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The equilibrium theory of a one-component nondegenerate charged gas, in a uniform continuum of a neutralizing charge, is studied on the basis of a linear-closure procedure for the BBGKY hierarchy which, in the present instance, corresponds to the use of the linearized Kirkwood superposition approximation, and leads to a second-order linear differential equation for the radial distribution function. Its solution is studied, numerically and analytically, for a wide range of the plasma parameter γ = e2/kTΛD, where ΛD is the Debye length. Results for γ ≪ 1 are compared with corrections to the Debye-Hückel theory derived by Abe and others. Solutions for γ ≫ 1, which approach an ordered state of the system as γ increases, are compared with two simple ``model'' calculations of the equilibrium theory of a body-centered cubic lattice of charges in a uniform background of compensating charge. The thermodynamic properties of these ``Coulomb lattice'' models are in qualitative agreement with those of the plasma as computed via our closure approximation for γ ≫ 1. The possibility is therefore suggested that, even for large γ, the linear closure approximation has a larger degree of validity than might be expected a priori.

A description is given of a general closure principle involving the minimization of the mean square error. The procedure based upon this principle can be applied to the truncation of the BBGKY hierarchy at various stages and to the approximation of unwanted terms arising in the equation of motion method by linear combinations of the observables to be retained. On a general level the significance of the closure principle is described in terms of the geometry of function space, and several useful general properties of the principle are derived. A discussion is devoted to the relation between the closure error (i.e., the least mean square error) and the error in the end result (e.g., the free energy, the radial distribution function, etc.); however, the results, while providing some insight, are not sufficiently refined to provide upper bounds to errors in all problems of statistical mechanics where the method is applicable. On the level of specific application it is shown that the principle yields results identical to the random phase approximation and to the linearized version of the Kirkwood superposition approximation in two special cases. Later sections of the paper describe in greater than usual generality, the formalism connecting thermodynamic properties and other equilibrium properties with the microscopic equations of motion in which closure approximations have been introduced. Two illustrative examples of the application of the over-all method were made to the case of a classical system of electrons in a uniform background of compensating charge, one leading to the well-known results of Debye and the other to a more accurate and elaborate theory developed in quantitative detail elsewhere.