= Two-phase scalar coexistence endpoint conditions
{title2=$V'=V''=V^{(3)}=0,\quad V^{(4)}>0$}
A smooth scalar <Landau free energy> can end a two-phase <phase coexistence curve> by merging two minima and their intervening maximum. At a generic stable endpoint, its first three derivatives vanish and its fourth derivative is positive. A field shift and two mixed control parameters reduce the local form to a quartic <Landau free energy>; the order-parameter jump then vanishes and the <magnetic susceptibility> diverges. This conclusion assumes there is no competing third phase and that the endpoint lies inside the control-parameter domain. It is not a general theorem about every first-order line: a sextic potential with fixed negative quartic coefficient has a symmetry-related ordered coexistence line that ends at a triple point with a finite jump.
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