Cavity averages for hard spheres in the presence of polydispersity and incomplete data

We develop a cavity-based method which allows to extract thermodynamic properties from position information in hard-sphere/disk systems. So far, there are available-volume and free-volume methods. We add a third one, which we call available volume after take-out, and which is shown to be mathematically equivalent to the others. In applications, where data sets are finite, all three methods show limitations, and they do this in different parameter ranges. We illustrate the principal equivalence and the limitations on data from molecular dynamics: In particular, we test robustness against missing data. We have in mind experimental limitations where there is a small polydispersity, say 4% in the particle radii, but individual radii cannot be determined. We observe that, depending on the used method, the errors in such a situation are easily 100% for the pressure and 10kT for the chemical potentials. Our work is meant as guideline to the experimentalists for choosing the right one of the three methods, in order to keep the outcome of experimental data analysis meaningful.