Collision density formulation, transport

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. 

In the collision density, the most commonly used integral transport formulation, the flux and adjoint equations can be written as follows ... 

The Boltzmann equation is considered valid as long as the density of the gas is sufficiently low and the gas properties are sufficiently uniform in space. Although an exact solution is only achieved for a gas at equilibrium for which the Maxwell velocity distribution is supposed to be valid, one can still obtain approximate solutions for gases near equilibrium states. However, it is evident that the range of densities for which a formal mathematical theory of transport processes can be deduced from Boltzmann s equation is limited to dilute gases, since this relation is reflecting an asymptotic formulation valid in the limit of no coUisional transfer fluxes and restricted to binary collisions only. Hence, this theory cannot without ad hoc modifications be applied to dense gases and liquids. 

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