= Solution
<Landau theory> expands a thermodynamic potential in an <order parameter>, constrained by <symmetry>. For a scalar with sign-reversal <symmetry>, a useful <free-energy density> is
$$
f=f_0+\frac{a(T-T_c)}2\eta^2+\frac b4\eta^4-h\eta,
\qquad a,b>0.
$$
Here $h$ is the <conjugate field>. At $h=0$, minimizing rather than merely finding stationary points gives
$$
\boxed{\eta_{\rm eq}=0\quad(T>T_c),\qquad
\eta_{\rm eq}=\pm\sqrt{\frac{a(T_c-T)}b}\quad(T<T_c).}
$$
The symmetry-related minima represent <spontaneous symmetry breaking>, as for a simple uniaxial <ferromagnet> with <magnetization> as <order parameter>. An infinitesimal field selects one sign. Above $T_c$, the susceptibility is $1/[a(T-T_c)]$; below it the stable-branch susceptibility is $1/[2a(T_c-T)]$. Minimization lowers $f$ by $-a^2(T-T_c)^2/(4b)$ below $T_c$, producing a heat-capacity <density> jump $T_ca^2/(2b)$ with continuous <entropy>.
\Image[/past-exam-of-the-thermal-and-statistical-physics-course-of-the-university-of-cambridge/2012/paper-1-tsp2012-landau.png]
{title=Landau minima}
{description=Original normalized quartic free-energy curves above, at and below a continuous transition, with the two symmetry-related equilibrium order-parameter branches.}
A negative quartic coefficient requires a stabilizing $+c\eta^6/6$ with $c>0$. It can produce a first-order transition before the quadratic coefficient changes sign: coexistence occurs at $a(T-T_c)=3b^2/(16c)$ and $\eta^2=-3b/(4c)$. Odd terms are allowed when the assumed <symmetry> is absent. <Landau theory> predicts mean-field exponents; spatial fluctuations can change critical behaviour, especially close to a <critical point> in low dimensions. A <gradient> term extends the model to nonuniform <order parameters> but does not by itself supply dynamical equations.
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