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In Christian Blatter's proof of Pick's theorem one assumes that at time $t=0$ a unit of heat is concentrated at each point of the plane with integer coordinates. This heat will be distributed over the whole plane by heat conduction, and at time $t=\infty$ it is equally distributed on the plane with density $1$.

I was wondering if and how one could show the behavior of the heat density over time using the heat equation. Unfortunately, I'm way too inexperienced with partial differential equations to know how to go about this. I'd be happy for any hints (or complete solutions). I'm mainly interested in an animated view of the heat flow, so a way to generate that in Mathematica would also be an acceptable solution.

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We are looking for the solution of the heat equation in the plane, with initial conditions $$\sum_{m,n=-\infty}^\infty\delta_{m,n}$$ (these are Dirac delta-functions). We get this as $$F(x,y,t)=\sum_{m,n=-\infty}^\infty E(x-m,y-n,t)$$ where $$E(x,y,t)=\frac1{4\pi t}\exp\left(\frac{-x^2-y^2}{4t}\right)$$ is the fundamental solution for the heat equation. So $$F(x,y,t)=\frac1{4\pi t}\Theta(x,t)\Theta(y,t)$$ where $$\Theta(x,t)=\sum_{m=-\infty}^\infty\exp(-(x-m)^2/(4t))$$ is some sort of theta function.

As for pictures, you'll have to ask someone else....

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