Split the difference

Electricity and Magnetism Level 2

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.
Josh Silverman by Josh Silverman
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