= Solution
<Landau theory> expands the <free energy> in an <order parameter>. For a scalar with sign-reversal symmetry and zero <conjugate field>, a useful <sextic even Landau potential> is
$$
f(m)=f_0+\frac r2m^2+\frac b4m^4+\frac c6m^6,
\qquad r=a(T-T_0),\quad a,c>0.
$$
Equilibrium selects the <global minimum>; other local minima can be metastable.
For $b>0$, the minimum changes continuously from $m=0$ to $m\simeq\pm\sqrt{-r/b}$ as $r$ passes below zero. This is a second-order transition at $T_0$: the <order parameter> is continuous, there is no <latent heat>, and thermodynamic derivatives such as <heat capacity> can be discontinuous.
For $b<0$, the sixth-order term stabilizes the <free energy>. Equality of the ordered and disordered minima, together with $f'(m)=0$, gives
$$
\boxed{m_t^2=-\frac{3b}{4c},\qquad r_t=\frac{3b^2}{16c}>0.}
$$
The <order parameter> jumps at $T_t=T_0+r_t/a$. The <entropy> generally jumps too, giving <latent heat>; metastable branches and barriers permit <hysteresis>, though equilibrium coexistence itself does not require <hysteresis>. The special case $b=0$ is tricritical. Mean-field exponents can fail where fluctuations dominate.
\Image[/past-exam-of-the-thermal-and-statistical-physics-course-of-the-university-of-cambridge/2014/paper-1-tsp2014-landau.png]
{title=Original even Landau free-energy curves: continuous quartic ordering and sextic first-order coexistence with separated equal-depth minima}
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