Marginal stability state

Thus for a fluid layer heated from below, a is real, and we can simply set a = 0 at the neutral state, and this greatly simplifies the problem of finding Rac corresponding to the marginal stability state, when a simple analytic solution is not possible. 

Fig. 5. Steady-state solution of deterministic rate equations to which a stochastic term has been added. Low noise level (mean absolute magnitude of fluctuations). Note increase in noise level near lower marginal stability point.

Thermodynamic equilibrium states correspond only to the stable or marginally stable states ( 0) of the mechanical analog. The first law of thermodynamics establishes the thermodynamic potential, while the second law of thermodynamics establishes the stability condition, as discussed in Chapter 5. 

In this equation, the values of r as coefficients of A vanish and correspond to a transition point. Beyond this point, the real part Ar of the roots A] and A2 changes its sign and hence the system becomes unstable, and at the marginal state, we have A, I A2 0. Using Eq. (e), the marginal stability condition becomes... 

Quality windows are also delineated by set values of the damping coefficients and attenuation factors that are computed in stability and frequency analysis. These too can be traced out efficiently in parameter space by augmented continuation schemes. The same is true of the turning points and bifurcation points in parameter space, points of marginal stability. These are the guides to situations in which there is more than one stable operating state. When such situations may arise, it becomes desirable to solve repeatedly the full equation system of flow for transient behavior in order to know how different start-up procedures and upsets select among the multiple stable states. 

The fact that an infinity of front velocities occurs for pulled fronts gives rise to the problem of velocity selection. In this section we present two methods to tackle this problem. The first method employs the Hamilton-Jacobi theory to analyze the dynamics of the front position. It is equivalent to the marginal stability analysis (MSA) [448] and applies only to pulled fronts propagating into unstable states. However, in contrast to the MSA method, the Hamilton-Jacobi approach can also deal with pulled fronts propagating in heterogeneous media, see Chap. 6. The second method is a variational principle that works both for pulled and pushed fronts propagating into unstable states as well as for those propagating into metastable states. This principle can deal with the problem of velocity selection, if it is possible to find the proper trial function. Otherwise, it provides only lower and upper bounds for the front velocity. 

Consider the 230 K curve in detail for a system initially in steady state A. As Iq increases increases until the marginal stability point B is reached where d jdlQ 00. Further increase of 1 requires an abrupt jump in I to point C if a steady state is to be maintained. Yet higher-power levels again lead to a gradual increase of. Now consider reducing the radiation... 

If uniformity were established one would still expect deviations from the ideal hysteresis loop. Fluctuations about the steady state could induce transitions from one branch to the other before the marginal stability point is reached. Such behavior is analogous to nucleation of a supercooled liquid freezing then occurs abruptly before the limit of metastability is reached. Referring to Fig. 7.9, one might wonder whether there could be an analog to the Maxwell construction in the van der Waals fluid. While a definitive answer is not yet available, the tentative answer is yes. Since... 

Fig. 1.2. Stationary states of the Schldgl model with fixed reactant and products pressures. Plot of the pressure of the intermediate px vs. the pump parameter (pa/pb). The branches of stable stationary states are labeled a and y and the branch of unstable stationary states is labeled p. The marginal stability points are at Fi and F3 and the system has two stable stationary states between these limits. The equistability point of the two stable stationary states is at F2...

At a critically steady (stationary) state the left and right marginal stability points coincide. 

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