On exact bounds for the cluster free energy in the three-dimensional lattice-gas model

The exact bounds by Lebowitz and Penrose for the cluster distribution of the simple cubic lattice-gas model with attractive interactions lead here to exact upper and lower bounds for the limiting cluster free energy or free energy per particle of a cluster of macroscopic size and, consequently, for the renormalized fugacity relating cluster distribution and partition functions. We also give practical realizations of the bounds for the cluster distribution. Our upper and lower bound expressions are always close enough to each other for 4J/kBT>1.5. The nature of the bounds at higher temperatures is also analyzed.