Characteristic timescale

Atmospheric turbulence is a dynamic process and the wavefront aberrations are constantly changing. A characteristic timescale may be defined as the time over which the mean square change in wavefront error is less than 1 rad. If the turbulence were concentrated in a single layer with Fried parameter ro moving with a horizontal speed of v ms then the characteristic time, ro, is given by... 

For a cylindrical vessel of width h, the characteristic timescale for the spin-up process, neglecting the effects at the periphery of the vessel is then... 

Figure 4.9 illustrates time-gated imaging of rotational correlation time. Briefly, excitation by linearly polarized radiation will excite fluorophores with dipole components parallel to the excitation polarization axis and so the fluorescence emission will be anisotropically polarized immediately after excitation, with more emission polarized parallel than perpendicular to the polarization axis (r0). Subsequently, however, collisions with solvent molecules will tend to randomize the fluorophore orientations and the emission anistropy will decrease with time (r(t)). The characteristic timescale over which the fluorescence anisotropy decreases can be described (in the simplest case of a spherical molecule) by an exponential decay with a time constant, 6, which is the rotational correlation time and is approximately proportional to the local solvent viscosity and to the size of the fluorophore. Provided that... 

The aim of this chapter is to describe approaches of obtaining exact time characteristics of diffusion stochastic processes (Markov processes) that are in fact a generalization of FPT approach and are based on the definition of characteristic timescale of evolution of an observable as integral relaxation time [5,6,30—41]. These approaches allow us to express the required timescales and to obtain almost exactly the evolution of probability and averages of stochastic processes in really wide range of parameters. We will not present the comparison of these methods because all of them lead to the same result due to the utilization of the same basic definition of the characteristic timescales, but we will describe these approaches in detail and outline their advantages in comparison with the FPT approach. 

It should be noted that besides being widely used in the literature definition of characteristic timescale as integral relaxation time, recently intrawell relaxation time has been proposed [42] that represents some effective averaging of the MFPT over steady-state probability distribution and therefore gives the slowest timescale of a transition to a steady state, but a description of this approach is not within the scope of the present review. 

However, mathematical evidence of such a definition of characteristic timescale has been understood only recently in connection with optimal estimates [54]. As an example we will consider evolution of the probability, but the consideration may be performed for any observable. We will speak about the transition time implying that it describes change of the evolution of the transition probability from one state to another one. 

The characteristic timescale of evolution of the mean coordinate in the general case may be easily obtained from (5.130) by substituting x for f(x). But for symmetric potentials [Pg.420]

In order to evaluate when steady-state conditions are relevant, as well as the relative importance of the various dynamic processes, it is useful to compare the characteristic timescale of each process (Table 5). It can be seen that the timescales of each of the main processes vary over several orders of magnitude, depending upon the precise nature of the solute, the accumulating surface and the physicochemistry and hydrodynamics of the medium. 

Table 5. Characteristic timescales of several important processes involved in biouptake as estimated by their residence time (physical processes) or by their half-reaction time (chemical reactions). See also [2,165]...

It is important to note that the reciprocal of the shear rate is the time taken for unit strain to occur in the material and is the characteristic timescale for our experiment. As long as the microstructure can reorganise by thermal motion in a shorter timescale, the value of De is less than 1 and the structure will remain in a configuration that is close to its... 

The phenomenological approach does not preclude a consideration of the molecular origins of the characteristic timescales within the material. It is these timescales that determine whether the observation you make is one which sees the material as elastic, viscous or viscoelastic. There are great differences between timescales and length scales for atomic, molecular and macromolecular materials. When an instantaneous deformation is applied to a body the particles forming the body are displaced from their normal positions. They diffuse from these positions with time and gradually dissipate the stress. The diffusion coefficient relates the distance diffused to the timescale characteristic of this motion. The form of the diffusion coefficient depends on the extent of ordering within the material. 

Thin-film ideal or Nemstian behavior is the starting point to explain the voltammetric behavior of polyelectrolyte-modified electrodes. This condition is fulfilled when (i) the timescale of the experiment is slower than the characteristic timescale for charge transport (fjD pp, with Ithe film thickness) in the film, that is all redox within the film are in electrochemical equibbrium at any time, (ii) the activity of redox sites is equal to their concentration and (iii) all couples have the same redox potential. For these conditions, anodic and cathodic current-potential waves are mirror images (zero peak splitting) and current is proportional to the scan rate [121]. Under this regime, there exists an analytical expression for the current-potential curve ... 

Here,/Qj ax( ) is a generalized Debye-Waller factor giving account for the decay of the correlations at faster times. In the framework of the MCT it is also called non-ergodicity factor. The characteristic timescale r (T) is the structural relaxation time and (3 is the stretching parameter (0[Pg.73]

Fig. 4.8 Temperature dependence of the dielectric characteristic times obtained for PB for the a-relaxation (empty triangle) for the r -relaxation (empty diamond), and for the contribution of the -relaxation modified by the presence of the a-relaxation (filled diamond). They have been obtained assuming the a- and -processes as statistically independent. The Arrhenius law shows the extrapolation of the temperature behaviour of the -relaxation. The solid line through points shows the temperature behaviour of the time-scale associated to the viscosity. The dotted line corresponds to the temperature dependence of the characteristic timescale for the main peak. (Reprinted with permission from [133]. Copyright 1996 The American Physical Society)...

Note that, in the two cases discussed above, chemical intuition was essential in properly defining superstates such that an appropriate separation of timescales was obtained. For more complex systems, intuition alone will not be sufficient and considerable effort might be required to identify an exploitable gap in the characteristic timescales of the system. There is thus a need to develop on-the-fly methods to appropriately define superstates based on MD data alone. Efforts toward this goal are presently under way [31]. 

If we consider the opposite extreme, where quadratic autocatalysis dominates, we should also change our characteristic timescale. So far we have based the wave velocity c on the cubic chemical time fch = /klal ... 

Table 9.1 First-order calculation for characteristic timescales (Gemmen and Johnson, 2005).

The coupled problem formulation is presented in Sect. 5.1. The loading rates for which thermal diffusion needs to be considered can be estimated for a one-dimensional problem as in [9,57]. This leads to the nondimensional quantity k which compares a characteristic timescale f0 associated with the loading conditions to the time for heat to diffuse over a characteristic length To as ... 

For k 1, isothermal conditions prevail, while /< -c 1 when the situation is adiabatic. The characteristic timescale to for the present study is defined as the time to attain the material toughness A[r for a given loading rate, i.e., to = Kf1/K. The characteristic length Lq is taken as the size of the plastic zone of a perfectly plastic material with yield stress s0 so that L0 = (fQ/so)2 [57]. For k 1, heat conduction needs to be accounted for and this condition results in the estimation of... 

The cohesive surface considered in the foregoing is based on observations made under quasistatic conditions. In particular, the incubation time for craze initiation is neglected and a critical stress state for craze nucle-ation is used (Eq. 11). For dynamic loading, a time-dependent craze initiation criterion is to be included in the kinetics, since the characteristic timescale associated with the loading can be comparable to that involved in the craze nucleation process. If the time for craze initiation is accounted for, another timescale is involved in the competition between crazing and shear yielding that determines whether or not crazing takes place. Therefore, a switch... 

The first step of the derivation involves the BO approximation separating the characteristic timescales of the electronic and nuclear motions in the system. In this step, the instantaneous free energy depending on the system nuclear coordinates q is defined by... 

TABLE 1 Characteristic Timescales for Diffusion in Porous Catalyst Particles. 

The growth of bubbles is controlled by the rates at which volatiles in the melt can diffuse towards the bubbles, and the opposing viscous forces. Near a bubble, volatiles are depleted such that melt viscosity increases dramatically, and diffusivities drop, making it harder for volatiles to diffuse through and grow the bubble. These opposing factors are described by the nondimensional Peclet number (Pe), which is the ratio of the characteristic timescales of volatile diffusion (T(1 = r lD, where r is the bubble radius and D the diffusion coefficient of the volatile in the melt) and of viscous relaxation (t = 17/AP where 17 is the melt dynamic viscosity and AP the oversaturation pressure, i.e., Pe = Dingwell... 

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