Mean field theories of phase transitions

In this paper we give an overview of the mean-field theory of phase transitions in coupled rotors with particular attention to the issues of reentiance, other quantum anomalies, and meta-stability. We comparatively analyze coupled planar rotors (two-dimensional model) and coupled linear rotors (three-dimensional). We show that the dipolar potential does not exhibit the reentrance anomaly, whereas the quadmpolar one does. The phase transition turns out to be second order in all cases except for the linear rotors in a quadmpolar potential where it is first order. We also investigate the effects of the crystal field in the case of the linear rotor model with quadmpolar potentials the crystal field causes the appearance of critical points which separate lines of the phase diagram where the transition is first order from regions where there is no... 

Gee RFI, Lacevic N, Fried LE (2005) Atomistic simulations of spinodal phase separation preceding poiymer crystaiiization. Nat Mater 5(l) 39-43 Gupta AM, Edwards SF (1993) Mean-field theory of phase transitions in liquid-crystalline polymers. J Chem Phys 98(2) 1588-1596... 

Gupta AM, Edwards SF. Mean-field theory of phase transitions in hquid-crystalhne polymers. J Chem Phys 1993 98 1588-1596. 

Like there always exists a vapor under the water, there are excitations on the ground of any condensate. They appear due to quantum and thermal fluctuations. In classical systems and also at not too small temperatures in quantum systems, quantum fluctuations are suppressed compared to thermal fluctuations. Excitations are produced and dissolved with the time passage, although the mean number of them is fixed at given temperature. Pairing fluctuations are associated with formation and breaking of excitations of a particular type, Cooper pairs out of the condensate. Fluctuation theory of phase transitions is a well developed field. In particular, ten thousands of papers in condensed matter physics are devoted to the study of pairing fluctuations. At this instant we refer to an excellent review of Larkin and Varlamov [15]. 

A major ingredient for an RG treatment is a simple and transparent characterization of the molecular forces driving phase separation. This situation calls for mean-field theories of the ionic phase transition. The past decade has indeed seen the development of several approximate mean-field theories that seem to provide a reasonable, albeit not quantitative, picture of the properties of the RPM. Thus, the major forces driving phase separation seem now to be identified. Moreover, the development of a proper description of fluctuations by GDH theory has gone some way to establish a suitable starting point for RG analysis. Needless to say, these developments are also of prime importance in the more general context of electrolyte theory. 

Obviously the enthalpy expression x a b in Eq. (3) neglects any correlation effects in the occupancy of lattice sites the probability that on neighboring lattice sites A-B-pairs occur is simply taken as the product < >A< >B of the respective volume fractions. This is a special case of a mean-field approximation (MFA), which is known to yield a critical behavior described by the Landau theory of phase transitions [100], which differs from the correct critical behavior expected [73,74] in the universal regime close to the critical point Xcrm 4>cri in Fig. 2. We shall discuss these various types of critical behavior in Sect. 2.2. 

It turns out [125] that the wetting transition separating the nonwet state of the surface is always first order when g > 0, while it may be second order for g < 0. Figure 55 shows the phase diagram [12 resulting from this simple mean field theory of wetting in polymer mixtures. A second order wetting transition occurs when the solution of Eq. (213) reaches the value The surface... 

Essential parts of the BCS theory can be taken over in the mean-field theory of the Peierls transition if one replaces the Debye energy tkoo in the superconductor by the Fermi energy Ep in the metal. Since Eplhrop, 10 -100, the Peierls phase transition temperatures are considerably higher than the critical temperatures of BCS superconductors, ii) The frequency of the phonons which are responsible for the Peierls transition has, for T > the temperature dependence... 

Theoretical treatments of liquid crystals such as nematics have proved a great challenge since the early models by Onsager and the influential theory of Maier and Saupe [34] mentioned before. Many people have worked on the problems involved and on the development of the continuum theory, the statistical mechanical approaches of the mean field theory and the role of repulsive, as well as attractive forces. The contributions of many theoreticians, physical scientists, and mathematicians over the years has been great - notably of de Gennes (for example, the Landau-de Gennes theory of phase transitions), McMillan (the nematic-smectic A transition), Leslie (viscosity coefficients, flow, and elasticity). Cotter (hard rod models), Luckhurst (extensions of the Maier-Saupe theory and the role of flexibility in real molecules), and Chandrasekhar, Madhusudana, and Shashidhar (pre-transitional effects and near-neighbor correlations), to mention but some. The devel-... 

As stated in Sec. 3, the retention of only the first term in Eq. [14] leads to the mean field theory of Maier and Saupe and the equivalent theory of the previous chapter. This version of the theory has been shown to provide a good qualitative picture of the nematic phase and its transition to the isotropic liquid. What, then, is it about the experimental facts that indicate the necessity of higher order terms in Fi ... 

The question of whether the normally first-order smectic-A to nematic phase transition can be second order in some materials is somewhat controversial. All of the published mean-field theories " of the smectic-A phase do exhibit second-order phase changes for certain values of the potential parameters. In both McMillan s theory and that of Lee et al, the second-order transition is predicted to occur at that end of homologous series having short chain lengths. More specifically, these models predict the second-order changes to occur when the ratio of transition temperatures Tan/Tni (or Tac/Tci) is at or below about 0.88 (see Fig. 5). 

Here we review the properties of the model in the mean field theory [328] of the system with the quantum APR Hamiltonian (41). This consists of considering a single quantum rotator in the mean field of its six nearest neighbors and finding a self-consistent condition for the order parameter. Solving the latter condition, the phase boundary and also the order of the transition can be obtained. The mean-field approximation is similar in spirit to that used in Refs. 340,341 for the case of 3D rotators. 

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