Calculation of the Multidimensional Distribution

The calculation of the time evolution operator in multidimensional systems is a fomiidable task and some results will be discussed in this section. An alternative approach is the calculation of semi-classical dynamics as demonstrated, among others, by Heller [86, 87 and 88], Marcus [89, 90], Taylor [91, 92], Metiu [93, 94] and coworkers (see also [83] as well as the review by Miller [95] for more general aspects of semiclassical dynamics). This method basically consists of replacing the 5-fimction distribution in the true classical calculation by a Gaussian distribution in coordinate space. It allows for a simulation of the vibrational... 

As noted above, solving the Boltzmann equation is problematic because of the multidimensionality of the problem. A promising approach to calculating the electron distribution function in low pressure plasmas is the so-called non-local approach to electron kinetics. This was proposed by Bernstein and Holstein [56] and popularized by Tsendin [57], who initially suggested this approach for the positive column of a DC discharge. Since then, the non-local approach has been applied to a variety of low pressure gas discharge systems [58]. 

Expressions (173) and (174) provide efficient formulations for the solution of deep-penetration problems whose geometrical irregularity is in the vicinity of the detector. The unperturbed flux distribution is calculated with a low-order, low-dimensional calculational method (such as a one-dimensional discrete-ordinates method). The irregularity in the geometry is treated as an alteration to the unperturbed system. The 5 or distribution is then obtained from the solution of Eq. (44a) or Eq. (163), respectively, for the local region of the detector and the alteration. High-order and multidimensional calculational methods (such as Monte Carlo) can be used for this local solution. Equation (173) provides an efficient formulation for the calculation of the response of many different detectors (such as different reaction rates and the spatial distribution of a given reaction rate) in a... 

If the data are assumed to come from a population having a multivariate normal distribution and it is furthermore assumed that the covariance matrix of each group is similar, then the conditional probability values can be calculated from the multidimensional normal distribution... 

The multidimensional theorem of averages can be used to calculate the higher-order joint distribution functions of derived sets of time functions, each of which is of the form... 

Here, V is vector notation for the set of all component energies Vy, and A, j gives the coefficient of Vy in the ith run. The Ay, without subscript i, indicate the values of A in the target ensemble. The histograms collected in the runs are multidimensional in that they are tabulated as functions of the component energies as well as the order parameter . Similarly, the final result of the WHAM calculation is a multidimensional probability distribution in V J and . 

The purpose of this chapter is a detailed comparison of these systems and the elucidation of the transition from regular to irregular dynamics or from mode-specific to statistical behavior. The main focus will be the intimate relationship between the multidimensional PES on one hand and observables like dissociation rate and final-state distributions on the other. Another important question is the rigorous test of statistical methods for these systems, in comparison to quantum mechanical as well as classical calculations. The chapter is organized in the following way The three potential-energy surfaces and the quantum mechanical dynamics calculations are briefly described in Sections II and III, respectively. The results for HCO, DCO, HNO, and H02 are discussed in Sections IV-VII, and the overview ends with a short summary in Section VIII. 

The hypergeometric distribution can be generalized to a multivariable form, the multivariate hypergeometric distribution, which can be used to extend Fisher s Exact Test to contingency tables larger than 2 by 2 and to multidimensional contingency tables. There is statistical software available to perform these calculations however, due to the complexity of the calculations and the large number of trial tables whose probability of occurrence must be calculated, this extension has received limited use. 

Middle atmosphere models calculate the spatial and temporal distribution of the net heating rate Q and the temperature not only as a function of altitude but also as a function of latitude, and even of longitude (local time). Such studies consider the multidimensional transport of heat and the solution of a thermodynamic equation like the one shown in Equation (3.10). The effect of waves should also be considered. Gravity wave dissipation, for example, may play an important role in the mesospheric heat budget. In the multidimensional models, the radiative scheme is often simplified and parameterized the most simple approach is to assume the cool-to-space approximation , in which it is assumed that exchange of heat between layers can be neglected in comparison to propagation out to space. 

The radiative transfer equation (RTE) is an integro-differential equation it is difficult to develop a closed-form solution to it in general multidimensional and nonhomogeneous media. After introducing a number of approximations, however, reasonably accurate models of the RTE can be obtained. In all models, the objective is to solve the RTE, or a modified form of it, in terms of radiation intensity or its moments (such as flux) and then calculate the distribution of the divergence of radiative flux V q everywhere in the medium. In this section, we will discuss the approximate models of the RTE which can be extended to multidimensional geometries. 

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