Abstract:
Mathematicians and physicists have long been interested in figuring out how heat settles to equilibrium in different media. Usual methods of solution approach this problem using deterministic methods—but is there another way?
Blog:
The Dirichlet problem asks for a function satisfying a partial differential equation on the interior of a region, while the value of the function along the boundary is pre-determined and fixed. Usually this question pops up naturally where the function represents heat energy or electric potential, because the solution tells you the steady state towards which your system eventually settles.
Almost every mathematics or physics undergraduate will encounter this problem at some point in their education, and run through the same typical routine—finding solutions on the interval, maybe a disc or a ball, then briefly touch on some Fourier analysis—and maybe a discussion of Green’s functions, if they’re lucky.
And all this might serve to give the student the impression that this problem is a relatively simple one, with some straightforward methods of solution, and not something that’s worth spending much more time thinking about. But this isn’t true! In fact, there’s an incredibly rich amount of mathematics here if you’re willing to look for it, but I’d like to quickly explain one perspective which I find particularly intuitive—because it’s random.
Pick a point inside your region, and imagine a jittery particle moving about randomly, starting from that point—we’d call this process ‘Brownian motion’. Track this particle until it eventually hits the boundary, and record the value of the function at that boundary point. Now do this over and over again, and compute the average over all hit boundary points, and then assign whatever this average value is to your original starting point. If you do this for every point in your region, it turns out this constructed function solves the Dirichlet problem!
This isn’t just a novel way of solving this problem—it really gets right to the heart of what we’re trying to model. If we think in terms the heat diffusion problem, what we call ‘heat energy’ is physically just some measure of how excited the particles in our medium are, and in reality, these particles jitter about much like Brownian motion. If we then imagine the fixed boundary values of our function as being a ‘heat input’, what we’re tracking with this game of averages is which points on the interior of our region are getting the most heat traffic from the boundary.
The last idea here relies on the idea of ‘reversibility’ of Brownian motion—really a symmetry which says the process looks the same backwards and forwards. Suppose we know which boundary points our particle hits starting from a fixed point. Then we also know from which boundary points our heat energy originates, if it passes through our point, because the distribution is exactly the same!
So not only is this approach (in my opinion) a much more intuitive way of understanding the Dirichlet problem, but it turns out that we can extend it to achieve solutions on much more complicated geometries, with the exact same idea!
Aidan Petrowski
The University of Melbourne
