Independent Gaussian approximation

Independent Gaussian approximation (IGA), direct molecular dynamics, Gaussian wavepacket propagation, 379-383... 

We have already argued (Section 7.4.2) that the Markovian nature of the system evolution implies that the relaxation dynamics of the bath is much faster than that of the system. The bath loses its memory on the timescale of interest for the system dynamics. Still the timescale forthe bath motion is not unimportant. If, for example, the sign of Rf) changes infinitely fast, it makes no effect on the system. Indeed, in order for a finite force R to move the particle it has to have a finite duration. It is convenient to introduce a timescale tb, which characterizes the bath motion, and to consider an approximate picture in which Rf) is constant in the interval [t, t -I- Tb], while Rff and Rff are independent Gaussian random variables if... 

In many biological applications of light scattering the solutions are sufficiently dilute to satisfy the assumption of particle independence but still sufficiently concentrated that N(0) SN(t)y can be neglected compared with <7V)>2. The Gaussian approximation is then valid. In most of this book we deal with this type of solution, and consequently automatically assume the validity of the Gaussian approximation unless otherwise stated. It is then necessary to present only F since F2 can be computed directly from Fi. 

We now consider the intensity of independent scattering from a random coil polymer molecule, consisting of (N + 1) beads connected by N bonds and obeying the Gaussian approximation (5.12). We assume that the volume of a bead is uu and the volume of the chain is v = (N + l)uu. Each bead thus contributes po u to the... 

Here k enumerates the field sources with concentrations All the sources are supposed to be independent and Gaussian approximation has been used as their concentration is sufficiently large. Latter situation is specific for relaxors. Vector Eok in Eq. (3.73) and tensor A in Eq. (3.74) define the contribution of fc-th type of sources into mean field and half-width of distribution function respectively. The calculations of these parameters have been performed for two types of the field sources k = 2 in Eq. 3.72), namely for point charges and dipoles with concentrations n , and na respectively. It has also been supposed, that the dipoles can have two orientations along +z and —z. The detailed calculations performed in [82, 89] lead to the following form of distribution function... 

The van Deemter approach deals with the effects of rates of nonequilibrium processes (e.g. diffusion) on the widths (ct ) of the analyte bands as they move throngh the column, and thus on the effective value of H and thns of N. Obviously, the faster the mobile phase moves through the column, the greater the importance of these dispersive rate processes relative to the idealized stepwise equilibria treated by the Plate Theory, since equilibration needs time. Thus van Deemter s approach discusses variation of H with u, the linear velocity of the mobile phase (not the volume flow rate (U), although the two are simply related via the effective cross-sectional area A of the column, which in turn is not simply the value for the empty tube but must be calculated as the cross-sectional area of the empty column corrected for the fraction that is occupied by the stationary phase particles). This approach identifies the various nonequilibrium processes that contribute to the width of the peak in the Gaussian approximation and shows that these different processes make contributions to Ox (and thus H) that are essentially independent of one another and thus can be combined via simple propagation of error (Section 8.2.2) ... 

We next studied tetra-alanine in solvation. We used the ECEPP/3 potential energy surface coupled with the volume method for calculating solvation energies using the Reduced Radius Independent Gaussian Sphere (RRIGS) approximation. 

If we assume that spot size is independent of profile shape when V= we can use the Gaussian approximation to check if the consequence of this assumption is consistent with the exact value of V. The normalized intensity is independent of profile shape only if the spot size has the same value tq for all profiles. Thus we set K = in the ratio S/ a N,as given by Table 15-2,and use the step profile with V = 2405 as a reference profile which determines the value of r . On rearranging... 

The power-law profiles satisfy Eq. (16-19b), and are therefore homogeneous function profiles. Consequently, the above expression is consistent with the more general result of Eq. (16-20). In the case of the infinite linear profile ( = 1), Eq. (17-6) reduces to the exact eigenvalue equation of Eq. (16-29), apart from a change in the multiplicative constant on the ri t from 1.018 to 3/(2 t ) S 1.024, a relative error of 0.6%. This excellent agreement is independent of and and demonstrates the accuracy of the Gaussian approximation for arbitrary eccentricity. 

For the fundamental modes on a circular fiber, is independent of 0. Consequently —tiy and p = p in Eq. (18-21). In other words, the propagation constants for the circular and elliptical fibers are identical for slight eccentricity, provided the core areas are equal [6]. The latter condition is equivalent to requiring equal profile volumes, as is clear from Eq. (17-13). Hence the present result is consistent with the more general result of Section 17-3, which showed that, within the Gaussian approximation, P = on an arbitrary, elliptical-profile fiber of slight eccentricity, provided the profile volumes are equal. 

Figure 1.8. Schematic frequency distributions for some independent (reaction input or control) resp. dependent (reaction output) variables to show how non-Gaussian distributions can obtain for a large population of reactions (i.e., all batches of one product in 5 years), while approximate normal distributions are found for repeat measurements on one single batch. For example, the gray areas correspond to the process parameters for a given run, while the histograms give the distribution of repeat determinations on one (several) sample(s) from this run. Because of the huge costs associated with individual production batches, the number of data points measured under closely controlled conditions, i.e., validation runs, is miniscule. Distributions must be estimated from historical data, which typically suffers from ever-changing parameter combinations, such as reagent batches, operators, impurity profiles, etc.

Like for the last example the optimum h goes as and the error as ryri). The convergence is slower than for the same function with an equidistant grid, but both h and e are (on this level of approximation) independent of a, i.e. essentially the same grid can be used for a very steep or a very flat Gaussian, there is only a shift via the a-dependence of yi and 1/2 ... 

To render the KP theory feasible for many-body systems with N particles, we make the approximation of independent instantaneous normal mode (INM) coordinates [qx° 3N for a given configuration xo 3W [12, 13], Hence the multidimensional V effectively reduces to 3N one-dimensional potentials along each normal mode coordinate. Note that INM are naturally decoupled through the 2nd order Taylor expansion. The INM approximation has also been used elsewhere. This approximation is particularly suited for the KP theory because of the exponential decaying property of the Gaussian convolution integrals in Eq. (4-26). The total effective centroid potential for N nuclei can be simplified as ... 

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