To clarify, this is not a grand canonical ensemble.

Let's rework everything using a grand canonical ensemble this time.

The grand canonical ensemble may also be used to describe classical systems, or even interacting quantum gases.

A generalization of this is the grand canonical ensemble, in which the systems may share particles as well as energy.

In other words, each single-particle level is a separate, tiny grand canonical ensemble.

This is the characteristic state function for the grand canonical ensemble.

The particle number and energy in the system have natural fluctuations in the grand canonical ensemble.

The usefulness of the grand canonical ensemble is illustrated in the examples below:

By abstracting away the reservoir, we will arrive at the grand canonical ensemble.

The grand potential is the characteristic state function for the grand canonical ensemble.