Ginzburg-Landau fluctuation theory

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. 

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-... 

To investigate these problems, we should first devise a Ginzburg-Landau free energy and then set up dynamic equations for network and solvent taking account of both nonlinear elasticity and inhomogeneous fluctuations. Therefore, the aims of this paper are firstly to introduce such a theory [19-21], secondly to review consequences of the theory obtained so far, and thirdly to give new results. Such efforts have just begun and many problems remain unsolved. 

In the previous section, we have seen that it cannot suffice to consider the order parameter alone. A crucial role is played by order parameter fluctuations that are intimately connected to the various singularities sketched in fig. 11. We first consider critical fluctuations in the framework of Landau s theory itself, and return to the simplest case of a scalar order parameter (j ) with no third-order term, and u > 0 [eq. (14)], but add a weak wavevector dependent field <5 H(x) = SHqexp(iq x) to the homogeneous field H. Then the problem of minimizing the free energy functional is equivalent to the task of solving the Ginzburg-Landau differential equation... 

This neglect of fluctuations in general is not warranted. One can recognize this problem in the framework of Landau s theory itself. This criterion named after Ginzburg (1960) considers the mean square fluctuation of the order parameter in a coarse graining volume Ul and states that Landau s theory is selfconsistent if this fluctuation is much smaller than the square of the order parameter itself,... 

In order to describe the diffusive dynamics of composition fluctuations in binary mixtures one can extend the time-dependent Ginzburg-Landau methods to the free energy functional of the SCF theory. The approach relies on two ingredients a free energy functional that accurately describes the chemical potential of a spatially inhomogeneous composition distribution out of equilibrium and an Onsager coefficient that relates the variation of the chemical potential to the current of the composition. 

Independently of the hydrodynamical context, the Ginzburg-Landau equation was derived by Graham and Haken (1968, 1970) in multimode lasers as a further development of the Haken-Sauermann theory (1963) it should be noted that fluctuations are included in most of their series of works. For various non-equilibrium phase transitions described by the Ginzburg-Landau-type equation, see the review article by Haken (1975 b) and his more recent monograph (1983). 

Extension of the classical Landau-Ginzburg expansion to incorporate nonclassical critical fluctuations and to yield detailed crossover functions were first presented by Nicoll and coworkers [313, 314] and later extended by Chen et al. [315, 316]. These extensions match Ginzburg theory to RG theory, and thus interpolate between the lower-order terms of the Wegner expansion at T -C Afa and mean-field behavior at f Nci-... 

The validity of the linear theory observed for the early stage of spinodal decomposition is chiefly related to the large size of the chain molecules. As shown above, characteristic quantities as the time t or the wavelength Am(0) of the fastest growing fluctuation are proportional to Ro and Rg, respectively. Furthermore, the Landau-Ginzburg criterion (cf. condition 2)) ensures that the mean-field regime is sufficiently extended. 

As a result of long-range fluctuations, the local density will vary with position in the classical Landau-Ginzburg theory of fluctuations this introduces a gradient term. A Ginzburg numberis defined (for a... 

In the first part of the tutorial, we provided a concise introduction to the theory of quantum phase transitions. We contrasted the contributions of thermal and quantum fluctuations, and we explained how their interplay leads to a very rich structure of the phase diagram in the vicinity of a quantum phase transition. It turns out that the Landau-Ginzburg-Wilson (LGW) approach, which formed the basis for most modern phase transition theories, can be generalized to quantum phase transitions by including the imaginary time as an additional coordinate of the system. This leads to the idea of the quantum-to-classical mapping, which relates a quantum phase transition in d-space dimensions to a classical one m d+1 dimensions. We also discussed briefly situations in which the LGW order parameter approach can break down, a topic that has attracted considerable interest lately. 

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