Time scale dominant

The concentration of some species evolves more rapidly than others for a certain initial period of time. When enough time passes, the terms of the slowest time scales begin to affect the system s behavior. Over time, the terms of slower time scales dominate the evolution of the species concentration. 

For these sequences the value of Gj, is less than a certain small value g. For such sequences the folding occurs directly from the ensemble of unfolded states to the NBA. The free energy surface is dominated by the NBA (or a funnel) and the volume associated with NBA is very large. The partition factor <6 is near unify so that these sequences reach the native state by two-state kinetics. The amplitudes in (C2.5.7) are nearly zero. There are no intennediates in the pathways from the denatured state to the native state. Fast folders reach the native state by a nucleation-collapse mechanism which means that once a certain number of contacts (folding nuclei) are fonned then the native state is reached very rapidly [25, 26]. The time scale for reaching the native state for fast folders (which are nonnally associated with those sequences for which topological fmstration is minimal) is found to be... 

The first chapter, on Conformational Dynamics, includes discussion of several rather recent computational approaches to treat the dominant slow modes of molecular dynamical systems. In the first paper, SCHULTEN and his group review the new field of steered molecular dynamics (SMD), in which large external forces are applied in order to be able to study unbinding of ligands and conformation changes on time scales accessible to MD... 

There are two different scales of deformation in any adhesive contact (1) the bulk scale of deformation which is characterized by the radius a of contact area over which the compressive forces are significant and (2) the zone of action of surface forces or the cohesive zone at the edge of the contact, characterized by the length d over which the tensile forces are dominant. When the contact boundary is moving with a speed u, the two scales of deformation translate into two time scales, one on the order of a/ v) and the other of the order of (d/v). 

It is often important to know how long an element spends in one environment before it is transported somewhere else in the Earth system. For example, if a time scale characterizing a chemical or physical transformation process in a region has been estimated, a comparison with the time scale characterizing the transport away from the region will tell which process is likely to dominate. 

Time scales of transport can also be applied to situations when no well-defined reservoirs can be defined. If the dominant transport process is advection by mean flow or sedimentation by gravity, the time scale characterizing the transport between two places is simply tadv = L/V where L is the distance and V the transport velocity. Given a t)q)ical wind speed of 20 m/s in the mid-latitude tropospheric westerlies, the time of transport around the globe would be about 2 weeks. 

To this point, we have emphasized that the cycle of mobilization, transport, and redeposition involves changes in the physical state and chemical form of the elements, and that the ultimate distribution of an element among different chemical species can be described by thermochemical equilibrium data. Equilibrium calculations describe the potential for change between two end states, and only in certain cases can they provide information about rates (Hoffman, 1981). In analyzing and modeling a geochemical system, a decision must be made as to whether an equilibrium or non-equilibrium model is appropriate. The choice depends on the time scales involved, and specifically on the ratio of the rate of the relevant chemical transition to the rate of the dominant physical process within the physical-chemical system. 

Longer ice-core records show that methane concentrations have varied on a variety of time scales over the past 220 000 years (Fig. 18-15) Qouzel et al, 1993 Brook et al, 1996). Wetlands in tropical (30° S to 30° N) and boreal (50° N to 70° N) regions are the dominant natural methane source. As a result, ice-core records for preanthropogenic times have been interpreted as records of changes in methane emissions from wetlands. Studies of modem wetlands indicate that methane emissions are positively correlated with temperature, precipitation, and net ecosystem productivity (Schlesinger, 1996). 

In this expression, w is a typical velocity scale and d a typical length scale, for example the diameter of a micro charmel. The Peclet number represents the ratio of the diffusive and the convective time-scales, i.e. flows with large Peclet numbers are dominated by convection. 

Often the electronic spin states are not stationary with respect to the Mossbauer time scale but fluctuate and show transitions due to coupling to the vibrational states of the chemical environment (the lattice vibrations or phonons). The rate l/Tj of this spin-lattice relaxation depends among other variables on temperature and energy splitting (see also Appendix H). Alternatively, spin transitions can be caused by spin-spin interactions with rates 1/T2 that depend on the distance between the paramagnetic centers. In densely packed solids of inorganic compounds or concentrated solutions, the spin-spin relaxation may dominate the total spin relaxation 1/r = l/Ti + 1/+2 [104]. Whenever the relaxation time is comparable to the nuclear Larmor frequency S)A/h) or the rate of the nuclear decay ( 10 s ), the stationary solutions above do not apply and a dynamic model has to be invoked... 

As seen in Eqs. (59)—(61), dephasing processes introduce two new time scales into the dynamics, in addition to the intermediate state lifetime that determines the structure of 8s in the isolated molecule case. One is the time scale of pure dephasing, and the other is the lifetime of the final state. Equation (64) illustrates that the Tff dependence of 8s is a condensed phase effect that vanishes in the limit of no dephasing. The more careful analysis later shows that the qualitative behavior of the channel phase is dominated by the rpd/rrr and Tpd / [ ratios, that is, by the rate of dephasing as compared to the system time scales. 

Let say we have a high order transfer function that has been factored into partial fractions. If there is a large enough difference in the time constants of individual terms, we may try to throw away the small time scale terms and retain the ones with dominant poles (large time constants). This is our reduced-order model approximation. From Fig. E3.3, we also need to add a time delay in this approximation. The extreme of this idea is to use a first order with dead time function. It obviously cannot do an adequate job in many circumstances. Nevertheless, this simple... 

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