The Heat Equation Sorts Waves By Size

The heat equation is usually introduced with a metal rod. If one part of the rod is hot and another is cold, temperature spreads out until the rod settles toward equilibrium. The equation is

The left side says the temperature is changing in time. The right side says the change is controlled by the second spatial derivative, the local concavity of the temperature graph.

That description is correct, but it can make the equation feel like a rule about nearby points exchanging heat. There is another viewpoint that is often more revealing: the heat equation is a machine that sorts waves by wavelength.

param time = 0.0
param high = 6
param mix = 0.62

background = BLACK
camera = Camera(4b)

let u = |x, time, high, mix|
    0.72 * (exp(-time) * sin(x) + mix * exp(-0.18 * high * high * time) * sin(high * x))

let HeatProfile = |time, high, mix| block {
    . stroke{GRAY, 1} Line(start: [-3.4, 0, 0], end: [3.4, 0, 0])
    . stroke{BLUE, 3} ExplicitFunc(|x| u(x, time, high, mix), [-PI, PI, 260])
    . color{WHITE} center{[0, 1.45, 0]} Text("small waves decay first", 1.04)
}

mesh profile = HeatProfile($time, $high, $mix)

slide "after a short time"
    time = 0.45
    play Lerp(1.4)

slide "after a longer time"
    time = 1.15
    play Lerp(1.4)

The blue graph begins as a mixture of a low-frequency sine wave and a high-frequency sine wave. As time increases, the high-frequency ripple vanishes much faster. What remains is the slow wave.

This is not an accident of the example. It is the central mechanism.

Sine waves are eigenvectors of heat

Start with the simplest possible temperature profile:

Taking two derivatives in $x$ gives

The second derivative does not change the shape of the wave. It only multiplies it by $-n^2$. That means a single sine wave stays a single sine wave under heat flow. Only its amplitude changes.

So try a solution of the form

Substituting into the heat equation gives

so

The important part is the exponent $n^2$. Doubling the frequency makes the decay four times faster. Tripling it makes the decay nine times faster.

Blur has a spectrum

Every reasonable temperature profile can be decomposed into waves:

Heat flow does not need to negotiate among these waves. It acts on each one independently:

This is why heat smooths. Sharp corners and tiny wrinkles require high-frequency ingredients. Those ingredients are exactly the ones multiplied by the smallest exponential factors.

param time = 0.0

background = BLACK
camera = Camera(4b)

let wave = |x, time|
    0.65 * (exp(-time) * sin(x) + 0.75 * exp(-5.0 * time) * sin(5 * x))

let Profile = |time| block {
    . stroke{GRAY, 1} Line(start: [-3.4, 0, 0], end: [3.4, 0, 0])
    . stroke{BLUE, 3} ExplicitFunc(|x| wave(x, time), [-PI, PI, 260])
    . color{WHITE} center{[0, 1.45, 0]} Text("heat filters frequency", 1.04)
}

mesh profile = Profile($time)

slide "frequency filter"
    time = 1.1
    play Lerp(2.0)

The equation is local in space, but its behavior is diagonal in frequency. That sentence is the whole trick. Local averaging and spectral filtering are the same process seen through different coordinates.

Why the second derivative appears

At a point where the temperature graph is concave up, the point is colder than the average of its neighbors, so it warms. At a point where the graph is concave down, it is hotter than the average of its neighbors, so it cools. The second derivative measures exactly this local imbalance.

For $\sin(nx)$, the imbalance is stronger when the wave is narrow. A high-frequency wave bends more sharply, so the local averaging pressure is larger. The factor $n^2$ is the analytic expression of that geometric fact.

This also explains why heat flow loses information. After enough time, many different initial profiles become nearly indistinguishable, because all their high-frequency differences have been suppressed. Running the heat equation backward would require resurrecting those tiny components by multiplying them by huge factors. Any measurement error would explode.

Heat is therefore not just blur. It is irreversible spectral bookkeeping.