Crossover time

The limitations and range of validity of the linear theory have been discussed in [17, 23, 24]- The linear approximation to equation (A3.3.54) and equation (A3.3.57) assumes that the nonlinear temis are small compared to the linear temis. As t[increases with time, at some crossover time i the linear... 

As is evident from the fomi of the square gradient temi in the free energy fiinctional, equation (A3.3.52). k is like the square of the effective range of interaction. Thus, the dimensionless crossover time depends only weakly on the range of interaction as In (k). For polymer chains of length A, k A. Thus for practical purposes, the dimensionless crossover time is not very different for polymeric systems as compared to the small molecule case. On the other hand, the scaling of to is tln-ough a characteristic time which itself increases linearly with k, and one has... 

In Fig. 20 we show the MSQ of a system of GM [66] with different mean chain lengths (depending on 7, cf. Eq. (12)) for three values of LO=l, 0.1, 0. 01. Since the individual chains have only transient identity, it is meaningless to discuss their center of mass diffusion. It is evident from Fig. 20 that the MSQ of the segments, g t) = ([x( ) - x(O)j ), follows an intermediate sub-diffusive regime, g(t) oc which is later replaced by conventional diffusion at some characteristic crossover time which grows... 

Fig. 24 NSE spectra from the PEP homopolymer (above) and the triblock (below) at 492 K. The solid lines are the result of a fit with the Ronca model [49] the dashed line presents the expected dynamic structure factor for Rouse relaxation corresponding to the highest Q-value. ( Q = 0.058 A"1 V Q = 0.068 A"1 Q = 0.078 A"1 A Q = 0.097 A"1 o Q = 0.116 A"1 Q = 0.116A 1). The arrows mark the crossover time xe. (Reprinted with permission from [39]. Copyright 1992 American Chemical Society, Washington)...

Q-dependence of the plateau levels. The results for the model parameters are given in Table 3. The crossover time xe = 15 ns (T = 492 K) calculated with Eq. (41b) agrees well with the observed spectral shape in dividing the initial fast decay from the plateau-like behavior at longer times. Figure 25 (lower part) compares the experimental spectra for times longer than xe with the local... 

We will take a somewhat different but equivalent criterion in order to describe the crossover. As the crossover time r, we take the Rouse relaxation time of a polymer section, spanning the tube diameter ... 

Fig. 4.16 Time evolution of the mean squared displacement (r ) (empty circle) at 363 K and the non-Gaussian parameter 2 obtained from the simulations at 363 K (filled circle) for the main chain protons of PL The solid vertical arrow indicates the position of the maximum of 2> At times r>r(Qinax)> the crossover time, a2 assumes small values, as in the example shown by the dotted arrows. The corresponding functions (r ) and a2 are deduced from the analysis of the experimental data at 320 K in terms of the jump anomalous diffusion model and are displayed as solid lines for (r )and dashed-dotted lines for a2- (Reprinted with permission from [9]. Copyright 2003 The American Physical Society)...

Characteristic Rouse times for chains with bending elasticity Terminal time for reptation Crossover time Rouse, local reptation Characteristic Rouse times for the all-rotational model Rouse time... 

Latora et al. [18] discussed a relation between the process of relaxation to equilibrium and anomalous diffusion in the HMF model by comparing the time series of the temperature and of the mean-squared displacement of the phases of the rotators. They showed that anomalous diffusion changes to a normal diffusion after a crossover time, and they also showed that the crossover time coincides with the time when the canonical temperature is reached. They also claim that anomalous diffusion occurs in the quasi-stationary states. 

These also show that our corrections are necessary to obtain lifetimes of ordered motions, because, without these corrections, the qualitative changes, and thus the clear crossover times, generally disappear. 

In this section, we have applied our novel method to the detecting ordered motions with the linear increase of singular values of stability matrixes. From the crossover times of local instabilities that change from linear to exponential increases, lifetimes of... 

X is the crossover time when the balance between the aggregation processes is established. The validity of this approach was confirmed by Montecarlo simulations of the full equation using a constant coagulation kernel and a breakup probability equal to k(i+j) where a and k were adjustable parameters [15]. Spatial fluctuations were compensated by cluster breakup and the generalized Smoluchowski equation had a critical dimension dc < 1. 

The maximum load current in our example is therefore 10 A, and is Idmax in Figure 5-2. If we plot the drain current and drain voltage with respect to time, we see that the crossover time, tcross, is 10 s here. Note that this time is by definition the time for both the voltage and the current to complete their transitions. 

If we now turn the mosfet OFF in the same way (with the crossover time kept fixed), we will get exactly the same energy loss term again, though this time with the voltage rising and the current falling. 

Note Here we seem to be indirectly suggesting that the drive resistance is the same for tum-on and turn-off. That need not be so. All the equations we will present can easily take any existing difference in the turn-on and turn-off drive resistances into account. So in general, we will have different crossover times for the turn-on and turn-off transitions. Also note, that in general, within a certain crossover interval (tum-on or turn-off), the actual time it takes for the voltage to transit need not be the same as the time the current takes (unlike the case of a resistive load). 

The crossover time, being the time during which both the current and voltage are transiting, is t2 + t3. As indicated, to know the driver dissipation, we need to consider the entire duration tl + t2 + t3 + t4. Note that by definition, at the end of t4, the gate voltage is at... 

One of the important issues is the possibility to reveal the specific mechanisms of subdiflFusion. The nonlinear time dependence of mean square displacements appears in different mathematical models, for example, in continuous-time random walk models, fractional Brownian motion, and diffusion on fractals. Sometimes, subdiffusion is a combination of different mechanisms. The more thorough investigation of subdiffusion mechanisms, subdiffusion-diffiision crossover times, diffusion coefficients, and activation energies is the subject of future works. 

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