The fundamental solution (Green's function) for the diffusion equation. A delta-function initial condition spreads into a Gaussian with variance σ²(t) = 2Dt — growing linearly in time. The peak height decays as 1/√t to conserve total concentration.
The diffusion equation governs heat conduction, mass diffusion, Brownian motion, and probability densities of random walks. Einstein (1905) showed that ⟨x²⟩ = 2Dt links the microscopic random walk to the macroscopic diffusivity D. Multiple sources superpose linearly.