Mechanical stability ($F^{\prime \prime }(x_{\ast })>0$)

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 $f=-F^{\prime }(x)$. At mechanical equilibrium, $f(x_{\ast })=0$, and a linearly restoring force has $f^{\prime }(x_{\ast })<0$, equivalently $F^{\prime \prime }(x_{\ast })>0$. 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.