Landau equation

With the fomi of free energy fiinctional prescribed in equation (A3.3.52). equation (A3.3.43) and equation (A3.3.48) respectively define the problem of kinetics in models A and B. The Langevin equation for model A is also referred to as the time-dependent Ginzburg-Landau equation (if the noise temi is ignored) the model B equation is often referred to as the Calm-Flilliard-Cook equation, and as the Calm-Flilliard equation in the absence of the noise temi. 

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... 

Figure C3.6.10 Defect-mediated turbulence in tire complex Ginzburg-Landau equation, (a) The phase, arg( ), as grey shades, (b) The amplitude [A], witli a similar color coding. In tire left panel topological defects can be identified as points around which one finds all shades of grey. Note tire apparently random spatial pattern of amplitudes.

In the previous sections, we briefly introduced a number of different specific models for crystal growth. In this section we will make some further simplifications to treat some generic behavior of growth problems in the simplest possible form. This usually leads to some nonlinear partial differential equations, known under names like Burgers, Kardar-Parisi-Zhang (KPZ), Kuramoto-Sivashinsky, Edwards-Anderson, complex Ginzburg-Landau equation and others. 

If we allow for exchange with the gas phase, then h x) does not obey a conservation condition and we have the simple Ginzburg-Landau equation ... 

For the explanation of macroscopic phenomena, the thickness of the phase boundary (interface) often plays no important role. As an example, we describe the movement of a phase boundary in two dimensions or the movement of a step edge on a crystal surface. We start with a Ginzburg-Landau equation [69]... 

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

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. 

The Landau equation in plasma theory is a nonlinear variant, but there P is a particle density rather than a probability. L.D. Landau, Physik. Z. Sovjetunion 10, 154 (1963) = Collected Papers (D. ter Haar ed., Pergamon, Oxford 1965) p. 163. The same is true for the nonlinear Fokker-Planck equation in M. Shiino, Phys. Rev. A 36, 2393 (1987). 

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... 

Fig. 2.47 Pseudostable perforated layer structure, observed following a quench from the lam to hex phase using a multimode analysis of the time-dependent Ginzburg-Landau equation, within the single-wavenumber approximation (Qi and Wang 1997). This structure results from the superposition of six BCC-type wavevectors.

The evolution Eq. 13 of the order parameter has a similar form to the time-dependent Landau equation [17], which is fundamental in nonequihbrium phase transitions. The asymptotic value of the order parameter 4>i,oo is determined as the zero of the velocity 4>i- The main difference from the standard model of phase transitions lies in the time dependence in the coefficients Ait) and B(t) induced by that of the achiral concentration ait) and the total chiral concentration qft). Because the concentrations a and q are nonnegative, A(t) cannot exceed Bit) Ait) < Bit). 

L. P. Gor kov, Macroscopic derivation of the Ginzburg-Landau equations in the theory of superconductivity. Sov. Physics JETP 36, 1364-1367 (1959). 

Vortex shedding behind a circular cylinder is explained theoretically as a Hopf bifurcation which is a consequence of linear temporal instability of the flow. In this point of view, the above temporal instability is moderated by nonlinearity of the system, that is quite adequately explained by Landau equation, as given in Landau (1944) and Drazin Reid (1981). Earlier numerical investigations by Zebib (1987), Jackson (1987) and Morzynski Thiele (1993) have identified the onset of vortex shedding to be at a critical Reynolds number (Rccr) between 45 and 46. 

It is also possible that one can re-interpret the Landau equation and write its amplitude variation by. 

Imbihl et al. was adapted to include modifications of oxygen adsorption on the 1 X 1 phase caused by defects or irregularities in the CO adlayer 317). Also the dynamics of the surface structure transition were descried differently, using a Ginzberg-Landau equation and a lattice model to calculate energies of the different states of the surface. These two modifications resulted in a better fit of the experimental data. 

M.M. Doria, J.E. Gubernatis, D. Rainer, Solving the Ginzburg-Landau equations by simulated annealing. Physical Review B Condensed Matter, 41 (1990) 6335. 

Patterns in fluids) Ahlers (1989) gives a fascinating review of experiments on one-dimensional patterns in fluid systems. In many cases, the patterns first emerge via supercritical or subcritical pitchfork bifurcations from a spatially uniform state. Near the bifurcation, the dynamics of the amplitude of the patterns are given approximately by tA — e4 - gA in the supercritical case, orx4 = e4-gA - kA in the subcritical case. Here Aft) is the amplitude, T is a typical time scale, and e is a small dimensionless parameter that measures the distance from the bifurcation. The parameter g > 0 in the supercritical case, whereas g < 0 and k > 0 in the subcritical case. (In this context, the equation iA = eA - gA is often called the Landau equation.)... 

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