A ferromagnet has magnetization order parameter $m$ and is at temperature $T$. The free energy is given by

$$F(T ; m)=F_{0}(T)+\frac{a}{2}\left(T-T_{c}\right) m^{2}+\frac{b}{4} m^{4}$$

where $a, b$ and $T_{c}$ are positive constants. Find the equilibrium value of the magnetization at both high and low temperatures.

Evaluate the free energy of the ground state as a function of temperature. Hence compute the entropy and heat capacity. Determine the jump in the heat capacity and identify the order of the phase transition.

After imposing a background magnetic field $B$, the free energy becomes

$$F(T ; m)=F_{0}(T)+B m+\frac{a}{2}\left(T-T_{c}\right) m^{2}+\frac{b}{4} m^{4}$$

Explain graphically why the system undergoes a first-order phase transition at low temperatures as $B$ changes sign.

The spinodal point occurs when the meta-stable vacuum ceases to exist. Determine the temperature $T$ of the spinodal point as a function of $T_{c}, a, b$ and $B$.