Scaling phenomena dynamic models

Thus, complex chemical processes are represented as a number of simple reactions that are very inhomogeneous on a time scale. Generally, it is impossible to separate the fast processes and the slow ones from each other, so that a continuous time monitoring of the total kinetic process is needed to understand the essence of the phenomenon. Mathematical models provide an adequate tool for the scanning of the kinetic curves. Fig. 1(a) shows a typical example of curves where two time scales are present. These time scales differ up to an order of 10 from each other. If one considers the process on the logarithmic scale, then just three different time scales may be identified, see Fig. 1(b). The presence of both fast and slow variables is explained by the occurrence of either large or small factors in the dynamical equations. For example, this is the case for so-called stiff systems of differential equations. 

A broad range of physical and chemical processes occur as a result of an encounter between pairs of particles. Depending on the physical phenomenon being investigated, the particles may range in size from the atomic scale to colloidal, while the bath density may range from that of a low-pressure gas to a dense liquid. The primary aim of this chapter is to show that a simple model for encounter dynamics can be derived from the Fokker-Planck equation (FPE), which applies, within limits, over the entire range of particle size and bath density. 

During ultrasonic irradiation of aqueous solutions, OH radicals are produced from dissociation of water vapor upon collapse of cavitation bubbles. A fraction of these radicals that are initially formed in the gas phase diffuse into solution. Cavitation is a dynamic phenomenon, and the number and location of bursting bubbles at any time cannot be predicted a priori. Nevertheless, the time scale for bubble collapse and rebound is orders of magnitude smaller than the time scale for the macroscopic effects of sonication on chemicals (2) (i.e., nanoseconds to microseconds versus minutes to hours). Therefore, a simplified approach for modeling the liquid-phase chemistry resulting from sonication of a well-mixed solution is to view the OH input into the aqueous phase as continuous and uniform. The implicit assumption in this approach is that the kinetics of the aqueous-phase chemistry are not controlled by diffusion limitations of the substrates reacting with OH. 

Abstract Pattern formation is a widespread phenomenon observed in different physical, chemical and biological systems on varions spatial scales, including the nanometer scale. In this chapter discussed are the universal features of pattern formation pattern selection, modulational instabilities, structure and dynamics of domain walls, fronts and defects, as well as non-potential effects and wavy patterns. Principal mathematical models used for the description of patterns (Swift-Hohenberg equation, Newell-Whitehead-Segel equation, Cross-Newell equation, complex Ginzburg-Landau equation) are introduced and some asymptotic methods of their analysis are presented. 

---