The <equilibrium magnetization> is a <global minimum> of the <Landau free energy>. Its candidates are the <stationary points>
$$
0=f'(m)=2m\left(a_2+2a_4m^2+3a_6m^4\right).
$$
Thus $m=0$, or $x=m^2\geq0$ is a nonnegative root of
$$
3a_6x^2+2a_4x+a_2=0.
$$
One retains candidates with nonnegative curvature $f''(m)$ and compares their free energies; the candidate with the smallest value is the equilibrium state. Equal global minima describe <phase coexistence>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 303 / 1 / a / Solution

Phase coexistence occurs when two or more thermodynamic phases have equal free energy under the same external conditions. A first-order transition crosses a coexistence locus, where the competing phases are simultaneously stable.


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