Landau order parameter

The range of coherence follows naturally from the BCS theory, and we see now why it becomes short in alloys. The electron mean free path is much shorter in an alloy than in a pure metal, and electron scattering tends to break up the correlated pairs, so dial for very short mean free paths one would expect die coherence length to become comparable to the mean free path. Then the ratio k i/f (called the Ginzburg-Landau order parameter) becomes greater than unity, and the observed magnetic properties of alloy superconductors can be derived. The two kinds of superconductors, namely those with k < 1/-/(2T and those with k > l/,/(2j (the inequalities follow from the detailed theory) are called respectively type I and type II superconductors. 

This fomi is called a Ginzburg-Landau expansion. The first temi f(m) corresponds to the free energy of a homogeneous (bulk-like) system and detemiines the phase behaviour. For t> 0 the fiinction/exliibits two minima at = 37. This value corresponds to the composition difference of the two coexisting phases. The second contribution specifies the cost of an inhomogeneous order parameter profile. / sets the typical length scale. 

An even coarser description is attempted in Ginzburg-Landau-type models. These continuum models describe the system configuration in temis of one or several, continuous order parameter fields. These fields are thought to describe the spatial variation of the composition. Similar to spin models, the amphiphilic properties are incorporated into the Flamiltonian by construction. The Flamiltonians are motivated by fiindamental synnnetry and stability criteria and offer a unified view on the general features of self-assembly. The universal, generic behaviour—tlie possible morphologies and effects of fluctuations, for instance—rather than the description of a specific material is the subject of these models. 

Analytic teclmiques often use a time-dependent generalization of Landau-Ginzburg ffee-energy fiinctionals. The different universal dynamic behaviours have been classified by Hohenberg and Halperin [94]. In the simple example of a binary fluid (model B) the concentration difference can be used as an order parameter m.. A gradient in the local chemical potential p(r) = 8F/ m(r) gives rise to a current j... 

Undoubtedly the most successful model of the nematic-smectic A phase transition is the Landau-de Gennes model [201. It is applied in the case of a second-order phase transition by combining a Landau expansion for the free energy in tenns of an order parameter for smectic layering with the elastic energy of the nematic phase [20]. It is first convenient to introduce an order parameter for the smectic stmcture, which allows both for the layer periodicity (at the first hannonic level, cf equation (C2.2A)) and the fluctuations of layer position ur [20] ... 

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-... 

We start from the time-dependent Ginzburg-Landau equation for a non-conserved order parameter 0... 

Transition matrix estimators have received less attention than the multicanonical and Wang-Landau methods, but have been applied to a small collection of informative examples. Smith and Bruce [111, 112] applied the transition probability approach to the determination of solid-solid phase coexistence in a square-well model of colloids. Erring ton and coworkers [113, 114] have also used the method to determine liquid-vapor and solid-liquid [115] equilibria in the Lennard-Jones system. Transition matrices have also been used to generate high-quality data for the evaluation of surface tension [114, 116] and for the estimation of order parameter weights in phase-switch simulations [117]. 

The construction of the phase diagram of a heteropolymer liquid in the framework of the WSL theory is based on the procedure of minimization of the Landau free energy T presented as a truncated functional series in powers of the order parameter with components i a(r) proportional to Apa(r). The coefficients of this series, known as vertex functions, are governed by the chemical structure of heteropolymer molecules. More precisely, the values of these coefficients are entirely specified by the generating functions of the chemical correlators. Hence, before constructing the phase diagram of the specimen of a heteropolymer liquid, one is supposed to preliminarily find these statistical characteristics of the chemical structure of this specimen. Here a pronounced interplay of the statistical physics and statistical chemistry of polymers is explicitly manifested. 

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

Define the order parameter V with conjugate field P and suppose that (3) is obtained by minimization of the Landau free energy ip(V, P,T), i.e. such that (3) is the equation that corresponds to dtp/dV = 0. Thus by integration of (3) at constant P and T,... 

To work in the opposite direction, minimization of the Landau free energy with respect to the order parameter, dip/dV = 0, clearly yields... 

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 ... 

Fig. 22. Finite-size scaling plot of the order parameter correlation function at 0 == 1/2 and temperatures in the regime Tj < T < T for the model of Fig. 20. Resulting estimates i((T) are indicated. (From Landau. )...

Such continuous phase transitions are conveniently described in a phenomenological Landau free-energy expansion of the order parameter. Since we... 

The excess thermodynamic properties correlated with phase transitions are conveniently described in terms of a macroscopic order parameter Q. Formal relations between Q and the excess thermodynamic properties associated with a transition are conveniently derived by expanding the Gibbs free energy of transition in terms of a Landau potential ... 

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