Equilibrium States With More Than One Variable

For a system of fixed composition, $\ensuremath{\delta}U(S,V)$ can be expanded¹

$$\input{equations/17-3A}$$ (24-1)

For a local equilibrium

$$\input{equations/17-3B}$$ (24-2)

so that

$$\input{equations/17-3C}$$ (24-3)

The matrix is called the Hessian of the system and for the inequality to be true it must be ``positive definite'' for a two-by-two matrix.

Necessary conditions for a local minimum are:

$$\input{equations/17-4A}$$ (24-4)

and

$$\input{equations/17-4B}$$ (24-5)

evaluated at the extrema.

Therefore:

$$\input{equations/17-4C}$$ (24-6)

$C_V > O$ for stability (If you add heat to a system, then its entropy must rise)

The second part (Eq. 24-5) that must also positive can be written in terms of the Jacobian

$$\input{equations/17-4D}$$ (24-7)

$$\input{equations/17-5A}$$ (24-8)

for a stable equilibrium.