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The corners cool off faster because there are more opportunities for “heat beads” to fall out of the material there.

Although this is exactly what the diffusion equation predicts, actually doing the math by hand is difficult.

Call that something “k” and you’ve got the diffusion equation, .

This is why the idea of “heat beads” is a useful intuition to use; the same math that describes the random motion of particles also describes how heat spreads through materials.

In one of his terribly clever 1905 papers, Einstein described how the random motion of individual atoms gives rise to diffusion. Adding up the probabilities from every possible starting position is the sort of thing integrals were made for: So far this is standard probability fare.

When it’s positive the function is curving up the way your hand curves when you palm is pointing up and when it’s negative the function is curving down.

By saying that the time derivative is proportional to the 2nd position derivative, you’re saying that “hills” will drop and “valleys” will rise.

It seems as though a bunch of water is a good place to store heat energy, but the more time something spends being hot, the more energy it drains into everything around it.

Answer Gravy: This gravy is just to delve into why picturing heat flow in terms of the random motion of hypothetical particles is a good idea.Several versions of the protocols find widespread use in applications such as web browsing, email, Internet faxing, instant messaging, and voice-over-IP (Vo IP).Websites are able to use TLS to secure all communications between their servers and web browsers.But again, heat isn’t actually particles, it’s just that picturing it as such leads to useful insights.While the equation and the intuition are straight forward, actually solving the diffusion equation in almost any real world scenario is a huge pain.So, if the surrounding air is 60°, then an 80° pool will shed heat energy twice as fast as a 70° pool.

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