The fundamental assumption in the description of phase transitions is that, at the transition point, the two phases of a two-phase system are in equilibrium. If G₁ and G₂ denote their thermodynamic potentials, then G₁ = G₂ at that point. This does not imply that the derivatives of the potentials are identical:

An important principle for understanding phase transitions is therefore already apparent: their characterization is based on the partial derivatives of thermodynamic potentials. Ehrenfest was the first to propose the following classification.

2.1.2 Phase Transitions According to Ehrenfest

Inequality (2.2) characterizes first-order phase transitions. Another broad class consists of phase transitions of order n. According to Ehrenfest, the nth partial derivatives of the thermodynamic potential differ at the transition point, whereas all lower derivatives must be equal:

Ehrenfest’s argument implies that canonically conjugate quantities change discontinuously at a first-order transition, and that the corresponding variable Aᵢ in Eq. (2.1) exhibits a discontinuity. For transitions of higher order, the first derivatives are equal; thermodynamic quantities such as temperature T or pressure p therefore vary continuously.

Experimental observations, however, show that critical quantities in thermodynamic systems tend to exhibit singularities rather than finite jumps. A classification strictly into first- through nth-order transitions is consequently not entirely satisfactory from a physical standpoint. It is now common to distinguish continuous and discontinuous phase transitions [5].

Original footnote 2: Paul Ehrenfest (1880–1933), physicist.