Callen H.B - Thermodynamics And An Introduction To Thermostatistics · Problem 8.1-1

FIRST-ORDER PHASE TRANSITIONS

Physics — Statistical Mechanics · stability-entropy-inequality

Problem

To establish the inequality 8.6 expand the left-hand side of 8.5 in a Taylor series to second order in

\Delta U

and

\Delta V

. Show that this leads to the condition

-S_{UU}(\Delta U)^2 + 2S_{UV}\Delta U\Delta V + S_{VV}(\Delta V)^2 \leq 0

Recalling that

S_{UU} \equiv \partial^2 S/\partial U^2 \leq 0

, show that this can be written in the form

(S_{UU}\Delta U + S_{UV}\Delta V)^2 + (S_{UU}S_{VV} - S_{UV}^2)(\Delta V)^2 \geq 0

and that this condition in turn leads to equation 8.6.