Infinity catastrophe

As 100%) RH is approached, the BET theory predicts that the amoimt of water adsorbed will be infinite. As the influence of the solid surface does not extend indeflnitely, it should be expected that a finite number of molecular layers would be adsorbed (presuming that the solid is not soluble.) Many authors attribute the infinity catastrophe of the BET theory to the lack of consideration of lateral adsorbate interactions. ° ... 

S(p) = e 2plL(p) similar to e 2, leading to the infinity catastrophe with increasing p. 

Therefore R(r) will become infinite as r goes to infinity and will not be quadrati-cally integrable. The only way to avoid this infinity catastrophe (as in the harmonic-oscillator case) is to have the series terminate after a finite number of terms, in which case the e factor will ensure that the wave function goes to zero as r goes to infinity. Let the last term in the series be Then, to have 6 +, b +2,... all vanish, the fraction multiplying bj in the recursion relation (6.86) must vanish when j = k we have... 

The sudden increase of D toward infinity at a certain critical value of nVSkT has been referred to as the 4rr//3 catastrophe arising from the presence of such a term in the expression for the electric field strength... 

The phenomena of ignition and extinction of a flame are typical examples of discontinuous change in a system under smooth variation of parameters. It is natural that they have played a substantial role in the formation of one of the branches of modern mathematics—catastrophe theory. In Ya.B. s work it is clearly shown that steady, time-independent solutions which arise asymptotically from non-steady solutions as the time goes to infinity are discontinuous. It is further shown that transition from one type of solution to the other occurs when the first ceases to exist. The interest which this set of problems stirred among mathematicians is illustrated by I. M. Gel fand s... 

When the condition (RIV) = 1 is fulfilled, the refractive index/ dielectric constant goes to infinity, and we have a NM-M transition. The Herzfeld criterion when applied to metal-ammonia solutions does indeed predict (67,93) that localized, solvated electrons are set free by mutual action of neighboring electrons at metal concentrations above 4-5 MPM, and measurements (Fig. 20) similarly indicate a dielectric catastrophe in this concentration range. The simple Herzfeld picture has recently been applied by Edwards and Sienko (70) to explain the occurrence of metallic character in the Periodic Table. 

Rayleigh found the v2 dependence, Jeans later supplied the rest). Their distribution function g(v) increases as v2, with no provision for a fall-off to zero as the frequency and the energy go to infinity ("ultraviolet catastrophe"). 

Figure 3 Schematic of a Morse function and the related harmonic, cubic, and quartic potentials (Eqs. [3] and [4]). When the bond length is increased beyond the point of the minimum, the harmonic potential rises too steeply. The cubic term corrects for the anharmonicity locally, but at longer distances turns and goes catastrophically to negative infinity. The quartic potential remains a good approximation over a relatively large range and is always attractive at large distances.

When a continuous change in the control parametr b results in exceeding the value b0 = 1/4, we have a loss of stability by the fixed point x (1). The new stable fixed point, x (2) = 1 — (1/4b), close to x,(1) for b close to b0, appears. Such a catastrophe is called bifurcation. Catastrophic behaviour of the process (3.83) for b > b0 is revealed in the fact that the solution (3.87) for b = b0 + e diverges to infinity for an arbitrarily small positive e. 

The analysis of linearized sytem thus allows, when conditions (1)—(3) are met, us to find the shape of phase trajectories in the vicinity of stationary (singular) points. A further, more thorough examination must answer the question what happens to trajectories escaping from the neighbourhood of an unstable stationary point (unstable node, saddle, unstable focus). In a case of non-linear systems such trajectories do not have to escape to infinity. The behaviour of trajectories nearby an unstable stationary point will be examined in further subchapters using the catastrophe theory methods. 

The theory now proceeds as developed in Sections V and VI, essentially unchanged. For example, P v) will have the same bimodal structure as shown in Fig. 14, but will now be continuous. Similar smoothing of all artificially introduced discontinuities will not affect the theory in any essential way. The loss of a sharp distinction between liquid- and solidlike cells could vitiate use of the percolation theory. The nonanalyticity in S will certainly be lost, leading to a communal entropy for which 9S/9p is always less than infinity. However, the first-order phase transition should be preserved, just as it was for most of the parameter space even when )3> 1. The discontinuity in p and v would be reduced, as would be the latent heat. One important effect of this smearing will be the appearance of a critical end point for the liquid, a temperature below which the liquid phase is no longer even metastable. The second-order transition, which is only a small region of parameter space for /8> 1, is now wiped out completely by the restoration of analyticity. Our theory thus leads to a first-order phase transition or no transition at all. However, the entropy catastrophe can be resolved within our theory only if a transition occurs. 

This unwelcome discovery is potentially catastrophic for our unified weak and electromagnetic gauge theory. There we have lots of gauge invariance, many conserved currents, both vector and axial-vector, and hence many Ward identities. Moreover the Ward identities play a vital role in proving that the theory is renormalizable. It is the subtle interrelation of matrix elements that allows certain infinities to cancel out and render the theory finite. Thus we cannot tolerate a breakdown of the Ward identities, and we have to ensure that in our theory these triangle anomalies do not appear. 

FIGURE 9.13 Early attempts at modeling the behavior of a blackbody included the Rayleigh-Jeans law. But as this plot illustrates, at one end of the spectrum the calculated intensity grows upward to infinity, the so-called ultraviolet catastrophe. 

A totally diflFerent situation becomes possible in the case where the system does not have a global cross-section, and when is not a manifold. In this case (Sec. 12.4), the disappearance of the saddle-node periodic orbit may, under some additional conditions, give birth to another (unique and stable) periodic orbit. When this periodic orbit approaches the stability boundary, both its length and period increases to infinity. This phenomenon is called a hlue-sky catastrophe. Since no physical model is presently known for which this bifurcation occurs, we illustrate it by a number of natural examples. 

Since the return time from/to the cross-section S (i.e. the period of L j) grows proportionally to cj(/i), it must tend to infinity as /i —H-oo (see Sec. 12.2 if L is a simple saddle-node, then the period grows as tt/V/IZ ). Since the vector field vanishes nowhere in 17, it follows that the length of must tend to infinity also. Since L, does not bifurcate when /x > 0, we have an example of the blue sky catastrophe [152]. 

The fourth and last situation corresponds to the blue sky catastrophe , i.e. when both period and length of the periodic orbit go to infinity upon approaching the stability boundary. This boundary is distinguished by the existence of a saddle-node periodic orbit under the assumption that all trajectories of the unstable set W ( ) return to as t -> -hoc, where W C ) n — 0. The tra-... 

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