Boltzmann H-theorem: S = −Σ p log p increases monotonically as a packet spreads.
The diffusion equation ∂p/∂t = D ∂²p/∂x² on a periodic 1D grid with initial Gaussian distribution. Boltzmann entropy S(t) = −Σᵢ pᵢ ln pᵢ increases monotonically → max value ln(N) at equilibrium (uniform distribution). This is the H-theorem: irreversibility emerges from reversible dynamics via coarse-graining. S grows as ½ ln(4πeDt + σ₀²) for Gaussian packets (analytically).