Past exam of the thermal and statistical physics course of the University of Cambridge / 2013 / Paper 1 / 4 / iii

For $a>0$ and $H>0$, the field tilts the single well toward positive M. Since $F^{\prime \prime }=a+3b{M}^2>0$ everywhere, there is exactly one minimum, at the positive solution $M_H$ of the equation of state. At that solution,This follows by eliminating ${\mu }_{0}H=a{M}_H+b{M}_H^3$ from F. At weak field, $M_H\simeq {\mu }_{0}H/a$ and $F_{\min }\simeq -{\mu }_0^2{H}^2/(2a)$.