I. Introduction
At any given temperature, the thermodynamic and transport properties of a material system are controlled by the interactions operating between the constituent particles of that system. These interactions are specific to each substance, and give rise to wide variations in crystal structures, and in the short-range order that X-ray and neutron diffraction experiments reveal to be present in liquids and amorphous solids. One of the most basic challenges perennially facing statistical mechanics is to provide logical and quantitative connections between interactions on the one hand, to long-range and short-range order in material systems on the other hand. Computer simulation has been used aggressively and productively for this purpose, while suffering obvious system-size and time-scale limitations. More analytic approaches in statistical mechanics that are designed to attain the same goals have a long history, marked by substantial but still incomplete success. The present paper intends to illuminate a small aspect of the general problem of interaction-order connections. In particular, attention focuses on the existence and nature of a family of continuous isothermal processes that simultaneously change the system density, as well as the interactions, in such a way that pair correlations remain invariant. The resulting theoretical analysis leads inevitably to conclusions about particle arrangements in space, and the way in which they can be produced, that in our opinion seem less than obvious. In particular, this line of investigation offers novel insights into the subject of hard-particle packings, e.g. for hard rods, disks, and spheres in 1,2, and 3 dimensions respectively. The following Section II includes a precise definition of the processes at issue, the “iso-g⁽²⁾ ” processes. Section II also illustrates these processes by exact low-density results, connects them to the well-known isothermal compressibility and Ornstein-Zernike relations, and stresses the relevance of the venerable HNC and PY approximate closures. For the purpose of providing a tangible example, Section III reports results for the one-dimensional rigid rod system, including calculation of an approximate effective pair potential that supplements the hard-core interaction during the iso-g⁽²⁾ process, and presents a Monte Carlo test of the accuracy of that effective potential. Section IV exhibits the corresponding effective pair potential for hard spheres in three dimensions. A final Section V presents discussion of several issues that extend beyond the scope of this initial investigation, but which represent natural research directions for further development of the concepts introduced and explored here.

Last updated 1-5-01