Superfluid transitions critical points

Figure 7.4 Phase diagram of 4He, showing the solid, gas, and two liquid phases (He-I, superfluid He-II), the A-line (dashed) of liquid-liquid transitions (upper terminus 1.76K, 29.8 atm lower terminus 2.17K, 0.0497 atm), and the gas-liquid critical point (circle-x 5.20K, 2.264 atm).

We note that even short-range interactions may, however, allow a mean-field scenario, if the system has a tricritical point, where three phases are in equilibrium. A well-known example is the 3He-4He system, where a line of critical points of the fluid-superfluid transition meets the coexistence curve of the 3He-4He liquid-liquid transition at its critical point [33]. In D = 3, tricriticality implies that mean-field theory is exact [11], independently from the range of interactions. Such a mechanism is quite natural in ternary systems. For one or two components it would require a further line of hidden phase transitions that meets the coexistence curve at or near its critical point. 

Helium-4 Normal-Superfluid Transition Liquid helium has some unique and interesting properties, including a transition into a phase described as a superfluid. Unlike most materials where the isotopic nature of the atoms has little influence on the phase behavior, 4He and 3He have a very different phase behavior at low temperatures, and so we will consider them separately Figure 13.11 shows the phase diagram for 4He at low temperatures. The normal liquid phase of 4He is called liquid I. Line ab is the vapor pressure line along which (gas + liquid I) equilibrium is maintained, and the (liquid + gas) phase transition is first order. Point a is the critical point of 4He at T= 5.20 K and p — 0.229 MPa. At this point, the (liquid + gas) transition has become continuous. Line be represents the transition between normal liquid (liquid I) and a superfluid phase referred to as liquid II. Along this line the transition... 

Not only do the thermodynamic properties follow similar power laws near the critical temperatures, but the exponents measured for a given property, such as heat capacity or the order parameter, are found to be the same within experimental error in a wide variety of substances. This can be seen in Table 13.3. It has been shown that the same set of exponents (a, (3, 7, v, etc.) are obtained for phase transitions that have the same spatial (d) and order parameter (n) dimensionalities. For example, (order + disorder) transitions, magnetic transitions with a single axis about which the magnetization orients, and the (liquid + gas) critical point have d= 3 and n — 1, and all have the same values for the critical exponents. Superconductors and the superfluid transition in 4He have d= 3 and n = 2, and they show different values for the set of exponents. Phase transitions are said to belong to different universality classes when their critical exponents belong to different sets. 

The relation A7 oc Rq obtained from the Ginzburg-Pitaevskii-Sobyanin theory for a finite system is related to the theory of second-order phase transitions with the experimental critical parameter, v = 0.67, for the superfluid fraction and for the correlation length scaling near the critical point of infinite systems [155, 193-197, 199]. This theory implies that the intensive properties of a system of size L(= Rq) depend on the ratio L/ T) Lf, where (T) = th bulk... 

Noise-induced transitions have been studied theoretically in quite a few physical and chemical systems, namely the optical bistability [12,13,5], the Freedricksz transition in nematics [14,15,16,5], the superfluid turbulence in helium II [17], the dye laser [18,19], in photochemical reactions [20], the van der Pol-Duffing oscillator [21] and other nonlinear oscillators [22]. Here I will present a very simple model which exhibits a noise-induced critical point. The so-called genetic model was first discussed in [4]. I will not describe its application to population genetics in this paper, see [5] for this aspect, but use a chemical model reaction scheme ... 

The understanding of continuous phase transitions and critical phenomena has been one of the important breakthrough in condensed matter physics in the early seventies. The concepts of scaling behavior and universality introduced by Kadanoff and Wi-dom and the calculation of non-gaussian exponents by Wilson and Fisher are undeniably brilliant successes of statistical physics in the study of low temperature phase transitions (normal to superconductor, normal to superfluid helium) and liquid-gas critical points. 

The number of states of matter is not well-defined compare a sharp transition such as melting with the gradual transition of liquid/vapour above the critical point. Sometimes superconductors and superfluids are considered different states of matter from the ordinary states, sometimes not. 

First indirect experimental observations of the critical Casimir force were made by Chan and Garcia [152]. They measured the thickness of He films on a copper substrate and detected a thinning of the films close to the critical point of transition to superfluidity, indicating an attractive critical Casimir force. For a He/ He mixture close to the tricritical point, the same authors found a repulsive critical Casimir force, which caused film thickening on the copper substrate [153] (for a later, refined theoretical analysis, see Ref [154]). The tricritical point is the point in the phase diagram where the superfluidity transition line terminates at the top coexistence line of He/He. 

Helium Purification and Liquefaction. HeHum, which is the lowest-boiling gas, has only 1 degree K difference between its normal boiling point (4.2 K) and its critical temperature (5.2 K), and has no classical triple point (26,27). It exhibits a phase transition at its lambda line (miming from 2.18 K at 5.03 kPa (0.73 psia) to 1.76 K at 3.01 MPa (437 psia)) below which it exhibits superfluid properties (27). 

It is demonstrated in [8] that the transport coefficients (thermal diffusivity, diffusion coefficient, fluidity, etc.) considered in the Fourier approximation are proportional to the stability coefficients. This makes it possible to determine whether we are dealing with a critical transition or a limited phase transition of the second kind and, in the latter case, which of the parameters are characteristic. In critical transitions, the transport coefficients decrease strongly, whereas in limited transitions of the second kind they tend to infinite values. This criterion shows that phase transitions of the second kind which occur in binary alloys, polymers, ferromagnets, ferroelectrics, liquid crystals, etc., are essentially transcritical transitions, which are sometimes close to the critical conditions because the values of the transport coefficients decrease strongly at the transition point. The occurrence of superfluidity in He H demonstrates that, even in the absence of a coordinate or a derivative which tends to zero, this substance is a superphase in the kinetic sense. 

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