On the Connections between Thermodynamics and Statistical Mechanics

The canonical, grand-canonical, and constant-pressure ensembles of statistical mechanics are related to thermodynamic functions by simple canonical forms. The basic such form is $S̃ = klnQ$⁠, where $S̃$ represents a Legendre transformation of the entropy and $Q$ is an appropriate partition function. This expression is of the Boltzmann form, which arises naturally in the microcanonical ensemble. The canonical structure, which is demonstrated, is pedagogically useful as (1) a memorization device, (2) an example of the utility of Legendre transformations in statistical mechanics, and (3) an aid in understanding the relationships of the various ensembles to one another.