Heat Kernel & Diffusion — Fundamental Solution

The heat equation ∂u/∂t = D∇²u has the fundamental solution G(x,t) = exp(−x²/4Dt)/√(4πDt). Any initial condition is a superposition u(x,t) = ∫G(x−y,t)u₀(y)dy. The heat kernel also describes random walks, Brownian motion, and diffusion on graphs and manifolds.

Settings

Diffusivity D: 1.0 Time t: 0.5 Mode: Number of sources: 3

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Width σ = √(2Dt) — the diffusion length. Amplitude decays as 1/√t to conserve total probability. On a graph, heat kernel H(t) = e^{-tL} where L is the graph Laplacian.