On a finite rod, both ends are held at fixed temperatures, and heat sloshes back and forth inside that little cage until it settles down. But picture something different: touch a soldering iron to the middle of an enormously long metal rail for one instant, then take it away. The rail is so long that, for a long while, its far-off ends might as well not exist — there is nothing for the heat to bounce off of at all. What happens to that one hot spot as time goes by?

Geophysicists ask a version of the same question about the ground itself: a shallow layer of soil heats and cools with the seasons, but hundreds of metres down the rock is, for all practical purposes, an endless slab stretching away in every direction — no wall down there for heat to reflect off. Whenever the boundary is so far away that it simply hasn't had time to matter yet, it is often easier — and, on an infinite domain, exactly correct — to drop the boundary altogether and solve the problem on the whole line.

On a genuinely infinite line there are no boundaries left to quantise anything, so the Fourier series that solved the finite rod breaks down — there's no discrete ladder of allowed frequencies to sum over. In its place comes the continuous Fourier transform. Transforming u_t = \alpha u_{xx} in x turns the PDE into a simple ODE in time for each frequency k:

Each frequency just decays at rate \alpha k^2 — high frequencies vanishing fastest, the same smoothing we saw on the interval. Transforming back gives the solution as a convolution — an entirely different, and beautifully clean, way of writing the answer.

The Gaussian that does everything

The inverse transform of e^{-\alpha k^2 t} is a Gaussian — the heat kernel:

It is the temperature from a unit "spike" of heat placed at the origin at t = 0 — exactly the soldering-iron touch from the opening paragraph, idealised down to an infinitely thin, infinitely hot instant. The solution for any initial profile f is then the kernel convolved with the data — superposing one spreading Gaussian for every starting point:

The kernel reveals two signatures of diffusion. Its width grows like \sqrt{t} (heat spreads, but ever more slowly), while its height falls like 1/\sqrt{t} so the total heat stays constant. And it is positive everywhere for any t > 0: a spike at the origin is instantly felt arbitrarily far away — diffusion has infinite propagation speed, the polar opposite of the wave equation's finite cone.

Worked example: how much does the spike actually flatten?

Take \alpha = 1 and watch the peak of the kernel, right at x = 0, where \Phi(0, t) = 1/\sqrt{4\pi t}. At t = 1,

Wait until t = 4 — four times as long — and

exactly half the peak height. That matches 1/\sqrt{t} precisely: multiplying time by 4 multiplies \sqrt{t} by 2, so the height is cut in half while the width doubles. No heat was lost — the same total area is now spread twice as wide and stands half as tall.

A spike spreading out

The heat kernel \Phi(x, t) with \alpha = 1. Advance time: the sharp spike slumps into an ever-wider, ever-flatter Gaussian, but the area beneath it (the total heat) never changes. Run t back toward zero and it sharpens toward an infinite spike — the initial point source, a so-called Dirac delta.

Worked example: smearing out a slab of hot metal

The kernel is not just for point spikes — it builds the solution for any starting temperature, one convolution at a time. Suppose the initial data is a "slab" of hot metal at temperature 1 between x = -1 and x = 1, and cold (temperature 0) everywhere else — no need for a Fourier series to describe such a blunt, non-periodic shape, which is exactly the point of working on the whole line.

Plugging that f into the convolution integral, and integrating the Gaussian kernel across the slab, produces the exact solution in terms of the error function:

The error function \operatorname{erf} is nothing exotic here — it is just the running total (the integral) of the Gaussian bell curve, so this formula is really saying "add up the kernel's contribution from every point of the slab, from -1 to 1," exactly the convolution recipe above, worked out in closed form.

At t = 0 this is exactly the sharp-edged slab; as t grows the sharp corners round off and the heat leaks out into the cold surroundings, exactly the same smoothing behaviour the kernel showed for a single spike — because under the hood, that is precisely what is happening: every point of the slab is quietly running its own spreading Gaussian, and the convolution integral is simply adding all of those Gaussians up.