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

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

To clarify, this is not a grand canonical ensemble.

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

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

In grand canonical ensemble V, T and chemical potential are fixed.

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

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

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

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

Thus each orbital is a grand canonical ensemble unto itself, one simple that its statistics can be immediately derived here.

Grand canonical ensemble - an ensemble of systems, each of which is again in thermal contact with a reservoir.

In this work statistical ensembles were averaged to obtain the sorption mechanism with the grand canonical ensemble.

For chemistry problems involving chemical reactions, the grand canonical ensemble provides the appropriate foundation, and there are two Lagrange multipliers.

The grand canonical ensemble can be seen as an extension of the canonical ensemble:

Notice that, for the microcanonical ensemble, plays the role of the partition function in the canonical and grand canonical ensembles.

However, there one would replace the canonical ensemble with the grand canonical ensemble, since there is exchange of particles between the system and the reservoir.

The Fermi-Dirac distribution, which applies only to a quantum system of non-interacting fermions, is easily derived from the grand canonical ensemble.

It generalizes the narrower concepts of the grand canonical ensemble and canonical ensemble in statistical mechanics.

It is the measure associated with the Boltzmann distribution, and generalizes the notion of the canonical ensemble to the grand canonical ensemble.

For example, one can specify the density operators describing microcanonical, canonical, and grand canonical ensembles of quantum mechanical systems, in a mathematically rigorous fashion.

The grand canonical ensemble can be derived from a microcanonical ensemble of a total system, where that total system is composed of an object together with a large reservoir.

In contrast to grand canonical ensemble, an open statistical ensemble satisfies the scale invariance requirement: general term of the included subsystem distribution corresponds to that of the original system.

The grand canonical ensemble is allowed to statistically fluctuate between various microscopic states (microstates) with differing energies and differing numbers of particles by exchanging these with a reservoir.

The grand canonical partition function applies to a grand canonical ensemble, in which the system can exchange both heat and particles with the environment, at fixed temperature, volume, and chemical potential.