# Is the reverse heat equation chaotic?

I was reading about heat flow, when this came up. Consider an isolated metal plate which had been heated locally in some small region and allowed to evolve thereafter. At some later time t, we measure the temperature $T(x,y,z)$ for all point $P(x,y,z)$ in the metal plate. This constitutes our initial conditions for the system. We wish to simulate a reverse heat flow with these initial conditions. Is this system chaotic? That is do you suppose that a small variation in the set of initial temperatures(caused by measurement errors) will lead to a huge difference in simulation of the system state at some earlier time?

• Heat eqn, $\partial_tV=\partial_x^2V$, is easily time-reversibe, no? Dec 24, 2016 at 18:35
• I don't know exactly. Justify your view please. The laplacian is basically an averaging instruction which is iterated over time all over the plate. Given an average value at a point, there exists an infinitude of distributions just before, which can lead to that value(right?) Therefore i was thinking this reverse simulation mightbehave in a funny way. Dec 24, 2016 at 18:55
• Well, what happens to the pde when $t\to -t$? What happens to your finite difference scheme? Dec 24, 2016 at 18:59
• @KyleKanos: For $t→∞$, we should have thermal equilibrium. So, it’s obviously not time-reversible (also see my answer). Dec 24, 2016 at 21:13
• @Wrzlprmft yes, I agree on that part, but what about $t\ll\tau$ where $\tau$ is the time of equilibrium? I still think it'd be time reversible (and I seem to recall doing some modeling showing as much at some point) Dec 24, 2016 at 22:16

This is discussed in this lecture and this MathSE question. The reverse time solution only exists for very smooth initial functions and may (must?) "blow up" (I think that is the mathematical terminology) in a finite time.

In "chaotic" behaviour we usually consider situations where long term solutions exist in a neighbourhood of (most) initial conditions. I don't think this is an example.

Note: When I talk about initial conditions, I refer to the time-forward problem. I do not think my answer would be intelligible otherwise.

Obviously, for $t→∞$, your plate would approach thermal equilibrium and thus every point would have the same temperature, irrespective of the location of local heating. Now, for a sufficiently high finite time, you can get arbitrarily close to the equilibrium. Assuming that the solution of the inverse problem is unique (which my intuition doubts), you can find arbitrarily close states that correspond to totally different initial states.

However (still assuming a unique solution), the dynamics (forward or inverted) is lacking another crucial property of chaos, namely topological mixing. The system does not return to states close to it’s initial state. If this were the case, we would have to observe the dynamics to go from:

• something near thermal equilibrium to
• a localised heat spot to
• something near thermal equilibrium
• and so on.

This obviously violates the second law of thermodynamics.