Bounds for the Derivatives of the Free Energy and the Pressure of a Hard-Core System near Close Packing

A simple argument shows how bounds to the derivatives of the free energy (or other thermodynamic potentials) may be derived from bounds to the free energy itself. The method is applied to study the divergence of the pressure p as a function of specific volume v for a system of particles with hard cores near close packing, Υ = [(v/vmin) − 1]→0. For a classical d-dimensional system with oriented ``cubical'' hard cores we prove that ξ1d/Υ≤p(v)vmin/kBT≤ξ2d/Υ,where the constants satisfy 0<ξ1<1<ξ2<∞. For a general hard-core system we establish only η1lnΥ−1≤p(v)vmin/kBT≤η2(lnΥ−1)d/Υfor any η1<1<η2.