Pattern formation transitions

This equation has been derived as a model amplitude equation in several contexts, from the flow of thin fluid films down an inclined plane to the development of instabilities on flame fronts and pattern formation in reaction-diffusion systems we will not discuss here the validity of the K-S as a model of the above physicochemical processes (see (5) and references therein). Extensive theoretical and numerical work on several versions of the K-S has been performed by many researchers (2). One of the main reasons is the rich patterns of dynamic behavior and transitions that this model exhibits even in one spatial dimension. This makes it a testing ground for methods and algorithms for the study and analysis of complex dynamics. Another reason is the recent theory of Inertial Manifolds, through which it can be shown that the K-S is strictly equivalent to a low dimensional dynamical system (a set of Ordinary Differentia Equations) (6). The dimension of this set of course varies as the parameter a varies. This implies that the various bifurcations of the solutions of the K-S as well as the chaotic dynamics associated with them can be predicted by low-dimensional sets of ODEs. It is interesting that the Inertial Manifold Theory provides an algorithmic approach for the construction of this set of ODEs. 

Kumai R, Okimoto Y, Tokura Y (1999) Current-induced insulator-metal transition and pattern formation in an organic charge-transfer complex. Science 284 1645-1647... 

Ray Kapral came to Toronto from the United States in 1969. His research interests center on theories of rate processes both in systems close to equilibrium, where the goal is the development of a microscopic theory of condensed phase reaction rates,89 and in systems far from chemical equilibrium, where descriptions of the complex spatial and temporal reactive dynamics that these systems exhibit have been developed.90 He and his collaborators have carried out research on the dynamics of phase transitions and critical phenomena, the dynamics of colloidal suspensions, the kinetic theory of chemical reactions in liquids, nonequilibrium statistical mechanics of liquids and mode coupling theory, mechanisms for the onset of chaos in nonlinear dynamical systems, the stochastic theory of chemical rate processes, studies of pattern formation in chemically reacting systems, and the development of molecular dynamics simulation methods for activated chemical rate processes. His recent research activities center on the theory of quantum and classical rate processes in the condensed phase91 and in clusters, and studies of chemical waves and patterns in reacting systems at both the macroscopic and mesoscopic levels. 

FIGURE 1.11. Pattern formation in a temperature gradient, using the set-up from Fig. 1.8, where the lower plate was set to a temperature T and the upper plate to T2 = T + AT. The transition of the film to columns (a) and stripes (b) was observed, often on the same sample. Adapted from [32]. 

When adsorbed (from ambient air), water molecules might act as plasticizers and alter the dynamics of polymers. Moreover, water has a strong dipole moment and, consequently, dielectric active relaxation processes, which could partially occlude significant parts of the dielectric spectra of interest. Special attention to this effect has to be paid when the dynamics of thin polymer films is investigated, for example in relation to phenomena like the glass transition, dewetting, pattern formation, surface mobility etc. 

Next, an example of CG-KMC from pattern formation on surfaces is presented. Another application to relatively thick membranes was given in Snyder et al. (2004). In the example considered here, atoms adsorb from a fluid reservoir on a flat surface. Subsequently, they may desorb back to the fluid, diffuse on the surface, or be annihilated by a first-order surface reaction, as shown in Fig. 11a. Attractive interactions between atoms trigger a phase transition from a dilute phase (a low coverage) to a dense phase (a high coverage) (Vlachos et al., 1991), analogous to van der Waals loops of fluid vapor coexistence. Surface reactions limit the extent of phase separation the competition between microphase separation and reaction leads to nanoscopic patterns by self-organization under certain conditions (Hildebrand et al., 1998). 

Figure 5, Measured particle traces superimposed onto the optically detected concentration pattern. Arrows mark the direction of particle movement during the early phase of the pattern formation (A), and during the transition to honeycomb patterns (B). Field of view 1.7 x 1,6 nmi. (Adapted with permission from reference 13. Copyright 2002 Oldenbourg Wissenschaftsverlag.)...

Hirotsu S, Kaneki A (1988) D5mamics of phase transition in polymer gels studies of spinodal decomposition and pattern formation. In Komrrra S, Frmrkawa H (eds) Dynamics of ordering process in condensed matter. Plenum, Kyoto, pp 481-486... 

Understanding the molecular basis of pattern formation and differentiation in frog embryos was previously hindered by the lack of a system for temporal and tissue-specific expression of wild-type and mutant forms of developmentally important genes. RNA injection, the most common transient expression method in Xenopus, has been effectively used to study maternally expressed genes. However, since RNAs are translated immediately after injection, this method is unfavorable for the study of zygotic gene products that are expressed only after the midblastula transition. Direct injection of DNA can be used to express genes behind temporal and tissue-specific promoters after the midblastula transition. 

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