Stability of stationary states

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. 

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For the purposes of fixing the stationary states we have up to this point only considered simply or multiply periodic systems. However the general solution of the equations frequently yield motions of a more complicated character. In such a case the considerations previously discussed are not consistent with the existence and stability of stationary states whose energy is fixed with the same exactness as in multiply periodic systems. But now in order to give an account of the properties of the elements, we are forced to assume that the atoms, in the absence of external forces at any rate always possess sharp stationary states, although the general solution of the equations of motion for the atoms with several electrons exhibits no simple periodic properties of the type mentioned (Bohr [1923]). 

Stability of stationary state bifurcations to periodic solutions... 

The second law for isolated systems shows that the excess entropy, A.V S SKI < 0, increases monotonically in time, d(AS)/dt > 0. Therefore, it plays the role of a Lyapunov function, and defines a global stability. So, dfi/dt is a Lyapunov function that guarantees the global stability of stationary states that are close to global equilibrium. 

In the linear nonequilibrium thermodynamics theory, the stability of stationary states is associated with Prigogine s principle of minimum entropy production. Prigogine s principle is restricted to stationary states close to global thermodynamic equilibrium where the entropy production serves as a Lyapunov function. The principle is not applicable to the stability of continuous reaction systems involving stable and unstable steady states far from global equilibrium. 

The stability of transport and rate systems is studied either by nonequilibrium thermodynamics or by conventional rate theory. In the latter, the analysis is based on Poincare s variational equations and Lyapunov functions. We may investigate the stability of a steady state by analyzing the response of a reaction system to small disturbances around the stationary state variables. The disturbed quantities are replaced by linear combinations of their undisturbed stationary values. In nonequilibrium thermodynamics theory, the stability of stationary states is associated with Progogine s principle of minimum entropy production. Stable states are characterized by the lowest value of the entropy production in irreversible processes. The applicability of Prigogine s principle of minimum entropy production is restricted to stationary states close to global thermodynamic equilibrium. It is not applicable to the stability of continuous reaction systems involving stable and unstable steady states far from global equilibrium. The steady-state deviation of entropy production serves as a Lyapunov function. 

THERMODYNAMIC CRITERIA OF ACHIEVEMENT AND STABILITY OF STATIONARY STATES... 

Thermodynamic Criteria of the Stability of Stationary States that... 

THERMODYNAMIC CRITERIA OF THE STABILITY OF STATIONARY STATES THAT ARE FAR FROM EQUILIBRIUM... 

A typical problem in thermodynamics of systems that are far from their equilibrium is the analysis of the stability of stationary states of the system. Thermodynamic criteria of the stability of stationary states are found the same way as for systems that are far from and close to thermodynamic equilibrium (see Section 2.4) by analyzing signs of thermodynamic fluxes and forces arising upon infinitesimal deviation of the system from the inspected stationary state. If the system is in the stable stationary state, then any infinitesimal deviation from this state must induce the forces that push it to return to the initial position. 

When a reactive system is far from its thermodynamic equilibrium, corol laries of the Prigogine theorem, which were derived for the case of the linear nonequilibrium thermodynamics, cannot be applied to analysis directly. Nevertheless, tools of thermodynamics of nonequilibrium pro cesses allow the deduction of some important conclusions on properties of the system, even though strongly nonequilibrium, including in some cases on the stability of stationary states of complex stepwise processes. For several particular cases, theorems similar to the Prigogine theorem can be proved, too. 

It follows from considerations of the stability of stationary states in sim pie kinetic schemes (Section 3.4) that the positively defined Lyapunov functional can be found for any stepwise catalytic reaction provided that it proceeds through transformations that are Hnear in respect to catalysis intermediates. It is important to note that the catalyst stationary state remains stable in the stepwise reactions that are autocatalytic in respect to the external reactants. 

The stability of stationary states may be investigated by linearizing the system (5.22) nearby an equilibrium state, see Section 5.1, equations (5.6). 

Fig. 6.15. (a) Two-dimensional bifurcation diagram for the stability of stationary states... 

This chapter focuses on the stability properties of networks or arrays of coupled monostable units or cells. We consider two types of coupling, namely diffusive coupling and photochemical coupling. The two main concerns are how the topology of the network connectivity and how spatial inhomogeneities in the array affect instabilities. Spatially discrete systems or networks of coupled cells are described by sets of ordinary differential equations. Methods to determine the stability of stationary states of ODEs are well developed. 

Preliminary conclusions on the stability of stationary states can be drawn from some qualitative physical considerations. They can be elucidated by means of the plot in Fig. 5.15. 

This equation serves as a necessary and sufficient criterion for the existence and stability of stationary states for non-autocatal3dic and auto-catalytic stationary states in multi-variable systems. 

The right hand side of (5.34) is negative semidefinite, so that the system tends towards the minimum of stable stationary state. Thus the function is a Liapunov function of the system. Further, satisfies the stationary solution of the master equation in the thermodynamic limit. All these properties assure that the function provides nessecary and sufficient conditions for the existence and stability of stationary states. 

In Chap. 5 we discussed reaction diffusion systems, obtained necessary and sufficient conditions for the existence and stability of stationary states, derived criteria of relative stability of multiple stationary states, all on the basis of deterministic kinetic equations. We began this analysis in Chap. 2 for homogeneous one-variable systems, and followed it in Chap. 3 for homogeneous multi-variable systems, but now on the basis of consideration of fluctuations. In a parallel way, we now follow the discussion of the thermod3mamics of reaction diffusion equations with deterministic kinetic equations, Chap. 5, but now based on the master equation for consideration of fluctuations. 

Liapunov functions) with physical significance, such as the excess work the work and power available from a transient decay to a stationary state macroscopic necessary and sufficient criteria of stability of stationary states thermodynamic criteria for bifurcations from one type of stationary state to another type thermodynamic criteria of relative stability, that is thermodynamic criteria of state selection and a connection of the thermodynamic theory to fluctuations. 

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