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Differential equations and their solutions for the formation of a high-energy plasma by the injection and trapping of energetic atoms in a magnetic-mirror geometry are discussed. The present approach differs from previous treatments in that the finite orbit size of the ions and the spatial dependence of the trapping process are included in the equations. By including the spatial dependence it was possible to evaluate the effect of charge exchange of trapped ions with neutral-beam atoms. Although not affecting the total number of trapped ions, this process results effectively in a change of the mean radial position of the ions. The effect on plasma growth rates of charge exchange of the trapped ions with beam atoms was found to be essentially negligible. The solutions for ion density vs time exhibit growth rates similar to those obtained when the plasma density is assumed to be spatially constant. Background gas variations during plasma buildup have also been investigated. Under the present set of conditions the effect of ``gas burnout'' and plasma pumping are small, even at high densities.

A general method of calculation is described for quantum statistical mechanics. It is based on a simplification of the Laplace transform of the density matrix which follows from a theorem due to Hugenholtz. The basic result is that an element of the density matrix can be written as a sum over graphs, with the contribution of each graph factored into contributions from connected or linked graphs. Applied to the grand partition function, the exponential formula of Bloch and deDominicis is obtained in a simple way. A similar formula is then derived for the canonical ensemble and the case of a nondegenerate gas.In this way the familiar result of Uhlenbeck and Beth is obtained for the second virial coefficient. Techniques are also introduced for evaluating ensemble averages of operators. In this connection, some care must be exercised in the case of diagonal operators. Finally, these methods are used to calculate the pair-correlation function for a system of fermions interacting through shortrange forces.