Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 46 / 1 / Solution

For the requested ordinary critical point take $u>0$, $r=r_{0}t$, and the quartic Landau free energy $U(M)=r{M}^2/2+u{M}^4/4-hM$. At $h=0,t<0$, minimization gives $M^2=-r/u$. Choosing the positive pure thermodynamic phase,For the magnetic susceptibility, differentiate the equation of state at fixed temperature:Both sides diverge as $|t{|}^{-1}$, with different amplitudes, soThe one-sided field limit below the transition selects a pure phase; differentiating the symmetric finite-volume mixture would instead include its macroscopic switching response.