= Hysteresis spinodals of a scalar quartic potential
{title2=$|h_{\rm sp}|=2|r|^{3/2}/(3\sqrt{3u})$}
For the <Landau free energy> $V(M)=rM^2/2+uM^4/4-hM$ with $r<0$ and $u>0$, stationary branches satisfy $h=rM+uM^3$. Their local stability requires $r+3uM^2>0$. At a <spinodal point>, $M_{\rm sp}=\pm\sqrt{|r|/(3u)}$ and $h_{\rm sp}=(2r/3)M_{\rm sp}$, giving the displayed field magnitude. Following local minima without <nucleation> produces a <hysteresis> loop switching at these limits. Equilibrium instead switches between the two global minima at $h=0$.
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