Intuition behind the paradox of instantaneous heat propagation

Question

On this MIT lecture, the difference between the heat equation and the wave equation includes signal travelling infinitely fast in the heat equation, while it has finite speed in the wave equation:

I guess I don't get the idea of "signal" because in the heat equation there is a partial derivative with respect to time of the function that assigns a temperature to each point in space at each time, and this partial derivative with respect to time, which is not infinite, would seem to be the speed.

So what is it exactly that travels infinitely fast. After a Google search it seems as though this is the "paradox of instantaneous heat propagation" and it gets into other equations, and even has a relativistic counterpart. In other words, I couldn't find any entry-level explanations.

Answer 1

Say you have a one dimensional heat problem. (Adding dimensions does not change the results, but does make drawings harder.) Initially, the line is at temperature $0$. At time $t = 0$, you drop some heat on the origin, say $f(x,0) = \delta(x)$ and then let the dynamics run. (Here, $\delta$ is the Dirac delta function. This is a distribution. It can be defined as the limit of a Gaussian distribution as the standard deviation decreases to zero -- the distribution is zero everywhere except at $x=0$ and the integral of the distribution over all of space is $1$.)

For all time afterwards, the distribution of heat is a Gaussian distribution with positive (and increasing) standard deviation. This distribution is nonzero everywhere. This means that immediately after time zero, the heat at every point in the universe jumps to a positive amount. That is, the information that heat was added at the origin is instantaneously transmitted to every point in the universe.

Answer 2

First the wave-equation: let's start with a small narrow pulse located close to $x=0$ ($u$ nonzero close to $x=0$ and zero elsewhere) and then let it go at $t=0$. Now let's see how long time does it take before any point $x_*$ sees the effect of this pulse. The solution $u(x,t) = f(x-vt) + f(x+vt)$ shows that it takes a time $t_* = \frac{x_*}{v}$ ($v$ being the wave speed) before the pulse has managed to travel all the way to the point $x_*$. If you are located at $x=x_*$ and observing the displacement $u(x_*,t)$ then before $t = t_*$ you would be completely unaware that we have a wave coming.

Secondly the heat equation: let $u$ denote the temperature and assume you have an initial smooth temperature distribution $u=0$ across space. We then add some heat close to $x=0$ at $t=0$ (we can represent this by $u(x,0) = \delta(x)$) and ask how long time does it take at a point $x_*$ before the addition of this temperature is felt. The solution is simple to compute: $u(x,t) = \frac{1}{\sqrt{4\pi t}}e^{-\frac{x^2}{4t}}$ and shows that even after the smallest fraction of a second the solution will have changed all across space. If you are observing the temperature at the point $x_*$ you will instantly know that heat has been added to the system no matter how far away you are.

Answer 3

One option is to recognize that the heat equation (or Fourier's law of heat conduction) is an approximation of a dissipative wave phenomenon. Consider the hyperbolic heat equation $$ \epsilon\theta_{tt}+\theta_t = \theta_{xx} $$ where $\epsilon \geq 0$ is a small parameter. For $\epsilon=0$ we get the (parabolic) heat equation, $\theta_t = \theta_{xx}$; otherwise, for $\epsilon>0$ we get the telegraph equation, a.k.a. the damped wave equation. Note that the absolute speed of sound equals ${\epsilon}^{-1/2}$, which becomes infinite as $\epsilon\to 0$.