5.3 Grand canonical ensemble ( μ,V,T) : Consider a system in contact with an energy reservoir as well as a particle reservoir and the system could exchange energy as well as particles (mass) with the reservoirs. Canonical ensemble theory has limitations in dealing these systems and needs generalization. It comes from the realization that not only the energy E but also the number of particles N of a physical system is difficult to measure directly. However, their average values, and , are measurable quantities. The system interacting with both energy and particle reservoirs comes to equilibrium when a common temperature T and a common chemical potential μ with the reservoir is established. In this ensemble, each microstate (r, s) corresponds to energy and number of particles in that state. If the system is in thermodynamic equilibrium at temperature T and chemical potential μ, the grand canonical partition function is given by

where is the fugacity of the system. In case of a system of continuous energy levels, the grand partition function can be written as

Note that division by is only for indistinguishable particles.

The expectation value of a thermodynamic quantity is given by

The grand potential is the appropriate potential or free energy to describe the thermodynamic system in equilibrium with temperature T and chemical potential μ. It can be shown that

All equilibrium thermodynamic properties can now be calculated by taking appropriate derivatives of the grand potential with respect to its parameters as given below.

In the table below, statistical quantities and the corresponding thermodynamic functions in cases of different ensembles are given. Thermodynamic quantities and response functions are different derivatives of these potential functions.

Ensemble	Statistical quantity	Thermodynamic functions
Microcanonical	Number of microstates:	Entropy:
Canonical	Canonical partition function:	Helmholtz free energy:
Grand Canonical	Grand partition function:	Grand potential: