Yesterday I wrote a post that explored the flow of heat both forwards and backwards in time. I used this as a venue to introduce the notion of entropy and to describe one extreme example of the butterfly effect—where small changes in initial data can create big changes in the final result. That’s all fine and good and I stand by that.

But I said that the reverse heat equation, which runs the flow of heat backwards in time, was an example of chaos. And as this reddit user points out, this is very wrong. I have now fixed the original post so that it doesn’t say anything wrong. But I owe you all an explanation here.

The Heat Equation is Not Chaotic

You can never, ever actually solve the reverse heat equation. It is an example of a so-called ill-posed problem. And understanding which problems are well-posed or ill-posed is a very important topic in both physics and mathematics. (This is actually the reason I’m interested in the reverse heat equation. It’s the archetypical ill-posed problem.)

Truly chaotic systems, on the other hand, are well-posed. Although they depend strongly on their initial conditions, meaning that finding exact solutions is difficult, they can be solved. To illustrate the difference, let’s look again at the reverse heat equation, shown in figure 2.

Temperature differences just build on themselves exponentially until the whole thing becomes completely unmanageable. And this is the problem. Now let’s look at a genuinely chaotic system: the flow of water in a very shallow pond, as shown in figure 3. (You can find another good video here.)

Notice the vortices that form? The precise initial configuration of the water dramatically changes the positions of the vortices. However, although the vortices merge, they don’t grow so much that we can’t make predictions any more. And this is the important difference. This property, called topological mixing, is also what keeps the heat equation from being chaotic.
(There are other technical reasons that the heat equation is not chaotic. But this is the big one, and it’s the thing that I really failed to emphasize in my last post. So I’m emphasizing it here.)

As an aside, notice how small vortices become bigger? This is a property of fluids that are tightly confined in one direction like in a shallow pond or on the surface of the Earth. It’s actually why hurricanes form. Small vortices merge to become big vortices. In fluids without the confinement, the process goes the other way, big vortices become small.

My Apologies

As a physicist—and not a mathematician—I believed that I knew the definition of mathematical chaos when I did not. And instead of checking my facts, I just blithely went ahead and wrote about it.

Many physicists don’t know about mathematical chaos; I’m not ashamed of my ignorance. But I am ashamed of not doing my homework before writing about a topic with which I am unfamiliar. Many of you trust me as an authority on math and physics, and in yesterday’s post, I failed to live up to that trust.

I promise to be more careful in the future.