Critical equilibrium state

The critical equilibrium states have been the subject of a large number of studies. Here, we shall consider only the two most common and simple cases, where the characteristic equation... 

The basic tools for studying critical cases include the method of reduction to the center manifold and the method of normal forms. The latter allows us to calculate the Lyapunov values that determine the stability of a critical equilibrium state. 

A system of differential equations near a critical equilibrium state can be written in the form... 

As shown in Chap. 5, the above critical equilibrium state lies in an invariant C -smooth center manifold defined by an equation of the form y = (a ), where (x) vanishes at the origin along with its first derivative. 

Our investigation of the stability of a critical equilibrium state will make use of Lyapunov functions. 

Prom the practical point of view, stability in the sense of Lyapunov is less important than asymptotic stability. In particular, it follows from simple continuity arguments that if a critical equilibrium state is asymptotically stable, then the trajectories of any nearby system will also converge to a small neighborhood of the origin where they will stay forever. The behavior of trajectories in this small neighborhood may be rather nontrivial. Nevertheless, any deviations from zero of trajectories of a nearby system must remain small because the equilibrium state is asymptotically stable at the critical parameter value. 

The answers to these questions are settled by the theory of bifurcations. In this chapter, we consider only local bifurcations, i.e. those which occur near critical equilibrium states, and near fixed points of a Poincare map. We restrict our study to the simplest but key bifurcations which have an immediate connection to the critical cases are discussed in the two last chapters. 

Khazin, L. G. and Shnol, E. E. [1991] Stability of Critical Equilibrium States (Manchester University Press). 

A homogeneous metastable phase is always stable with respect to the fonnation of infinitesimal droplets, provided the surface tension a is positive. Between this extreme and the other thennodynamic equilibrium state, which is inhomogeneous and consists of two coexisting phases, a critical size droplet state exists, which is in unstable equilibrium. In the classical theory, one makes the capillarity approxunation the critical droplet is assumed homogeneous up to the boundary separating it from the metastable background and is assumed to be the same as the new phase in the bulk. Then the work of fonnation W R) of such a droplet of arbitrary radius R is the sum of the... 

As with experimental work on polymer adsorption, experiments in the area of dispersion stability in the presence of polymers require detailed characterisation of the systems under study and the various controlling parameters (discussed above) to be varied in a systematic way. One should seek the answer to several questions. Is the system (thermodynamically) stable If not, what is the nature of the equilibrium state and what are the kinetics of flocculation If it is stable, under what critical conditions ( s, T, x> p etc.) can flocculation be induced ... 

For liquid-vapor interfaces, the correlation length in the bulk is of t he order of atomic distance unless one is close to the critical point Hence the concept of local equilibrium is well justified in most practical circumstances For. solid surfaces above the roughening temperature, the concept also makes sense. Since the surface is rough adding (or removing) an atom to a particular part of the surface docs not disturb the local equilibrium state very much, and this sampling procedure can be used to determine the local chemical potential. This is the essence of the Gibbs-Thomson relation (1). 

For the kinetics of a reaction, it is critical to know the rough time to reach equilibrium. Often the term "mean reaction time," or "reaction timescale," or "relaxation timescale" is used. These terms all mean the same, the time it takes for the reactant concentration to change from the initial value to 1/e toward the final (equilibrium) value. For unidirectional reactions, half-life is often used to characterize the time to reach the final state, and it means the time for the reactant concentration to decrease to half of the initial value. For some reactions or processes, these times are short, meaning that the equilibrium state is easy to reach. Examples of rapid reactions include H2O + OH (timescale < 67 /is at... 

The steady-state solution that is an extension of the equilibrium state, called the thermodynamic branch, is stable until the parameter A reaches the critical value A,. For values larger than A<, there appear two new branches (61) and (62). Each of the new branches is stable, but the extrapolation of the thermodynamic branch (a ) is unstable. Using the mathematical methods of bifurcation theory, one can determine the point A, and also obtain the new solution, (i.e., the dissipative structures) in the vicinity of A, as a function of (A - A,.). One must emphasize that... 

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