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

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- Solution a

Solution

The equilibrium magnetization is a global minimum of the Landau free energy. Its candidates are the stationary points

0 = f^{'} (m) = 2 m (a_{2} + 2 a_{4} m^{2} + 3 a_{6} m^{4}) .

Thus

m = 0

, or

x = m^{2} \geq 0

is a nonnegative root of

3 a_{6} x^{2} + 2 a_{4} x + 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.

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