Editing Boltzmann distribution (section)

== Generalized Boltzmann distribution ==
A distribution of the form
:<math>\Pr\left(\omega\right)\propto\exp\left[\sum_{\eta=1}^{n}\frac{X_{\eta}x_{\eta}^{\left(\omega\right)}}{k_{B}T}-\frac{E^{\left(\omega\right)}}{k_{B}T}\right]</math>
is called '''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>

The Boltzmann distribution is a special case of the generalized Boltzmann distribution. The generalized Boltzmann distribution is used in statistical mechanics to describe [[canonical ensemble]], [[grand canonical ensemble]] and [[isothermal–isobaric ensemble]]. The generalized Boltzmann distribution is usually derived from the [[principle of maximum entropy]], but there are other derivations.<ref name="Gao2019" /><ref name="Gao2022">{{cite journal |last1= Gao |first1= Xiang |date= March 2022 |title= The Mathematics of the Ensemble Theory |url= https://www.sciencedirect.com/science/article/pii/S2211379722000390|journal= Results in Physics|volume= 34|pages= 105230|doi= 10.1016/j.rinp.2022.105230 |bibcode= 2022ResPh..3405230G |s2cid= 221978379 |arxiv= 2006.00485 }}</ref>

The generalized Boltzmann distribution has the following properties:
* It is the only distribution for which the entropy as defined by [[Entropy (statistical thermodynamics)#Gibbs entropy formula|Gibbs entropy formula]] matches with the entropy as defined in [[Entropy (classical thermodynamics)|classical thermodynamics]].<ref name="Gao2019" />
* It is the only distribution that is mathematically consistent with the [[fundamental thermodynamic relation]] where state functions are described by ensemble average.<ref name="Gao2022" />