Pressure-dependent partition functions

It is shown that the partition functions relating to statistical ensembles can be classified as cumulative, distributive or differential with regard to any of the extensive thermodynamic variables. The Laplace transforms involved in the general formulation of transformed partition functions, when extensive parameters are replaced by their conjugate intensive variables, are two-sided and appear in different forms according to the classification mentioned. The petit partition function at constant pressure can be defined for both classical and quantal systems as a Stieltjes integral or dimensionless Laplace transform over the volume-dependent partition function. This expression contains no arbitrary external parameters and satisfies all the conditions formulated by Münster; it can be interpreted as the ratio of two partition functions, the one relating jointly to the system and a macroscopic boundary, and the second to the boundary alone. It is known that the constant pressure grand partition function as defined by Guggenheim diverges for all thermodynamically compatible values of its independent variables. It is shown that the Gibbs-Duhem relation, and hence all the statistical properties of the system, except its size, can be derived by differentiating the reciprocal of this, or a similarly defined, function, not its logarithm as with other partition functions. The mathematical differences arising out of the various proposed definitions are illustrated in a number of simple examples.