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When you see phase diagrams for states of matter, they typically look something like this:

enter image description here

This shows discrete changes from one state to the next. I am wondering though if it is actually a discrete change, or just a continuous change within a measurably small window. That is, even though it looks instantaneous, at a smaller timescale/resolution, you would see intermediate phases, or a sort of in-between phase. This sort of seems like a ridiculous question, because of course the material is going to not be heated exactly uniformly, for example, so it seems like it must be in a "partial state", like ice just halfway melted. So it seems like atoms/electrons at one point in the material are in liquid form, and atoms in another point touching it are in solid form. But perhaps there is more to it. Maybe there is an in between state to the atoms so that it blends more naturally. If not, wondering why not.

Wondering if one could explain how that works, and if there is only these few states of matter, what is actually happening at the boundary between the two.

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This is related to the definitions involved. Phase transitions are defined as discontinuities in one or more thermodynamic parameters of the system. The nonequilibrium dynamics of the system are in fact not instantaneous, but in the thermodynamic limit (as the number of particles $N\rightarrow\infty$) they are.

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