We know already how to treat big systems in diffusive contact, they will reach their most probable configuration described by the minimum of F. What about small systems, in diffusive contact with big reservoir ? We must be able to take into account their discrete states. To do this we define the Gibbs factor, and Gibbs sum.
In our exercise above we assume implicitly constant volume for both small system and reservoir, so it will not be explicitly written.
We get the Gibbs factor describing the probability that the system is in a given state with certain energy and number of particles. The chemical potential here is this of the reservoir