Entropy
Equivalence of definitions

Proofs of equivalence between the entropy in statistical mechanics — the Gibbs entropy formula: S = − k B ∑ i p i ln p i {\displaystyle S=-k_{\mathsf {B}}\sum _{i}{p_{i}\ln {p_{i}}}} and the entropy in classical thermodynamics: d S = δ Q r e v T {\displaystyle \mathrm {d} S={\frac {\delta Q_{\mathsf {rev}}}{T}}} 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 generalized Boltzmann distribution and the identification of the thermodynamic internal energy as the ensemble average U = ⟨ E i ⟩ {\textstyle U=\left\langle E_{i}\right\rangle }. 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 generalized Boltzmann distribution.

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:
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