Entropy Concavity and Stability

Stability in the entropy representation is equivalent to entropy being concave in the extensive variables.

Entropy Concavity and Stability

Definition (global concavity)

Let the fundamental relation in the entropy representation be $S=S(U,V,N)$ for a single-phase system, with $(U,V,N)$ being extensive variables . The entropy is concave if for any two states $x_1=(U_1,V_1,N_1)$ and $x_2=(U_2,V_2,N_2)$ and any $\lambda\in[0,1]$ ,

S\!\big(\lambda x_1 + (1-\lambda)x_2\big) \ge \lambda S(x_1) + (1-\lambda)S(x_2).

Physically, this expresses that “mixing” or composing subsystems at fixed total $(U,V,N)$ cannot reduce total entropy ; it is the mathematical backbone of why the equilibrium state of an isolated system is a stable entropy maximum, i.e. a core part of thermodynamic stability .

Local (differential) criterion

If $S$ is twice differentiable, concavity is equivalent to the Hessian (matrix of second partial derivatives ) with respect to $(U,V,N)$ being negative semidefinite. For example, restricting attention to $(U,V)$ at fixed $N$ , one requires

\left(\frac{\partial^2 S}{\partial U^2}\right)_{V,N} \le 0, \qquad \det \begin{pmatrix} S_{UU} & S_{UV}\\ S_{VU} & S_{VV} \end{pmatrix} \ge 0,

with $S_{UU}$ etc. denoting the corresponding second derivatives at fixed $N$ .

Consequences for response functions

Concavity translates into positivity of familiar response functions :

Using the defining relation $1/T = (\partial S/\partial U)_{V,N}$ for the temperature $T$ , one finds
$\left(\frac{\partial^2 S}{\partial U^2}\right)_{V,N} ={} -\frac{1}{T^2}\left(\frac{\partial T}{\partial U}\right)_{V,N} ={} -\frac{1}{T^2\,C_V},$
where $C_V$ is the constant-volume heat capacity . Thus $S_{UU}\le 0$ implies $C_V\ge 0$ (strict concavity corresponds to $C_V>0$ ).
Through Legendre transforms to potentials such as the Helmholtz free energy $F(T,V,N)$ and Gibbs free energy $G(T,P,N)$ (a thermodynamic instance of the Legendre transform ), entropy concavity enforces mechanical stability conditions like $(\partial P/\partial V)_{T,N}<0$ , i.e. a positive isothermal compressibility .
A standard identity relating measurable susceptibilities,
$C_P - C_V = \frac{T V \alpha^2}{\kappa_T},$
shows (for $T>0$ ) that positivity of the isothermal compressibility and the thermal expansion coefficient implies the constant-pressure heat capacity satisfies $C_P\ge C_V$ .