How is a phase equilibrium defined for a one-component system?

Question

A question on this site asked whether a one-component system is at equilibrium when melting or boiling, and the disparate answers were somewhat dependent on the definition of phase equilibrium. Another question asks whether two phases can ever exist in equilibrium at the boiling point. The term "phase equilibrium" seems to imply that under certain conditions, a one-component system with two phases is indeed at equilibrium.

Searching for the definition of phase equilibrium, I found these two:

Definition 1

Phase equilibrium is the study of the equilibrium which exists between or within different states of matter namely solid, liquid and gas. Equilibrium is defined as a stage when chemical potential of any component present in the system stays steady with time.

Source: https://www.wattco.com/casestudy/phase-equilibrium

Definition 2

Phase equilibrium is the state of thermodynamic system, in which the different phases of the substance having common boundary surfaces do not vary quantitatively.

Source: https://www.elsevier.com/books/computational-materials-science/ovrutsky/978-0-12-420143-9

Implications of the distinct definitions

According to definition 1, water present as two phases at the melting point would be at equilibrium even if slowly melting isothermally. According to definition 2, the same situation would not be called equilibrium because the amount of ice is decreasing over time (and thermodynamic parameters such as entropy and inner energy are changing over time).

Phase rule

For the mentioned system, the phase rule specifies one independent degree of freedom (you can change one intensive parameter, say temperature, and all other intensive parameters, in this case just pressure, will be determined, given that the system is supposed to be at equilibrium). Because it is a one-component system, the concentrations (or activities) are constant and are not a degree of freedom. If you are used applying the equilibrium concept to solution chemistry, this is somewhat unusual.

Question: What is the official definition of phase equilibrium, and how does it apply to a one-component system where the chemical potentials are independent of the mole ratios?

Answer 1

Melting is a process. A process is not a state. Since only states can be "at equilibrium", it doesn't make sense to speak of a real melting substance as being in an equilibrium state. However, melting is deemed reversible if performed infinitely slowly, and during such a transition the system passes through a series of equilibrium (or "quasi-equilibrium") states. But such a process is an idealization. In reality, you can't "melt ice isothermally". Real transitions are therefore irreversible because system and surroundings are not actually in thermal equilibrium during heat transfer$^\dagger$.

Note that if you place in contact two of the intermediate states sampled during the transition neither will change, that is, each remains in its original equilibrium state.

The second definition encompasses the first one (equality of chemical potential across phases). The second definition therefore better captures what is meant by phase equilibrium. The conditions for phase equilibrium are those used to derive the phase rule, namely mechanical, chemical and thermal balance between phases. A phase is a region of uniform composition, temperature and pressure.

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$^\dagger$ We say that it is possible to melt a pure substance using an ideal reversible process that leads the system infinitely slowly through a series of equilibrium states (each intermediate stage in that reversible process is an equilibrium state). In order to move between intermediate states, an infinitesimally small amount of heat has to be transferred from the infinitesimally warmer surroundings (exploiting, say, something like a Maxwell demon). In what is a logical sleight of hand, we say that the temperature difference is so small however, that the system and surroundings are effectively isothermal. This does not reflect what happens in a real process.