I have checked several references for the derivation of the probability function of the canonical ensemble. I have seen two (essentially similar) approaches. Both assume a system is placed in a large reservoir:
- Study the probability that the system is in a given microstate. Assert that this probability is proportional to $\Omega_S$ where $\Omega_S$ is the multiplicity of the system. For example, https://www2.oberlin.edu/physics/dstyer/StatMech/CanonicalEnsemble.pdf.
- Study the probability that the reservoir is in a given microstate. Assert that this probability is proportonal to $\Omega_R$ where $\Omega_R$ is the multiplicity of the reservoir.
Both of these seem unsatisfactory to me. In both cases it seems like the probability should be proportional to $\Omega_R \cdot \Omega_S.$ Why is it okay to neglect this multiplication?