Marginal stability model

The marginal stability model is attractive in its explanation of the great variety of snowflake shapes. It is somewhat less convincing in explaining the repetitiveness of the minute variations in all six directions since the micro-environmental changes may occur also across the snowflakes themselves and not only between the spaces assigned to different snowflakes. 

According to Langcr s marginal stability model [2-15], the snow crystal may start with a relatively stable shape. The crystal may, however, be easily destabilized by a small perturbation. A rapid process of crystallization from the surrounding water vapor ensues. The rapid growth gradually transforms the... 

In Example 6.4, when there was no model uneertainty, K for marginal stability was 8, and for a gain margin of 6dB, K was 4. In this example with model uneertainty, from equation (9.154) marginal stability oeeurs with K = 3.5, so this is the maximum value for robust stability. For robust performanee, equation (9.150) applies. For a speeifie step input let lV(s) = 1 /s now... 

U-shaped energy landscape arise according to the tube model of Banavar and Maritan (2003a, b) from a novel phase of matter in the vicinity of a phase transition in which the folds arise (Banavar and Maritan, 2003b see also Chapter 12, this volume). The tube model also implies that few other polymers may exist that will exhibit discrete, stable, folded conformations associated with their characteristic marginal stability (J. R. Banavar, personal communication). 

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...

That means one has a marginal spin-glass transition in the SK model (Bray and Moore 1979), and the marginal stability of the spin-glass phase holds for all temperatures below Tf (Sompolinsky 1981). Sompolinsky and Zippelius (1981,1982) obtain an exponent v which decreases with temperatures below T. ... 

Driven nonlinear systems often tend to develop spatially periodic patterns. The underlying mathematical models usually permit a continuous set of linearly stable solutions. As a possible mechanism of selecting a specific pattern the principle of marginal stability is presented, being applicable to situations, where a propagating front leaves a periodic structure behind. We restrict our discussion to patterns on interfaces which are more easily accessible than three-dimensional structures, for example in hydrodynamic flow. As a concrete system a recently analyzed model for dendritic solidification is discussed. 

If it is supposed for convenience that e is an externally variable parameter of the model, then for n > nc fhe transition between stable branches of the deterministic solution should occur discontinuously at the points of marginal stability (arrows. 

Frequency methods can give us the relative stability (the gain and phase margins). In addition, we could construct the Bode plot with experimental data using a sinusoidal or pulse input, i.e., the subsequent design does not need a (theoretical) model. If we do have a model, the data can be used to verify the model. However, there are systems which have more than one crossover frequency on the Bode plot (the magnitude and phase lag do not decrease monotonically with frequency), and it would be hard to judge which is the appropriate one with the Bode plot alone. 

Hydrate formation from gas dissolution of rising water Hyndman and Davis (1992) proposed that as methane-unsaturated water rises, it becomes saturated at lower pressures. As the saturated (or supersaturated) water passes through the phase stability zone, hydrate formation occurs without a free gas zone. This model results in a maximum hydrate concentration at the three-phase (BSR) boundary with a successively lower hydrate amounts above the BSR as was shown to be the case in Cascadia Margin ODP Drill Sites 889 and 890 by Hyndman et al. (1996). 

Quantitative models and numerical computations are and will continue to be central in the implementation of model-predictive feedback controllers. Unfortunately, numerical computations alone are weak or not robust in answering questions such as the following How well is a control system running Are the disturbances normal Why is derivative action not needed in a loop What loops need dead time compensation, and should it be increased or reduced Have the stability margins of certain loops changed and, if so, how should the controllers be automatically retuned ... 

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