= Mechanical stability
{title2=$F''(x_*)>0$}
For a mechanical coordinate $x$ at fixed <temperature>, total <volume> and <particle> numbers, the relevant <thermodynamic potential> is the <Helmholtz free energy> $F(x)$. Its generalized <force> is $f=-F'(x)$. At <mechanical equilibrium>, $f(x_*)=0$, and a linearly restoring <force> has $f'(x_*)<0$, equivalently $F''(x_*)>0$. A <Taylor expansion> then proves a strict local free-energy minimum. If $F$ is a <strictly convex function> throughout the accessible domain, the stationary state is the unique global minimum.
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