at a point in 2 the T p-plane R+. Combining (26) and (27), we can obtain the following theoretical T -p phase diagram, see Fig. 3. 5116 JIAYAN YANG AND DONGPEI ZHANG

Figure 2. The diagram for critical control parameters (Tc, pc)

Figure 3. The theoretical T -p phase diagram for 4He

We should point out that in Fig. 3 the line segment BC is the critical line for normal liquid and superﬂuidity and curve segment EC is the critical curve for solid and superﬂuidity. For convenience, we set 2 D = {(T, p) ∈ R+ | the region that represents the Liquid He I in Fig.3}, 2 H = {(T, p) ∈ R+ | the region that represents the solid phase in Fig.3}, 2 M = {(T, p) ∈ R+ | the region that represents the Liquid He II in Fig.3}.

Thus, when λ1 crosses the line segment BC to enter into M from D (see Fig. 3), the PES for λ1 is given as followings,  < 0, as (T, p) ∈ D,  λ1 = 0, as (Tc, pc) ∈ BC-line, (29)  > 0, as (T, p) ∈ M.

Besides, when λ2 crosses the curve segment EC to enter into M from H in Fig. 3, the PES for λ2 is  < 0, as (T, p) ∈ H,  λ2 = 0, as (Tc, pc) ∈ EC-curve, (30)  > 0, as (T, p) ∈ M.

4.3. The superfluid phase transition theorems of liquid 4He. Basing on the PES (29) and (30), in this section, we give the superfluid phase transition SUPERFLUIDITY PHASE TRANSITIONS FOR LIQUID 4HE SYSTEM 5117 theorems of the dynamic equations (14) and (19). First, we give the superfluid phase transition theorem (i.e. λ-transition) near the BC-line in Fig. 3. Theorem 4.1. (λ-transition theorem) Assume that (28) holds. For the dynamic equation (14), the following assertions hold true: (1) As (T, p) ∈ D, the steady state solution of (14), ψ = 0, is asymptotically stable, which implies the system (14) is in the normal liquid phase; (2) As the control parameters (T, p) enter into M from D crossing the BC-line λ1 in Fig. 3, equation (14) exhibits a stable superfluid phase ρs = with λ1 α1 denoted as (15), which implies there exists a superfluid phase transition near BC-line in Fig. 3. Moreover, the phase transition is continuous, i.e., second- order with Ehrenfest classification.

Proof. From (29), λ1 < 0 as (T, p) ∈ D. Then the steady state solution ψ = 0 is asymptotically stable. Then Assertion (1) is true. Then, we prove Assertion (2). By (29) and Lemma 2.2, the equation (14) undergoes a dynamical phase transition. As (T, p) enter into M from D, see Fig. 3, λ1 changes into positive from negative, and the steady state solution ψ = 0 loses its stability and becomes unstable. Mean- 2 λ1 λ1 while, the equation (14) bifurcates a positive solution |ψ| = , i.e., ρs = > 0, α1 α1 which means the superfluid phase exhibits. Additionally, the bifurcated solution |ψ|2 is asymptotically stable. λ1 Also, it is easy to derive that the bifurcated solution ρs = satisfies the follow- α1 ing limit λ lim 1 = 0, (T,p)→(Tc,pc) α1 where λ1 in (15), α1 in (16) and (Tc, pc) satisfy (26). Based on the item (1) in Principle 2.2, the phase transition is continuous. More- over, according to Lemma 2.3, the phase transition is of second-order with Ehrenfest classification. The Assertion (2) is proved. Theorem 4.2. (Solid-liquid transition theorem) Assume that (28) holds. For the equations (19)-(20), we have the following assertions: (1) As (T, p) ∈ H, the steady state solution (23) of (19) is asymptotically stable, which implies the system (19) is in solid phase. (2) The equation (19) exists a phase transition from solid state to superfluid phase as the control parameters (T, p) enter into M from H crossing the EC-curve, see Fig. 3. Moreover, the phase transition is random and of first-order with Ehrenfest classification. Proof. Here, we only study the phase transition of equation (24) for it is equivalent with that of (19). By (30), the steady state solution ρn = 0 of (24) is asymptotically stable as (T, p) ∈ H. Furthermore, the phase of 4He is solid. As the control parameters (T, p) enter into M from H crossing the EC-curve (see Fig. 3), λ2 > 0. By Lemma 2.2, the equation (24) undergoes a dynamical phase transition. Then, we discuss the bifurcated solutions of (24). First, we prove that the topology of steady state solutions for (24) is given as From (24), we have the steady state equation given by 0 2 3 λ2ρn − 3b1ρnρn − b1ρn = 0, (31) 5118 JIAYAN YANG AND DONGPEI ZHANG

Figure 4. The topology of steady state solutions for (24)

0 where b1 in (22), ρn in (23), λ2 in (25). For equation (31), by eliminating a ρn, the discriminant of the corresponding 0 2 quadratic equation can be expressed as 4 = (3b1ρn) + 4b1λ2. As (T, p) ∈ H, λ2 < 0 and 4 < 0, (31) has only one trivial solution. When 4 > 0, it is clear that equation (31) owns two nontrivial solutions −3b ρ0 + p(3b ρ0 )2 + 4b λ ρ+ = 1 n 1 n 1 2 , n 2b 1 (32) 0 p 0 2 − −3b1ρn − (3b1ρn) + 4b1λ2 ρn = < 0. 2b1 Further, we have

 < 0, as (T, p) ∈ H,  +  ρn = 0, as (Tc, pc) ∈ EC-curve,  > 0, as (T, p) ∈ M. Moreover, as the control parameters (T, p) change, there exists a couple (T ∗, p∗) such that the discriminant 4(T ∗, p∗) = 0. That is, the equation (31) has two equal 0 ∗ ∗ solutions ρfn = −3ρn(T , p ). According to the continuous dependence of λ2 on control parameters (T, p), the following limits about the solutions in (32) can be got + lim ρn = 0, (T,p)→(Tc,pc) − 0 lim ρn = −3ρn(Tc, pc) 6= 0, (33) (T,p)→(Tc,pc) + − 0 ∗ ∗ lim ρn = lim ρn = ρn = −3ρn(T , p ), (T,p)→(T ∗,p∗) (T,p)→(T ∗,p∗) f 0 where ρn in (23), (Tc, pc) satisfy (27). Based on the above discussion, Fig. 4 is obtained. Second, we prove the stability of the solutions in (32). + − By the stability analysis, ρn (as (T, p) in M) and ρn are stable, which can be seen in the following Fig. 5. Note that, all of the above conclusions hold true for equation (19). Without loss + − ∗ ∗ of generality, we still let ρn (as (T, p) ∈ M) and ρn (as T < T and p < p ) be the two bifurcated stable solutions of equation (19). Then, we prove the phase transition is from solid phase to superﬂuidity. SUPERFLUIDITY PHASE TRANSITIONS FOR LIQUID 4HE SYSTEM 5119

Combined with (20), as 4He is in solid phase, we can get that g0 ρs = 0 ⇐⇒ ρn = . g2 − While, as (19) bifurcates a stable solution ρn , we have g2 − ρs = (ρn − ρn ) > δ > 0. (34) g1 (34) indicates that the superfluidity suddenly appears and the phase transition (19) undergoes is from solid state to superfluidity. Based on the item (3) in Principle 2.2, (33) shows that the phase transition is random. According to Lemma 2.3, we can derive that this random transition is of first-order. The proof is completed.

Remark 3. For equation (19) and a given low temperature T , the p-ρn dynamical phase diagram for 4He is shown in the following ﬁgure

Figure 5. The p-ρn dynamical phase diagram

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