Einstein’s cute trick was to say “Listen, I don’t know what ϕ() is a probability distribution (and the sum of probabilities over all possibilities is 1).
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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.
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.
: I’m having a debate with my wife that I think you can help us resolve. It has an electric heater, which we set to keep the pool water at 85 degrees Fahrenheit. My wife says we should turn the heater off while we’re away to save energy. The one and only gain from leaving a pool heater on is that it will be warm when you get in.
I say that it takes less energy to maintain the pool at 85 while we’re away then to let it drop about ten degrees (summer evenings can get quite cool where we live in upstate New York) and then use the heater to restore 85. And what variables are relevant to the calculation? The same is true of all heaters (pool, car, space, whatever).So, if the surrounding air is 60°, then an 80° pool will shed heat energy twice as fast as a 70° pool.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 advanced calculus. A difference in temperature leads to a net flow of heat energy.