Perturbation theory transport equations

Let us assume (a) that the energy levels of our molecules are E0, Elt E%, . En (b) that the fraction of molecules in the with state at time t is xm(t) (c) that the transition probabilities per unit time Wnm from state m to n can be computed in terms of the interaction of the molecules with a heat bath (which is postulated to remain at temperature T) by application of quantum-mechanical time dependent perturbation theory (the fVBra s being proportional to squares of absolute values of the matrix elements of the interaction energy) and (d) that the temporal variations of the level concentrations are described through the transport equation... 

A primary objective of this work is to provide the general theoretical foundation for different perturbation theory applications in all types of nuclear systems. Consequently, general notations have been used without reference to any specific mathematical description of the transport equation used for numerical calculations. The formulation has been restricted to time-independent and linear problems. Throughout the work we describe the scope of past, and discuss the possibility for future applications of perturbation theory techniques for the analysis, design and optimization of fission reactors, fusion reactors, radiation shields, and other deep-penetration problems. This review concentrates on developments subsequent to Lewins review (7) published in 1968. The literature search covers the period ending Fall 1974. 

Perturbation theory in reactor physics is usually formulated and used in the integrodifferential formulation of transport theory (or in the diffusion approximation to this formulation). This formulation is convenient for use because (1) the perturbations in the operators of the Boltzmann equation are linear with the physical perturbations in the system, and (2) most computer codes in use solve the integrodifferential (or the diffusion) equations. 

The perturbation theory we consider is for the static reactivity pertaining to a perturbation of the reactor from a reference critical state. We use the general form of the transport equations with spelled-out notations and a continuous representation of the phase-space variables (r, E, 1). The subscript 0 is omitted from the parameters corresponding to the critical reactor. Instead, a bar denotes perturbed parameters. We take 7=1. 

Different perturbation theory expressions for reactivity are obtained from different formulations of the neutron transport equation. 

An alternative to using equilibrium MD for computing transport coefficients is to use nonequilibrium molecular dynamics (NEMD) in which a modified Elamiltonian is used to drive the system away from equilibrium. By monitoring the response of the system in the limit of a small perturbation, the transport coefficient associated with the perturbation can be calculated. There is a rich literature on the use of NEMD to calculate transport coefficients the interested reader is referred to the excellent monograph by Evans and Morriss and the review article by Cummings and Evans.The basic idea behind the technique is that a system will respond in a linear fashion to a small perturbation. The following linear response theory equation is applicable in this limit ... 

The Statistical Rate Theory (SRT) is based on considering the quantum-mechanical transition probability in an isolated many particle system. Assuming that the transport of molecules between the phases at the thermal equilibrium results primarily from single molecular events, the expression for the rate of molecular transport between the two phases 1 and 2 , R 2, was developed by using the first-order perturbation analysis of the Schrodinger equation and the Boltzmann definition of entropy. 

In using Eqs. 19 and 34 as starting points for the development of the theory of transport processes, several tasks remain. The first and perhaps simplest is to develop expressions for the transport coefficients in terms of the single and pair densities /(1> and /< >. Next the continuity equations must be solved to give explicit expressions for the densities when the system is perturbed from its equilibrium state by the transport process under consideration. Finally, it is required to obtain the frictional coefficient f in terms of the intermolecular potential energy function according to Eq. 33. 

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). 

As a basis, Schoeller et al. took a semi-classical Master equation approach to calculate the transport properties. Thereby, they assumed an incoherent tunneling, which was treated as a perturbation, while the Coulomb interaction between charged nanopartides was taken into account nonpertubatively within a capacitance model. However, in contrast to the standard orthodox theory, they explidtly considered the discrete nature of the electronic spectrum of the nanopartides. In the calculated /(V)... 

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