Landau-Ginzburg coefficients

More precisely, Brezin, Le Guillou, and Zinn-Justin showed25 that the coefficients >p of the expansion of any physical quantity associated with the Landau-Ginzburg model... 

Barium titanate is also the first material with the ferroelectricity studied theoretically by Devonshire (1949). Theoretical description is based on the idea of the thermodynamic potentials known from the Landau-Ginzburg theory of phase transitions (Devonshire 1949, 1951). Barium titanate is mainly used in multilayer ceramic capacitors and in positive temperature coefficient (PTC) elements (Bhalla et al. 2000). 

Landau-Ginzbuig coefficients 282 Landau-Ginzburg Hamiltonian, critical 303 Landau-Khalatnikov mechanism 564 Landau-Lifshitz theory, phase transitions 366 Landau-Peierls instabilities 285, 647 Langmuir-Blodgett film, atomistic simulations 85 Laplace equation 445... 

The theoretical prediction is supported by the experiments. Patterns that spontaneously form from the uniform state have multiple domains with different characteristic angles. The root-mean-square width A rms of the observed angular distribution function changes with the bifurcation parameter in qualitative accord with theory compare Figure 8b with Figure 8a [13]. A quantitative comparison of experiment and theory would require an evaluation of the coefficients in the Landau-Ginzburg equation from the chemical kinetics and diffusion coefficients of the reactants [47]. 

This complex Ginzburg-Landau equation describes the space and time variations of the amplitude A on long distance and time scales detennined by the parameter distance from the Hopf bifurcation point. The parameters a and (5 can be detennined from a knowledge of the parameter set p and the diffusion coefficients of the reaction-diffusion equation. For example, for the FitzHugh-Nagumo equation we have a = (D - P... 

Here, the final three terms are a Ginzburg-Landau expansion in powers of i j. The coefficient t varies as a function of temperature and other control variables. When it decreases below a critical threshold, the system undergoes a chiral symmetry-breaking transition at which i becomes nonzero. The membrane then generates effective chiral coefficients kHp = k n>i f and kLS = which favor membrane curvature and tilt modulations, respec-... 

It should be emphasized that this way of including fluctuations has no other justification than that it is convenient and bypasses a description of the noise sources, compare IX.4. It may provide some qualitative insight into the effect of noise, but does not describe its actual mechanism. For instance, fluctuations in the pumping should give rise to randomness in the coefficient a, rather than to an additive term. Yet the equation (7.6) has been the subject of extensive study and it is famous in statistical mechanics under the name of generalized Ginzburg-Landau equation. It may well serve us as an illustration for a stochastic process.510... 

Here M is a mobility coefficient, which is assumed to be constant and r/(r.t) is the random thermal noise term, which for a system in equilibrium at temperature T satisfies the fluctuation-dissipation theorem. The free energy functional is taken to be of a Ginzburg-Landau form. In the notation of Qi and Wang (1996,1997) it is given by... 

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. 

The composition dependence accounts for the fact that currents of A- and B-densities have to exactly cancel in order to fulfill the incompressibility constraint. This local Onsager coefficient completely neglects the propagation of forces along the backbone of the chain and monomers move independently. Such a local Onsager coefficient is often used in calculations of dynamic models based on Ginzburg-Landau type energy functionals for reasons of simplicity [85-87]. 

In these equations k, a, ct, d ( = Pu in eq 10.29), and g are system-dependent coefficients with g being related to the inverse of the Ginzburg number AIq. Slightly different versions for the crossover function R q) have also been used. In the critical limit 0 one recovers the linear-model parametric equation in Section 10.2.2 with coefficients a and k. In the classical limit q-rcc. Ad becomes an analytic function of AT and Ap. For a comparison of this phenomenological parametric crossover equation with the crossover Landau models the reader is referred to some previous publications. " " ... 

The values are only representative because the parameters for alloys and compounds depend on how the material is fabricated. The penetration depth h.Qi(0) is given as the coefficient of the Ginzburg-Landau temperature dependence as X,gj(T) = A,qj(0)(1-likewise for the coherence length where i T) = c (0)(l — T/T )The upper critical field is given at zero temperature as well as the energy gap A . 

We have seen in Sects. 3.5 and 4.2 that the Ginzburg-Landau equation is appropriate as a model reaction-diffusion system for which the method of phase description is demonstrated. Some coefficients of the expansion of Qy//bt where then calculated to give, see (3.5.13a,b and 4.2.27),... 

If the amplitude of the complex order parameter ij/ is constant, (10.8) describes changes of the free energy which are due to compression or dilation of the layers (i.e., deviations of Q from Qo ) or due to phase shifts of ij/, i.e., displacements of the smectic layers. In contrast to the gradient term appearing in the Ginzburg-Landau ansatz [44], two coefficients, C and C , appear because of the anisotropy of the liquid crystal. In a one-constant approximation, C = C = C, (10.8) is reduced to... 

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