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 l ength scale problem arises quite generally when it is necessary (for w hatever reason) to apply semiclassical statistical mechanical theory i n configuration space alone, rather than in the full phase space of th e system. We find that the length scale, derived by earlier workers co ncerned primarily with systems in the thermodynamic limit, is not suit able for application of the constant pressure ensemble to small system s such as clusters in nucleation theory or mesodomains in microemulsio n theory. We discuss some of the well-known deficiencies of the conven tional 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 deve lopments but which are not always understood and distinguished by work ers in the field. We derive the proper length scale applicable to syst ems of any size and remark that when it is used for small systems, the constant pressure ensemble partition function can no longer be derive d from that for the canonical ensemble by simple Laplace transformatio n. 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 o f small systems that may be conceptual rather than real, and that, for this reason, the ensemble should be placed on a firm fundamental foun dation. 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 differenc es between small and large systems.