Non-monotonic behavior

Let us remember that Eqns. (12.22) and (12.23) have to be coupled to the diffusion equations in the a and 0 phases in order to complete the total set of kinetic equations for the phase transformation (Le., the advancement of the interface). This set is very complicated and nonlinear and may lead to non-monotonic behavior of vb and the chemical potentials of the components in space and time, as has been observed experimentally (Figs. 10-13 and 12-9). Coherency stresses and other complications such as plastic flow have been neglected in this discussion. 

The superconducting properties induced in the normal metal manifest themselves in many different ways, including energy-dependent transport properties and a modification of the local density of states. For instance, the conductance of a normal conductor connected to a superconducting electrode shows a striking re-entrant behavior [4]. At non-zero temperature and/or bias, the conductance of the normal metal is enhanced as compared to the normal-state. At zero temperature and zero bias, the expected conductance coincides with the normal-state value. The conductance has therefore a non-monotonous behavior. 

Consider now the frequency dependence of the third cumulant. We will be interested in the case of a good conductor where the charge-relaxation time tq is much shorter than the dwell time td- Unlike the second cumulant of current, the third cumulant P3(wi,w2) in general exhibits a strong dispersion at wy2 1/td [11], For symmetry reasons, this dispersion vanishes for symmetric cavities and cavities with two tunnel or two ballistic contacts. The shape of P3(wi, w2) essentially depends on the parameters of the contacts. In particular, for a cavity with one tunnel and one ballistic contact with equal conductances Gl = Gr = G it exhibits a non-monotonic behavior as one goes from lv1 = uj2 = 0 to high frequencies. A relatively simple analytical expression for this case may be obtained if td 3> tq and one of the frequencies is zero ... 

To conclude, in a BF mixture model of superconductivity we have extracted an analytic relation between a BEC Tc and the electron chemical potential that varies with the degree of CP formation. The BEC Tc formula contains contributions from both varying electron number as well as from the redistribution of ffee-electron states caused by boson formation. As a result Tc vs X exhibits non-monotonic behavior, so that superconductivity emerges only for a limited range of coupling-parameter and doping-concentration values. 

In both backtransformed EXAFS functions, it can be seen that the envelope of the EXAFS does not drop continuously with k, as was observed with backscat-terers of small atomic number (as oxygen, see Fig.4j, k). Here, we have elements with higher atomic numbers as backscatterers, where Ramsauer-Townsend resonances introduce non-monotonic behavior into the backscattering amplitude functions (see Sect 3.1). 

Later, Falkenhagen and co-workers and Onsager and Fuoss established a method of calculating parameter A starting from the Debye-Huckel theory. However, the above equation is only valid for concentrations up to about 0.01 mol/L. According to the above equation the relative viscosity should always increase with concentration. However, experiments show non-monotonic behavior for several electrolytes such as most of the potassium halides, and several mbidium and cesium halides [12]. 

The transition temperature non-monotonic behavior, namely minimum at thickness /zmin = l pRzjg, followed by increase at the smallest thicknesses for high values of flexoelectric coefficient /n are related to the third term in Eq. (4.19) that is inversely proportional to if-. Despite the term is negligible at higher thicknesses, its contribution to the transition temperature dominates over the second... 

Transition dipole moments are a lot more sensitive to the level of theory. Basis sets are important for the transition dipole moments however, the effect is more difficult to monitor. Plotting the transition dipole moments versus 1/n in most cases gives non-monotonic behavior, so an extrapolation is not possible. Diffuse functions are also... 

Fig. 3.4. The density of states for a non-ideal glassy structure. Non-monotonic behavior is expected.

Norton and Vlachos [30] analyzed the variation in the Nu profile along the axial length of a parallel plate microburner. Figure 10.9 shows the Nu profile for a stoichiometric propane-air mixture. Nu displays a non-monotonic behavior, with a decrease near the entrance and a jump or a discontinuity at the light-off point Such a discontinuity is often termed a new entrance effect because it qualitatively mimics the entrance effect, i.e. an exponential decrease until Nu reaches an asymptotic limit followed by a jump. These results are similar to those reported in [31] with a thin-wall approximation. The thin-wall approximation is unsuitable in a microbumer because the walls play a major role via axial heat recirculation and affect the gas-phase transport due to the close gas-solid coupling. 

Figure 11 Time dependence of Cartesian component coupiing of Rouse modes for p= 1 (a) and p= 9 (b). An aitemative definition C shows more pronounced non-monotonic behavior for p= 9 (c).

An interesting effect has been reported at very dilute electrolyte concentration. Ray and Jones observed a non-monotonous behavior of the... 

Comparing the results of the simulations corresponding to the several values of A with the results of the monomeric shoulder-dumbbell simulations, we see an unexpected non-monotonic behavior. The effect of the introduction of a rather small anisotropy due to the dimeric nature of the particle (small values of the interparticle separation A) leads to the increase of the size of the regions of anomalies (de Oliveira et al., 2010). Nevertheless, the increase of A shrinks those regions. 

For modification of the surface potential the surface of the Silicon wafer was replaced by Mica surface. This increases the surface potential from about —70 to about —150 mV. This increases the amplitude of the force oscillation. Due to the electrostatic repulsion between the nanoparticles from the outer walls the strength of particle ordering increases. Monte Carlo simulations showed that the picture is not as simple. A non-monotonous behavior was observed as shown in Fig. 5. 

The binodals of the liquid-vapor phase coexistence as a function of molar fraction and temperature resemble the binodals of a one-component system as a function of density and temperature At high temperature, there is a critical point. Upon decreasing temperature the polymer-rich phase becomes more concentrated in polymer, while the solvent concentration increases in the vapor phase. The spinodal of the polymer liquid, however, exhibits a non-monotonous temperature dependence of the composition. This dependence is parallel to the non-monotonous behavior of the nucleation barrier as we increase temperature. In fact, at the pressure considered, and even more so at lower pressures (cf. Fig. 20), there exists an extended temperature region, where the polymer-fraction at the spinodal of the liquid decreases upon increasing temperature. 

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