Variables fast-relaxing

Obviously, the above algorithms are not suitable when transients of the finer scale model are involved (Raimondeau and Vlachos, 2000), as, for example, during startup, shut down, or at a short time after perturbations in macroscopic variables have occurred. The third coupling algorithm attempts fully dynamic, simultaneous solution of the two models where one passes information back and forth at each time step. This method is computationally more intensive, since it involves continuous calls of the microscopic code but eliminates the need for a priori development of accurate surfaces. As a result, it does not suffer from the problem of accuracy as this is taken care of on-the-fly. In dynamic simulation, one could take advantage of the fast relaxation of a finer (microscopic) model. What the separation of time scales between finer and coarser scale models implies is that in each (macroscopic) time step of the coarse model, one could solve the fine scale model for short (microscopic) time intervals only and pass the information into the coarse model. These ideas have been discussed for model systems in Gear and Kevrekidis (2003), Vanden-Eijnden (2003), and Weinan et al. (2003) but have not been implemented yet in realistic MC simulations. The term projective method was introduced for a specific implementation of this approach (Gear and Kevrekidis, 2003). 

Throughout this volume we shall apply the Zwaimg approach mainly to cases where the a time scale is fairly well separated from the b time scale. In other words, the variables b will be assumed to be fast-relaxing, that is, pre-... 

Following the notation of Section III we can identify as a fast-relaxing variable and, by solving Eq. (3.7), determine ... 

In this problem, in ad tion to discussed in Section IV, we have another time scale, Let us assume that our experimental apparatus only allows us to observe long-time regions corresponding to r yr/, where is the mechanical time scale introduced in provision (i) of Section V.A. In consequence, the dynamics induced by the interplay of inertia and external noise belongs to the short-time region. In such a limit T,y - Ty, both and v play the role of fast-relaxing variable while x is our variable of interest. By applying our AEP, we then recover the corrections to the diffusional approximation due to the non-white external noise. 

Analytic solutions of Eq. (1.16) cannot be obtained, and thus one must make some approximations. In many physical systems, y and X are very large (y 00, X oo) therefore both v and f can be considered as fast relaxing variables and eliminated by an adiabatic elimination procedure (AEP). The AEP of Chapter II allows us to derive from Eq. (1.16) the equation of motion of a(x t) regarded as being the contracted distribution ... 

The naive approximation on Eq. (S.9) fails because v(t) cannot be considered such a fast relaxing variable with respect to x(t) as to assume v = 0 in Eq. (5.7). The presence of a mixing term - yxv would lead us to employ AEP with some caution. We can get rid of v(t) as promised, but only when D can be considered small. The condition of a large y imposed by Smoluchowski is no longer enough for treating our system. 

If the momentum L, itself is considered as a fast relaxing variable, that is, the motion of the solute is supposed to be completely diffusive, then it... 

Check it 5.2.1.6 also shows that it is possible to include relaxation induced effects into simulations which might be a source of perturbation in a real experiment. The overall pulse sequence time, especially in the long sequences used in nD experiments, becomes a very important, restrictive variable in the development of new pulse sequences. For fast relaxing molecules such as macromolecules the overall pulse sequence time is crucial because if the nuclei start to relax before the data acquisition there will be a loss in signal intensity. It is also important that for maximum signal intensity the spin system returns to thermal equilibrium between scans. 

Other experimental techniques have been used to study the very fast relaxation of dye molecules in solution. Ricard and Ducuing studied rhodamine molecules in various solvents and observed vibrational rates ranging from 1 to 4 ps for the first excited singlet state. Their experiment consisted of two pulses with a variable delay time between them the first excites molecules into the excited state manifold and the second measures the time evolution of stimulated emission for different wavelengths. Ricard found a correlation between fast internal conversion and vibrational relaxation rates. Laubereau et al. found a relaxation time of 1.3 0.3 ps for coumarin 6 in CCI4. They used an infrared pulse to prepare a well-defined vibrational mode in the ground electronic state, and monitored the population evolution with a second pulse that excited the system to the lowest singlet excited state, followed by fluorescence detection. 

Let us now compare the theory as set forth above with experiment. Using the parameter values typical of the squid giant axon gNa = 0.12 mmho/cm, Tm = 2 ms, Co = 1 yxF/cm, = 30 mV, at 18.5 C the parameter a is 0.0446. The experimental value V = 0.274 found for this a value is shown with an asterisk in Figure 10. The value of V calculated from Eq. (48) is 0.3 and that found from Eq. (51) is 0.39. If we compare these figures with V = 1 calculated for the fast relaxation of the variable m, it will become clear that the limiting case of slow relaxation is much closer to reality than rapid relaxation. 

Correct interpretation of variable temperature relaxation time measurements at more than one spectrometer operating frequency can lead to characterization of a dynamical process (e.g. for fast self diffusion, a continuous increase is seen in T2 at high temperatures and T2 7 above the minimum in 7, i.e. for D > 10 cm s ). Results for phases with slower motions can be ambiguous due to sample decomposition at higher temperatures. 

Transverse relaxation of musculature is relatively fast compared with many other tissues. Measurements in our volunteers resulted in T2 values of approximately 40 ms, when mono-exponential fits were applied on signal intensities from images recorded with variable TE. More sophisticated approaches for relaxometry revealed a multi-exponential decay of musculature with several T2 values." Normal muscle tissue usually shows lower signal intensity than fat or free water as shown in Fig. 5c. Fatty structures inside the musculature, but also water in the intermuscular septa (Fig. 5f) appear with bright signal in T2-weighted images. 

The constant-pressure formulation results in a high-index problem even with implementation of instead of A as the dependent variable. This results from the absence of any time derivatives for u. Attempts to solve the methane-ignition problem with the constant-pressure formulation were generally unsuccessful, except by significantly relaxing the error control on A and u. Even then, while the solutions appear generally correct, they exhibit unstable behavior near fast transients, particularly on A. 

Whether the relaxation is fast or slow at 4.2 K can be checked experimentally, using the following arguments. Figures 2.6b shows a 4.2 K Mossbauer spectrum recorded for B = 8.0 T. The solid line outlines the contribution of Z. The remaining absorption pattern, well understood, originates from the [Feni(H20)6]3 + contaminant. The sharpness of the absorption lines of Z shows that the intermediate has nearly axial symmetry thus, we can set EID = 0 in Equation 2.2 and Ax = Ay in Equation 2.3. We have recorded, and published, spectra in variable applied fields and at different temperatures.11 Since the spectra are a bit noisy, it will aid the reader if we... 

Here, w = m, n, and S. V represents the membrane potential, n is the opening probability of the potassium channels, and S accounts for the presence of a slow dynamics in the system. Ic and Ik are the calcium and potassium currents, gca = 3.6 and gx = 10.0 are the associated conductances, and Vca = 25 mV and Vk = -75 mV are the respective Nernst (or reversal) potentials. The ratio r/r s defines the relation between the fast (V and n) and the slow (S) time scales. The time constant for the membrane potential is determined by the capacitance and typical conductance of the cell membrane. With r = 0.02 s and ts = 35 s, the ratio ks = r/r s is quite small, and the cell model is numerically stiff. The calcium current Ica is assumed to adjust immediately to variations in V. For fixed values of the membrane potential, the gating variables n and S relax exponentially towards the voltage-dependent steady-state values noo (V) and S00 (V). Together with the ratio ks of the fast to the slow time constant, Vs is used as the main bifurcation parameter. This parameter determines the membrane potential at which the steady-state value for the gating variable S attains one-half of its maximum value. The other parameters are assumed to take the following values gs = 4.0, Vm = -20 mV, Vn = -16 mV, 9m = 12 mV, 9n = 5.6 mV, 9s = 10 mV, and a = 0.85. These values are all adjusted to fit experimentally observed relationships. In accordance with the formulation used by Sherman et al. [53], there is no capacitance in Eq. (6), and all the conductances are dimensionless. To eliminate any dependence on the cell size, all conductances are scaled with the typical conductance. Hence, we may consider the model to represent a cluster of closely coupled / -cells that share the combined capacity and conductance of the entire membrane area. 

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