Equilibrium is a difficult concept for me. I know several statistical ensembles or quantum states characterized entirely by a few thermodynamic quantities, eg. the canonical ensemble characterized entirely by the temperature via the Boltzmann weights. A system whose instantaneous macroscopic quantities, eg. pressure, are equal to a corresponding thermal average in some ensemble is said to be in equilibrium.

I'm interested in how systems approach equilibrium. Is there a quantity that measures "distance from equilibrium"?

In some sense entropy defines such a quantity, since it always increases until it is at a maximum at equilibrium. In another sense free energy defines such a quantity, since it is a minimum at equilibrium. The first I suppose is equilibrium defined relative to the microcanonical ensemble, where energy is fixed, while the second is defined relative to the canonical ensemble, which may exchange energy with a reservoir. Similarly the Gibbs free energy defines another kind of distance from equilibrium quantity. I am confused by the dependence on the ensemble here.

I found this wonderful nobel lecture by Ilya Prigogine http://www.nobelprize.org/nobel_prizes/chemistry/laureates/1977/prigogine-lecture.pdf

He explains how AN equilibrium is characterized by A Lyapunov function (free energy functional) which decreases towards its corresponding equilibrium *within some region of the equilibrium*. Far from any equilibrium, these all compete with eachother and the equilibrium that one eventually attains can be very sensitive to these initial conditions (or boundary conditions when we're thinking about a system in contact with a thermodynamic reservoir).