Length Scale for the Constant Pressure Ensemble:  Application to Small Systems and Relation to Einstein Fluctuation Theory

In this paper we address the constant pressure ensemble and the volume scale that must be introduced in order to represent the corresponding partition function as a dimensionless integral. The volume scale or length scale problem arises quite generally when it is necessary (for whatever reason) to apply semiclassical statistical mechanical theory in configuration space alone, rather than in the full phase space of the system. We find that the length scale, derived by earlier workers concerned primarily with systems in the thermodynamic limit, is not suitable for application of the constant pressure ensemble to small systems such as clusters in nucleation theory or mesodomains in microemulsion theory. We discuss some of the well-known deficiencies of the conventional representation of the constant pressure ensemble and some which are not so well-known. Also the close connection between the constant pressure ensemble and Einstein fluctuation theory is emphasized, and we clarify the two types of fluctuation that are relevant to both developments but which are not always understood and distinguished by workers in the field. We derive the proper length scale applicable to systems of any size and remark that when it is used for small systems, the constant pressure ensemble partition function can no longer be derived from that for the canonical ensemble by simple Laplace transformation. We emphasize the fact that although the constant pressure ensemble has only found modest application in the statistical thermodynamics of macroscopic systems, it is being increasingly applied in the theory of small systems that may be conceptual rather than real, and that, for this reason, the ensemble should be placed on a firm fundamental foundation. In particular, we illustrate the relevance of the so-called “shell molecule”. Finally we apply our development to fluctuations in small systems to illustrate the qualitative and quantitative differences between small and large systems.