Diffusion Equation & Green's Function

Heat equation ∂T/∂t = D∇²T — Green's function, superposition, and temperature evolution

Parameters

Theory

∂T/∂t = D · ∂²T/∂x²
G(x,t) = exp(-x²/4Dt) / √(4πDt)
T(x,t) = ∫ G(x-x',t) T(x',0) dx'
The Green's function is a Gaussian that broadens as √(4Dt). Each initial condition evolves by convolution with G. The width grows as σ(t)=√(2Dt) — diffusive spreading. Color shows temperature; white curve = T(x,t); blue dashed = G centered at x=0.

Live Stats

Width σ = √(2Dt) grows as t^(1/2). Total heat ∫T dx is conserved. Entropy S = ½ln(4πeDt) increases monotonically — irreversibility!