A Generalized Perturbation Theory

In applying this method in the discussion of the wave equation H+(x) = W4(x), (27-1) 

The functions Fn(x) are conveniently taken as the members of a complete set of orthogonal functions of the variables x it is not necessary, however, that they be orthogonal in the same configuration space as that for the system under discussion. Instead, we assume that they satisfy the normalization and orthogonality conditions 

For an arbitrary choice of the functions Fn(x) Equation 27-5 represents an infinite number of equations in an infinite number of unknown coefficients An. Under these circumstances questions of convergence arise which are not always easily answered. In special cases, however, only a finite number of functions Fn(x) will be needed to represent a given function p(x) in these cases we know that the set of simultaneous homogeneous linear equations 27-5 has a non-trivial solution only when the determinant of the coefficients of the A s vanishes that is, when the condition 

Our problem is now in principle solved We need only to evaluate the roots of Equation 27-7 to obtain the allowed energy values for the original wave equation, and substitute them in the set of equations 27-5 to evaluate the coefficients An and obtain the wave functions. 

The relation of this treatment to the perturbation theory of Chapter VI can be seen from the following arguments. If the functions F (x) were the true solutions (x) of the wave equation 27-1, the determinantal equation 27-7 would have the form 

---

A generalized perturbation theory will be discussed in Section 27a. 

Doppler reactivity is the other important parameter in FBR cores because it assures the intrinsic safety characteristics of the core. The present cross-section adjustment method cannot treat the Doppler reactivity which is dominated by resonance-peak broadening of cross-sections. We have launched a new study to extend the applicability of the cross-section adjustment and design accuracy evaluation system to the Doppler effect. The basic method to evaluate sensitivity of self-shielding factors has been successfully derived from a generalized perturbation theory, and a prototype system to calculate the Doppler sensitivity is now under verification. 

In fact, our interest in the present formulation, the use ofNSS s andLKD s, has been aroused when studying the integrals over Cartesian Exponential Type Orhitals [la,b] and Generalized Perturbation Theory [ld,ej. The use of both symbols in this case has been extensively studied in the above references, so we will not repeat here the already published arguments. Instead we will show the interest of using nested sums in a wide set of Quantum Chemical areas, which in some way or another had been included in our research interests [Ic]. 

In this context equations (50) and (53) can be considered forming a completely general perturbation theory for nondegenerate systems, although a recent development permits to extend the formalism to degenerate states [lej. 

Hence, it is believed that general anesthetics exert most, if not all, of their effects by binding to one or more neuronal receptors in the CNS. This idea is a departure from the general perturbation theory described earlier that is, that the inhaled anesthetics affected the lipid bilayer rather than a specific protein. Continued research will continue to clarify the mechanism of these drugs, and future studies may lead to more agents that produce selective anesthetic effects by acting at specific receptor sites in the brain and spinal cord. 

For a comprehensive account and many remarkable developments of general perturbation theory, see Lowdin (1962,1963,1964,1965,1966,1968,1982a) and Lowdin and Goscinski (1971). 

As before, a general thermodynamic theory of stability formulation is quadratic in the perturbations of 8T. 8 V, and 8Nk, because the forces and flows vanish at equilibrium... 

To address the hmitations of ancestral polymer solution theories, recent work has studied specific molecular models - the tangent hard-sphere chain model of a polymer molecule - in high detail, and has developed a generalized Rory theory (Dickman and Hall (1986) Yethiraj and Hall, 1991). The justification for this simplification is the van der Waals model of solution thermodynamics, see Section 4.1, p. 61 attractive interactions that stabilize the liquid at low pressure are considered to have weak structural effects, and are included finally at the level of first-order perturbation theory. The packing problems remaining are attacked on the basis of a hard-core model reference system. 

The code FORMOS A-P has been developed over a number of years at North Carolina State University [2-4] for the purpose of automating the process of determining the family of near optimum fuel and BP LPs, while taking into account, with a minimum of assumptions, the complexities of the reload design problem. FORMOSA-P couples the stochastic optimization technique of Simulated Annealing (SA) [5] with a computationally efficient neutronics solver based on second-order accurate, nodal generalized perturbation theory (GPT) [6-7] for evaluating core physics characteristics over the cycle. 

The method of domain perturbations was used for many years before its formal rationalization by D. D. Joseph D. D. Joseph, Parameter and domain dependence of eigenvalues of elliptic partial differential equations, Arch. Ration. Mech. Anal. 24, 325-351 (1967). See also Ref. 3f. The method has been used for analysis of a number of different problems in fluid mechanics A. Beris, R. C. Armstrong and R. A. Brown, Perturbation theory for viscoelastic fluids between eccentric rotating cylinders, J. Non-Newtonian Fluid Mech. 13, 109-48 (1983) R. G. Cox, The deformation of a drop in a general time-dependent fluid flow, J. Fluid Mech. 37, 601-623 (1969) ... 

The results we have obtained are valid for a single frequency and, if we wish to be able to perform time-dependent perturbation theory for a general perturbation of the form... 

The recent expansion of the application of perturbation theory formulations is mainly due to the development of the generalized perturbation theory (GPT). Several versions of GPT formulations have been described. They are characterized by their form and their method of derivation. They are also distinguished by the form of the integral parameters to which they apply and by the method they use to allow for the flux and adjoint perturbation. A unified presentation of GPT is given in Section V, together with an elucidation of problems of accuracy and range of applicability of different formulations. Also outlined in Section V is a perturbation theory for altered systems. 

The development and application of generalized perturbation theory (GPT) has made considerable progress since its introduction by Usachev (i(S). Usachev developed GPT for a ratio of linear flux functionals in critical systems. Gandini 39) extended GPT to the ratio of linear adjoint functionals and of bilinear functionals in critical systems. Recently, Stacey (40) further extended GPT to ratios of linear flux functionals, linear adjoint functionals, and bilinear functional in source-driven systems. A comprehensive review of GPT for the three types of ratios in systems described by the homogeneous and the inhomogeneous Boltzmann equations is given in the book by Stacey (41). In the present review we formulate GPT for composite functionals. These functionals include the three types of ratios mentioned above as special cases. The result is a unified GPT formulation for each type of system. 

Generalized perturbation theory for two special cases of composite functionals are presented and discussed in some detail GPT for reactivity (Section V,B), and GPT for a detector response in inhomogeneous systems (Section V,E). The GPT formulation for reactivity is equivalent to a high-order perturbation theory, in the sense that it allows for the flux perturbation, GPT for a detector response in inhomogeneous systems 42, 43) is, in fact, the second-order perturbation theory known from other derivations I, 44, 45). These perturbation theory formulations provide the basis for new methods for solution of deep-penetration problems. These methods are reviewed in Section V,E,2. 

The generalized perturbation theory expressions presented in this section for systems described by the homogeneous Boltzmann equation (excluding Section V,B,2) are in the form proposed by Stacey (40, 41). Had we assumed that the overall alteration in the reactor retains criticality, we would have achieved the Usachev-Gandini version of GPT. Stacey s version is often associated (41, 46, 48, 62) with the variational perturbation theory as distinguished from the GPT of Usachev-Gandini. Does the variational approach provide a different perturbation theory than the GPT derived (35,39) from physical considerations Is one of these versions of perturbation theory more general or more accurate than the other What does the term GPT stand for ... 

It might be useful if a unified terminology were established for what is becoming an important field of perturbation theory. We propose that the term generalized perturbation theory be used for all perturbation theory formulations in which the flux and adjoint perturbations are allowed for... 

The design and analysis of realistic power systems increasingly involves the representation of nonlinear models with progressively higher demands on the accuracy of computations. The difficulties of nonlinear problems are well known, and approximation methods, of which perturbation theory is one, are to be welcomed. Of course, perturbation theory has been applied to problems, such as fuel burnup, where the properties are a function of the neutron flux it has been customary, however, to linearize the problem around the unperturbed problem. When the accuracy obtained by this or similar devices is inadequate, there is a case for considering a more general perturbation theory for nonlinear systems. 

D. SARGIS, GAPER - A Transport Perturbation Theory Program, GA-8667, Gulf General Atomic (April 1968). 

This is a formal property. It was derived in a very general fashion. It sometimes happens, however, that when we calculate a p>articular M (/9, a) in perturbation theory it diverges, and in removing the divergence there may be difficulties in satisfying (9.5.10) or analogous relations, as we shall presently see. 

---