Integral transport theory, formulations

Many methods and computer codes have been developed for the solution of Eq. (47) for the purpose of reactivity calculations. The majority of them are based on integral transport theory formulations. They include the methods of Karam et al. (20), Collins and Palmer (27), Kier and Salvatores (22,23), McGrath and Foell (24), Fischer (25), Oosterkamp (26), and McGrath and Fischer (27). 

System perturbations can usually be expressed as perturbations in the macroscopic cross sections. The operators of the perturbation integrals of the different formulations depend on these cross-section perturbations the functional dependence varies with the formulation. The simplest dependence is found in the integrodifferential formulation in which the perturbation operators are the cross-section perturbations. Conversely, the integral transport theory formulations include kernel perturbations that do not depend linearly on cross-section perturbations. Consequently, it is necessary to evaluate the perturbation in the kernels before applying integral pertur-... 

Different first-order approximations can be derived from each integral transport theory formulation. These approximations differ in the distribution (density and adjoint) functions they use. 

The first-order approximations in the integral transport theory formulations partially account for the effect of the perturbation on the neutron distribution. 

All first-order approximations (pertaining to integral transport theory) considered are equivalent, in accuracy, either to Pid[x where (j) stands for one of the three approximations, < fl> bd> or ( fd> fo the perturbed flux distribution and stands for either ( fl or ( bd-cf> is a better approximation to compared to better approximation to compared to we conclude that all first-order perturbation expressions in integral transport theory formulations considered in this work are equivalent, in accuracy, to some high-order approximation to Pji>. This higher accuracy can be computed, in integral formulations, using the flux and source-importance functions for the unperturbed reactor. 

Little evidence is available on the accuracy of the different integral transport theory formulations and the calculational effort required to implement them. It would be useful to establish the relative accuracy and practicability of the different formulations and approximations by numerical investigation. 

Interest in integral transport theory methods for reactivity calculations has recently increased, mainly because of two features peculiar to integral formulations. 

The collision probability is one of several possible formulations of integral transport theory. Three other formulations are the integral equations for the neutron flux, neutron birth-rate density, and fission neutron density. Oosterkamp (26) derived perturbation expressions for reactivity in the birth rate density formulation. The fission density formulation provides the basis for Monte Carlo methods for perturbation calculations (52, 55). 

This section presents perturbation theory expressions and adjoint functions that correspond to the collision probability, flux, birth-rate density, and fission density formulations [see also reference (54)]. The functional relation between different first-order approximations of perturbation theory in integral and in integrodifferential formulations is established. Specifically, the approximation of the integrodifferential formulation that is equivalent, in accuracy, to each of the first-order approximations of the integral theory formulations is identified. The physical meaning of the adjoint functions corresponding to each of the transport theory formulations and their interrelation are also discussed. 

We shall restrict the comparison to the flux and birth-rate density formulations of integral transport theory and to two sets of distribution functions one consists of the flux and source importance function, and the other set consists of the solutions of the transport equations in the formulation under consideration. 

The formulation dependence of first-order perturbation theory of integral transport is illustrated by comparing the four approximations - ... 

Following Fey nman s original work, several authors pmsued extensions of the effective potential idea to construct variational approximations for the quantum partition function (see, e g., Refs. 7,8). The importance of the path centroid variable in quantum activated rate processes was also explored and revealed, which gave rise to path integral quantum transition state theory and even more general approaches. The Centroid Molecular Dynamics (CMD) method for quantum dynamics simulation was also formulated. In the CMD method, the position centroid evolves classically on the efiective centroid potential. Various analysis and numerical tests for realistic systems have shown that CMD captures the main quantum effects for several processes in condensed matter such as transport phenomena. 

Autonomous phenomena in multicellular systems are considered in the section on bioelectric patterning in cell aggregates. A mathematical model is considered which describes the cell aggregate via macroscopic variables--concentrations and voltage. Cells are "smeared out" to formulate the theory in terms of continuum variables. A nonlinear integral operator is introduced to model the intercellular transport that corrects the cruder diffusion-like terms usually assumed. This transport term is explicitly related to the properties of cell membranes. 

It is noted that the integrals evaluated to arrive at Eq. (2) result in self-truncaticm of the flux and cross-section moments, thus making Eq. (1) theoretically consistent with the transport equation. In addition, although the components of the diffusion coefficient in the x and y directions are different, the formulation is such that it is compatible with the numerical algorithms existing in most 2-D diffusion theory codes. 

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