Jump discontinuity

This shows very clearly that the specific heat has a jump discontinuity at J= T ... 

There is jump discontinuity in the specific heat as the temperature passes from below to above the critical temperature. 

A function is said to be piecewise continuous on an intei val if it has only a finite number of finite (or jump) discontinuities. A function/on 0 < f < oo is said to be of exponential growth at infinity if there exist constants M and Ot such that l/(t)l < for sufficiently large t. 

Generalization to piecewise smooth surfaces meeting at vertices or on edges on which the derivative suffers finite jump discontinuities may be made in the usual way. 

Fig. 17 shows the adsorption isotherms of all (undimerized and dimerized) particles. Except for a very fast increase of adsorption connected with filling of the first adlayer, the adsorption isotherm for the system A3 is quite smooth. The step at p/k T 0.28 corresponds to building up of the multilayer structure. The most significant change in the shape of the adsorption isotherm for the system 10, in comparison with the system A3, is the presence of a jump discontinuity at p/k T = 0.0099. Inspection of the density profiles attributes this jump to the prewetting transition in the... 

If point F in Fig. 2 is reached without physical burn-out occurring, then, as shown by Nukiyama, a further increase in heat flux will raise the surface temperature in the direction of E until physical burn-out does occur. If, however, the heat flux at point F is decreased, the surface temperature does not revert to the value at C, but moves along the curve towards D. On reaching D, it was observed by Nukiyama that the surface temperature undergoes another jump discontinuity along the dotted line DG, and stabilizes at G in the nucleate-boiling region. Both the transition lines CF and DG can be passed only in the direction shown by the arrows in Fig. 2. 

Slow burn-out tends to be associated with high-quality burn-out conditions and to produce a not unduly excessive wall-temperature rise. In fact, there appears to be an extreme condition in which the temperature rise may hardly be noticeable, and it becomes difficult to say whether burn-out has occurred. These circumstances probably coincide with the jump discontinuity in Fig. 3 ceasing to exist for certain values of system parameters. The condition is effectively one in which, at the burn-out point, the heat-transfer coefficient is the same whether the surface is vapor-blanketed or liquid-wetted. 

As a result, there is a jump discontinuity in the temperature at Z=0. The condition is analogous to the Danckwerts boimdary condition for the inlet of an axially dispersed plug-flow reactor. At the exit of the honeycomb, the usual zero gradient is imposed, i.e. 

Boltzmann s W-function is not monotonic after we perform a velocity inversion of every particle—that is, if we perform time inversion. In contrast, our -function is always monotonic as long as the system is isolated. When a velocity inversion is performed, the 7f-function jumps discontinuously due to the flow of entropy from outside. After this, the 7f-function continues its monotonic decrease [10]. Our -function breaks time symmetry, because At itself breaks time symmetry. 

When the lamellae are annealed at a given temperature, they thicken with time. The thickening is usually continuous and L increases logarithmically with time. However, there are several examples, where the lamellar thickness increases in a stepwise manner. For example, the initial lamella may contain chains each with four folds (five stems). As thickening process continues, the lamellar thickness jumps discontinuously to three folds, and so on. This phenomenon is referred to as quantized thickening [25]. 

These arguments do not hold in a CSTR because the conversion and temperature jump discontinuously fiom X = 0, T = Tq to X, T in the reactor and at the exit. Trajectories are continuous curves for the PFTR but are only single points for the CSTR. We wiU examine this in more detail in the next chapter. 

We conclude that the growth of a new phase is controlled by the rate of dissipation at a moving kink. This dissipation is taking place at the microlevel and must be prescribed in order for the macro-description to be complete. The incompleteness of the continuum model manifests itself through the sensitivity of the solution to the singular (measure-valued) contributions describing fine structure of the subsonic jump discontinuities (kinks). 

Evans C.M. Ablow, ChemRevs 61, 135-37 (1961) (The usual way of treating shocks is to idealize them to jump discontinuities, in this way taking into account the effect of... 

Accdg to Ref 66, p 135> the usual way of treating shocks is to idealize them to jump discontinuities, in this way taking into account the> effect of the irreversible process caused by friction and heat conditions. It is assumed that the flow involving such a discontinuous process is completely determin ed by the three lawstof conservation of mass, momentum and energy and the condition that the entropy does not decrease in the discontinuous process. Outside of the transition zone the flow is determined by the differential eqs 2.1.1, 2.2.2 2.2.3 listed on p 132 of Ref 66... 

A prediction of theory (Chapter 4) is that when an insulator-metal transition of either band-crossing or Mott type occurs through a change of composition in an alloy, the zero-temperature conductivity should jump discontinuously from zero to a finite value. This seems to be the case for the alloys with Ti203. The alloys with titanium have a conductivity when metallic of order 104 1 cm-1 at... 

The situation for a flaw initially at 75° to the draw direction is similar in that R achieves a maximum value as shown in Figure 8. However, the new orientation p of this flaw is continuous with draw ratio as shown in Figure 9 and does not display a jump discontinuity as does the flaw with P = 90°. Furthermore, the flaw never becomes circular regardless of draw ratio (R /R 5 in Figure 8). 

The front of the detonation is a jump discontinuity and therefore can be handled in the same manner as the one we used with nonreactive shock waves. 

The continuity requirement given by condition (3) is merely a restatement of the every day experience in the laboratory that, as a chemical reaction proceeds, the compositions change smoothly from one composition to another and do not jump discontinuously from one composition to another composition differing greatly from the first. Mathematically it is the requirement that a continuous line remains unbroken, and neighboring pomts remain neighboring points as the reaction proceeds. 

The -model possesses a line of glass transitions where the long time limit f = jumps discontinuously it obeys the equivalent equation to (21). The glass transition line is parameterized by (vJ, V2) = ((2A — with 0.5 < A < 1, and... 

If we try to avoid this non-uniqueness by restricting 0 to the range -zr < 0 < zr, then the velocity vector jumps discontinuously at the point corresponding to 0 = zr. Try as we might, there s no way to consider 0 = 0 as a smooth vector field on the entire circle. 

In practice, the voltage appears to jump discontinuously back to zero, but that is to be expected because [ln(/ - 7 )] has infinite derivatives of all orders at f (See Exercise 8.5.1.) The steepness of the curve makes it impossible to resolve the continuous return to zero. For instance, in experiments on pendula, Sullivan and Zimmerman (1971) measured the mechanical analog of the 7 - V curve—namely, the curve relating the rotation rate to the applied torque. Their data show a jump back to zero rotation rate at the bifurcation. 

Fig. 54. Schematic phase diagrams for wetting and capillary condensation in the plane of variables temperature and chemical potential difference, (a) Refers to a case in which the semi-infinite system at gas-liquid condensation (ftaKX — d = 0) undergoes a second-order wetting transition at T = 7V The dash-dotted curves show the first-order (gas-liquid) capillary condensation at p = jt(I), T) which ends at a capillary critical point T v, for two choices of the thickness D. For all finite D the wetting transition then is rounded off. (b), (c) refer to a case where a first-order wetting transition exists, which means that ps remains finite as T - T and there jumps discontinuous towards infinity. Then for /iaKX - /i > 0 a transition may occur during which the thickness of the layer condensed at the wall(s) jumps from a small value to a larger value ( prewelting ). For thick capillaries, this transition also exists (c) but not for thin capillaries because then /Jcnn - (D,T) simply is loo large.

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