Thermodynamics

Stability and Response Functions

Heat capacities and compressibilities come out positive. You can read why off the curvature of the potentials.

Updated July 2026

Push on a gas and it pushes back. Warm it and it takes up heat rather than running away to a different temperature on its own. These sound like separate facts about matter, but they are the same statement seen twice, and it is a statement about curvature.

A system in equilibrium sits at a minimum of the relevant potential. A minimum is not just a place where the first derivative vanishes; it also curves upward. That upward curvature, written out, is exactly the requirement that the response functions are positive.

Response functions

A response function measures how much one quantity moves when you nudge another. The two that matter here are the heat capacity and the compressibility:

C_{V} = T (\frac{\partial S}{\partial T})_{V}, κ_{T} = - \frac{1}{V} (\frac{\partial V}{\partial P})_{T} .

$C_{V}$ is how much heat it costs to raise the temperature by a degree at fixed volume. $κ_{T}$ is how much the volume shrinks per unit of pressure applied. Both are things you can measure without ever mentioning entropy.

Stability forces their sign

Take the entropy as a function of energy and expand a small subsystem against the rest. For the total entropy to be at a maximum, $S$ must be concave in its extensive variables. Translated into the energy picture, $U$ is convex, and convexity of $U$ in $S$ and $V$ is the pair of inequalities

C_{V} \geq 0, κ_{T} \geq 0.

Consequence

What a negative sign would mean

Suppose $C_{V}$ were negative. A patch of the system that happened to fluctuate warmer would, on absorbing heat, get colder still, and the fluctuation would run away instead of relaxing. Positive $C_{V}$ is precisely the condition that such a fluctuation is restored. The same argument on volume gives positive $κ_{T}$ : a region that compressed slightly would otherwise keep collapsing.

This is Le Chatelier's principle in its exact form. A stable system responds to a disturbance in the direction that opposes it, and the response functions are positive because that is what opposing the disturbance means.

The two heat capacities

Heat capacity depends on what you hold fixed while heating. At fixed pressure the system is free to expand and do work, so it takes more heat to warm. The difference has a clean form:

C_{P} - C_{V} = \frac{T V α ^{2}}{κ _{T}},

where $α$ is the thermal expansion coefficient. Every factor on the right is a square or a positive quantity, so

C_{P} \geq C_{V},

always, for any substance. The gap vanishes only when $α = 0$ , that is, when the material does not expand on heating and there is no work to be done.

The whole subject has this shape. The laws fix what is conserved and which way things run; stability fixes the signs of how they respond, and it does it not by adding a principle but by insisting that equilibrium really is a minimum and not a saddle.