On the Equation of State of Small Systems

Lebowitz and Percus recently described a method by which a thermodynamic function $F$ in a canonical ensemble could be expressed in terms of a differential operation on $F$ as described in a grand canonical ensemble. The result of such a transformation was to derive an explicit expression for the dependence of the virial coefficients in the expansion of the canonical pressure on the number of molecules in the periodic system. The purpose of this article is to elucidate the methods and assumptions used in this transformation and to derive the expression for the canonical pressure by applying this method to the $\mathrm{(N–P–T)}\mathrm{ensemble}\mathrm{\hspace{0.17em}\to \hspace{0.17em}(N–V–T)}$ ensemble transformation.