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.

  • $\begingroup$ 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. $\endgroup$
    – StephenG
    May 17 '18 at 22:29
  • $\begingroup$ @StephenG Sure. $\endgroup$
    – foneyoscar
    May 17 '18 at 22:33
  • $\begingroup$ @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. $\endgroup$
    – David Z
    May 18 '18 at 6:00

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