Work relation for determining the mixing free energy of small-scale mixtures.

In thermodynamically characterizing a mixture comprising a finite number of molecules, we consider two kinds of alchemical protocol for producing a mixture from a pure substance. The first is a simple alchemical protocol, whereas the second is a protocol comprising a series of processes with feedback control and mixing with semipermeable membranes. A comparison of the two numerically determined free-energy changes provides a combinatorial factor that indicates the indistinguishability of the molecules. The comparison also uncovers a work relation for determining the mixing free energy, which generates the basic properties of mixtures. We demonstrate a numerical calculation of applying the work relation to a mixture of argon and krypton. The mixing free energy clearly shows the characteristics of liquid--vapor transition.