Hang a weight on a rubber band, stretched maybe 50% past resting length. Shine a heat lamp on the band. Naive prediction: the band warms, expands a little, the weight drops. What actually happens: the band contracts, and lifts the weight.
The same band warms when you stretch it — snap a fresh one against your lip, you can feel it — and cools when you let it relax. Steel does the opposite. These are one effect, two readings.
The spring constant of a steel wire is the spring constant of a billion atomic bonds wired in parallel. Each bond stores potential energy as it pulls away from its rest length. Heat the wire, the bonds vibrate harder, the wire expands. Most solids work that way.
Rubber doesn't. Rubber is long polymer chains tangled in a network. At room temperature each chain is a random walk — most microstates available to it are coiled, very few are extended. Stretching the band picks out the rare extended microstates. Work done on the band lowers its entropy.
The thermodynamic restoring force for an elastomer is
For an ideal rubber the first term — the energetic contribution — is approximately zero. All the restoring force comes from the second. The chains pull back not because their bonds are stretched but because they want more microstates.
So when you stretch the band at constant T you push ΔS negative. The energy you put in as work has nowhere to go in the bonds — for ideal rubber there is no stored potential energy — and it comes back out as heat. The band warms.
Heat the band at fixed length and you raise T, raising the magnitude of the entropic restoring force at every strain. The chains pull harder. If the load is fixed — which a hanging weight does — the band must shorten to balance. Stiffer when hot. That's the falsifiable signature.
If the elasticity is purely entropic, force at fixed strain should be linear in absolute temperature. Double T, double f. That's not a hand-wave; it falls straight out of f = −T(∂S/∂L). Measured tensile stress vs T for natural rubber, 10°C to 70°C, lands inside experimental error of the line.
The prediction bites the other way too. Find a regime where force at fixed strain isn't linear in T and the elasticity isn't ideal-entropic there. The known offenders are strain-induced crystallization (at large extensions natural rubber starts ordering into crystallites — that's energetic, and it shows as a knee in the stress-strain curve) and finite-extensibility (chains running out of microstates because they've reached their contour length). Each leaves a fingerprint where the linear law breaks.
I had the postcard: rubber band contracts when heated, weird, entropy. What I didn't have was the partition. Force decomposes into an energetic part and an entropic part. “Ideal rubber” just means the energetic part is negligible. The temperature scaling falls straight out, as a clean prediction. The regimes where rubber stops behaving entropically are exactly the ones where the linear law breaks.
The general move: an “entropic force” sounds mystical until you write the partition. Then it's the term that survives when (∂U/∂L) goes to zero. Nothing mystical. It's the term that's left.