Hill's Small Systems Nanothermodynamics: A Simple Macromolecular Partition Problem with a Statistical Perspective

Using a simple example of biological macromolecules which are partitioned between bulk solution and membrane, we investigate T.L. Hill's phenomenological nanothermodynamics for small systems. By introducing a {\em systems size dependent} equilibrium constant for the bulk-membrane partition, we obtain Hill's results on differential and integral chemical potentials $\mu$ and $\hat{\mu}$ from computations based on standard Gibbsian equilibrium statistical mechanics. It is shown that their difference can be understood from an equilibrium re-partitioning between bulk and membrane fractions upon a change in system's size; it is closely related to the system's fluctuations and inhomogeneity. These results provide a better understanding of the nanothermodynamics and clarify its logical relation with the theory of statistical mechanics.