Critical-Like Behavior

The fact that the decay becomes arbitrarily slow as p —> 0 indicates the presence of a critical point, p = 0. (Another example of critical-like behavior in a probabilistic system is given later in this subsection). A more detailed discussion of various critical-like properties of CA appears in chapter 7. 

Note that the quantity (1- Q D)) can be thought of as a disorder parameter distinguishing the different regimes of behavior. Numerical measurements of this disorder parameter around the transition point at. which the zigzag pattern collapses spontaneously (i.e. around etc 3.92) suggest a critical-like behavior [kaneko89a] ... 

The concentration fluctuations in our system can possibly further be subdivided into a solute (surfactant aggregates) concentration fluctuation and a solvent concentration fluctuation. The solute concentration fluctuation is similar to that of critical-like behavior observed in many W/0 microemulsion systems (28), while the solvent... 

The most pronounced evidence of the critical-like behavior in these compounds can be found by using the nonlinear dielectric effect (NDE). NDE is coupled via the 4-point correlation to multimolecular heterogeneities -fluctuations. It was shown in refs. that ... 

Because the latent heat at the N-I transition is very small and since the orientational interactions between liquid crystal molecules are short range, one might expect many physical properties to display critical-like behavior, described by power laws with the appropriate critical exponents. These fluctuation effects are, indeed, observed in many properties including the specific heat capacity. Figure 9 gives the temperature dependence of Cp near the N-I transition for the same substance as in Fig. 8. No data points are displayed in the coexistence region, whose width is indicated by the narrow box around the arrow marking N-I-... 

Possible gene-interactive alterations in 5-HT function and BDNF expression was recently evaluated in mice with a combined manipulation of the genes for 5-HTT and BDNF. Male but not female 5-FlTT -BDNF DKO mice showed further decreases in brain 5-HT concentrations as well as further increases in anxiety-like behavior and stress reactivity compared to 5-HTT -BDNF controls (Murphy et al. 2003). These findings support the notion of critical role of gene-gene interaction in brain plasticity related to anxiety and related disorders. 

Pawlak R, Magarinos AM, Melchor J, McEwen B, Strickland S (2003) Tissue plasminogen activator in the amygdala is critical for stress-induced anxiety-like behavior. Nat Neurosci 6 168-174... 

The same equations, albeit with damping and coherent external driving field, were studied by Drummond et al. [104] as a particular case of sub/second-harmonic generation. They proved that below a critical pump intensity, the system can reach a stable state (field of constant amplitude). However, beyond the critical intensity, the steady state is unstable. They predicted the existence of various instabilities as well as both first- and second-order phase transition-like behavior. For certain sets of parameters they found an amplitude self-modula-tion of the second harmonic and of the fundamental field in the cavity as well as new bifurcation solutions. Mandel and Erneux [105] constructed explicitly and analytically new time-periodic solutions and proved their stability in the vicinity of the transition points. 

Figure 2.10a displays details of some Van der Waals PV isotherms in the near-critical region. As shown in the figure, the isotherms below Tc exhibit pronounced oscillatory loops, quite unlike the experimental plateau-like behavior shown in Fig. 2.8. Indeed, no simple cubic equation could exhibit true plateau-like behavior, and the near-horizontal looping pattern in Fig. 2.10a is apparently the best that a cubic polynomial can do to represent such a flattened region. [The fact that the Van der Waals P = P(V) isotherm is a cubic polynomial will be apparent from expanding (2.13).]...

The transition to the continuum fluid may be mimicked by a discretization of the model choosing > 1. To this end, Panagiotopoulos and Kumar [292] performed simulations for several integer ratios 1 < < 5. For — 2 the tricritical point is shifted to very high density and was not exactly located. The absence of a liquid-vapor transition for = 1 and 2 appears to follow from solidification, before a liquid is formed. For > 3, ordinary liquid-vapor critical points were observed which were consistent with Ising-like behavior. Obviously, for finely discretisized lattice models the behavior approaches that of the continuum RPM. Already at = 4 the critical parameters of the lattice and continuum RPM agree closely. From the computational point of view, the exploitation of these discretization effects may open many possibilities for methodological improvements of simulations [292], From the fundamental point of view these discretization effects need to be explored in detail. 

The control of electron transfer is a critical issue in the fabrication of molecular electronic devices from the viewpoint of electronic circuit formation however, electron transfer processes of redox polymer-coated electrodes fabricated using a conventional polymer-coating method usually shows a diffusion-like behavior because the redox sites are randomly distributed in the polymer film (Fig. la) 17-20 consequently, it is difficult to control the electron transfer direction in three dimensions. 

Figure 6-5 shows the evolution of the dynamic moduli for a LM pectin/caldum system in the vicinity of the gel point as a function of the aging time. The evolution of the dynamic moduli was similar to that one observed as a function of the calcium concentration. In the initial period of aging the system showed the typical liquid-like behavior. Then both moduli increased with time, G increasing more rapidly than G" and with lower dependence on frequency. After 15 hr of aging, the system was just above the critical gel point, with a viscoelastic exponent A in the range of0.65-0.68. After the gel point, G passed beyond G", first in the lower frequency range where one can observe the initial formation of the elastic plateau. 

Figure 3. The linearized derivative-based plot (see eq. (3)) showing the validity of the critical-like MCT behavior for dc conductivity (the main part of the plot) and dielectric relaxation time (the inset). Values of critical MCT exponents are given in the plot. For both magnitudes the singular temperature TyqQp =215K + 3K. ...

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