Critical simplification dynamics

The example of critical simplification analyzed in this chapter may serve as a good subject for some generalization. Let us assume that in our system there are three independent variables (chemical concentrations), one mass conservation balance, and one infinitely fast reaction between two surface species. As usual, our system is governed by the mass-action law and there is no question of autocatalysis. In this case, the steady-state value of one variable will be zero (or rather, negligible) and the dynamics of the system will be two-dimensional. Consequently, critical simplification will be observed for this system as well. 

Dynamic simulations are also possible, and these require solving differential equations, sometimes with algebraic constraints. If some parts of the process change extremely quickly when there is a disturbance, that part of the process may be modeled in the steady state for the disturbance at any instant. Such situations are called stiff, and the methods for them are discussed in Numerical Solution of Ordinary Differential Equations as Initial-Value Problems. It must be realized, though, that a dynamic calculation can also be time-consuming and sometimes the allowable units are lumped-parameter models that are simplifications of the equations used for the steady-state analysis. Thus, as always, the assumptions need to be examined critically before accepting the computer results. 

At this point we introduce the fvst approximation. We split the correlation functionmode-mode coupling theory, which has been applied with great success to binary mixtures near their critical point by Kawasaki S and Ferrell. However, for the present problem, the idea (in a different language) goes back to Kirkwood and Risemarm. s We make a further (minor) simplification. We assume that the essential time dependence is contained in the (tm) part of the correlation, while the (cc) part may be taken at equal times. This may be justified by a detailed study of the (cc) dynamics, along the lines of Section VI.2.2. 

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