Dynamic model and phase transitions for liquid helium

JOURNAL OF MATHEMATICAL PHYSICS 49, 073304 2008 Dynamic model and phase transitions for liquid helium Tian Ma 1 and Shouhong Wang 2,a 1 Department of Mathematics, Sichuan University, Chengdu, 610064, People s Republic of China 2 Department of Mathematics, Indiana University, Bloomington, Indiana 47405, USA Received 7 May 2008; accepted 24 June 2008; published online 25 July 2008 This article presents a phenomenological dynamic phase transition theory modeling and analysis for liquid helium-4. First we derive a time-dependent Ginzburg Landau model for helium-4 by 1 separating the superfluid and the normal fluid densities with the superfluid density given in terms of a wave function and 2 using a unified dynamical Ginzburg Landau model. One the one hand, the analysis of this model leads to phase diagrams consistent with the classical ones for liquid helium-4. On the other hand, it leads to predictions of 1 the existence of a metastable region H, where both solid and liquid He II states are metastable and appear randomly depending on fluctuations and 2 the existence of a switch point M on the curve, where the transitions changes types. It is hoped that these predictions will be useful for designing better physical experiments and lead to better understanding of the physical mechanism of superfluidity. 2008 American Institute of Physics. DOI: 10.1063/1.2957943 I. INTRODUCTION Superfluidity is a phase of matter in which unusual effects are observed when liquids, typically of helium-4 or helium-3, overcome friction by surface interaction when at a stage, known as the lambda point for helium-4, at which the liquid s viscosity becomes zero. Also known as a major facet in the study of quantum hydrodynamics, it was discovered by Kapitsa, Allen, and Misener in 1937 and was described through phenomenological and microscopic theories. Helium atoms have two stable isotopes 4 He and 3 He. 4 He consists of two electrons, two protons, and two neutrons, which are six fermions. Therefore, 4 He has an integral spin and obey the Bose Einstein statistics. Liquid 4 He, called the Bose liquid, displays a direct transition from the normal liquid state liquid He I to the superfluid state liquid He II at temperature T =2.19 K, which can be considered as the condensation of particles at a quantum state. The main objectives of this article are 1 to establish a time-dependent Ginzburg Landau model for liquid 4 He and 2 to study its dynamic phase transitions, leading to two specific physical predictions. Hereafter, we shall present briefly the main ingredients of the study presented in this article. The ideas and method in this article can be used to study 3 He and its mixture with 4 He; this study will be reported elsewhere. To derive the model, we recall that, in the late 1930s, Ginzburg Landau proposed a mean-field theory of continuous phase transitions. With the successful application of the Ginzburg Landau theory to superconductivity, it is natural to transfer something similar to the superfluidity case, as the superfluid transitions in liquid 3 He and 4 He are of similar quantum origin as superconductivity. Unfortunately, as indicated by Ginzburg, 1 a satisfactory Ginzburg Landau theory for liquid helium-4 is still lacking and, in fact, one cannot derive the classical phase diagram and stability of transition solutions from the known Ginzburg Landau free energy; see among others Onuki. 12 a Electronic mail: showang@indiana.edu. URL: http://www.indiana.edu/fluid. 0022-2488/2008/49 7 /073304/18/$23.00 49, 073304-1 2008 American Institute of Physics

073304-2 T. Ma and S. Wang J. Math. Phys. 49, 073304 2008 The basic ingredients of the modeling in this article are as follows. First, we separate the normal and superfluid densities and write the total density as the sum of superfluid density s and the normal fluid density n. This separation of densities provides a simple way to identify different phases. Second, the Ginzburg Landau free energy is derived using 1 the mean-field theory, 2 the classical phase diagram for helium-4, and 3 the insights from the dynamic transition theory, which, as we shall discuss below, is used to analyze the model introduced here. In simple words, we retain the terms in the mean-field theory expansion of the free energy such that the phase transition obeys the classical phase diagram and the stability of the transition solutions. Third, with this Ginzburg Landau free energy, a time-dependent model is then derived based on a unified time-dependent Ginzburg Landau model for equilibrium phase transitions, proposed recently by the authors. This unified model is based on the Le Châtelier principle and some general characteristics of pseudogradient flows. An important feature of this unified model is to have added terms, i and j in 2.8 and 2.14, to the gradient flow of the energy functional. As we shall explain in Sec. II, these added terms are determined by physical laws and the constraints given in the model, as in the case for the classical time-dependent Ginzburg Landau model for superconductivity. In addition, as we shall see, these added terms vanish for the steady state solutions, and consequently the steady solutions are critical points of the energy functional, as suggested by classical theories. The model is analyzed using a dynamic transition theory developed recently by the authors. 7 11 With this theory, we derive a new dynamic phase transition classification scheme, which classifies phase transitions into three categories: type I, type II, and type III. As we know, in physics, equilibrium phase transitions are classified by either the Ehrenfest or the modern classification scheme, based on the lowest derivative of the free energy that is discontinuous at the transition. The superfluid phase transitions have been regarded as continuous phase transitions, also called second-order phase transitions. However, many important issues are still not clear; see among many others Ginzburg, 1 Reichl, 13 and Onuki. 12 In comparison to a classical classification scheme, the dynamic classification is for both equilibrium and nonequilibrium phase transitions and provides dynamical behavior of the transitions. In addition, as we shall see, the states after the transition in our theory are given by local attractors, representing complete physical reality after the transition. It has been applied to study a number of problems including ferromagnetism, 10 binary systems, 8 PVT systems, 9 superconductivity, 4 and classical and geophysical fluid dynamics. 3,5,6 For the phase transition problem of helium-4 studied in this article, we focus on our analysis on the homogeneous case where the pressure p is treated as a constant, and the resulting model is reduced to a system of two ordinary differential equations. More general case where the model is a partial differential equation system will be studied in a forthcoming paper. In fact, the homogeneous case is important from the physical point of view. The analysis in this homogeneous case not only gives us a clear understanding of the mechanism of the related physical issues but also leads to important physical predictions. Also, the dynamical transition theory offers great insights not only for the modeling as mentioned earlier but also for deriving the theoretical results and predictions from the model, even in the homogeneous case where the model is a system of ordinary differential equations. It is fair to say that it is not clear at all to see these results and the physical predictions from the classical mathematical theory. The results in this article lead to two physical predictions: 1 the existence of a metastable region H, where both solid and liquid He II states are metastable and appear randomly depending on fluctuations, and 2 the existence of a switch point M, where the transitions, between the superfluid state liquid He II and the normal fluid state liquid He I, change from first order type II with the dynamic classification scheme to second order type I. Of course, these predictions need to be verified by experiments, and it is hoped that the new model, the ideas, and methods introduced in this article will lead to some improved understanding of superfluidity. This article is organized as follows. First, the new dynamic transition theory is recapitulated in

073304-3 Phase transitions for liquid helium J. Math. Phys. 49, 073304 2008 Sec. II, and a dynamic phase transition model for 4 He is introduced in discussed in Sec. III. The dynamic transitions of the model are analyzed in Sec. IV. Section V gives some physical conclusions and predictions. II. GENERAL PRINCIPLES OF PHASE TRANSITION DYNAMICS In this section, we recapitulate a general program of studies for dynamic phase transition in nonlinear sciences, initiated recently by the authors, which will be used in the study of phase transitions for liquid helium-4 in this article. The part related to this article includes 1 a dynamic transition theory, leading to a dynamic classification of both equilibrium and nonequilibrium phase transitions, and 2 a unified time-dependent Ginzburg Landau model. A. Dynamic transition theory In sciences, nonlinear dissipative systems are generally governed by differential equations, which can be expressed as an abstract evolution equation du dt = L u + G u,, u 0 =, 2.1 where X and X 1 are two Banach spaces, X 1 X is a compact and dense inclusion, u: 0, X is unknown function, and R 1 is the system parameter. Assume that L :X 1 X is a parametrized linear completely continuous field depending continuously on R 1, which satisfies L = A + B a sectorial operator, A:X 1 X a linear homeomorphism, 2.2 B :X 1 X a linear compact operator. In this case, we can define the fractional order spaces X for R 1. Then we also assume that G, :X X is C r r 1 bounded mapping for some 0 1, depending continuously on R 1, and G u, = o u X, R 1. 2.3 Hereafter we always assume conditions 2.2 and 2.3, which represent that the system 2.1 has a dissipative structure. A state of the system 2.1 at is usually referred to as a compact invariant set. In many applications, is a singular point or a periodic orbit. A state of 2.1 is stable if is an attractor; otherwise is called unstable. Definition 2.1: We say that the system (2.1) has a phase transition from a state at = 0 if is stable on 0 (or on 0 ) and is unstable on 0 (or on 0 ). The critical parameter 0 is called a critical point. In other words, the phase transition corresponds to an exchange of stable states. Obviously, the attractor bifurcation of 2.1 is a type of transition. However, bifurcation and transition are two different but related concepts. Let j C j N be the eigenvalues counting multiplicity of L and assume that Re i 0 if 0,, =0 if = 0, 1 i m, 2.4 0 if 0 Re j 0 0, j m +1. 2.5

073304-4 T. Ma and S. Wang J. Math. Phys. 49, 073304 2008 The following theorem is a basic principle of transitions from equilibrium states, which provides sufficient conditions and a basic classification for transitions of nonlinear dissipative systems. This theorem is a direct consequence of the center manifold theorems and the stable manifold theorems; we omit the proof. Theorem 2.1: Let conditions (2.4) and (2.5) hold true. Then, the system (2.1) must have a transition from u, = 0, 0, and there is a neighborhood U X ofu=0 such that the transition is one of the following three types. 1 2 3 Continuous transition: There exists an open and dense set Ũ U such that for any Ũ, the solution u t, of (2.1) satisfies lim lim sup u t, X =0. 0 t Jump transition: For any 0 0 + with some 0, there is an open and dense set U U such that for any U, lim sup u t, X 0 t for some 0 is independent of. Mixed transition: For any 0 0 + with some 0, U can be decomposed into two open sets U 1 and U 2 (U i not necessarily connected): Ū=Ū 1 +Ū 2,U 1 U 2 =, such that lim lim sup u t, X =0, U 1, 0 t lim sup u t, X 0, U 2. t With this theorem in our disposal, we are in position to give a new dynamic classification scheme for dynamic phase transitions. Definition 2.1: (Dynamic classification of phase transition) The phase transitions for (2.1) at = 0 is classified using their dynamic properties: continuous, jump, and mixed as given in Theorem 2.1, which are called type I, type II, and type III, respectively. An important aspect of the transition theory is to determine which of the three types of transitions given by Theorem 2.1 occurs in a specific problem. A corresponding dynamic transition has been developed recently by the authors for this purpose; see Refs. 7 and 11. We refer interested readers to these references for details of the theory. Hereafter we recall a few related theorems in this theory used in this article. First, we remark here that one crucial ingredient for applications of this general dynamic transition theory is the reduction of 2.1 to the center manifold function. In fact, by this reduction, we know that the type of transitions for 2.1 at 0, 0 is completely dictated by its reduction equation near = 0, dx dt = J x + g x, for x R m, 2.6 where g x, = g 1 x,,...,g m x,, and m g j x, = G x i e i + x,,,e, j 1 j m. 2.7 i=1 Here e j and e j 1 j m are the eigenvectors of L and L, respectively, corresponding to the eigenvalues j as in 2.4, J is the m m order Jordan matrix corresponding to the eigenvalues given by 2.4, and x, is the center manifold function of 2.1 near 0. Second, let 2.1 be a gradient-type equation. Under conditions 2.4 and 2.5, in a neighborhood U X of u=0, the center manifold M c in U at = 0 consists of three subsets,

073304-5 Phase transitions for liquid helium J. Math. Phys. 49, 073304 2008 M c = W u + W s + D, where W s is the stable set, W u is the unstable set, and D is the hyperbolic set of 2.6. Then we have the following theorem. Theorem 2: Let (2.1) be a gradient-type equation, and conditions (2.4) and (2.5) hold true. If u=0 is an isolated singular point of (2.1) at = 0, then we have the following assertions. 1 2 The transition of (2.1) at u, = 0, 0 is continuous if and only if u=0 is locally asymptotically stable at = 0, i.e., the center manifold is stable: M c =W s. Moreover, (2.1) bifurcates from 0, 0 to minimal attractors consisting of singular points of (2.1). If the stable set W s of (2.1) has no interior points in M c, i.e., M c =W u +D, then the transition is jump. Third, we also have a dynamic transition theorem of 2.1 from a simple critical eigenvalue, which has been used in analyzing PVT systems and the ferromagnetic systems. 9,10 We refer the interested readers to these references for details for this theorem. More general transition theorems are derived and used in many problems in sciences; see Ma and Wang. 7,11 B. A unified time-dependent Ginzburg Landau models for equilibrium phase transitions In this section, we recall a unified time-dependent Ginzburg Landau theory for modeling equilibrium phase transitions in statistical physics, derived and used recently by the authors; see, e.g., Refs. 9 11. Consider a thermal system with a control parameter. The classical Le Châtelier principle says that if an external stress is applied to a system at equilibrium, the system will adjust itself to minimize that stress. It allows us to predict the direction a reaction 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 the Le Châtelier principle, for a system with thermodynamic potential H u,, the governing equations are essentially determined by the functional H u,. When the order parameters u 1,...,u m are nonconserved variables, i.e., the integers, u i x,t dx = a i t const, then the time-dependent equations are given by u i t = i H u, + i u, u, for 1 i m, u i u =0 or u =0, 2.8 n where / u i are the variational derivative, i 0, and i satisfy i H u, dx =0. i u i 2.9 Condition 2.9 is required by the Le Châtelier principle. We remark here that following the Le Châtelier principle, one should have an inequality constraint. However physical systems often obey most simplified rules, as many existing models for specific problems are consistent with the equality constraint here. This remark applies to the constraint 2.15 below as well. Two remarks for this model are now in order.

073304-6 T. Ma and S. Wang J. Math. Phys. 49, 073304 2008 First, in practical applications of the above unified model, the terms i can be determined by physical laws and 2.9. For example, the time-dependent Ginzburg Landau equations for superconductivity with an applied field can be easily derived using 2.8 with the Ginzburg Landau free energy, 9 and the corresponding functions 1 and 2 are uniquely determined by 1 the Maxwell equation and 2 condition 2.9. As we know, the time-dependent Ginzburg Landau model for superconductivity was first proposed Schmid 14 and subsequently validated by Gor kov and Eliashberg 2 in the context of the microscopic Bardeen Cooper Schrieffer theory of superconductivity. Because of gauge invariance, the generalization is nontrivial. In addition to the order parameter and the vector potential, a third variable is needed to complete the description of the physical state of the system in a manner consistent with the gauge invariance. As we can see from Ref. 9, our derivation using the unified model is much simpler and transparent. Second, as we shall see near the end of this section, the steady state solutions of 2.8 with 2.9 are critical points of the energy functional H, and this is in agreement with the classical steady state approaches. When the order parameters are the number density and the system has no material exchange with the external, then u j 1 j m are conserved, i.e., This conservation law requires a continuity equation u j x,t dx = const. 2.10 u j t = J j u,, where J j u, is the flux of component u j. In addition, J j satisfy J j = k j j i, i j where l is the chemical potential of component u l, j i = H u, j u, u,, i j u j 2.11 2.12 2.13 and j u, is a function depending on the other components u i i j. Thus, from 2.11 2.13 we obtain the dynamical equations as follows: where j 0 are constants and j satisfy u j t = j u j H u, j u, u, for 1 j m, u =0, u =0, 2.14 n n j H u, dx =0. j u j 2.15 When m=1, i.e., the system is a binary system, consisting of two components A and B, then the term j =0. The above model covers the classical Cahn Hilliard model. It is worth mentioning that for multicomponent systems, these j play an important rule in deriving time-dependent models. We shall address this issue in a forthcoming paper. If the order parameters u 1,...,u k are coupled to the conserved variables u k+1,...,u m, then the dynamical equations are

073304-7 Phase transitions for liquid helium J. Math. Phys. 49, 073304 2008 u i t = i H u, + i u, u, for 1 i k, u i u j t = j u j H u, j u, u, for k +1 j m, u i =0 or u i =0 for 1 i k, n u j =0, u j = 0 for k +1 j m. 2.16 n n Here i and j satisfy 2.9 and 2.15, respectively. The model 2.16 we derive here gives a general form of the governing equations to thermodynamic phase transitions and will play a crucial role in studying the dynamics of equilibrium phase transition in statistic physics. Finally we remark that the steady state solutions of 2.16 are simply the critical points of the energy functional H, satisfying the following steady state equations: i u i H u 0, =0, 1 i k, j u j H u 0, =0, k +1 j m, u i =0 or u i =0, 1 i k, n u j =0, u j =0, k +1 j m. 2.17 n n Hence the unified model 2.16 is consistent with classical models in the steady state case. To derive the above steady state equations, we only have to show that a steady state solution u 0 of 2.16 satisfies i u 0, u 0, =0, 1 i k, To see this, let u 0 satisfy j u 0, u 0, =0, k +1 j m. 2.18 i u i H u 0, i u 0, u 0, =0, 1 i k, j H u 0, j u 0, u 0, =0, k +1 j m. 2.19 u j Multiplying i u 0, u 0, and j u 0, u 0, on the first and the second equations of 2.19, respectively, and integrating them, then we infer from 2.9 and 2.15 that

073304-8 T. Ma and S. Wang J. Math. Phys. 49, 073304 2008 FIG. 1. Classical PT phase diagram of 4 He. 2 i u 0, u 0, dx =0, i j u 0, u 0, 2 dx =0, j which imply that 2.18 holds true. III. DYNAMIC MODEL FOR LIQUID 4 He 4 He was first liquidized at T=4.215 K and p=1 10 5 Pa by Onnes in 1908. In 1938, Kapitza found that when the temperature T decreases below T C =2.17 K, liquid 4 He will transit from normal liquid state to superfluid state, in which the fluid possesses zero viscosity, i.e., the viscous coefficient =0. The liquids with =0 are called the superfluids, and the flow without drag is called the superfluidity. The superfluid transition is called the -phase transition, and its phase diagram is illustrated by Fig. 1. A. Ginzburg Landau free energy The order parameter describing superfluidity is characterized by a nonvanishing complex valued function : C as in the superconductivity, originating from the quantum Bose Einstein condensation. In the two-fluid hydrodynamic theory of superfluidity, the density of 4 He is given by = s + n, 3.1 where s is the superfluid density and n is the normal fluid density. The square 2 is proportional to s, and without loss of generality, we take as 2 = s. 3.2 Based on the quantum mechanics, ih represents the momentum associated with the Bose Einstein condensation. Hence, the free energy density contains the term 1 2m ih 2 = h2 2m 2, where h is the Planck constant and m is the mass of atom 4 He. Meanwhile the superfluid state does not obey the classical thermodynamic laws, which the normal liquid state obeys. Therefore, in the free energy density, satisfies the Ginzburg Landau expansion

073304-9 Phase transitions for liquid helium J. Math. Phys. 49, 073304 2008 k 1 h 2 2m 2 + 1 2 2 + 2 4 4, and n has the expansion as in the free energy for PVT systems 9 with a specific treatment on the pv term in the Gibbs free energy, where p is the pressure and V is the volume. For simplicity we ignore the entropy and consider the coupling action of and n, i.e., add the term 1 2 n 2 in the free energy density. Thus the Ginzburg Landau free energy for liquid 4 He near the superfluid transition is given by G, n = k 1h 2 2m 2 + 1 2 2 + 2 4 4 + 3 2 n 2 + k 2 2 n 2 + 1 2 2 n + 2 3 3 n + 3 4 4 n p n + 0 2 n 2 dx. 3.3 Here we retain the terms in the mean-field theory expansion of the free energy such that the phase transition obeys the classical phase diagram and the stability of the transition solutions. In other words, the derivation of the Ginzburg Landau free energy is based on 1 the mean-field theory, 2 the classical phase diagram for helium-4, and 3 the theoretical results and physical predictions derived in this article for the corresponding dynamic model. B. Dynamic model governing the superfluidity By 2.8, we derive from 3.3 the following time-dependent Ginzburg Landau equations governing the superfluidity of liquid 4 He: t = k 1h 2 m 1 2 2 3 n, n t = k 2 n 1 p 0 n 2 n 2 3 n 3 3 2 2 + p. 3.4 It is known that the following problem has a solution n 0 H 2 H 0 1 standing for the density of liquid He I for any p L 2 : k 2 n 0 + 1 p 0 n 0 + 2 n 0 2 + 3 n 0 3 = p, 0 n =0. 3.5 n Here the coefficient 3 0 as will be explained below. In the case where the pressure p is a constant, the steady state solution 0 n is treated as a constant as well, solving the following algebraic equation: 1 p 0 n 0 + 2 n 0 2 + 3 n 0 3 = p. To derive the nondimensional form of 3.4, let 3.6 x,t = lx, t,, n = 0, 0 n + n 0, = ml2 h 2 k 1, a 1 = 3 0, = k 2 l 2, a 2 = 2 0 2,

073304-10 T. Ma and S. Wang J. Math. Phys. 49, 073304 2008 b 1 = 3 0 2 2 0, b 2 = 0 3 n 0 3 + 2, b 3 = 0 2 3, 1 = 1 + 3 n 0, 2 = 3 n 0 2 3 +2 n 0 2 + 0 p 1, where n 0 is the solution of 3.5. Thus, suppressing the primes, Eq. 3.4 is rewritten as t = + 1 a 1 n a 2 2, n = n + 2 n b 1 2 b 2 2 t n b 3 3 n. 3.7 The boundary conditions associated with 3.7 are n =0, n n =0 on. 3.8 When the pressure p is independent of x, then the problem 3.7 and 3.8 can be approximately replaced by the following systems of ordinary differential equations for superfluid transitions: d dt = 1 a 1 n a 2 2, d n = 2 n b 1 2 b 2 2 dt n b 3 3 n. 3.9 In this article, we shall focus on this model for the homogeneous case where the pressure p is treated as a constant. There are two main reasons behind the study of this simplified model. First, the case represents a physically important situation in phenomenological studies of phase transitions of liquid helium. Second, with this simplified case, we can see clearly the mechanism of the related phase transitions, leading to specific physical predictions. Of course, the partial differential equation model 3.7 will be needed in the more general case and will be studied in a forthcoming paper. By multiplying to both sides of the first equation of 3.9 and by 3.2, Eq. 3.9 is reduced to d s dt = 1 s a 1 n s a 2 s 2, d n dt = 2 n b 1 s b 2 n 2 b 3 n 3, for s 0 and x 0 0. s 0, n 0 = x 0,y 0, 3.10

073304-11 Phase transitions for liquid helium J. Math. Phys. 49, 073304 2008 FIG. 2. a The curve of 1 =0, b the curve of 1 =0, c the curve of 1 =0, and d the curve of 2 =0. We need to explain the physical properties of the coefficients in 3.4 and 3.7. It is known that the coefficients i 1 i 3 and j 0 j 3 depend continuously on the temperature T and the pressure p, i = i T,p, j = j T,p, 1 i 3, 0 j 3. From the both mathematical and physical points of view, the following conditions are required: In addition, by the Landau mean-field theory we have 2 0, 3 0, 3 0, T,. 3.11 1 0 if T,p A, 1, 3.12 0 if T,p A 2 1 0 if T,p B, 1, 3.13 0 if T,p B 2 where A i, B i i=1,2 are connected open sets such that Ā 1 +Ā 2 =B 1+B 2=R + 2 and Ā 1 Ā 2,B 1 B 2 are two simple curves in R + 2 ; see Figs. 2 a and 2 b. In particular, in the PT-plane, 1 T 0, 1 p 0, 1 T 0, 1 0. p 3.14 By 3.11 the following nondimensional parameters are positive: a 1 0, a 2 0, b 1 0, b 3 0. 3.15 By 3.12 3.14, the following two critical parameter equations: 1 = 1 T,p =0,

073304-12 T. Ma and S. Wang J. Math. Phys. 49, 073304 2008 FIG. 3. ACF is the curve of 2 =0 and BCD is the curve of 1 =0. 2 = 2 T,p =0, 3.16 give two simple curves l 1 and l 2, respectively, in the PT-plane R + 2 ; see Figs. 2 c and 2 d. The parameters 2 and b 2 depend on the physical properties of the He atom and satisfy the following relations: b 2 T,p = 0 2 +3 n 0 3 0 if and only if sol l at 2 T,p =0, b 2 T,p = 0 2 +3 n 0 3 0 if and only if sol l at 2 T,p =0, 3.17 where sol and l are the densities of solid and liquid and n 0 is the solution of 3.5 representing the liquid density. These relations in 3.17 can be deduced by the dynamic transition theorem of 2.1 from a simple critical eigenvalue, Theorem A.2 in Ma and Wang. 10 IV. DYNAMIC PHASE TRANSITION FOR LIQUID 4 He In order to illustrate the main ideas, we discuss only the case where the pressure p is independent of x, i.e., we only consider Eqs. 3.9 and 3.10 ; the general case can be studied in the same fashion and will be reported elsewhere. A. PT -phase diagram Based on physical experiments together with 3.12 3.14, the curves of 1 T, p =0 and 1 T, p =0 in the PT-plane are given by Figs. 2 a and 2 b, respectively. By the formulas 1 T,p = 1 T,p + 3 n 0, 2 T,p = 1 T,p 0 p 2 n 0 2 3 n 02 3, together with 3.11 and 3.17, the curves l 1 and l 2 given by 3.16 in the PT-plane are illustrated in Figs. 2 c and 2 d. Let D 1 = T,p R + 2 1 T,p 0, D 2 = T,p R + 2 1 T,p 0, E 1 = T,p R + 2 2 T,p 0, E 2 = T,p R + 2 2 T,p 0. Let the curve of 1 T, p =0 intersect with the curve of 2 T, p =0 at a point C; see Fig. 3.

073304-13 Phase transitions for liquid helium J. Math. Phys. 49, 073304 2008 When 1 crosses curve segment CB to enter into D 1 from D 2 see Fig. 3, we have 2 0 and 1 satisfies 1 T,p 0 if T,p D 2, =0 if T,p CB, 0 if T,p D 1. In this case, by Theorem 2.1, Eq. 3.10 have a phase change that describes the transition from liquid He I to liquid He II, and the second equation of 3.10 can be equivalently rewritten as d n = 2 2b 2 dt n 3b 3 n 2 n b2 +3b 3 n n 2 b 3 n 3, 4.1 where n= n n, n satisfies the equation 2 n b 2 n 2 b 3 n 3 = b 1 s, and s 0 is the transition solution of 3.10. Since n is the density deviation of normal liquid, by 3.1 we have n = n 0 s. On the other hand, n 0 represents the density of liquid He I. Thus, n s 0. From the PT-phase diagram of 4 He Fig. 1, we see that sol l for 4 He near T=0; therefore by 3.17, b 2 0. Hence we derive that 2 2b 2 n 3b 3 n 2 0, 4.2 for T, p in the region D 1 \H, as shown in Fig. 3. It follows from 4.1 and 4.2 that when T, p is in the region D 1 \H, liquid 4 He is in the superfluid state. When 2 crosses curve segment CA, as shown in Fig. 3, to enter into E 1 from E 2, then 1 0 and 2 T,p 0 if T,p E 2, =0 if T,p CA, 0 if T,p E 1. In this case, Eq. 3.10 characterizes the liquid-solid phase transition, and the first equation of 3.10 is equivalently expressed as d s = 1 a 1 n s a 2 2 dt s, 4.3 where n 0 is the transition state of solid 4 He. By 3.15 we have 1 T,p a 1 n 0, 4.4 for any T, p in the region E 1 \H, as shown in Fig. 3. From 4.3 and 4.4 we can derive the conclusion that as T, p in E 1 \H, s =0 is stable, i.e., 4 He is in the solid state. However, the shadowed region H in Fig. 3 is a metastable region for the solid and liquid He II states, where any of these two phases may appear depending on the random fluctuations. Thus, from the discussion above, we can derive the theoretical PT-phase diagram given by Fig. 2, based on Eq. 3.10. In comparison with the experimental PT-phase diagram Fig. 1, a slight difference in Fig. 4 is that there exists a metastable region H, where the solid phase and the He II phase are possible to occur. This metastable region in Fig. 4 corresponding to the coexistence curve CE in Fig. 1 is a theoretical prediction that needs to be verified by experiments.

073304-14 T. Ma and S. Wang J. Math. Phys. 49, 073304 2008 FIG. 4. Theoretical PT-phase diagram: H is a metastable region where both solid and liquid He II states appear randomly depending on fluctuations. Point M is a point where the transitions between the superfluid state liquid He II and the normal fluid state liquid He I change from type II to type I; namely, the transition crossing CM is type II and the transition crossing MB is type I. B. States in the metastable region We consider the dynamical properties of transitions for 3.10 in the metastable region. It is clear that at point C= T C, p C, and the metastable region H satisfies that 1 T C,p C =0, 2 T C,p C =0, 4.5 H = T,p R + 2 1 T,p 0, 2 T,p 0. To study the structure of flows of 3.10 for T, p H, it is necessary to consider Eq. 3.10 at the point C= T C, p C and by 4.5, which are given by d s dt = a 1 n s a 2 s 2, d n = b 1 s b 2 2 dt n b 3 3 n. Due to 3.17 and sol l, we have 4.6 b 2 0 for T,p H. 4.7 Under condition 4.7, Eq. 4.6 has the following two steady state solutions: Z 1 = s, n = 0, b 2 /b 3, Z 2 = s, n = a 1 a 2,, = b 2 2b 3 1+4a1 b 1 b 3 /a 2 b 2 2 1. By direct computation, we can prove that the eigenvalues of the Jacobian matrices of 4.6 at Z 1 and Z 2 are negative. Hence, Z 1 and Z 2 are stable equilibrium points of 4.6. Physically, Z 1 stands for solid state, and Z 2 for superfluid state. The topological structure of 4.6 is schematically

073304-15 Phase transitions for liquid helium J. Math. Phys. 49, 073304 2008 FIG. 5. The flow structure in the metastable regions. illustrated in Fig. 5 a ; the two regions R 1 and R 2 divided by curve AO in Fig. 5 b are the basins of attraction of Z 1 and Z 2, respectively. We note that in H, 1 and 2 are small, i.e., 0 1 T,p, 2 T,p 1 for T,p H, and 3.10 can be considered as a perturbed system of 4.6. Thus, for T, p H the system 3.10 has four steady state solutions Z i=z T, p 1 i 4 such that lim Z 1 T,p,Z 2 T,p,Z 3 T,p,Z 4 T,p = Z 1,Z 2,0,0, T,p T C,p C and Z 1 and Z 2 are stable, representing the solid state and liquid He II state, respectively; Z 3 and Z 4 are two saddle points. The topological structure of 3.10 for T, p H is schematically shown in Fig. 5 c, and the basins of attraction of Z 1 and Z 2 are R 1 and R 2 as illustrated by Fig. 5 d. In summary, with the above analysis and the dynamic transition theory, we arrive at the following theorem. Theorem 4.1: There exist four regions Ẽ 1, D 1,D 2, and H in the PT -plane (see Fig. 4), which are defined by Ẽ 1 = T,p R + 2 1 T,p 0, 2 T,p 0, D 1 = T,p R + 2 1 T,p 0, 2 T,p 0,

073304-16 T. Ma and S. Wang J. Math. Phys. 49, 073304 2008 D 2 = T,p R + 2 1 T,p 0, 2 T,p 0, such that the following conclusions hold true. H = T,p R + 2 1 T,p 0, 2 T,p 0, 1 2 3 4 If T, p Ẽ 1, the phase of 4 He is in solid state. If T, p D 1, the phase is in superfluid state. If T, p D 2, the phase is in the normal fluid state. If T, p H, there are two regions R 1 and R 2 in the state space s, n such that under a fluctuation that is described by the initial value x 0,y 0 in (3.10), if x 0,y 0 R 1 then the phase is in solid state, and if x 0,y 0 R 2 then it is in superfluid state. In general, the observed superfluid transitions of 4 He are of the second order, i.e., type I continuous with the dynamic classification. However, from 3.10 we can prove the following theorem, which shows that for a higher pressure the superfluid transitions may be of zeroth order, i.e., type-ii jump transition with the dynamic classification scheme. Theorem 4.2: Let T 0, p 0 satisfy that 1 T 0, 0 =0, 2 T 0, p 0 0. Then, (3.10) have a superfluid transition at T 0, p 0 from D 2 to D 1. In particular, the following assertions hold true. 1 Let A = a 1b 1 a 2 at T,p = T 0,p 0. 4.8 2 Then if A 0 the superfluid transition is type I, and if A 0 the superfluid transition is type II. 2 The equation A = a 1b 1 a 2 =0 2 determines a point M on the CB coexistence curve in Fig. 4, where the transition changes type from type II to type I. Proof: Step 1. By the assumption, it is clear that 1 T,p 0 if T,p D 2, =0 if T,p = T 0,p 0, 0 if T,p D 1, where D 1,D 2 are as in Fig. 2 c. Hence, by Theorem 2.1, 3.10 has a transition at T 0, p 0.By 2 T 0, p 0 0, the first eigenvalue of 3.10 is simple, and its eigenvector is given by e= e s,e n = 1,0. The reduced equation of 3.10 on the center manifold reads dx dt = 1x a 1 xh x a 2 x 2 for x 0, where h x is the center manifold function. By 3.10, h can be expressed as h x = b 1 x + o x 2 for x 0. 2 Thus the reduced equation of 3.10 is given by

073304-17 Phase transitions for liquid helium J. Math. Phys. 49, 073304 2008 dx dt = 1x + Ax 2 + o x 2 for x 0, 4.9 where A is as in 4.8. The theorem follows from 4.9 and Theorem 2.2. Step 2. By the nondimensional form, we see that a 1 b 1 = 1 2 3 2 0 2 2, where 3 is the coupled coefficient of n and s. Physically, 3 is small in comparison with 2, namely, On the other hand, we know that 0 a 1 b 1 a 2. 2 T 0,p 0 0 as T 0,p 0 T C,p C, where C= T C, p C is as in 4.5. Therefore we deduce that there exists a pressure p b p p C such that 2 a 1 b 1 A = 2 T 0,p 0 a 0 0 if 0 p. 0 p if p p 0 p C Thus, when the transition pressure p 0 is below some value p 0 p, the superfluid transition is of second order, i.e., is continuous with the dynamic classification scheme, and when p 0 p the superfluid transition is of first order, i.e., jumps in the dynamic classification scheme. The proof is complete. V. PHYSICAL CONCLUSIONS AND PREDICTIONS As we know, the classical phase transition is illustrated in Fig. 1. In this phase diagram, the coexistence curve CE separates the solid state and the superfluid state liquid He II, and curve CB is the coexistence curve between the superfluid state liquid He II and the normal fluid state liquid He I ; the critical point is the triple point. It is considered by the classical theory that the transition crossing CB is second order with the Ehrenfest or modern classification scheme type I with the dynamic classification scheme. However, in the phase transition diagram Fig. 4 derived based on Theorems 4.1 and 4.2, there is a metastable region H where both solid and liquid He II states are metastable and appear randomly depending on fluctuations. Point M is a switch point where the transitions between the superfluid state liquid He II and the normal fluid state liquid He I change from first order type II with the dynamic classification scheme to second order type I ; namely, the transition crossing CM is second order type II and the transition crossing MB is first order type I. In summary, with the results obtained in this article, we are able to predict the existence of the metastable region H and the existence of the switch point M. It is hoped that these predictions can be verified by experiments. ACKNOWLEDGMENTS The authors are grateful for the insightful comments made by an anonymous referee. This work was supported in part by the Office of Naval Research and by the National Science Foundation. 1 V. L. Ginzburg, On superconductivity and superfluidity what Ihave and have not managed to do, as well as on the physical minimum at the beginning of the xxi century, Phys. Usp. 47, 1155 2004. 2 L. Gor kov and G. M. Eliashberg, Generalization of the Ginzburg-Landau equations for non-stationary problems in the case of alloys with paramagnetic impurities, Sov. Phys. JETP 27, 328 1968. 3 C.-H. Hsia, T. Ma, and S. Wang, Stratified rotating Boussinesq equations in geophysical fluid dynamics: Dynamic bifurcation and periodic solutions, J. Math. Phys. 48, 065602 2007.

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