The Heat Kernel
On a finite rod, separation gave a Fourier series. On an infinite rod
there are no boundaries to quantise the modes, so the series becomes a
Fourier transform.
Transforming u_t = \alpha u_{xx} in x turns
the PDE into a simple ODE in time for each frequency k:
\hat{u}_t = -\alpha k^2\,\hat{u} \;\Longrightarrow\; \hat{u}(k, t) = \hat{u}(k, 0)\,e^{-\alpha k^2 t}.
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.
The Gaussian that does everything
The inverse transform of e^{-\alpha k^2 t} is a Gaussian — the
heat kernel:
\Phi(x, t) = \frac{1}{\sqrt{4\pi\alpha t}}\,e^{-x^2 / (4\alpha t)}.
It is the temperature from a unit "spike" of heat placed at the origin at
t = 0. The solution for any initial profile
f is then the kernel convolved with the data —
superposing one spreading Gaussian for every starting point:
u(x, t) = \int_{-\infty}^{\infty} \Phi(x - y, t)\,f(y)\,dy.
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.
- Fourier transform turns the heat equation into \hat{u}_t = -\alpha k^2\hat{u}; each mode decays as e^{-\alpha k^2 t}.
- The kernel \Phi(x, t) = (4\pi\alpha t)^{-1/2} e^{-x^2/4\alpha t} spreads like \sqrt{t} and shrinks like 1/\sqrt{t} (total heat conserved).
- General solution: u = \Phi * f (convolution with the initial data).
- Diffusion has infinite speed — the kernel is positive everywhere for every t > 0.
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.