Abstract
A variational principle yielding upper bounds to the chemical potential is derived. Based upon the grand canonical distribution, it exploits the familiar variational principle yielding upper bounds to the Helmholtz function. Simple systems which involve only pairwise particle interactions and which are describable by classical statistical mechanics are examined in a manner which exploits the discrete nature of the variable
N
which appears in the grand canonical ensemble. A typical system is partitioned to comprise a "test cluster" of particles and a remainder. The potential energy of the former is dealt with exactly. The potential energy of each particle of the remainder is represented by an effective potential upon which the variational principle imposes a self-consistent (field) condition. For any finite test cluster, regardless of the range of the interparticle forces, the effective potential is short-range in the usual thermodynamic limit and independent of the number of particles in the cluster. For ionic systems the Poisson-Boltzmann equation is a consequence of the variation principle on the chemical potential.