Average over time

Thus, unlike molecular dynamics or Langevin dynamics, which calculate ensemble averages by calculating averages over time, Monte Carlo calculations evaluate ensemble averages directly by sampling configurations from the statistical ensemble. 

Since the ligand is in fast exchange with respect to its bound and free state, every molecule experiences an averaging over time where the fraction L... 

The potential perturbation in SWV is a stair-like square wave as shown in Fig. 64. Taking an average over time, the potential change in SWV can be considered as linear and the concept of scan rate still applies. In practice, the scan rate of an S W voltammogram is often described in terms of the frequency of the square wave, /, which can be converted to the scan rate by the following equation (see Fig. 64) ... 

The intermediates of the tricarboxylic acid cycle are present in the mitochondria only in very small quantities. After the oxidation of acetyl-CoA to CO2, they are constantly regenerated, and their concentrations therefore remain constant, averaged over time. Anabolic pathways, which remove intermediates of the cycle (e.g., gluconeogenesis) would quickly use up the small quantities present in the mitochondria if metabolites did not reenter the cycle at other sites to replace the compounds consumed. Processes that replenish the cycle in this way are called anaplerotic reactions. 

Two alternative assumptions have been made for the manner in which the area variation occurs. The more realistic postulates that all elements of the surface remain in the surface throughout an oscillation cycle. Increasing surface area stretches the surface (A3, B5) and causes a velocity normal to the surface which increases the diffusion rate. For a surface of area Aq suddenly exposed at t = 0, the mass transfer product averaged over time is given by... 

In a typical scattering experiment, a detector measures the intensity of the scattered radiation over a period of time, say, tn, in discrete steps of At (see Figure 5.16a). As shown in the figure, the intensity i(s,t) fluctuates around an average value because of the random motions of the scatterers. Until now, we have denoted the average by simply i(s) for convenience. To be precise, however, the average should be denoted by, say, i(s), and this is an average over time, t, defined as... 

As noted above, London dispersive interactions occur even between molecules of apolar compounds like alkanes, that on average over time exhibit a rather smooth distribution of electrons throughout the whole molecular structure. This interaction occurs in all chemicals because there are momentary (order of femtosecond timescales) displacements of the electrons within the structure such that short-lived electron-rich and electron-poor regions temporarily develop. This continuous movement of electrons implies the continuous presence of short-lived dipoles in the structure. This fleeting dipole is felt by neighboring molecules whose electrons respond in a complementary fashion. Consequently, there is an intermolecular attraction between these molecular regions. In the next moment, these attractive interactions shift elsewhere in the molecule. 

We define a field n which has a gradient along the x-axis on the average over time and which is independent of t, y, and z ... 

All atoms and molecules experience London dispersion forces, which result from the motion of electrons. Take even a simple nonpolar molecule like Br2, for instance. Averaged over time, the distribution of electrons throughout the molecule is symmetrical, but at any given instant there may be more electrons at one end of the molecule than at the other, giving the molecule a short-lived dipole. This instantaneous dipole on one molecule can affect the electron distributions in neighboring molecules and induce temporary dipoles in those neighbors (Figure 10.5b). As a result, weak attractive forces develop and Br2 is a liquid at room temperature rather than a gas. 

Fig. 58 Rod-bundle phase formed by compounds 184 and 185 with two branched and semiperfluorinated lateral chains and their organization in the LC rod-bundle phases the local cross-section of the columns is thought to be elliptical, but circular when averaged over time and space, hence leading to 2D hexagonal symmetry [342, 343], Reproduced with permission [342], opyright 2008, American Chemical Society (ACS)...

The momentary noise disturbance is averaged over time to obtain the noise disturbance . We will not use the term noise loudness because the value of y is taken such that the subjective quality model is optimized in that case does not necessarily represent noise loudness. The logarithm (log 10) of the noise disturbance is defined as the perceptual audio quality measure (PAQM). 

A rigorous analyses shows that at spectral function due to planar libration involves the averaging over time during a half-period. 

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