Ergodicity boundary reaction

Figure 8 Two basic examples of ergodicity boundary reaction (a) Connection between ergodic components and (b) Connection from one ergodic component to element that is connected to the both ergodic components by oriented paths. In both cases, for = 0, the ergodic components are >42) and A3. ...

If the ergodicity boundary reaction starts in the ergodic component Gi and ends at B which does not belong to the "opposite" basin of attraction II2, then T >S> This is the first possible obstacle. 

Let the ergodicity boundary reaction start at A e Gi and end at B e ll2- We define the maximal linear chain of reactions in dominant system with start at B . .. This chain belongs to IJ2. Let us extend this chain from the left by the... 

Nevertheless, sometimes it is possible to point the reaction rate constant that is limiting for the relaxation in the following sense. For known topology of reaction network and given ordering of reaction rate constants we find such a constant (ergodicity boundary) that... 

We study a multiscale system with a given reaction rate constants ordering, kj >kj > >kj. Let us suppose that the network is weakly ergodic (when there are several ergodic components, each one has its longest relaxation time that can be found independently). We say that kj, l r n is the ergodicity boundary k if the network of reactions with parameters kj, kj, .., kj (when. . kj — 0)... 

Let us describe the possible location of the ergodicity boundary in the general multiscale reaction network (ifl. After deletion of reactions with constants fcy,j,kj from the network two ergodic components (minimal sinks) appear, Gi and Gz- The ergodicity boundary starts in one of the ergodic components, say Gi, and ends at the such a reagent B that another ergodic component, Gz, is reachable by B (there exists an oriented path from B to some element of Gz). 

Thus, the well-known concept of stationary reaction rates limitation by "narrow places" or "limiting steps" (slowest reaction) should be complemented by the ergodicity boundary limitation of relaxation time. It should be stressed that the relaxation process is limited not by the classical limiting steps (narrow places), but by reactions that may be absolutely different. The simplest example of this kind is an irreversible catalytic cycle the stationary rate is limited by the slowest reaction (the smallest constant), but the relaxation time is limited by the reaction constant with the second lowest value (in order to break the weak ergodicity of a cycle two reactions must be eliminated). 

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