# Split the difference

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

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

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

Assumptions

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