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 the- ory 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 transi- tion, 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 ap- proach using the thermodynamic potential, but is not a mean field theory approach. And the thermodynamic potential (Gibbs free energy) applied in this paper is pro- posed 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 math- ematical 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!1 (3) there is an open and dense set O of initial data in the space of state functions, such that for any u0 2 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 de- termined 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 thermody- namic system is given by (see [20]) 5110 JIAYAN YANG AND DONGPEI ZHANG 8 @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 8 du < = L u + G(u; λ); dt λ (4) : u(0) = '; 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 8 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 ' 2 UnΓλ the solution uλ(t; ') of (4) satisfies that 8 lim sup kuλ(t; ')kH ≥ δ(λ) > 0; < t!1 : 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.