Call that something “k” and you’ve got the diffusion equation, .
The second derivative, , is a way to describe how a function is curving.
You can actually derive useful math from it, which is a damn sight better than most science metaphors (E.g., “space is like a rubber sheet” is not useful for actual astrophysicists).
In very much the same way that a concentrated collection of beads will spread themselves uniformly, hot things will lose heat energy to the surrounding cooler environment.
This is why coffee/tea/soup will be hot for a little while, but tepid for a long time; it cools faster when it’s hotter. Since you lose more heat energy from a hot pool than from a cool pool, the most efficient thing you can do is keep the temperature as low as possible for as long as possible.
It’s all well and good to talk about how heat beads randomly walk around inside of a material, but if that material isn’t uniform or has an edge, then suddenly the math gets remarkably nasty.
Fortunately, if all you’re worried about is whether or not you should leave your heater on, then you’re probably not sweating the calculus.
It’s well worth taking a stroll through statistical mechanics every now and again.
The diffusion of heat is governed, not surprisingly, by the “diffusion equation”.