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I am reading in some book: "The glass transition is similar in appearance to a second-order phase transition, but it is not a true thermodynamic phase transition. This is because the transition temperature depends on the rate at which we do the experiment."

What's the matter with that rate? Is it only a question of mathematical definition of thermodynamic phase or is there any physical reason to exclude rate-dependent transitions?

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Thermodynamic phase transitions are defined as discontinuities in (derivatives) of the free energy. Thus, they are about differences in equilibrium properties, even though often we end up measuring a non-equilibrium property because it's easier. But nowhere in this framework does a rate-dependent, i.e. non-equilibrium, situation occur, and therefore the glass transition is not a "proper" phase transition in that sense.

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