Editing Grand canonical ensemble (section)

== Basics ==

In simple terms, the grand canonical ensemble assigns a probability {{math|''P''}} to each distinct [[microstate (statistical mechanics)|microstate]] given by the following exponential:
<math display="block">P = e^{{(\Omega + \mu N - E)}/{(k T)}},</math>
where {{math|''N''}} is the number of particles in the microstate and {{math|''E''}} is the total energy of the microstate. {{math|''k''}} is the [[Boltzmann constant]].

The number {{math|Ω}} is known as the [[grand potential]] and is constant for the ensemble. However, the probabilities and {{math|Ω}} will vary if different {{math|''µ'', ''V'', ''T''}} are selected. The grand potential {{math|Ω}} serves two roles: to provide a normalization factor for the probability distribution (the probabilities, over the complete set of microstates, must add up to one); and, many important ensemble averages can be directly calculated from the function {{math|Ω(''µ'', ''V'', ''T'')}}.

In the case where more than one kind of particle is allowed to vary in number, the probability expression generalizes to
<math display="block">P = e^{{(\Omega + \mu_1 N_1 + \mu_2 N_2 + \dots + \mu_s N_s - E)}/{(k T)}},</math>
where {{math|''µ''<sub>1</sub>}} is the chemical potential for the first kind of particles, {{math|''N''<sub>1</sub>}} is the number of that kind of particle in the microstate, {{math|''µ''<sub>2</sub>}} is the chemical potential for the second kind of particles and so on ({{math|''s''}} is the number of distinct kinds of particles). However, these particle numbers should be defined carefully (see the [[#Meaning of chemical potential, generalized "particle number"|note on particle number conservation]] below).

The distribution of the grand canonical ensemble is called [[Boltzmann distribution#Generalized Boltzmann distribution|generalized Boltzmann distribution]] by some authors.<ref name="Gao2019">{{cite journal |last1= Gao |first1= Xiang |last2= Gallicchio |first2= Emilio |first3= Adrian |last3= Roitberg  |date= 2019 |title= The generalized Boltzmann distribution is the only distribution in which the Gibbs-Shannon entropy equals the thermodynamic entropy |url= https://aip.scitation.org/doi/abs/10.1063/1.5111333|journal= The Journal of Chemical Physics|volume= 151|issue= 3|pages= 034113|doi= 10.1063/1.5111333|pmid= 31325924 |arxiv= 1903.02121 |bibcode= 2019JChPh.151c4113G |s2cid= 118981017 |access-date= }}</ref>

Grand ensembles are apt for use when describing systems such as the [[electron]]s in a [[Electrical conductor|conductor]], or the [[photons]] in a cavity, where the shape is fixed but the energy and number of particles can easily fluctuate due to contact with a reservoir (e.g., an electrical ground or a [[black body|dark surface]], in these cases). The grand canonical ensemble provides a natural setting for an exact derivation of the [[Fermi–Dirac statistics]] or [[Bose–Einstein statistics]] for a system of non-interacting quantum particles (see examples below).

; Note on formulation :
: An alternative formulation for the same concept writes the probability as <math>\textstyle P = \frac{1}{\mathcal Z} e^{(\mu N-E)/(k T)}</math>, using the [[Partition function (statistical mechanics)|grand partition function]] <math>\textstyle \mathcal Z = e^{-\Omega/(k T)}</math> rather than the grand potential. The equations in this article (in terms of grand potential) may be restated in terms of the grand partition function by simple mathematical manipulations.