Time scales fluctuations

Under the conditions just discussed, large time-scale fluctuations of the density of pairs were neglected. In order to take into account such large-scale fluctuations we introduce an average of the pair operator over a small volume, p. This small volume is much smaller than the total volume of the system, however, it is large enough to smear out small-scale fluctuations. We call the quantity p the smoothed pair operator. Then it is useful to define... 

Unlike a glass, a liquid does not have essentially static structures that give rise to inhomogeneous broadening. Nonetheless, liquids can have fast time scale fluctuations that give rise to homogeneous broadening and much... 

An important discovery by Yavor Shopov in the mid-1980 s was that the luminescence also exhibits a banding, in this case, on a scale of tens of micrometers. These bands take the form of alternating bright and less bright luminescence intensity. The thickness of the luminescence bands is comparable to annual growth rates. It is reasonably well established that the smallest scale luminescence bands represent a seasonal record (Baker et al., 1993 Shopov et al., 1994). Superimposed on the annual records are longer time scale fluctuations that must represent longer scale variations in depositional conditions (Fig. 18). 

The simulations also revealed that flapping motions of one of the loops of the avidin monomer play a crucial role in the mechanism of the unbinding of biotin. The fluctuation time for this loop as well as the relaxation time for many of the processes in proteins can be on the order of microseconds and longer (Eaton et al., 1997). The loop has enough time to fluctuate into an open state on experimental time scales (1 ms), but the fluctuation time is too long for this event to take place on the nanosecond time scale of simulations. To facilitate the exit of biotin from its binding pocket, the conformation of this loop was altered (Izrailev et al., 1997) using the interactive molecular dynamics features of MDScope (Nelson et al., 1995 Nelson et al., 1996 Humphrey et al., 1996). 

In a simulation it is not convenient to work with fluctuating time intervals. The real-variable formulation is therefore recommended. Hoover [26] showed that the equations derived by Nose can be further simplified. He derived a slightly different set of equations that dispense with the time-scaling parameter s. To simplify the equations, we can introduce the thermodynamic friction coefficient, = pJQ. The equations of motion then become... 

The second example of an air pollutant that affects the total body burden is carbon monoxide (CO). In addihon to CO in ambient air, there are other sources for inhalation. People who smoke have an elevated CO body burden compared to nonsmokers. Individuals indoors may be exposed to elevated levels of CO from incomplete combustion in heating or cooking stoves. CO gas enters the human body by inhalation and is absorbed directly into the bloodstream the total body burden resides in the circulatory system. The human body also produces CO by breakdown of hemoglobin. Hemoglobin breakdown gives every individual a baseline level of CO in the circulatory system. As the result of these factors, the body burden can fluctuate over a time scale of hours. 

The fact that there are no characteristic length scales immediately implies a similar lack of any characteristic time scales for the fluctuations. Consider the effect of a single perturbation of a random site of a system in the critical state. The perturbation will spread to the neighbors of the site, to the next nearest neighbors, and so on, until, after a time r and a total of / sand slides, the effects will die out. The distribution of the life-times of the avalanches, D t), obeys the power law... 

Development in recent years of fast-response instruments able to measure rapid fluctuations of the wind velocity (V ) and of fhe tracer concentration (c ), has made it possible to calculate the turbulent flux directly from the correlation expression in Equation (41), without having to resort to uncertain assumptions about eddy diffusivities. For example, Grelle and Lindroth (1996) used this eddy-correlation technique to calculate the vertical flux of CO2 above a foresf canopy in Sweden. Since the mean vertical velocity w) has to vanish above such a flat surface, the only contribution to the vertical flux of CO2 comes from the eddy-correlation term c w ). In order to capture the contributions from all important eddies, both the anemometer and the CO2 instrument must be able to resolve fluctuations on time scales down to about 0.1 s. 

Many important erosion-related phenomena are episodic and infrequent, such as flash floods, landslides, and glaciations, while others such as orogenesis and soil formation involve time scales that exceed those of major climate fluctuations. In either case, the time scale of human existence is too short to make adequate observations. Consequently, it is difficult to directly estimate the rates or characterize the effects of such phenomena on erosion products. The key to understanding weathering and erosion, on a continental scale, is to decipher the relationship between landforms, the processes that produce them, and the chemistry and discharge of river-borne materials. 

There may also be natural fluctuations within this cycle that occur over time scales ranging from thousands of years (glacial-interglacial) to millions of years (Follmi, 1996). Overall, the... 

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... 

The fluctuations of the magnetization direction around an easy axis, known as collective magnetic excitations, can be considered fast compared to the time scale of Mossbauer spectroscopy because there are no energy barriers between magnetization directions close to an easy direction, and the magnetic splitting in the... 

Since MPC dynamics yields the hydrodynamic equations on long distance and time scales, it provides a mesoscopic simulation algorithm for investigation of fluid flow that complements other mesoscopic methods. Since it is a particle-based scheme it incorporates fluctuations, which are essential in many applications. For macroscopic fluid flow averaging is required to obtain the deterministic flow fields. In spite of the additional averaging that is required the method has the advantage that it is numerically stable, does not suffer from lattice artifacts in the structure of the Navier-Stokes equations, and boundary conditions are easily implemented. 

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