Spatio-temporal pattern

One may also observe a transition to a type of defect-mediated turbulence in this Turing system (see figure C3.6.12 (b). Here the defects divide the system into domains of spots and stripes. The defects move erratically and lead to a turbulent state characterized by exponential decay of correlations [59]. Turing bifurcations can interact with the Hopf bifurcations discussed above to give rise to very complicated spatio-temporal patterns [63, 64]. 

Several methods have been employed to study chemical reactions theoretically. Mean-field modeling using ordinary differential equations (ODE) is a widely used method [8]. Further extensions of the ODE framework to include diffusional terms are very useful and, e.g., have allowed one to describe spatio-temporal patterns in diffusion-reaction systems [9]. However, these methods are essentially limited because they always consider average environments of reactants and adsorption sites, ignoring stochastic fluctuations and correlations that naturally emerge in actual systems e.g., very recently by means of in situ STM measurements it has been demon-... 

The HS model exhibits a rich variety of spatio-temporal patterns. During the oscillatory behavior, if the simulation starts with an empty grid in the hexagonal phase the only possible event is CO adsorption. Consequently, when a certain CO coverage is reached, the surface starts to convert into the 1 X 1 phase. Oxygen cannot adsorb yet, due to the lack of empty sites. 

D. Walgraef. Spatio-Temporal Pattern Formation. New York Springer Verlag, 1997, pp. 1-301. 

R. J. Gelten et al. Monte Carlo simulation of a surface reaction model showing spatio-temporal pattern formations and oscillations. J Chem Phys 705 5921-5934, 1998. 

K. Asakura, J. Lanterbach, H.H. Rothermund, and G. Ertl, Spatio-temporal pattern formation during catalytic CO oxidation on a Pt(100) surface modified with submonolayers of Au, Surf. Sci. 374, 125-141 (1997). 

Figure 2.16. Computer simulation of spatio-temporal pattern formation in CO oxidation on a surface. [Adapted from R.J. Celten, A.P.J. Jansen, R.A. van Santen, j.j. Lukkien, j.P.L Segers and P.A.j. Hilbers,j. Chem. Phys. 108 (1998) 5921.]...

Figure 11.1 Concept of dissipative structures for the emergence of spatio-temporal patterns.

This chapter is organized as follows. We first present a short description of the criteria used for selecting the data used for driving the hydrological simulations at the basin scale. Subsequently, we briefly describe Soil and Water Assessment Tool (SWAT), the hydrological model adopted for this study, and the setup thereof. Later on, we continue with a brief review of the main spatio-temporal patterns of climate,... 

In the microscopic techniques discussed above, the challenge was to visualize the atomic detail. However, in catalysis one also encounters phenomena that occur on the scale of micrometers or millimeters which ask for imaging. In particular, the ordering of adsorbates in large islands and the development of spatio-temporal patterns in oscillating reactions [8], This spectacular phenomenon has stimulated the exploration of imaging techniques that provide information on patterns on the micrometer to millimeter scale. 

Jimenez, A. J., Garcia-Fernandez, J. M., Gonzalez, B. and Foster, Ik. G. (1996) The spatio-temporal pattern of photoreceptor degeneration in the aged rd/rd mouse retina. Cell Tissue Res 284, 193-202. 

H.H. Rotermund, W. Engel, M. Kordesh, G. Ertl, Imaging of Spatio Temporal Pattern Evolutionduring Carbon Monoxide Oxidation on Platinum, Nature 343 (6256) (1990) 355. 

Larson, IRG l99A).ln.Spatio-Temporal Patterns, Cladis PE, Palffy-Muhoray P (eds), SFI Studies in the Sciences of Complexity, Proceedings, Vol XXI, Addison-Wesley, Reading, MA. Larson RG (1996). Rheol Acta 35 150. 

The kinetics of chemical reactions on surfaces is described using a microscopic approach based on a master equation. This approach is essential to correctly include the effects of surface reconstruction and island formation on the overall rate of surface reactions. The solution of the master equation using Monte Carlo methods is discussed. The methods are applied to the oxidation of CO on a platinum single crystal surface. This system shows oscillatory behavior and spatio-temporal pattern formation in various forms. 

In the following, we will discuss DMC simulations on the CO oxidation on the Pt(lOO) surface, that were done in our laboratories. The simulations show oscillations in the CO2 production rate as well as several types of spatio-temporal pattern formation. In essence, it is an extension of the ZGB model with desorption and diffusion of A, finite reaction rates and surface reconstruction. We will discuss it to illustrate the complexity of the models with which DMC simulations can be done nowadays. For clarity, we will stick to the A and B2 notation employed in the previous section. Species A corresponds to CO and B2 corresponds to 02- Furthermore, we will speak in terms of reaction rates instead of relative reaction probabilities. This terminology is entirely justified in the DMC approach that we used. 

In section 3.2.2 we have discussed four regimes of front generation. Spatio-temporal pattern formation is observed in two of these regimes. The type of pattern formation that is observed in the third regime was already discussed above. Figure 6A shows another example of the cellular patterns. 

Under conditions where the front generation is slow, i.e. in the second regime of section 3.2.2, spatio-temporal pattern formation is observed in several forms. Target patterns, rotating spirals and turbulent structures are the observed forms. When turbulent patterns are present, sometimes small fragments of reaction fronts exhibit solitonic behavior. Figure 6 shows the four main forms of pattern formation that we have observed in our simulations. 

Figure 6. Four examples of spatio-temporal pattern formation in our simulations (A) cellular structures, (B) target patterns, (C) a double rotating spiral, (D) turbulent patterns.

We have also cast the DMC model in a set of ordinary differential equations, thus translating it to a mean-field approach with the site-approximation. Only the kinetic oscillations can be modeled in this way. To model the spatio-temporal pattern formations, diffusion terms would have to be added to the mean-field description, in order to account for the spatial dependence of the reactant concentrations. 

The main application that was discussed was a microscopic model for the oxidation of CO, catalyzed by a Pt(lOO) single crystal surface. The simulations show kinetic oscillations as well as spatio-temporal pattern formation in the form of target patterns, rotating spirals and turbulent patterns. Finally, mean-field simulations of the same model were compared with the Monte Carlo simulations. When diffusion is fast and the simulation grids are small, the results of Monte Carlo simulations approach those of the mean-field simulations. 

R. J. Gelten, A. P. J. Jansen, R. A. van Santen, J. J. Lukkien, J. P. L. Segers, and P. A. J. Hilbers, Monte Carlo simulations of a surface reaction model showing spatio-temporal pattern formations and oscillations, J. Chem. Phys., 108 (1998) 5921. 

Mining and analysing spatio-temporal patterns of gene expression in an integrative database framework. J Integr Bioinform 7 128... 

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