TFAE: Thermodynamic Stability Criteria

Consider a simple macroscopic system in thermodynamic equilibrium with a differentiable fundamental relation written in the entropy representation $S=S(U,V,N)$ (or equivalently the energy representation $U=U(S,V,N)$). The following are equivalent local stability criteria (for fixed $N$ one may drop $N$ from the notation).

1. Entropy maximum principle (microcanonical stability).
   Among nearby equilibrium states with the same $(U,V,N)$, the equilibrium state locally maximizes entropy :

   $$\delta^2 S \le 0 \quad \text{at fixed } (U,V,N).$$

2. Concavity of the entropy.
   The function $S(U,V,N)$ is concave in its extensive variables, i.e. for $0\le \lambda \le 1$,

   $$S(\lambda x + (1-\lambda)y) \ge \lambda S(x) + (1-\lambda)S(y),$$

   where $x=(U,V,N)$ and $y=(U',V',N')$.
   Equivalently, the Hessian matrix of second derivatives of $S$ is negative semidefinite (when it exists).

3. Convexity of the internal energy.
   The internal energy $U(S,V,N)$ is convex in $(S,V,N)$, equivalently its Hessian is positive semidefinite.
   (This is the Legendre-dual statement to concavity of $S$; see Legendre transform .)

4. Minimum principle for appropriate thermodynamic potentials.
   At fixed intensive controls, equilibrium minimizes the corresponding potential. In particular:

   • At fixed $(T,V,N)$, equilibrium minimizes the Helmholtz free energy $F(T,V,N)=U-TS$.
   • At fixed $(T,P,N)$, equilibrium minimizes the Gibbs free energy (if used in your convention).

5. Positivity of response functions.
   Stability is equivalent to nonnegative susceptibilities such as:

   • Heat capacity at constant volume:
     

     $$C_V = \left(\frac{\partial U}{\partial T}\right)_{V,N} \ge 0,$$

     matching heat capacity at constant volume .

   • Isothermal compressibility:
     

     $$\kappa_T = -\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_{T,N} \ge 0,$$

     equivalently $(\partial P/\partial V)_{T,N} \le 0$ where pressure is $P$.

6. Second-derivative tests for the Helmholtz free energy.
   Using $S=-(\partial F/\partial T)_{V,N}$ and $P=-(\partial F/\partial V)_{T,N}$, stability is equivalent to

   $$\left(\frac{\partial^2 F}{\partial T^2}\right)_{V,N} \le 0 \quad\text{and}\quad \left(\frac{\partial^2 F}{\partial V^2}\right)_{T,N} \ge 0,$$

   which are the differential forms of $C_V\ge 0$ and $\kappa_T\ge 0$.

Prerequisites and context: thermodynamic stability , entropy , internal energy , Helmholtz free energy , heat capacity , Legendre transform .