Heat Kernel Diffusion

K(x,y,t) — heat spreading from a point source on different geometries

Geometry:

Time t

Source position

Eigenmode terms

K(x,y,t) = Σₙ φₙ(x)φₙ(y)e^{−λₙt}

Line: K = (4πt)^{-1/2}e^{-|x-y|²/4t}
Circle: K = (1/2π)Σ e^{-n²t}e^{in(θ-θ₀)}
Torus: product of two circles

The heat kernel K(x,y,t) gives the temperature at point x at time t, given unit heat injected at y at t=0. It satisfies the heat equation ∂K/∂t = ΔK where Δ is the Laplace-Beltrami operator. The eigenfunction expansion K = Σ φₙ(x)φₙ(y)e^{−λₙt} reveals how different frequency modes decay exponentially. On a circle (compact manifold), heat "wraps around" and eventually distributes uniformly. On ℝ (non-compact), it spreads indefinitely — the famous Gaussian spreading √(4πt). The heat kernel encodes deep geometric information: trace(K) = Σ e^{-λₙt} is the heat trace, related to spectral geometry.