Interested in corrections to Boltzmann distribution for finite reservoirs

Background

In statistical mechanics, the way we usually get the probability distribution is as follows:

1. We pick a subsystem $S$, coupled to a reservoir $R$.
2. We suppose that $p(E_s,N_s) \propto \exp{(S_R(E -E_s)/k)}$, where $E$ is the total energy of $S + R$, and $E_s$ is the energy of $S$.
3. We expand the reservoir entropy $S_R(E-E_s)$ around $E_s = 0$.
4. We take the reservoir to be infinite in energy and particle number, allowing us to neglect higher-order terms in that expansion.

This leads to the result that the probability $p(E_s,N_s)$ that $S$ is found in a particular state $(E_s, N_s)$ is $$p(E_s,N_s) \propto \exp{(-(\beta E_s + \mu N_s))}.$$ The particularly simple form of $p(E_s,N_s)$ stems from the fact that the second-order (and higher) terms in the expansion of the reservoir entropy $S_R(E - E_s)$ vanish in the thermodynamic limit.

My question

Has anyone has explored corrections to this equation for small systems which may be interacting with reservoirs $R$ which are

1. finite (as in, have a specified energy of particle number) or

2. may be influenced by $S$ in a non-negligible way?

My specific interest is in non-quantum systems, and reservoirs with small particle number.

• If you are seeking resources (e.g. books or papers) could you make that clearer what type of resource you need and (again if so) please add the "resource recommendation" tag. May 17 '18 at 22:29
• @StephenG Sure. May 17 '18 at 22:33
• @StephenG Actually, I really don't think this should be a resource-recommendations question. It's too specific. I think it should be edited to ask about the concept, and then people can provide references in addition to direct answers. May 18 '18 at 6:00