= Solution
<Critical exponents> describe leading singular powers near a <continuous phase transition>. For <reduced temperature> $t=(T-T_c)/T_c$, common definitions are the spontaneous <order parameter> $M\sim(-t)^\beta$, <magnetic susceptibility> $\chi\sim|t|^{-\gamma}$, singular heat capacity $c_s\sim|t|^{-\alpha}$, <correlation length> $\xi\sim|t|^{-\nu}$, and the critical equation of state $M\sim\operatorname{sgn}(h)|h|^{1/\delta}$. At criticality the <connected correlation function> has decay power $D-2+\eta$.
In <Landau theory>, solve the stationary equation, evaluate its stable minimum, and differentiate with respect to the conjugate variables. For a scalar quartic theory with $r=r_0t$, $r_0>0$, the equation of state is $h=rM+uM^3$. Its ordered solution and differentiated response give $\beta=1/2$ and $\gamma=1$, calculated explicitly in the unheaded solution below. At $r=0$ the same equation gives $\delta=3$. The minimized zero-field singular potential is zero above the transition and $-r^2/(4u)$ below, giving a heat-capacity jump and $\alpha=0$. Expanding the gradient functional about a stable minimum gives inverse quadratic response $\kappa q^2+U''(M)$, so $\xi=\sqrt{\kappa/U''(M)}\sim|t|^{-1/2}$ and the Gaussian critical decay has $\eta=0$.
These are <mean-field critical exponents>. <Renormalization group> methods obtain fluctuation-corrected powers from fixed-point scaling dimensions; they need not equal the Landau values below the <upper critical dimension>. At a marginal dimension logarithms can supplement the leading powers.
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