Grand canonical ensemble
The grand canonical ensemble or μVT ensemble extends the canonical ensemble by allowing the exchange of both energy and particles with an external reservoir, so that the conserved state functions are the temperature , the volume , and the chemical potential .
The derivation is practically identical to that of the canonical ensemble. We refer to the system of interest as , and the reservoir as . In total, has energy and population .
Let be the number of -microstates with energy . Then the probability that is in a specific microstate is as follows:
Then, as for the canonical ensemble, we assume and , and approximate by Taylor-expanding around and . The resulting probability distribution is known as the Gibbs distribution, with :
Where the normalizing grand partition function is defined as follows:
In contrast to the canonical ensemble, whose thermodynamic potential was the Helmholtz free energy , the grand canonical ensemble instead minimizes the grand potential :
So . This is proven in the same way as for in the canonical ensemble.
- H. Gould, J. Tobochnik, Statistical and thermal physics, 2nd edition, Princeton.