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All the phase transition cases I came across so far have this property: the lower temperature phase has lower symmetry than the higher temperature one. But it is nowhere explicitly said that, lower temperature phase always has lower symmetry than the higher temperature phase. So, I was thinking, is there any counter-example to show that higher temperature phase in a phase transition can have lower symmetry too?

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Water at 20C and atmospheric pressure has a higher symmetry than ice VII at 100C and 10GPa pressure.

You may well point out this is cheating because the pressure is different in the two cases. However this makes the point that temperature is not the only variable. If you're looking at a phase transition between a disordered and ordered phase then you need to consider the Gibbs free energies of the two phases. The phase with the lower Gibbs free energy is the one that will form.

The Gibbs free energy is defined as:

$$ G = H - TS $$

and a negative Gibbs free energy change means the phase change occurs (give or take a few kinetic barriers). Generally speaking, for a change from ordered to disordered the entropy increases. i.e. $\Delta S$ is positive. If you consider an isothermal change you get a negative $-T\Delta S$ contribution that gets more negative as the temperature gets higher, so you would expect that in general changes from ordered to disordered will occur at increased temperature.

This doesn't mean it's impossible to get a disorder to order transition with increasing temperature, but the enthalpy, $H$, would have to have an odd temperature dependance to outweight the entropic effect.

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I guess you could use the example of those theories where Lorentz invariance is broken at short distances/high temperatures and restored at low temperature, although I've never really understood the theories where this supposedly works. – Michael Brown Feb 12 '13 at 10:10
Thanks a lot for the answer, I am looking at the phase diagram of ice now. – user20719 Feb 13 '13 at 2:09


Thus, temperature has the almost universal effect that if a symmetry is spontaneously broken at low temperature, it is restored at temperatures above a certain critical value. Qualitatively, it can be understood as follows. Temperature, particularly high temperature, provides a lot of thermal energy to a physical system to wash out any structure in the zero temperature potential which may be responsible for symmetry breaking. There is, however, one class of symmetries where temperature has the inverse effect, namely, in a supersymmetric theory, a symmetric phase at low temperature goes to a broken phase at high temperature. (Of course, if supersymmetry is broken at low temperature, it continues to be broken even at high temperature.)


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