Multiple stable stationary state systems

For systems with multiple stable stationary states there arises the issue of relative stability of such states. As in the previous section we treat systems with a single intermediate and stoichiometric changes in X are limited to 1. 

In Chap. 2 we analyzed single variable linear and non-linear systems with single and multiple stable stationary states by use of the deterministic equations of chemical kinetics. We introduced species-specific affinities and the concept of an excess work with these we showed the existence of a thermodynamic state function 4> and compiled its many interesting properties, see (2.15 2.19), including its relation to fluctuations as given by the stationary solution of the master equation, (2.34). We continue this approach here by turning to systems with more than one intermediate, [1]. 

So far we have considered only homogeneous reaction systems in which concentrations are functions of time only. Now we turn to inhomogeneous reaction systems in which concentrations are functions of time and space. There may be concentration gradients in space and therefore diffusion will occur. We shall formulate a thermodynamic and stochastic theory for such systems [1] first we analyze one-variable systems and then two- and multi-variable systems, with two or more stable stationary states, and then apply the theory to study relative stabihty of such multiple stable stationary states. The thermodynamic and stochastic theory of diffusion and other transport processes is given in Chap. 8. 

Front Propagation and Relative Stability in Systems with Multiple Stable Stationary States (Nodes)... 

Thermodynamic Prediction of Relative Stability for the Systems WITH Multiple Stable Stationary States... 

This is a known condition for the existence of multiple roots of equation (6.37). When A < 0, equation (6.37) has three stationary states of which yu y3 are stable on the other hand, when A > 0, there is only on stable stationary state. The system crossing the sensitive state changes its properties from a bistable state it becomes a monostable state. 

The region of multiple stationary states extends for the pump parameter (equal to pa/pb) from Fi to F3 the line segments with positive slope, marked a and y, are branches of stable stationary states, the line segment with negative slope, marked P, is a branch of unstable stationary states. A system started at an unstable stationary state will proceed to a stable stationary state along... 

Second, we discuss front propagation in systems with multiple stationary states, again far from equilibrium. Consider a chemical system with two stable stationary states at given external constraints. Contact of two such systems, one in each of the two stable stationary states, leads to a front propagation of transition from the less to the more stable stationary state. We report on studies of this process by means of numerical solutions of reaction diffusion equations, experiments and a thermodynamic analysis of stability and relative stability based on the concept of excess work. 

Let us return to Fig. 1.12(c), where there are multiple intersections of the reaction rate and flow curves R and L. The details are shown on a larger scale in Fig. 1.15. Can we make any comments about the stability of each of the stationary states corresponding to the different intersections What, indeed, do we mean by stability in this case We have already seen one sort of instability in 1.6, where the pseudo-steady-state evolution gave way to oscillatory behaviour. Here we ask a slightly different question (although the possibility of transition to oscillatory states will also arise as we elaborate on the model). If the system is sitting at a particular stationary state, what will be the effect of a very small perturbation Will the perturbation die away, so the system returns to the same stationary state, or will it grow, so the system moves to a different stationary state If the former situation holds, the stationary state is stable in the latter case it would be unstable. 

Another useful rule which can frequently guide us to situations where oscillatory solutions will be found is the Poincare-Bendixson theorem. This states that if we have a unique stationary state which is unstable, or multiple stationary states all of which are unstable, but we also know that the concentrations etc. cannot run away to infinity or become negative, then there must be some other non-stationary atractor to which the solutions will tend. Basically this theorem says that the concentrations cannot just wander around for an infinite time in the finite region to which they are restricted they must end up somewhere. For two-variable systems, the only other type of attractor is a stable limit cycle. In the present case, therefore, we can say that the system must approach a stable limit cycle and its corresponding stable oscillatory solution for any value of fi for which the stationary state is unstable. 

Fig. 6.9. Domains of attraction for the two stable branches of stationary-state solution systems which have initial extents of conversion lying within the shaded region evolve to the lower branch those in the unshaded region approach the highest extent of reaction. In the region of multiple stationary states, the middle branch acts as a separatrix.

It may also seem sensible, if there are multiple solutions, to ask which of the states is the most stable In fact, however, this is not a valid question, partly because we have only been asking about very small disturbances. Each of the two stable states has a domain of attraction . If we start with a particular initial concentration of A the system will move to one or other. Some initial conditions go to the low extent of reaction state (generally those for which 1 — a is low initially), the remainder go to the upper stationary state. The shading in Fig. 6.9 shows which initial states go to which final stationary state. It is clear from the figure that the middle branch of (unstable) solutions plays the role of a boundary between the two stable states, and so is sometimes known as a separatrix (in one-dimensional systems only, though). 

The previous two chapters have considered the stationary-state behaviour of reactions in continuous-flow well-stirred reactions. It was seen in chapters 2-5 that stationary states are not always stable. We now address the question of the local stability in a CSTR. For this we return to the isothermal model with cubic autocatalysis. Again we can take the model in two stages (i) systems with no catalyst decay, k2 = 0 and (ii) systems in which the catalyst is not indefinitely stable, so the concentrations of A and B are decoupled. In the former case, it was found from a qualitative analysis of the flow diagram in 6.2.5 that unique states are stable and that when there are multiple solutions they alternate between stable and unstable. In this chapter we become more quantitative and reveal conditions where the simplest exponential decay of perturbations is replaced by more complex time dependences. 

In previous chapters we have dealt only with systems which have one or two independent concentrations. This has been sufficient for a wide range of intricate behaviour. Even with just a single independent concentration (one variable), reactions may show multiple stationary states and travelling waves. Oscillations are, however, not possible. To understand the latter point we can think in terms of the phase plane or, more correctly for a one-dimensional system, the phase line (Fig. 13.1(a)). As the concentration varies in time, so the system moves along this line. Stationary-state solutions are points on the line the arrows indicate the direction of motion along the line, as time increases, towards stable states and away from unstable ones. Figure 13.1(b) shows this motion and phase line behaviour represented in terms of some potential, with stable states a minima and an unstable (saddle) solution as a maximum. 

In 1977. Professor Ilya Prigogine of the Free University of Brussels. Belgium, was awarded Ihe Nobel Prize in chemistry for his central role in the advances made in irreversible thermodynamics over the last ihrec decades. Prigogine and his associates investigated Ihe properties of systems far from equilibrium where a variety of phenomena exist that are not possible near or al equilibrium. These include chemical systems with multiple stationary states, chemical hysteresis, nucleation processes which give rise to transitions between multiple stationary states, oscillatory systems, the formation of stable and oscillatory macroscopic spatial structures, chemical waves, and Lhe critical behavior of fluctuations. As pointed out by I. Procaccia and J. Ross (Science. 198, 716—717, 1977). the central question concerns Ihe conditions of instability of the thermodynamic branch. The theory of stability of ordinary differential equations is well established. The problem that confronted Prigogine and his collaborators was to develop a thermodynamic theory of stability that spans the whole range of equilibrium and nonequilibrium phenomena. 

Figure 4a shows Xq the stationary state value of X for the nonlinear system as a function of P. With this set of parameters no multiple steady states exist but the steady states marked by the broken line are unstable and evolve to stable limit cycles. The amplitude of the limit cycles is shown in Figure 4B. 

Stability analysis could prove to be useful for the identification of stable and unstable steady-state solutions. Obviously, the system will gravitate toward a stable steady-state operating point if there is a choice between stable and unstable steady states. If both steady-state solutions are stable, the actual path followed by the double-pipe reactor depends on the transient response prior to the achievement of steady state. Hill (1977, p. 509) and Churchill (1979a, p. 479 1979b, p. 915 1984 1985) describe multiple steady-state behavior in nonisothermal plug-flow tubular reactors. Hence, the classic phenomenon of multiple stationary (steady) states in perfect backmix CSTRs should be extended to differential reactors (i.e., PFRs). 

A system in which the dependent variables are constant in time is said to be in a steady or stationary state. In a chemical system, the dependent variables are typically densities or concentrations of the component species. Two fundamentally different types of stationary states occur, depending on whether the system is open or closed. There is only one stationary state in a closed system, the state of thermodynamic equilibrium. Open systems often exhibit only one stationary state as well however, multistability may occur in systems with appropriate elements of feedback if they are sufficiently far from equilibrium. This phenomenon of multistability, that is, the existence of multiple steady states in which more than one such state may be simultaneously stable, is our first example of the universal phenomena that arise in dissipative nonlinear systems. 

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