Complex systems locally stable equilibrium

Thus, on the one hand, the foam film type (and therefore the type of stabilisation due to long-range and short-range forces) is the determining factor for the course of tp(Ap0) dependences. On the other hand, in some systems an avalanche-like destruction occurs at A/ cr. A reasonable question arises as to why a foam built up of different types of foam films destructs before reaching Aplr. If the foam is considered as a system built up of equilibrium foam films, then they should be infinitely stable. However, the foam is a more complex system, built up of foam films and borders, subjected to the effect of several other factors (gas diffusion transfer, coalescence of bubbles and changes in the foam film size, external actions, local stretching, collective effects of destruction, etc.) which can lead to its destruction. 

In the case of the fast binary reaction we could eliminate the reaction term from the reaction-diffusion-advection equation. But in general this is not possible. In this chapter we consider another class of chemical and biological activity for which some explicit analysis is still feasible. We consider the case in which the local-reaction dynamics has a unique stable steady state at every point in space. If this steady state concentration was the same everywhere, then it would be a trivial spatially uniform solution of the full reaction-diffusion-advection problem. However, when the local chemical equilibrium is not uniform in space, due to an imposed external inhomogeneity, the competition between the chemical and transport dynamics may lead to a complex spatial structure of the concentration field. As we will see in this chapter, for this class of chemical or biological systems the dominant processes that determine the main characteristics of the solutions are the advection and the reaction dynamics, while diffusion does not play a major role in the large Peclet number limit considered here. Thus diffusion can be neglected in a first approximation. 

In the original proof the system under consideration was assumed to be analytic. Later on, other simplified proofs have been proposed which are based on a reduction to a non-local center manifold near the separatrix loop (such a center manifold is, generically, 3-dimensional if the stable characteristic exponent Ai is real, and 4-dimensional if Ai = AJ is complex) and on a smooth linearization of the reduced system near the equilibrium state (see [120, 147]). The existence of the smooth invariant manifold of low dimension is important here because it effectively reduces the dimension of the problem. ... 

The proposed structure of the complex does not assume a static distribution of the sequences. The system is of course a dynamic one, but we study it at equilibrium. A given COOH group, involved in a complex at the moment t, may be free or in the carboxylate from at t + dt. However the average number of complexed sequences remains invariant with time for a fixed composition of the system. The situation can be compared with the behaviour of macromolecules adsorbed at a solid-liquid interface their mean conformation is stable even if locally an adsorption/desorption equilibrium occurs. 

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