Widely separated time scales

This is possible when one has widely separated time scales such that the relaxation time is by far the longest characteristic time involved in the time evolution of the system. 

Using Eq. (43) with a suitable distribution function, time constants of the p-process can be extracted from experimental susceptibility spectra in the glassy state (T < Tg). However, above Tg, where both a- and p-process are present, the spectral shape analysis becomes more involved. Taking into account that also fast (ps) relaxational and vibrational dynamics are present (cf. Section IV.B), the correlation function of a type B glass former near Tg is a three-step function, reflecting the dynamics occurring on different widely separated time scales. This is schematically shown in Fig. 34. 

This analysis shows that the limit cycle has two widely separated time scales. the crawls require Ar O(/r) and the jumps require Ar Oiji ). Both time scales are apparent in the waveform of x(r) shown in Figure 7.5.2, obtained by numerical integration of the van der Pol equation for /r=10 and initial condition (Xo.yo) = (2.O). 

Traditionally the two-pulse photon-echo of a two-level system is described in terms of dynamics where there is a separation of the frequency fluctuations into two widely separated time scales, one of which is much faster and the other much slower than the time that characterizes the inhomogeneous distribution of frequencies. This gives rise to a hxed distribution of homogeneously broadened transitions for each spectral transition of the solute. The echo electric held generated from two very short pulses interacting with a molecule but separated by an interval x is, apart from constant factors, given by... 

Coming to the role of the intrinsic parameters, the general trend is that bimodality is enhanced for the parameter range for which the deterministic evolution displays two widely separated time scales. On the other hand -and this leads us to the role of initial conditions- to "probe such a time scale difference the system has to start from a state located sufficiently before the inflexion point of the deterministic potential (cf. Fig. 4) Otherwise it undergoes a rapid relaxation to the final state following essentially the deterministic path. 

Often different motions with widely separated time scales are present. The fastest motions, for which the extreme narrowing limit is fulfilled, then lead to preaveraged line shapes, which are still sensitive to additional, now somewhat slower, motions with correlation times of the order of the inverse width of the preaveraged spectrum. A typical example, where such stepwise averaging is assumed, is the surfactant/water system. Local motions lead to strongly narrowed lines (by a factor of 10 for surfactant molecules and a factor of 100 for water), which can be narrowed further by diffusion along the hydrophobic-hydrophilic interface if this involves reorientations. This explains why the spectra of lamellar and hexagonal phases differ substantially in width, by a factor of approximately two Diffusion... 

This complex system would be difficult to solve directly. However, the problem is separable by taking advantage of the widely different time scales of conversion and deactivation. For example, typical catalyst contact times for the conversion processes are on the order of seconds, whereas the time on stream for deactivation is on the order of days. [Note Catalyst contact time is defined as the volume of catalyst divided by the total volumetric flow in the reactor at unit conditions, PV/FRT. Catalyst volume here includes the voids and is defined as WJpp — e)]. Therefore, in the scale of catalyst contact time, a is constant and Eq. (1) becomes an ordinary differential equation ... 

The main contributions to the frequency-time correlation function are assumed to be, as in the earlier works [123, 124], from the vibration-rotation coupling and the repulsive and attractive parts of the solvent-solute interactions. In several theories, the (faster) repulsive and the (slower) attractive contributions are assumed to be of widely different time scales and are treated separately. However, this may not be true in real liquids because the solvent dynamic interactions cover a wide range of time scales and there could be a considerable overlap of their contributions. The vibration-rotation coupling contribution takes place in a very short time scale and by neglecting the cross-correlation between this mechanism and the atom-atom forces, they... 

The basic assumptions in fluid mechanics are thus that for lengths and time scales much larger than the characteristic molecular lengths and times, the continuum representation provides a quantitative correct description of the fluid dynamic behavior of the system. In general, the differential description is useful for processes where there is a wide separation of scales between the smallest macroscopic scales of interest and the microscopic scales associated with the internal structure of the fluid. The mean free path which is of the order of for gases is commonly used as a suitable characteristic... 

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-... 

Essentially, each of the above systems has two widely different time scales. If the initial transient is not of interest, the systems can be projected onto a one-dimensional subspace. The subspace is invariant in that no matter where one starts, after a fast transient, all trajectories get attracted to the subspace in which A and B are algebraically related to each other. In essence, what one achieves is dimension reduction of the reactant space through time scale separation. For large, complex systems sueh as oil refining, it is difficult to use the foregoing ad hoc approaches to reduce system dimensionality manually. Computer codes are available for mechanism reduction by means of the QSA/QEA and sensitivity analysis. ... 

The properties of membranes commonly studied by fluorescence techniques include motional, structural, and organizational aspects. Motional aspects include the rate of motion of fatty acyl chains, the head-group region of the phospholipids, and other lipid components and membrane proteins. The structural aspects of membranes would cover the orientational aspects of the lipid components. Organizational aspects include the distribution of lipids both laterally, in the plane of the membrane (e.g., phase separations), and across the membrane bilayer (phospholipid asymmetry) and distances from the surface or depth in the bilayer. Finally, there are properties of membranes pertaining to the surface such as the surface charge and dielectric properties. Fluorescence techniques have been widely used in the studies of membranes mainly since the time scale of the fluorescence lifetime coincides with the time scale of interest for lipid motion and since there are a wide number of fluorescence probes available which can be used to yield very specific information on membrane properties. 

Delgado et aL recently demonstrated that time-scale separation is an effective way to localize metabolic control to only a few enzymes. They considered model pathways in which the eigenvalues of the Jacobian of the system are widely separated (i.e., systems with time-scale separation). Their treatment assumes the system possesses a unique, asymptotically stable steady-state and that the reaction steps of the system under analysis are... 

Analytical chromatographic options, based on linear and nonlinear elution optimization approaches, have a number of features in common with the preparative methods of biopolymer purification. In particular, both analytical and preparative HPLC methods involve an interplay of secondary equilibrium and within the time scale of the separation nonequilibrium processes. The consequences of this plural behavior are that retention and band-broadening phenomena rarely (if ever) exhibit ideal linear elution behavior over a wide range of experimental conditions. First-order dependencies, as predicted from chromatographic theory based on near-equilibrium assumptions with low molecular weight compounds, are observed only within a relatively narrow range of conditions for polypeptides and proteins. 

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