Cumulant approximation

At this point it might be helpful to summarize what has been done so far in terms of effective potentials. To obtain the QFH correction, we started with an exact path integral expression and obtained the effective potential by making a first-order cumulant expansion of the Boltzmann factor and analytically performing all of the Gaussian kinetic energy integrals. Once the first-order cumulant approximation is made, the rest of the derivation is exact up to (11.26). A second-order expansion of the potential then leads to the QFH approximation. 

This loss of information is easier to see after applying a cumulant approximation (61). This approximation is equivalent to assuming that the distribution of Sco is always Gaussian and simplifies Equation (3) to... 

For more complex systems with intermediate modulation or multiple modulation time scales, a cumulant approximation is useful [see Equation (4)] (34) ... 

In pure dephasing, the loss of coherence is caused by time-dependent perturbations of the vibrational frequency. Pure dephasing has been assumed in the earlier portions of this chapter and, in general, dominates vibrations in liquids. Within the cumulant approximation, both the Raman echo and FID decays can be calculated from Cffl(t) [Equations (4) and (13)]. Relating these measurement to properties of the solvent and solute requires a molecular model for Co,(t). 

Note that Eq. (A.61) is valid once the factorization approximation is made irrespective of the second-order cumulant approximation. After performing a lengthy but straightforward algebra using Eqs. (A.59)-(A.61), we obtain the result of Q in the fast modulation regime given as Eq. (4.77). 

Figure 5.12. The compressibility factor of a hard-sphere fluid. The solid squares, circles, and diamonds indicate values obtained from the computer simulations of Erpenbeck and Wood [26], Rotenberg [27] and Woodcock [28], respectively. Curve 1 is a plot of the seventh-order virial polynomial. Curve 2 is the Shinomoto, second-order cumulant approximation of Eq. (5.204). Finally, 3 is a plot of the RG seventh-order cumulant approximation of Eq. (5.205) and of the Camahan-Starling equation (5.183) as well, since the two are indistinguishable on the scale of this figure.

Figxire 4 shows the predicted cxirves from these models for an intermediate value of T j, together with the second cumulant approximation to these models. The difference between them is striking, with no deviation between the different L values for the Fokker-Planck model, whose curves lie below the lowest cumulant approximation, and a distinct fanning out of the J diffusion curves. Comparing these with the results from a low torque CBr simulation we found reasonable agreement with the J diffusion model (6). 

Figure 5. Comparison of best fit second order truncated memory function predictions (dashed lines) with simulation results for a high torque state of CBr (6) and the lowest cumulant approximation (dotted line).

In a second class, a density expansion is used to obtain a series approximation for G (e) that is accurate at short times and small ncent rat ions. One approach is to construct a Fade approximant for G (e) from this series that causes G (t) to decay to zero at long times. A second approach is to construct cumulant approximants for the series by inverting the Laplace transform of the density exMnsion and re-expressing the series as a short-time expansion for fn[G (t)]. 

Two-Particle Cumulant Approximation. The preceding discussion on the three particle Fade and the other approximations is not limited to the case of ideal chain statistics, but it is only in this... 

The general expression for the two-particle cumulant approximation for a translationally invariant pol)nneric material that contains a small chromophore concentration is [10,13]... 

X Bq of Pu has been released, mostiy from bum-up of the nuclear powered sateUite SNAP-9a and that 3.7 X 10 Bqof + ° Pu was released by the Chernobyl accident (167,168). Many studies have been done to determine the cumulative fallout on sods, plants, bodies of water, animals, and humans. For example, the cumulative Pu fallout ia forest and grasslands and ia the Hver of elderly humans ia Bavaria, Germany are approximately... 

FIG. 9-23 Cumulative probability of a given net present value or less for a project showing normal and Gompertz approximations. 

Cumulative volume over the range of 1 to 50 percent can also be shown to vary approximately as D. This is equiv ent to finding that the number of droplets of a given size is inversely proportional to the droplet area or the surface energy of the droplet. 

When a distribufion of particle sizes which must be collected is present, the aclual size distribution must be converted to a mass distribution by aerodynamic size. Frequently the distribution can be represented or approximated by a log-normal distribution (a straight line on a log-log plot of cumulative mass percent of particles versus diameter) wmich can be characterized by the mass median particle diameter dp5o and the standard statistical deviation of particles from the median [Pg.1428]

In order to be consistent with normal usage, the particle-size distribution when this parameter is used should Be a straight line between approximately 10 percent cumulative weight and 90 percent cumulative weight. By giving the coefficient of variation ana the mean particle diameter, a description of the particle-size distribution is obtained which is normally satisfactory for most industrial purposes. If the product is removed from a mixed-suspension ciystallizer, this coeffi-... 

The molecular polarizability can be considered to be the cumulation of individual bond polarizabilities. The bond polarizability is known (in simple cases) to be an approximately linear function of bond length for small amplitudes of vibration. That is, polarizability is essentially a bond property and consequently is independent of direction along any axis (or independent of sense ). 

Figure 6 shows a cumulative probability plot of both the maximum dally and hourly NO2 averages In cities for the 1980-84 time period. The plotted values can be directly compared to the WHO guideline values of 150/tg/m3 for the maximum 24-hour level and 400/tg/m3 for the maximum 1-hour level. In both cases, about 25% of the cities worldwide exceed the guideline values. Based on these proportions of cites with NO2 concentrations above the short-term guideline values. It Is estimated that approximately 15-20 percent of urban residents In North America and Europe are at Increased risk to short-term high NO2 exposures. 

Radial motion of fluid can have a significant, cumulative effect on the convective diffusion equations even when Vr has a negligible effect on the equation of motion for V. Thus, Equation (8.68) can give an accurate approximation for even though Equations (8.12) and (8.52) need to be modified to account for radial convection. The extended versions of these equations are... 

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