Scaling phenomena regime

Frechet-Minkowski distance measure at the small scale, this distance approaches a constant value as the two lumps approach each other. This infinitesimal regime signals the devitation from the Euclidean geometry on small scales. In the asymptotic distance regime the Frechet-Minkowski distance becomes Euclidean distance. This phenomenon may play a significant role in the Planck regime. 

Realization of this three-layer regime for decision-making about an approaching natural disaster depends on the agreement between the spatiotemporal scales of the monitoring system and the respective characteristics of the natural phenomenon. Most difficult for decision-making are delayed action natural catastrophes which... 

Figure 14 Ti (p, T) vs. temperature for the solvent ethane at fixed density (the critical density) and the theoretically calculated curve. The scaling factor, frequency co, and the hard sphere diameters are the same as those used in the fit of the 34°C density-dependent data, i.e., there are no free parameters in this calculation. Notice the presence of an inverted regime, i.e., a range of temperatures for which the lifetime increases with temperature, contrary to expected behavior. The lifetime peaks at 375 K before decreasing with temperature. Remarkably, the theory captures this phenomenon, though it overestimates the drop in the lifetime with temperature after 375 K.

The phenomenon of CR is due to the presence of two different characteristic time scales in the system which are affected by noise in different manner. One is the time during which the system just fluctuates around the stationary state, which is needed to activate an excitation. We call this the activation time The second time scale, tlie excur.sion time te, is the typical duration of an excitation loop, compare Fig. 1.0. Noise of low intensity does not affect tg, compare panels B.l and B.2 of Fig. 1.4. Consequently, assuming std[te) small and ta tg excitations are I are events), we can write that c sTd Q)/ (Q). In this regime, the spikes are completely random events, so that sjd(ta)/(f ) = 1, and, as noise in-... 

The results for i p(r) obtained for different values of A, see Fig. 1.11, demonstrate that under a random telegraph signal the coherence of noise-induced excitation is enhanced by an optimal choice of the correlation time. Here, the optimal correlation time Topt decreases as the noise amplitude A increases. Further simulations not shown here, confirm that this phenomenon holds for a wide range of the bifurcation parameter o, covering almost the whole excitable regime. We emphasize that for not well separated time scales, noise-induced excitations are possible even if both cp-... 

To understand properly the relationship between the glass transition phenomenon observed in computer-simulated systems and that observed in laboratory systems, it is necessary to be familiar with the temperature dependence of the relaxation time. The point to be made is that the transition, which is the thermodynamic manifestation of a failure to maintain equilibrium during cooling, occurs sharply in laboratory systems but diffusely in simulated systems, primarily because of a great difference in relaxation time temperature (or volume) dependence in the time-scale regimes in which the processes are observed in the two cases. 

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