A Statistical Mechanical Approach to the Problem of a Fluid in an External Field

A rigorous, equilibrium statistical mechanical treatment of a fluid in a weak external field is given. The technique involves a cell division which leads to upper and lower bounds for the free energy density. Under a suitable double limiting procedure these limits coalesce, yielding a free energy consisting of a field‐free term plus a field‐dependent term. The cell division allows a direct physical definition of the local pressure p(s) and the local density ρ(s). This treatment provides a rigorous derivation of the thermodynamics of a fluid in a weak external field and, in particular, the hydrostatic equation gradp = − ρ gradφ.