Thermodynamic stability

Thermodynamic stability means that allowed small perturbations do not lower the appropriate thermodynamic potential from an equilibrium state. The constraints select that potential: an isolated system maximizes entropy, a system at fixed temperature and volume minimizes Helmholtz free energy, and one at fixed temperature and pressure minimizes Gibbs free energy.

For an unconstrained scalar order parameter $\varphi$, a stationary state has ${\partial }_{\varphi }F=0$, and positive curvature ${\partial }_{\varphi }^{2}F>0$ gives local stability. A distant lower minimum can still make it metastable rather than globally stable. In an ordinary additive fluid, positive isothermal compressibility expresses resistance to density perturbations: increasing pressure should reduce volume. Loss of the corresponding curvature can mark a spinodal. These statements depend on the ensemble and permitted perturbations; mechanical equilibrium alone does not establish thermal, chemical or global stability.

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.