Fluctuations in the μpT ensemble

We work out the statistical ensemble of an equilibrium macroscopic system composed of particles interacting via short-range forces in a completely open situation in which heat, mechanical work, and matter can be exchanged with the surroundings. Attention is focused on variances and covariances of extensive properties, with particular emphasis on their connection with thermodynamic response functions like the isothermal compressibility or the isobaric thermal expansivity. The exact formulas we derive prove that, with full generality, response functions quantify variances and covariances of ratios between extensive properties such as the number density, the enthalpy per particle, or the "Hill energy" per unit volume. Results for grand canonical or isothermal-isobaric ensembles are recovered when either the volume or the number of particles are fixed, while consistency with Einstein's thermodynamic theory is also encountered. A water-like Ising model illustrates how the standard mean-field approach proceeds in the ensemble, while it validates the correctness of our fluctuation formulas numerically. A potential usefulness of the ensemble to deepen on the phenomenological pattern of behavior of thermodynamic systems shows up.