Ginzburg-Landau equation time-dependent

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. 

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

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.

Mesoscopic methods include several field-based approaches such as cell dynamical systems (CDS), mesoscale density functional theory (DFT), and self-consistent field (SCF)" theory. Most of these methods are related to the time-dependent Ginzburg-Landau equation (TDGL) ... 

As in the case of homogeneous systems, there are two kinds of stochastic descriptions for reaction-diffusion systems as well the master equation approach and the stochastic differential equation method. Until now we have dealt with the first approach however, stochastic partial differential equations are also used extensively. Most often partial differential equations are supplemented with a term describing fluctuations. In particular, time-dependent Ginzburg-Landau equations describe the behaviour of the system in the vicinity of critical points (Haken, 1977 Nitzan, 1978 Suzuki, 1984). A usual formulation of the equation is ... 

One is the time-dependent Ginzburg-Landau equation which is described by a complex order parameter and vector potential. The other is the Langevin-type stochastic equation of motion for magnetic vortices in two and three dimensions, which is described in terms of vortex position variables. 

Close to the bifurcation point, one can generally derive a contracted description of the dynamics only in terms of the order parameter which is associated to the unstable mode. The corresponding equation of motion in many cases takes a form similar to the time-dependent Ginzburg-Landau equation (T.D.G.L.) familiar from the theory of equilibrium phase transition ... 

Equation (60) is called the time-dependent Ginzburg-Landau (TDGL) equation. 

Dynamical Self-Organization. When the parameter X passes slowly through X (l),the bifurcation picture of the previous section accurateiy describes the system. However, in Fucus, and probably in many other examples, this time scale separation between the characteristic time on which X varies and the time to obtain the patterned state does not hold. Thus a dynamical theory allowing for the interplay of these two time scales is required to characterize the developmental scenario. A natural formalism to describe this process is that of time dependent Ginzburg-Landau (tdgl) equations used successfully in other contexts of nonequilibrium phase transitions (27). 

MSI) that uses the same time-dependent Ginzburg Landau kinetic equation as CDS, but starts from (arbitrary) bead models for polymer chains. The methods have been summarized elsewhere. Examples of recent applications include LB simulations of viscoelastic effects in complex fluids under oscillatory shear,DPD simulations of microphase separation in block copoly-mers ° and mesophase formation in amphiphiles, and cell dynamics simulations applied to block copolymers under shear. - DPD is able to reproduce many features of analytical mean field theory but in addition it is possible to study effects such as hydrodynamic interactions. The use of cell dynamics simulations to model non-linear rheology (especially the effect of large amplitude oscillatory shear) in block copolymer miscrostructures is currently being investigated. ... 

To elucidate the spatiotemporal emergence of crystalline structure and liquid-hquid phase separation in these polyolefin blends, we employ the time dependent Ginzburg-Landau (TDGL) equations pertaining to the conserved concentration order parameter and the nonconserved crystal order parameter. The spatiotemporal evolution of the nonconserved order parameter i/f, known as TDGL model-A equation (31,32), may be expressed as... 

Basically, the time-dependent Ginzburg-Landau (TDGL) equation [12] relates the temporal change of a phase order parameter to a local chemical potential and a nonlocal interface gradient. With respect to a non-conserved phase field order parameter, the TDGL model A equation is customarily described as ... 

The dynamical behavior of Ginzburg-Landau models is described by Langevin equations. In the simplest case, the equation of motion for a conserved order parameter field 0(r, /), which now depends on time / in addition to r, reads... 

The time evolution of the density field pi r) can be described by a time dependent Landau-Ginzburg type equation (11). The boundary conditions that are used on the simulation box are periodic boundary conditions. For the diffusion flux in the vicinity of the filler particles, rigid-wall boundary conditions are used. A simple way to implement these boundary conditions in accordance with the conservation law is to allow no flux through the filler particle surfaces, i.e.,... 

To elucidate the time evolution of the concentration and orientation fluctuations during a SD, we introduce kinetic equations based on the time-dependent Landau-Ginzburg (TDLG) equations for concentration and orientational order parameters. 

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