the decay of Fourier modes
Pluck a temperature profile apart into its pure wavy ingredients — a long smooth wave, a wave with two bumps, a wave with three, and so on — and feed them to the heat equation. The wiggly ones fade fast, the smooth ones linger. The decay of Fourier modes is this fact: under the heat equation every sine wave decays exponentially, and the more wiggly it is, the faster it dies.
On an interval of length L with ends held at zero, separation of variables tells you the building-block solutions are sin(n pi x / L) e^(-lambda_n t), where the decay rate is lambda_n = k (n pi / L)^2. Here is the key: lambda_n grows like n^2. So the fundamental mode (n = 1) decays slowly, the second harmonic decays four times as fast, the third nine times as fast. A general solution is a Fourier sine series u(x,t) = sum b_n sin(n pi x / L) e^(-lambda_n t); as time goes on, the high-n terms vanish almost immediately and what survives is dominated by the slowest, smoothest mode n = 1. Walk through it: you found the b_n from the initial data, each rides its own decaying exponential independently (superposition), and the n^2 in the exponent is the engine that crushes the fine detail first.
This single fact explains the heat equation's whole personality. The smoothing effect is exactly the rapid death of high-frequency modes (sharp features are made of high harmonics, so they smooth instantly). The approach to steady state is the slow survival of the lowest mode. And running time backward is catastrophic because the exponentials become e^(+lambda_n t), so high modes explode — that is why the backward heat equation is ill-posed. Caveat: the n^2 law is for these clean trigonometric eigenmodes on a nice interval; on other domains the eigenvalues differ in detail, but the qualitative ordering (rougher decays faster) always holds.
Start with u(x,0) = sin(pi x/L) + 0.5 sin(5 pi x/L). The five-bump part decays 25 times faster, so almost at once you see a clean single hump that then slowly fades to zero.
Decay rate scales as n^2: the n=5 mode dies 25 times faster than n=1.
The exponent carries n^2, not n — that quadratic is why diffusion erases fine detail so violently and why time-reversal blows up. A common slip is to think the decay is merely exponential in n; it is exponential in n-squared times t.