Landau theory

A Landau theory for blue phase was proposed by Brazovskii, Dmitriev, Homreich, and Shtrik-man [7-10]. In this theory, the free energy of the blue phase is expressed in terms of a tensor order parameter which is expanded in Fourier components. The free energy is then minimized with respect to the order parameter with the wave vector in various cubic symmetries. In a narrow temperature region below the isotropic transition temperature, the stmctures with certain cubic symmetries have free energy lower than both the isotroic and cholesteric phases. 

De Gennes used Landau theory to describe the isotropic-nematic transition. In his theory, he used a scalar order parameter S defined by 

This tensor order parameter is traceless and symmetric and vanishes in the isotropic phase. The anisotropic physical properties of the liquid crystal are closely related to the tensor order parameter. For example, the dielectric tensor of the hquid crystal is 

For a cholesteric hquid crystal with the chirality q and helical axis along the z direction, the tensor order parameter is 

Because the optical properties of blue phases are of great importance, we choose the traceless 

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Gompper G and Zsohooke S 1991 Elastio properties of interfaoes in a Ginzburg-Landau theory of swollen mioelles, droplet orystals and lamellar phases Euro. Phys. Lett. 16 731... 

Johnson D, Aiiender D, Dehoff D, Maze C, Oppenheim E and Reynoids R 1977 Nematio-smeotio A-smeotio C poiyoritioai point Experimentai evidenoe and a Landau theory Phys.Revs B 16 470-5... 

The first step in studying phenomenological theories (Ginzburg-Landau theories and membrane theories) has usually been to minimize the free energy functional of the model. Fluctuations are then included at a later stage, e.g., using Monte Carlo simulations. The latter will be discussed in Sec. V and Chapter 14. 

Ginzburg-Landau theories of amphiphiles have been reviewed at various places [1,25], among others, in Chapter 14 of this book. Hence we shall be brief in this subsection. 

The basic idea of a Ginzburg-Landau theory is to describe the system by a set of spatially varying order parameter fields, typically combinations of densities. One famous example is the one-order-parameter model of Gompper and Schick [173], which uses as the only variable 0, the density difference between oil and water, distributed according to the free energy functional... 

Random interface models for ternary systems share the feature with the Widom model and the one-order-parameter Ginzburg-Landau theory (19) that the density of amphiphiles is not allowed to fluctuate independently, but is entirely determined by the distribution of oil and water. However, in contrast to the Ginzburg-Landau approach, they concentrate on the amphiphilic sheets. Self-assembly of amphiphiles into monolayers of given optimal density is premised, and the free energy of the system is reduced to effective free energies of its internal interfaces. In the same spirit, random interface models for binary systems postulate self-assembly into bilayers and intro-... 

For a recent review on Ginzburg-Landau theories, see G. Gompper. Ber. Bunsenges Phys Chemie i00 264-271, 1996. 

G. Gompper, S. Zschocke. Elastic properties of interface in a Ginzburg-Landau theory of swollen micells, droplet crystals and lamellar phases. Euro-phys Lett 16 13 -136, 1991. 

G. Gompper, M. Kraus. Ginzburg-Landau theory of ternary amphiphilic systems. II. Monte Carlo simulations. Phys Rev E 47 4301- 312, 1993. 

Figure 5. Temperature development of the electronic density of states in fee FeaNi with the temperature dependent input taken from the Ginzburg-Landau theory (magnetic moments are given per atom).

Toledano J-C, Toledano P (1987) The Landau theory of phase transitions. World Scientific, Singapore... 

Kuchanov SI, Panyukov SV (2006) A new look at the Landau theory of phase transitions in polydisperse heteropolymer liquids (paper to be published)... 

The properties of the two helium isotopes in the liquid state are strongly influenced by quantum effects. In Fig. 2.8, the specific heat of 3He, calculated from the ideal gas Fermi model (Tp = 4.9 K) with the liquid 3He density, is compared with the experimental data. The inadequacy of this model is evident. A better fit, especially at the lower temperatures, is obtained by the Landau theory [25]. 

The Landau theory applies in the vicinity of a critical point where the order parameter is small and assumed continuous. The Gibbs function... 

Ginzburg Landau theory of superconductivity Beyond the post Gaussian approximation... 

The Landau theory predicts the symmetry conditions necessary for a transition to be thermodynamically of second order. The order parameter must in this case vary continuously from 0 to 1. The presence of odd-order coefficients in the expansion gives rise to two values of the transitional Gibbs energy that satisfy the equilibrium conditions. This is not consistent with a continuous change in r and thus corresponds to first-order phase transitions. For this reason all odd-order coefficients must be zero. Furthermore, the sign of b must change from positive to negative at the transition temperature. It is customary to express the temperature dependence of b as a linear function of temperature ... 

Orientational disordering of the carbonate groups in CaC03 above 1260 K may serve as an example of application of Landau theory. Below the transition temperature, alternate layers of planar CO3 groups point in opposite directions. In the high-temperature modification they are free to rotate and become equivalent. The sym- ... 

The Ginzburg-Landau theory for the CFL phase has been derived in Refs. [10, 16-19], The authors of Ref. [19] have taken into account the rotated electromagnetism and the rotation of the star. They conclude that ordinary quantized magnetic vortices are unstable in the CFL phase, but the rotational vortices are topologically stable. They have not considered boundary problems and concluded that the CFL condensate in an external magnetic held behaves like a type I superconductor. 

In this paper we study the distribution of the magnehc held of a neutron star with superconducting CFL quark matter core in the framework of the Ginzburg-Landau theory. We solve the Ginzburg-Landau equations with proper boundary conditions. 

Fig. 29). Using Landau theory, Bak et al. (BMVW) have shown that it is the wall crossing energy A which determines the symmetry of the weakly incommensurate phase and the nature of the phase transition ... 

The application of Landau theory to rock-forming minerals has been promoted by Ekhard Salje and his coworkers in an attempt to achieve better quantification of complex transition phenomena (mainly in feldspars, but also in pyroxenes and spinels Salje etal., 1985 Salje, 1985,1988 Carpenter and Salje, 1994a,b Carpenter, 1988). 

We will see detailed application of Landau theory to complex superimposed transition phenomena when we treat the energetics of feldspars in chapter 5. 

Salje (1985) interpreted overlapping (displacive plus Al-Si substitutional) phase transitions in albite in the light of Landau theory (see section 2.8.1), assigning two distinct order parameters Q n and to displacive and substitutional disorder and expanding the excess Gibbs free energy of transition in the appropriate Landau form ... 

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