Thermodynamic Stability

Definition and physical interpretation

A thermodynamic equilibrium state is stable if small allowed variations of the state variables do not drive the system away from equilibrium; equivalently, the equilibrium extremum is not merely stationary but locally “curved the right way.”

Which quantity must be extremized depends on the constraints imposed by the surroundings:

• For an isolated system (fixed $U,V,N$), stability means the entropy $S$ is maximized at fixed internal energy $U$, volume $V$, and particle number $N$ (the second law viewpoint).

• For a system in contact with a thermal reservoir at fixed $T$ (and fixed $V,N$), stability means the Helmholtz free energy $F$ is minimized.

• For fixed $T$ and pressure $P$ (and fixed $N$), stability means the Gibbs free energy $G$ is minimized.

Mathematically, these stability statements translate into sign conditions on second variations (or Hessians), and are tightly connected to convexity/concavity properties of the fundamental relation.

Key local consequences (signs of response functions)

For a simple compressible single-phase system in stable equilibrium, one expects measurable susceptibilities to have their “physical” sign, for example:

• Positive heat capacities, such as the constant-volume heat capacity $C_V$ and the constant-pressure heat capacity $C_P$:

  $$C_V = \left(\frac{\partial U}{\partial T}\right)_{V,N} > 0, \qquad C_P = \left(\frac{\partial H}{\partial T}\right)_{P,N} > 0.$$

• Mechanical stability against volume fluctuations, often expressed as

  $$\left(\frac{\partial P}{\partial V}\right)_{T,N} < 0,$$

  which is equivalent to positivity of the isothermal compressibility .

These conditions are compactly encoded by the representation-dependent criteria entropy concavity (in $S(U,V,N)$) and energy convexity (in $U(S,V,N)$). In practice, Maxwell relations are often used to rewrite stability criteria in terms of experimentally accessible derivatives.