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Superheating of a liquid past its boiling point (retarded boiling) is a well-known phenomenon. However, is there such a thing as supercooling of gas past its condensation point?

Conversely, supercooling of a liquid (retarded freezing) is also well-known. Does the reverse ever occur: can I superheat a solid up past its melting point?

Nanite
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Superheating and supercooling should be possible whenever there is a first order phase transition. This includes a wide variety of magnetic, structural, and electronic phase transitions. The physical mechanism is the same as in the water-ice transition: in a first order phase transition. the energy functional is still a local minimum at the old phase, there is a barrier to the formation of the new phase and so an adiabatically cooled sample will remain in the old phase past the point when the old phase is a global minimum.