Split the difference

Suppose that the free energy of a system is described by F=(TTc)m2+14m4\mathcal{F}=(T-T_c)m^2+\frac14 m^4 where TT is the temperature, and mm represents some property of the system, such as its overall magnetization. As TT rises and falls, but does not cross TcT_c, the allowed physical value(s) of mm, mm^* change somewhat (see Assumptions).

When T\displaystyle T crosses TcT_c, however, there is a qualitative shift in the allowed values of mm^*, and the system undergoes a phase transition.

Suppose we're working with the system at a temperature T\displaystyle T below TcT_c such that TTc=10T-T_c=-10, find m\lvert m^*\rvert.

Assumptions

  • The system always resides in states mm^* which minimize its free energy. These are the "physical" states of the system.
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