Isothermal-isobaric ensemble for small systems

The application of the isothermal-isobaric $(N-P-T)$ ensemble to small systems is considered. In the small system limit, which is currently gaining in scientific and technological significance, a volume scale must be introduced in order to obtain a partition function that is dimensionless. The volume scale, however, must be carefully chosen since it depends upon the nature of the boundary separating the system from the surroundings. If the incorrect volume scale is used, the resulting $N-P-T$ ensemble partition function will not rigorously describe the small system of interest. Although volume scales become inconsequential in the thermodynamic limit, care must be exercised in formulating the ensembles used to study small systems.