Editing Entropy (section)

=== Equivalence of definitions ===
Proofs of equivalence between the entropy in statistical mechanics — the [[Entropy (statistical thermodynamics)#Gibbs entropy formula|Gibbs entropy formula]]:<math display="block">S = - k_\mathsf{B} \sum_i{p_i \ln{p_i}}</math>and the entropy in classical thermodynamics:<math display="block">\mathrm{d} S = \frac{\delta Q_\mathsf{rev}}{T}</math>together with the [[fundamental thermodynamic relation]] are known for the [[microcanonical ensemble]], the [[canonical ensemble]], the [[grand canonical ensemble]], and the [[isothermal–isobaric ensemble]]. These proofs are based on the probability density of microstates of the generalised [[Boltzmann distribution]] and the identification of the thermodynamic internal energy as the ensemble average <math display="inline">U = \left\langle E_i \right\rangle </math>.<ref>{{cite book |last= Callen|first= Herbert|date= 2001|title= Thermodynamics and an Introduction to Thermostatistics (2nd ed.)|publisher= John Wiley and Sons|isbn= 978-0-471-86256-7}}</ref> Thermodynamic relations are then employed to derive the well-known [[Gibbs entropy formula]]. However, the equivalence between the Gibbs entropy formula and the thermodynamic definition of entropy is not a fundamental thermodynamic relation but rather a consequence of the form of the [[Boltzmann distribution#Generalized Boltzmann distribution|generalized Boltzmann distribution]].<ref>{{cite journal |last1= Gao |first1= Xiang |last2= Gallicchio |first2= Emilio |first3= Adrian |last3= Roitberg |year= 2019 |title= The generalized Boltzmann distribution is the only distribution in which the Gibbs-Shannon entropy equals the thermodynamic entropy |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 }}</ref>

Furthermore, it has been shown that the definitions of entropy in statistical mechanics is the only entropy that is equivalent to the classical thermodynamics entropy under the following postulates:<ref name="Gao2022">{{cite journal |last1= Gao |first1= Xiang |date= March 2022 |title= The Mathematics of the Ensemble Theory |journal= Results in Physics|volume= 34|pages= 105230|doi= 10.1016/j.rinp.2022.105230 |bibcode= 2022ResPh..3405230G |s2cid= 221978379 |doi-access= free |arxiv= 2006.00485 }}</ref>
{{ordered list
| The probability density function is proportional to some function of the ensemble parameters and random variables.
| Thermodynamic state functions are described by ensemble averages of random variables.
| At infinite temperature, all the microstates have the same probability.
}}