Thermodynamic Properties of Small Systems

We investigate the dependence of the pressure of a homogeneous system, at a given density $\rho$ and temperature $T$, on the number of particles $N$. The particles of the system are assumed to interact via forces of finite range $a$ and are confined to a periodic cube of volume $L^3$, $\rho \equiv \frac{N}{L^3}$. We find that there are generally two types of $N$ dependencies in the pressure and other intensive properties of the system. There is a simple dependence which goes essentially as a power series in ($\frac{1}{N}$) and may be computed explicitly in terms of the grand-ensemble averages of these properties where it is absent. The other, more complex, dependence comes from the volume dependence of those cluster integrals which are large enough to wind at least once around the periodic torus. These do not appear in a virial expansion for terms $k\le {\left(\frac{N}{\rho a^3}\right)}^{\frac{1}{3}}$. They play however a dominant role in the $N$ dependence observed by Alder and Wainwright in their machine computations on a hard-sphere gas. While the explicit calculation of these terms is very difficult and has been carried through only in a few special cases, they may be related, approximately at least, to the radial distribution function in an infinite system. We also find an expression for the correlation between the particles of an ideal gas represented by a microcanonical ensemble.